Points $A$ and $B$ are 10 units apart. Points $B$ and $C$ are 4 units apart. Points $C$ and $D$ are 3 units apart. If $A$ and $D$ are as close as possible, then the number of units between them is ．(1996 AJHSME, Question 8)

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities

If $A B=10$ and $B C=4$, then $(10-4) \leq A C \leq(10+4)$ by the triangle inequality. In the triangle inequality, the equality is only reached when the "triangle" $A B C$ is really a degenerate triangle, and $A B C$ are collinear. Simplifying, this means the smallest value $A C$ can be is 6 . Applying the triangle inequality on $A C D$ with $A C=6$ and $C D=3$, we know that $6-3 \leq A D \leq 6+3$ when $A C$ is minimized. If $A C$ were larger, then $A D$ could be larger, but we want the smallest $A D$ possible, and not the largest. Thus, $A D$ must be at least 3 , but cannot be smaller than 3 . Therefore, $B$ is the answer.


There are $20$ balls of the same size in a box.  Lucas says: "Half of them are red."  Peter says: "The number of red balls is $5$ times that of green balls."  Claire says: "There are three colors of balls in the box: red, light blue, and green."  How many dark blue balls are there in the box?

Overseas Competition->Knowledge Point->Calculation Modules->Fractions


Overseas Competition->Knowledge Point->Calculation Modules


Suppose that $x$ and $y$ are nonzero real numbers such that $\frac{x+y}{x}=2$, what is the value of $\frac{x}{y}$?

Rearranging, we find $x+y=2x, x=y, \frac xy = 1$.


Cathy wants to cut a wooden stick. In how many places does she need to break a wooden stick in order to get $7$ pieces?


Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Operating Directly

$2◆3=2\times2+3=7$  So the answer is $\rm A$.


Starting with some gold coins and some empty treasure chests, I tried to put $9$ gold coins in each treasure chest, but that left $2$ treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had $3$ gold coins left over. How many gold coins did I have? ( 2017 AMC8, Questions \#17)

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Linear Equations with Multiple Variables

We can represent the amount of gold with $g$ and the amount of chests with $c$. We can use the problem to make the following equations: $$ \begin{gathered} 9 c-18=g \textbackslash\textbackslash{} 6 c+3=g \end{gathered} $$ We do this because for $9$ chests there are $2$ empty and if $9$ were in each $9$ multiplied by $2$ is $18$ left.  Therefore, $6 c+3=9 c-18$. This implies that $c=7$. We therefore have $g=45$. So, our answer is (C) 45


\textbf{A stationery store owner calculated the mean and median price of all types of pen: 5 and 3, respectively. He plans to have a 20\% off sale. What are the new mean and median of pens in that store?}

\textbf{mean: 4; median: 2.4}

\textbf{mean: 5; median: 2.4}

\textbf{~mean: 4; median: 3}

\textbf{~mean: 5; median: 3}

\textbf{mean: 2.4; median: 4}

\textbf{If you times some number to the data set,~ the mean or median needs to be times with the same number. So for here, everything times with 0.8.}


Which of the following is the same as~$44\times25$? (~\uline{~~~~~~~~~~}~)

$$4\times 25+4\times 25$$

$$44\times 2+44\times 5$$

$$4\times 2+4\times 5$$

$$40\times 25+4\times 25$$

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Distributive Law of Whole Numbers


$$50\textbackslash\%$$ of $$30\textbackslash\%=15\textbackslash\%$$of ．

Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Operations of Fractions

$$50\textbackslash\%\times 30\textbackslash\%=$$ half of $$30\textbackslash\%=15\textbackslash\%=15\textbackslash\%$$ of $$100\textbackslash\%$$.


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction


How many polynomials are there in the following expressions:  $$-4{{x}^{2}}+2$$，$$-\frac{1}{3}mn$$，$$ \pi $$，$$\frac{{{(2x-y)}^{2}}}{3}$$，$$32\frac{1}{4}$$~ ~．

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions


What value does the $4$ represent in the number $55.431$?

Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals

The number $$4$$ is located on the tenth place, thus representing $$0.04$$.  Check Lesson 4 Concept 1 on textbook


Assuming $a \neq 3, b \neq 4$, and $c \neq 5$, what is the value in simplest form of the following expression? (Adapted From 2020 AMC 10A Problems, Question \#3)  $$ \frac{2a-6}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{2c-10}{4-b} $$

$\frac{a b c}{15}$

$\frac{4}{a b c}-\frac{1}{15}$

$\frac{1}{15}-\frac{1}{a b c}$

If $x \neq y$, then $\frac{x-y}{y-x}=-1$. We use this fact to simplify the original expression: $$ \frac{2a-6}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{2c-10}{4-b} = -2\times 2= 4$$.


The zeroes of the function $f(x)=x^{2}-a x+2 a$ are integers. What is the sum of the possible values of $a$? (2015 AMC 10A Problems, Question \#23)

By Vieta\textquotesingle s Formula, $a$ is the sum of the integral zeros of the function, and so $a$ is integral.  Because the zeros are integral, the discriminant of the function, $a^{2}-8 a$, is a perfect square, say $k^{2}$. Then adding $16$ to both sides and completing the square yields $$ (a-4)^{2}=k^{2}+16 $$.  Therefore, $(a-4)^{2}-k^{2}=16$ and $$((a-4)-k)((a-4)+k)=16$$.  Let $(a-4)-k=u$ and $(a-4)+k=v$; then, $a-4=\frac{u+v}{2}$ and so $a=\frac{u+v}{2}+4$.  Listing all possible $(u, v)$ pairs such that $a$ is integral: $(2,8),(4,4),(-2,-8),(-4,-4)$ (not counting transpositions because this does not affect $u+v$). Then, $a=9,8,-1,0$.  These $a$ sum to $16$.


Alice, Tom, Chloe and Susan ate chicken nuggets together. Everyone ate at least 2 pieces. The person who ate the least ate 4 pieces less than the person who ate the most. How many pieces of chicken nuggets did they each eat? (adapted from $$2021$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$8$$）

$8, 2, 4, 3$

$4, 1, 5,9$

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Subtraction of Whole Numbers->Subtraction in Horizontal Form

In B, C, D, the smallest number less than $2$. In the E, the difference between the largest number and the smallest number is $6$.


Which of the following quotients is $$1$$ more than $$162\div18$$?

$$128\div 16$$

$$120\div 15$$

$$132\div 12$$

$$110\div 11$$

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division

Since $$162 \div18 =9$$, we want a quotient of $$10$$. That\textquotesingle s $$\text{D}$$.


A box contains five cards, numbered $1,2,3,4$, and $5$. Three cards are selected randomly without replacement from the box. What is the probability that $4$ is the largest value selected? (2017 AMC 8 Problems, Question \#10)

There are $\_5C\_3 = 10$ possible groups of cards that can be selected. If $4$ is the largest card selected, then the other two cards must be either $1$,$2$, or $3$, for a total $\_3C\_2 = 3$ groups of cards. Then, the probability is just $(\text{C}) \frac{3}{10}$.


The distance between A and B is $$350$$ $\text{km}$. Kin and Mary drive away from A and B respectively at $8$ a.m. and go towards each other at same time. Kin drives $$40$$ $\text{km/h}$, and Mary drives $$50$$ $\text{km/h}$. Mary rested for $$2$$ hours on her way and then continues driving. They will meet at~\uline{~~~~~~~~~~}~.

$$10$$ a.m.

$$11$$ a.m.

$$12$$ p.m.

$$1$$ p.m.


Bob sees that the license plate number of the car consists of $5$ digits: $1$, $2$, $6$, $7$, $9$. If these $5$ digits are filled in the square~$$\huge\square+\square =\square +\square $$, which number is not used?~(adapted from $$2017$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$8$$)

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Addition of Whole Numbers->Addition in Horizontal Form


Tiantian drank $$6\frac{3}{8}$$ litres of water. Matthew drank $$4\frac{5}{6}$$ litres of water. How many litres of water did the both of them drink altogether? Give your answer as a mixed number in the simplest form.


None of the above


In the diagram below, a circle centered at $O$ has radius $4 \text{cm}$. It is divided into $4$ regions by two chords that are perpendicular to each other at point $N$. It is known that $OM=1 \text{cm}$, $MN=2\text{cm}$. Find the value of: Area of $(I + III) -$ Area of $(II+IV)$

Overseas Competition->Knowledge Point->Calculation Modules->Unit Conversion->Converting between Units of Area


Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump $5$ pads to the right or $3$ pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located $2023$ pads to the right of her starting position?


Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Grouping in Fast Addition and Subtraction of Whole Numbers


Overseas Competition->Knowledge Point->Calculation Modules->Operations through Formulas-> Difference of Two Squares Formula


Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 5 cups. What percent of the total capacity of the pitcher did each cup receive?~ (2020 AMC 8, Question \#5)

Overseas Competition->Knowledge Point->Calculation Modules->Percentage Calculation

The pitcher is $\frac{3}{4}$ full, i.e. $75 \textbackslash\%$ full. Therefore each cup receives $\frac{75}{5}=(\mathbf{C}) 15$ percent of the total capacity.


no solution

$$x=2$$ or $$x=3$$

$$\begin{eqnarray}\frac{16}{4-{{x}^{2}}}+\frac{x-2}{x+2}\&=\&\frac{x+2}{x-2}\textbackslash\textbackslash{} -16+{{(x-2)}^{2}}\&=\&{{(x+2)}^{2}}\textbackslash\textbackslash{} x\&=\&-2\end{eqnarray}$$  After verification, $$x=-2$$ is not a solution.  There is no solution.


The sum of $$5$$ consecutive numbers is $$500$$. Among all five numbers, what is the smallest value?

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Arithmetic Sequences


Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns of Number Tables->Triangle Number Table


In multiplying a number by $\dfrac{1}{100}$, the result may be obtained by moving the decimal point of that number．

two places to the left

one place to the left

two places to the right

one place to the right

Multiplying a number by $\dfrac{1}{100}$, is the same as dividing by $100$. When dividing by $100$, move the decimal point $2$ places left.


If $$3$$ apples weigh as much as $$4$$ pears, and $$2$$ pears weigh as much as $$5$$ plums, then $$9$$ apples weigh as much as~ \uline{?~} plums.

If $$9$$ apples weigh as much as $$12$$ pears, and $$12$$ pears weigh as much as $$30$$ plums, then $$9$$ apples weigh as much as $$30$$ plums.


Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-19$, the remainder is 99 , and when $P(x)$ is divided by $x-99$, the remainder is 19. What is the remainder when $P(x)$ is divided by $(x-19)(x-99)$? (1999 AHSME Problems, Question \#17)

Overseas Competition->Knowledge Point->Calculation Modules

Since the divisor $(x-19)(x-99)$ is a quadratic, the degree of the remainder is at most linear.  We can write $P(x)$ in the form $$ P(x)=Q(x)(x-19)(x-99)+c x+d $$ where $c x+d$ is the remainder.  By the Remainder Theorem, plugging in 19 and 99 gives us a system of equations:  $$ \begin{aligned} \& 99 c+d=19 \textbackslash\textbackslash{} \& 19 c+d=99\end{aligned} $$  Solving gives us $c=-1$ and $d=118$, thus, our answer is $(\text{C})-x+118$.


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations


Given that $$a\Delta b=a+b-4$$, for example, $$3\Delta 2 = 3 +2-4$$, what is $$3\Delta4$$?

Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition


\textbf{Which of the following is/are categorical variable(s)?}  \textbf{I. The mean of Grade 1 boys' heights}  \textbf{II. The colors of jackets in the class}  \textbf{III. The types of pens in a stationary store}

I

II

III

I, II

II, III

\textbf{A categorical variable is defined by the set of groups or categories (qualitative values) that individuals are placed into; it is not a numerical value. The mean of~ Grade 1 boys' heights is a number. The colors of jackets in the class and the types of pens in a stationary store cannot be described by numbers.~}


A movie ticket costs $17$ dollars and the tax is $a$ dollars. George purchases a movie ticket and writes on his ledgar : " the ticket cost $17+a$ dollars". Is this description accurate?

yes

no

When writing an algebraic expression with units, we need to write it in parentheses.


Overseas Competition->Knowledge Point->Calculation Modules->Fractions


Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers->Exponentiation of Powers


The next number in the sequence $$1, 3, 7, 13, 21, \cdots$$ is．

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns in Number Sequences

Difference between the numbers is $$2$$, $$4$$, $$6$$, $$8$$. Next one is $$21+10$$.


Given that the points $A(-2, y\_1)$, $B(0, y\_2)$ and $C(3, y\_3)$ are all on the graph of $f(x)=-2x^{2}+4x+m$, then the relationship among $y\_1$, $y\_2$ and $y\_3$ is:

$y\_2 \textgreater{} y\_3\textgreater y\_1$

$y\_1~\textgreater{} y\_3\textgreater y\_2$

$y\_2 \textgreater{} y\_1\textgreater y\_3$

$y\_3~\textgreater{} y\_2\textgreater y\_1$

The parabola opens down with axis of symmetry $x=1$. Therefore, point $B$ is closer the axis of symmetry than point $C$, and point $C$ is closer the axis of symmetry than point $A$


Which of the following fraction is smaller than $$\frac{1}{9}$$?

With same numerator, the one we split into more portion is smaller


\textbf{A statistical test involves the following null and alternative hypotheses.~}  \textbf{H0: $\mu$ = 64}  \textbf{Ha: $\mu$ \textgreater{} 64}  \textbf{Which of the following describes a Type II error?}

\textbf{Failing to reject the null hypothesis when the population mean is 64}

\textbf{Failing to reject the null hypothesis when the population mean is greater than 64}

\textbf{Rejecting the null hypothesis when the population mean is 64}

\textbf{Rejecting the null hypothesis when the population mean is greater than 64}

\textbf{Failing to reject the null hypothesis when the p-value is less than the significance level}

\textbf{Type II error ($\beta$): the error of failing to reject the null hypothesis when it is false.~}


The sum of the recurring decimals $$0.\overline {62}$$ and $$0.\overline {16}$$ is~\uline{~~~~~~~~~~}~.

$$\frac {78}{100}$$

Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals


Three members of the Euclid Middle School girls\textquotesingle~softball team had the following conversation.  Ashley: I just realized that our uniform numbers are all $$2-$$digit primes.  Bethany: And the sum of your two uniform numbers is the day of my birthday earlier this month.  Caitlin: That\textquotesingle s funny. The sum of your two uniform numbers is the day of my birthday later this month.  Ashley: And the sum of your two uniform numbers is today\textquotesingle s date.  What number does Caitlin wear?  ($$2014$$ AMC $$8$$ Problem, Question \#$$23$$)

The maximum amount of days any given month can have is $$31$$, and the smallest, two-digit primes are $$11$$, $$13$$, and $$17$$. There are a few different sums that can be deduced from the following numbers, which are $$24, 30$$, and $$28$$, all of which represent the three days. Therefore, since Brittany says that the other two people\textquotesingle s uniform numbers add up to her birthday eadier in the month, that means Caitlin and Ashley\textquotesingle s numbers must add up to $$24$$. Similarly, Caitlin says that the other two people\textquotesingle s uniform numbers add up to her birhday later in the month, so the sum must add up to $$30$$. This leaves $$28$$ as today\textquotesingle s date. From this, Caitlin was referring to the uniform numbers $$13$$ and $$17$$ telling us that her number is $$11$$, giving our solution as $$(\text{A})=11$$.


Given that $$23+25+27 +\ldots+(2k-1)=m^{2}$$, where $$k$$ and $$m$$ are whole  numbers, $$k\textgreater30$$, find the value of $$m$$.

None of the above


Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Complex Fractions


Which of the following is not equal to $$10$$?

$$100\div 10$$

$$10 \div1$$

$$100\times 10$$

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division

Since $$100 \times 10 = 1000$$, choice $$\text{D}$$ is not equal to $$10$$.


What is the remainder when $$16+16+16 +16$$ is divided by $$4$$?

Since $$16\div4$$ has remainder $$0$$, the remainder is $$0 + 0+0+0 = 0$$.


An amusement park has a collection of scale models, with ratio $1: 45$, of buildings and other sights from around the country. If the height of the One World Trade Center is $1770$ feet. What is the height in feet of its replica to the nearest whole number? (adapted from 2018 AMC 8, Question 1)

Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions

You can see that since the ratio of real building\textquotesingle s heights to the model building\textquotesingle s height is $1: 45$. If the height of the center is $1770$ feet, to find the height of the model, we divide by $45$ . That gives us $39.3$ which rounds to $39$ . Therefore, to the nearest whole number, the duplicate is (D) $39$ feet.


$$24\times 26\times 28\times 30\times 32=48\times 52\times 56\times 60\times $$．

$$24\times 26\times 28\times 30\times 32=\left(24\times 26\times 28\times 30\right)\times \left(2\times 2\times 2\times 2\right)\times \underline{2}$$.


How many positive integers can fill the blank in the sentence below? ``One positive integer is~\uline{~~~~~~~~~~}~more than twice another, and the sum of the two numbers is $28$.''


$$5^{1}\cdot5^{4}=5^{1+4}=5^{5}$$,  \uline{Teacher should introduce the formula of $$a^{m}a^{n}=a^{m+n}$$}.


A soccer player computes his win ratio by dividing the number of matches he has won by the total number of matches he has played. At the start of a weekend, his win ratio is exactly~$0.5$. During the weekend, he plays five games, winning three and losing two. At the end of the weekend, his win ratio is greater than~$0.505$. What\textquotesingle s the largest number of matches he could\textquotesingle ve won before the weekend began?

Assume she won~$x$~games before the weekend, we obtain the inequality~$\frac{x+3}{2x+5}\gt0.505$~. Solve the inequality, we get~$x\lt47.5$


Three positive integers are equally spaced on a number line. The middle number is $15$, and the largest number is $4$ times the smallest number. What is the smallest of these three numbers?


Eddie has a card. The number on the card is a neighbouring number of 10, but is not a neighbouring number of 12. What is the number on Eddie\textquotesingle s card?~\uline{~~~~~~~~~~}~

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers


If $$20\times 30$$ is divided by $$40$$, the remainder is．


Which of the following numbers\textquotesingle{} value does not change after removing all ``$$0$$''s ．

Only $$22.990$$\textquotesingle s $$"0"$$ can be removed without causing other digits to change their places from the original number.


Mary cooked $9$ cakes. She cut two of them into $3$ pieces. How many cakes does she have now?


Camila writes down five positive integers. The unique mode of these integers is 2 greater than their median, and the median is 2 greater than their arithmetic mean. What is the least possible value for the mode?


If $$a$$@$$b=\frac {a\times b}{a+b}$$ for positive integers $$a$$ and $$b$$ , what is $$5$$@$$10$$?

$$\frac 3{10}$$

$$\frac {10}3$$

Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Operating Directly->Ordinary Type

Substitute $$a=5$$ and $$b=10$$ into the expression for $$a$$@$$b$$ to get: $$5$$@$$10=\frac {5\times 10}{5+10}=\frac {50}{15}=\frac {10}3$$.  Thus, the answer choice $$\frac {10}3$$ is correct.


Is it true that If an odd function has zero in its domain, then it must pass through the origin. (Hint: $0$ is in the domain, we can apply $f(-x) = -f(x)$ at $x=0$)

True

False

By definition of an odd function, $f(-x) = -f(x)$ for all $x$ in the domain. Since $x = 0$ is in the domain, we have $f(-0) = -f(0)$, which means $f(0) = -f(0)$, or $2f(0) = 0$, and thus $f(0) = 0$ (this represents the origin).


Dividing a certain two-digit number by $$10$$ leaves a remainder of $$9$$, and dividing it by $$9$$ leaves a remainder of $$8$$. The sum of the number\textquotesingle s two digits is．

Dividing a certain two-digit number by $$10$$ leaves a remainder of $$9$$, so it is $$19$$, $$29$$, $$39$$, $$49$$, $$59$$, $$69$$, $$79$$, $$89$$, or $$99$$. The only number listed with remainder $$8$$ when divided by $$9$$ is $$89$$, so the number is $$89$$ and $$8+9=17$$.


Lucy writes numbers from $1$ to $200$ on the blackboard. Then, she plays a game with her friend Mike. Each time they take turn to delete the first two numbers in the sequence and write their sum by the end of the sequence. For example, Lucy deletes $1$ and $2$, then writes $3$ behind $200$ for the first time. They play this game until there is only one number on the blackboard. What is the number?

The result is equal to the sum of all numbers from $1$ to $200$, which is $(1+200)\times200\div2=20100$.


Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns in Number Sequences->Pyramid Sequences


How many types of vitamins are there in the fruits below? There are~\uline{~~~~~~~~~~}~types


Which of the following expression is a perfect square?

$98! \cdot 99!$

$98! \cdot 100!$

$99! \cdot 100!$

$99! \cdot 101!$

$100! \cdot 101!$


Lee has $1$ red cube, $1$ yellow cube, $2$ blue cubes and $3$ green cubes. How many ways are there for Lee to arrange his cubes in a row if cubes of the same color is indistinguishable?


How many minutes are there in $$2$$ weeks?

One week has $$7$$ days. Two weeks has $$7+7$$  One day has $$24$$ hours.  One hour has $$60$$ minutes.


One ticket to a mini concert costs $\textbackslash$20$ at full price. Nicole buys $4$ tickets using a coupon that gives her a $25\textbackslash\%$ discount. Bel buys $5$ tickets using a coupon that gives her a $30\textbackslash\%$ discount. How many more dollars does Nicole pay than Bel?


The students in a class sit in rows. There is the same number of students in each row. There are $2$ rows of students in front of Robert and $1$ row of students behind him. In his row, there are $3$ students on his left and $5$ students on his right. How many students are there in this class? (2022 Math Kangaroo Problem, Level 3-4, Question \#15)

There are $3+5+1=9$ students in each row, and there are $2+1+1=4$ rows. Thus, there are $9\times4=36$ students in this class.


Avril has a card. The number on the card is a neighbouring number of 10, but is not a neighbouring number of 12. What is the number on Eddie\textquotesingle s card?~\uline{~~~~~~~~~~}~


In a competitive market, a producer is able to sell their good for $\textbackslash$ 10$ per unit, while the cost of producing each unit is $\textbackslash$ 8$. What is the producer surplus in this scenario?

$\textbackslash$ 2$

$\textbackslash$ 8$

$\textbackslash$ 10$

$\textbackslash$ 12$

$\textbackslash$2$, as the producer surplus is calculated as the difference between the price the good is sold for and the cost of production, which is $\textbackslash$10$ - $\textbackslash$8$ = $\textbackslash$2$.


The real numbers $c, b, a$ form an arithmetic sequence with $a \geq b \geq c \geq 0$. The quadratic $a x^{2}+b x+c$ has exactly one root. What is this root?

$-7-4 \sqrt{3}$

$-7+4 \sqrt{3}$

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Unary Quadratic Equations

It is given that $a x^{2}+b x+c=0$ has 1 real root, so the discriminant is zero, or $b^{2}=4 a c$. Because $a, b, c$ are in arithmetic progression, $b-a=c-b$, or $b=\frac{a+c}{2}$. We need to find the unique root, or $-\frac{b}{2 a}$ (discriminant is 0 ). From $b^{2}=4 a c$, we can get $-\frac{b}{2 a}=-\frac{2 c}{b}$ Ignoring the negatives(for now), we have $\frac{2 c}{b}=\frac{2 c}{\frac{a+c}{2}}=\frac{4 c}{a+c}=\frac{1}{\frac{1}{\frac{4 c}{a+c}}}=\frac{1}{\frac{a+c}{4 c}}=\frac{1}{\frac{a}{4 c}+\frac{1}{4}}$. Fortunately, finding $\frac{a}{c}$ is not very hard. Plug in $b=\frac{a+c}{2}$ to $b^{2}=4 a c$, we have $a^{2}+2 a c+c^{2}=16 a c$, or $a^{2}-14 a c+c^{2}=0$, and dividing by $c^{2}$ gives $\left(\frac{a}{c}\right)^{2}-14\left(\frac{a}{c}\right)+1=0$, so $\frac{a}{c}=\frac{14 \pm \sqrt{192}}{2}=7 \pm 4 \sqrt{3}$. But $7-4 \sqrt{3}\textless1$, violating the assumption that $a \geq c$. Therefore, $\frac{a}{c}=7+4 \sqrt{3}$. Plugging this in, we have $\frac{1}{\frac{a}{4 c}+\frac{1}{4}}=\frac{1}{2+\sqrt{3}}=2-\sqrt{3}$. But we need the negative of this, so the answer is (D).


The sum of five consecutive natural numbers is equal to $$2005$$. The greatest of these numbers is:~\uline{~~~~~~~~~~}~. (2005 Math Kangaroo Problem, Level 5-6, Question \#17)

Based on the middle term rule, we can find the middle number is $$401$$, and the greatest number is $$401+1+1=403$$.


Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers


In December, Tom-the-cat slept for exactly 3 weeks. Which calculations should we do in order to find how many minutes he stayed awake during this month?

(31 - 7) x 3 x 24 x 60

(31 - 7) x 24 x 60

(31 -7 x 3) x 24 x 60

(31 -7 x 3) x 24 x 60 x 60

(30 -7 x 3) x 24 x 60

Overseas Competition->Knowledge Point->Calculation Modules->Unit Conversion

The cat slept for exactly 3 weeks for 7 x 3 days.  The cat was awake for 31 - 7 x 3 days.  A day has 24 hours, and an hour has 60 minutes.  Therefore, the cat was awake in (31- 7 x 3) x 24 x 60


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Operation Strategy in Addition and Subtraction of Rounding Whole Numbers


If $$2$$ dogs weigh as much as $$3$$ cats, and $$2$$ cats weigh as much as $$15$$ mice, how many dogs weigh as much as $$45$$ mice?

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions


What is the simplest form of $$2$$ yards $$:30$$ feet? (Note that $1$ yard is equal to $3$ feet)．

Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions->Ratio

We need to make the units same first. $$2$$ yards equal to $$6$$ feet. Now we could remove the same unit, feet. We get $$6:30$$ and simplify it to $$1:5$$.


The sum of the first $$100$$ positive even whole numbers is $$10100$$.  What is the sum of the first $$101$$ positive even whole numbers?

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Arithmetic Sequences


The ones digit of $$106\times107\times108\times109\times110$$ is．

The ones digit is the same as the ones digit of $$6 \times7\times8\times9 \times0$$. `


Overseas Competition->Knowledge Point->Calculation Modules->Operations through Formulas-> Difference of Two Squares Formula


What is the tens digit of $7^{2011}$? (2011 AMC 8 Problems, Question \#22)

Overseas Competition->Knowledge Point->Calculation Modules

Since we want the tens digit, we can find the last two digits of $7^{2011}$. We can do this by using modular arithmetic.  $$ \begin{aligned} 7 \equiv 07 \&(\bmod 100) \textbackslash\textbackslash{} 7^{2} \equiv 49 \&(\bmod 100) \textbackslash\textbackslash{} 7^{3} \equiv 43 \&(\bmod 100) \textbackslash\textbackslash{} 7^{4} \equiv 01 \&(\bmod 100) \end{aligned} $$  We can write $7^{2011}$ as $\left(7^{4}\right)^{502} \times 7^{3}$. Using this, we can say: $$ 7^{2011} \equiv\left(7^{4}\right)^{502} \times 7^{3} \equiv 7^{3} \equiv 343 \equiv 43 \quad(\bmod 100) . $$  From the above, we can conclude that the last two digits of $7^{2011}$ are $43$. Since they have asked us to find the tens digit, our answer is (D) $4$


A palindrome is a number that can be read the same forward and backward. For example, the numbers 99, 252 and 4884 are palindromes. How many 3. digit palindrome numbers are there?

None of the above

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables


Given the equation $3x+4y=5$, find the slope and $y$-intercept.

$m=-3, b=4$

$m=-3, b=5$

$m=\frac{3}{4}, b=\frac{5}{4}$

$m=-\frac{3}{4}, b=\frac{5}{4}$

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Linear Equations with one Variable


Store $A$ is selling watermelon at the price of $32$ dollars per pound. Store $B$ is selling the same watermelon at the price of $30$ dollars per $16$ ounces. Which store has a better buy?

$A$

$B$


The number of hours in $$10$$ days $$=$$ the number of minutes inhours.

Overseas Competition->Knowledge Point->Calculation Modules->Unit Conversion->Converting between Units of Time

The number of hours in $$10$$ days is $$240$$; $$240$$ minutes is $$240\div60 =4$$ hours.


Divide through by $$2$$ then multiply by $$7$$ to get $$4:14=2:7=14:49$$.


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division


If $ a◆b$ means$(a\times b)+b$ , then $2◆3$ has the value．

Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Operating Directly

If $a◆b$~ represents$(a\times b)+b$ , $2◆3=(2\times3)+3=9$ .  So the answer is $\rm B$.


There are $$30$$ students in a classroom. They sit at desks in pairs in such a way that every boy sits with girl, and only half of girls sit with a boy. How many boys are in the classroom?

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers


Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Operations of Fractions


The product of any whole number and $$2$$ is always．

prime

composite

odd

even

The product of a whole number and an even number must be even.


The arrows on the two spinners shown below are spun. Let the number $N$ equal $10$ times the number on Spinner $A$, added to the number on Spinner $B$. What is the probability that $N$ is a perfect square number? (2022 AMC 8 Problems, Question \#12)

First, we calculate that there are a total of $4 \cdot 4=16$ possibilities. Now, we list all of two-digit perfect squares. $64$ and $81$ are the only ones that can be made using the spinner. Consequently, there is a $\frac{2}{16}=$ (B) $\frac{1}{8}$ probability that the number formed by the two spinners is a perfect square.


If $16^{-2}=\left(\frac{1}{4}\right)^{}x$, what is the value of $x$?

Convert both sides into powers with the same bases:  $4^{-4}=4^{-x}$  $x=4$.


The 2000 Census identified the ethnic breakdown of the state of California to be approximately as follows: White:46％, Latino:32\%, Asian:11\%, Blcak:7\%,and Other:4\%.~Assuming that these are mutually exclusive categories (this is not a realistic assumption), what is the probability that a randomly selected person from the state of California is of Asian or Latino descent?

The correct answer is(d). Because ethnic group categories are assumed to be mutually exclusive, P(Asian or Latino)=P(Asian)+P(Latino)=32\%+11\%=43\%


Kitty writes down a sequence of five integers. The rule she uses is, "after the first two terms, each term is the sum of the two previous terms." She sequence is ---, ---, ---, ~$$18$$, $$29$$.  What is her first term? ．

$$0 $$

$$ 3 $$

$$ 4 $$

$$ 5 $$

$$ 7$$

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns in Number Sequences

Let the first six terms of Kitty\textquotesingle s sequence be $$a$$, $$b$$, $$c$$, $$18$$ and $$29$$ respectively.  Then $$c+ 18= 29$$, so $$c= 11$$. Hence $$b+11= 18$$, so $$b=7$$.  Therefore, $$a+7=11$$, so $$a=4$$.


How many multiples of $$7$$ are between $$20$$ and $$100$$?

$$3\times7=21$$, start from the $$3^{}\rm{rd}$$  $$14\times7=98$$, end at the $$14^{}\rm{th}$$  Thus, the number of term remain: $$14-2=12$$


Half of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 4 cups. What percent of the total capacity of the pitcher did each cup receive?~ (adapted from 2020 AMC 8, Question \#5)

Overseas Competition->Knowledge Point->Calculation Modules->Percentage Calculation

The pitcher is half full, i.e. $50 \textbackslash\%$ full. Therefore each cup receives $\frac{50}{4}=(\mathbf{C}) 12.5$ percent of the total capacity.


\textbf{The distribution of the number of hours worked by volunteers last year at a large hospital is approximately normal with mean 80 and standard deviation 7. Volunteers in the top 20 percent of hours worked will receive a certificate of merit. If a volunteer from last year is selected at random, which of the following is closest to the probability that the volunteer selected will receive a certificate of merit given that the number of hours the volunteer worked is less than 90?}


None of the above


In a fruit market, $$1\textasciitilde\text{kg}$$ of apples is priced at $$\textbackslash$a$$. $$1\textasciitilde\text{kg}$$ of strawberries costs $$\textbackslash$b$$ more than $$2$$ times $$a$$. How much will $$3\textasciitilde\text{kg}$$ of strawberries cost?

$$1\textasciitilde\text{kg}$$ of strawberries costs $$2a+b$$,  so $$3\textasciitilde\text{kg}$$ of strawberries costs $$3\times (2a+b)=6a+3b$$.  So, the answer is C.


We want to know the $2021^{th}$ term and the $2022^{th}$ term to get the median. We know that $44^{2}=1936\textless2021$, and $45^{2}=2025\textgreater2021$.  So, the number $1^{2}$, $2^{2}$, $3^{2}$, $\cdots $, $44^{2}$ are between $1$ to $1936$. $1936+44=1980$, which mean that $1936$ is the $1980^{th}$ number.  Thus, the $2021^{th}$ term will be $1936+41=1977$, and similarly the $2021^{th}$ term will be $1978$.  So, the answer is $1977.5$.


If the degree measures of the angles of a triangle are in the ratio $3: 3: 4$, what is the degree measure of the largest angle of the triangle?

Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions

The sum of the ratios is 10 . Since the sum of the angles of a triangle is $180^{\circ}$, the ratio can be scaled up to $54: 54: 72(3 \cdot 18: 3 \cdot 18: 4 \cdot 18)$. The numbers in the ratio $54: 54: 72$ represent the angles of the triangle. The question asks for the largest, so the answer is


The GPAs (grade point averages) of students who take the AP Statistics exam are approximately normally distributed with a mean of 3.4 and a standard deviation of 0.3. What is the probability that a student selected at random from this group has a GPA lower than 3.0?


Let $a$ and $b$ be two consecutive odd integers. If $a$ is three times $b$, what is their sum $a+b$?


Overseas Competition->Knowledge Point->Calculation Modules->Comparing, Ordering and Estimating

Sugar water theory. 1 gram of sugar added each time, and the sugar water gets sweeter.


What is the smallest whole number larger than the perimeter of any triangle with a side of length $12$ and a side of length $13$? (adapted from 2015 AMC8, Question 8)

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities

We know from the triangle inequality that the last side, $s$, fulfills $s\textless12+13$. Adding $12+13$ to both sides of the inequality, we get $s+12+13\textless50$, and because $s+12+13$ is the perimeter of our triangle, (B) 50 is our answer.


Which of the following groups has equivalent ratios?

$1$ to $5$, $\frac{0.5}{1}$

$$\frac{1}{5}$$, $1:5$

$\frac{1}{5}$, $5$ to $10$

$1:5$, $\frac{5}{10}$

Overseas Competition->Knowledge Point->Calculation Modules->Operations through Formulas


Which of the following numbers\textquotesingle{} value does not change after removing all "$$0$$" s ．

Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals


Match the linear equation, $6x-3y=2$, with its corresponding $y$-intercept.


Given that $$a\Psi b=a\times b+2$$, for example, $$3\Psi 1 = 3\times1 +2$$, what is $$3\Psi4$$?

Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition


Real numbers $x$ and $y$ satisfy $x+y=4$ and $x \cdot y=-2$. What is the value of $$ x+\frac{x^{3}}{y^{2}}+\frac{y^{3}}{x^{2}}+y? $$

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Proportional Equations


The ratio of $A$ to $B$ is $3:4$. If we multiply $A$ by $3$, $B$ should~\uline{~~~~~~~~~~}~.

multiply by $4$

divide by $3$

add by $8$


One apple, one banana, and two peaches together weigh $12$ lbs. One apple and one peach together weigh $5$ lbs. One banana and $2$ peaches together weigh $5$ lbs more than one apple and one peach weigh together. Each peach weighs the same. How many pounds does one banana weigh?

$3$ lbs

$4$ lbs

$5$ lbs

$6$ lbs

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution


$$\frac{5}{2} \sqrt{2}$$

$$\frac{7}{2} \sqrt{2}$$

Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers


Let $a, b$, and $c$ be positive integers with $a \geq b \geq c$ such that $a^{2}-b^{2}-c^{2}+a b=2011$ and $a^{2}+3 b^{2}+3 c^{2}-3 a b-2 a c-2 b c=-1997$. What is $a$?

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Addition and Subtraction of Equations

Add the two equations. $$ 2 a^{2}+2 b^{2}+2 c^{2}-2 a b-2 a c-2 b c=14 . $$ Now, this can be rearranged and factored.  $$ \begin{aligned} \&\left(a^{2}-2 a b+b^{2}\right)+\left(a^{2}-2 a c+c^{2}\right)+\left(b^{2}-2 b c+c^{2}\right)=14 \textbackslash\textbackslash{} \&(a-b)^{2}+(a-c)^{2}+(b-c)^{2}=14 \end{aligned} $$.  $a, b$, and $c$ are all integers, so the three terms on the left side of the equation must all be perfect squares. We see that the only is possibility is $14=9+4+1$ $(a-c)^{2}=9 \Rightarrow a-c=3$, since $a-c$ is the biggest difference. It is impossible to determine by inspection whether $a-b=1$ or 2 , or whether $b-c=1$ or 2 . We want to solve for $a$, so take the two cases and solve them each for an expression in terms of $a$. Our two cases are $(a, b, c)=(a, a-1, a-3)$ or $(a, a-2, a-3)$. Plug these values into one of the original equations to see if we can get an integer for $a$. $a^{2}-(a-1)^{2}-(a-3)^{2}+a(a-1)=2011$, after some algebra, simplifies to $7 a=2021$. 2021 is not divisible by 7 , so $a$ is not an integer. The other case gives $a^{2}-(a-2)^{2}-(a-3)^{2}+a(a-2)=2011$, which simplifies to $8 a=2024$. Thus, $a=253$.


There are four soccer teams that are competing in a round-robin format. In the event of a draw, $$1$$ point would be awarded to both teams. $$3$$ points would be given to the team that wins and $$0$$ points would be given to the team that loses. The scores of Team $$A$$, $$B$$, $$C$$ and $$D$$ are $$7$$, $$4$$, $$4$$ and $$1$$ point respectively. How many matches ended in a draw?

Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Finding Patterns

We can first assume that all games ended in a $win-lose$ scenario.  Number of games played $=3+2+1=6$  Maximum total score $=6\times3=18$  Everytime a $win-lose$ scenario changes to a $draw-draw$ scenario, the total score decreases by $3-2=1$  Difference in score $=18-16=2$  Number of matches that ended in a draw $=2\div1=2$


The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$?

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Unary Quadratic Equations

The sum of the first $m$ odd integers is given by $m^{2}$. The sum of the first $n$ even integers is given by $n(n+1)$. Thus, $m^{2}=n^{2}+n+212$. Since we want to solve for $n$, rearrange as a quadratic equation: $n^{2}+n+\left(212-m^{2}\right)=0$. Use the quadratic formula: $n=\frac{-1+\sqrt{1-4\left(212-m^{2}\right)}}{2}$. Since $n$ is clearly an integer, $1-4\left(212-m^{2}\right)=4 m^{2}-847$ must be not only a perfect square, but also an odd perfect square for $n$ to be an integer.  Let $x=\sqrt{4 m^{2}-847}$; note that this means $n=\frac{-1+x}{2}$. It can be rewritten as $x^{2}=4 m^{2}-847$, so $4 m^{2}-x^{2}=847$. Factoring the left side by using the difference of squares, we get $(2 m+x)(2 m-x)=847=7 \cdot 11^{2}$. Our goal is to find possible values for $x$, then use the equation above to find $n$. The difference between the factors is $(2 m+x)-(2 m-x)=2 m+x-2 m+x=2 x$. We have three pairs of factors, $847 \cdot 1,121 \cdot 7$, and $77 \cdot 11$. The differences between these factors are 846,114 , and 66 - those are all possible values for $2 x$. Thus the possibilities for $x$ are $423$, $57$, and $33$. Now plug in these values into the equation $n=\frac{-1+x}{2}$, so $n$ can equal $211$, $28$, or $16$, hence the answer is $255$.


Adding five of the six fractions $$\frac{1}{2}$$, $$\frac{2}{3}$$, $$\frac{1}{4}$$, $$\frac{1}{6}$$, $$\frac{1}{9}$$ and $$\frac{1}{18}$$ gives a total of $$1.5$$.  Which of the fractions is not used?


One tour minibus can seat no more than $$8$$ people. What is the smallest number of minibuses needed to take $$75$$ people? (Adapted from 2000 Math Kangaroo Problem, Level 3-4, Question \#6)

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division->Division of Whole Numbers->Division with Remainders


Let $A$ and $B$ be positive whole numbers. $A$ is a $2$-digit number and $B$ is a $4$-digit number. If $A+B$ and $A^{2}+B^{2}$ are both multiples of $7$, find the largest possible value of $B-A$.

None of the above

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Calculation of Multi-digit Numbers


How many millimetres are there in $$0.08$$ kilometres?

$$\rm 80 mm$$

$$\rm 800 mm$$

$$\rm 8000 mm$$

$$\rm 80000 mm$$

Overseas Competition->Knowledge Point->Calculation Modules->Unit Conversion->Converting between Units of Length


Two integers are inserted into the list $3,3,8,11,28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?

Overseas Competition->Knowledge Point->Calculation Modules


The hundreds digit of the product $$2014\times400$$ is．

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division


Function $f$ is defined for the set of all natural number as follows:  $$f(x)=\begin{cases}\dfrac{x}{2},(\text{when }x\text{ is even}) \textbackslash\textbackslash{} x+1,(\text{when }x\text{ is odd}) \textbackslash\textbackslash{} \end{cases}$$.  For example: $$f(3)=3+1=4$$, $$f(2)=2\div 2=1$$.  What is the value of $$f(f(f(f(f(17)))))$$?

Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Number Machine

$$\textasciitilde\textasciitilde\textasciitilde\textasciitilde f\left( f\left( f\left( f\left( f\left( 17 \right) \right) \right) \right) \right)$$  $$=f\left( f\left( f\left( f\left( 18 \right) \right) \right) \right)$$  $$=f\left( f\left( f\left( 9 \right) \right) \right)$$  $$=f\left( f\left( 10 \right) \right)$$  $$=f\left( 5 \right)$$  $$=6$$．


Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Multiplication and Division of Decimals


Preview Question:  Which of the following is not an expression?

$$a$$

$$a+b=a+b$$

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions


Which one of the following fractions is larger than $\dfrac{1}{4}$?

Overseas Competition->Knowledge Point->Calculation Modules->Comparing, Ordering and Estimating->Comparing and Ordering


Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Complex Fractions


What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$?


\textbf{Ten percent of all trucks undergoing a certain inspection will fail the inspection. Assume that trucks are independently undergoing this inspection one at a time. The expected number of trucks inspected before a truck fails inspection is}

\textbf{This follows a geometric distribution.}  $$\mu = \frac{1}{p} = \frac{1}{0.10} = 10$$


Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Arithmetic Sequences->Sum of Terms in Arithmetic Sequences


3, 6, , 12, 15.~~Which number should be filled in the bracket?

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Arithmetic Sequences


\textbf{Which of the following events are independent?}  \textbf{P(A) = 0.7, P(B) = 0.2, P(C) = 0.1, P(B\textbar A) = 0.5, P(B\textbar C) = 0.2, P(A∩C) = 0.0}

\textbf{A and B only}

\textbf{A and C only}

\textbf{A, B, and C}

\textbf{B and C only}

\textbf{None are independent}

\textbf{P(B\textbar A)\ne P(B) → A and B not independent}  \textbf{P(B\textbar C) =P(B) → B and C independent}  \textbf{P(A\textbar C)=P(A∩B)/P(C)=0.05/0.1=0.5\ne P(A)→ A and C not independent}


Peter bought three apples of $5$ dollars, and $4$ bananas of $4$ dollars. How much did he spend in total?~(adapted from $$2011$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$7$$)

$$6$$ dollars

$$7$$ dollars

$$8$$ dollars

$$9$$ dollars

$$10$$ dollars

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Addition of Whole Numbers->Addition in Horizontal Form


A class had more boys than girls. After winter transfer, $3$ boys and $3$ girls joined the class. What is the relationship between boys and girls in the class now?

boys $\textgreater$ girls

boys $=$ girls

boys $\textless$ girls

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities

When add the same number to both sides of the inequality, the equation is still true


If $$2$$ hoots $$=1$$ holler, then $$(10\times 1)$$ hollers $$=(10\times 2)$$ hoots.


Which three numbers can be put in the blanks to make the statement correct?  $6 \times$~\uline{~~~~~~~~~~}~$-$~\uline{~~~~~~~~~~}~$=$ $21 +$~\uline{~~~~~~~~~~}~$\times 2$

$5$, $4$ and $2$

$6$, $2$ and $7$

$7$, $5$ and $8$

$8$, $7$ and $11$

$9$, $7$ and $12$


The original price of a product was $$80$$ dollars, and it\textquotesingle s on sale for 30\% off, this product isdollars cheaper than before．

the new price is $$70\textbackslash\%$$ of the original price, so the new price is $$80\times 70\textbackslash\%=56$$；  $$80-56=24$$．  so choose $$\text{B}$$．


A recipe that makes~$5$~servings of hot chocolate requires~$2$~squares of chocolate,~$\dfrac{1}{4}$~cup sugar,~$1$~cup water and~$4$~cups milk. Jordan has~$5$~squares of chocolate,~$2$~cups of sugar, lots of water, and~$7$~cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make?

Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions

Assuming excesses of the other ingredients, the chocolate can make~$\dfrac{5}{2}\cdot5=12.5$~servings, the sugar can make~$\dfrac{2}{1/4}\cdot5=40$~servings, the water can make unlimited servings, and the milk can make~$\dfrac{7}{4}\cdot5=8.75$~servings. Limited by the amount of milk, Jordan can make at most~$\boxed{\left( D\right)\textbackslash{} 8\dfrac{3}{4}}$~servings.


Bob is standing in the sixth place from the front to the back. The teacher distributes the apples from front to back according to the pattern as $1$, $3$, $5$ and so on, which means the first student has $1$ apple, and the second student has $3$ apples\ldots How many apples can Linda get if Linda is standing behind to Bob?~(adapted from $$2006$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$1$$)

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns in Number Sequences


In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term in the sequence is $4000$. What is the first term?


Compare the following pair of fractions using an inequality sign.  $$A=\frac{773}{778}$$, $$B=\frac{884}{889}$$

$$\frac{773}{778}\textgreater\frac{884}{889}$$

$$\frac{773}{778}\textless{}\frac{884}{889}$$

Overseas Competition->Knowledge Point->Calculation Modules->Fractions


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction


Alysha and Julia have some biscuits. Altogether they have $28$ biscuits.  Alysha has $4$ more biscuits than Julia.  How many biscuits does Alysha have?


Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3 x+y}{x-3 y}=-2$. What is the value of $\frac{x+3 y}{3 x-y}$? (2017 AMC 10B Problems, Question \#4)

Rearranging, we find $3 x+y=-2 x+6 y$, or $5 x=5 y \Longrightarrow x=y$. Substituting, we can convert the second equation into $$\frac{x+3 x}{3 x-x}=\frac{4 x}{2 x}=\text { (D) } 2$$.


Which set of numbers go into the missing blanks respectively to make the equation true?  $7=14$ $\div$~\uline{~~~~~~~~~~}~$=\frac{14}{\square}=\frac{14}{1}\times\frac{1}{\square}$

$2, 1, 2$

$1, 2, 2$

$2, 2, 1$

$2, 2, 2$

Dividing a term by $2$ is the same as multiplying it by half ($\frac{1}{2}$)!


The sixteenth term of an arithmetic progression is 40. The fifty-fifth term of this arithmetic progression is 157. Find the eighty-first term of this arithmetic progression.

By the formula~$d=\dfrac{a\_{m}-a\_{n}}{m-n}\textbackslash{} \Rightarrow\textbackslash{} d=\dfrac{a\_{55}-a\_{16}}{55-16}=\dfrac{157-40}{39}=3.$  By the formula~$a\_{n}=a\_{m}+\left( n-m\right)d\textbackslash{} \textbackslash{} \textbackslash{} \Rightarrow\textbackslash{} \textbackslash{} a\_{81}=a\_{55}+\left( 81-55\right)\times3=235.$


Fido the Dog, Philemon the Cat and $4$ monkeys together weigh $24$ lbs. Fido and one monkey together weigh $11$ lbs. Philemon and $2$ monkeys together weigh $1$ lb less than Fido and one monkey weigh together. Each monkey weighs the same. How much does Philemon weigh? (2011 Math Kangaroo Problem, Level 1-2, Question \#21)

$3$ lbs

$4$ lbs

$5$ lbs

$6$ lbs

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution


Which of the following is not an expression?

$$z$$


David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? (2014 AMC 10A Problems, Question \#15)

Note that he drives at $50$ miles per hour after the first hour and continues doing so until he arrives. Let $d$ be the distance still needed to travel after $1$ hour. We have that $\frac{d}{50}+1.5=\frac{d}{35}$, where the $1.5$ comes from $1$ hour late decreased to $0.5$ hours early. Simplifying gives $7 d+525=10 d$, or $d=175$. Now, we must add an extra $35$ miles traveled in the first hour, giving a total of (C) $210$ miles.


\textbf{A grocer would like to determine the proportion of milk cartons that have expired within 0.05 of the true proportion with a 95 percent confidence interval. What is the minimum required sample size?}

\textbf{$n \geq (\frac{Z\_{\alpha/2}}{ME})^{2} {p(1-p)}$}  \textbf{Since p is unknown, then use p = ½. $n \geq (\frac{Z\_{\alpha/2}}{2ME})^{2}$}  \textbf{$Z\_{\alpha/2}=1.96$ The ME is how far off from the true proportion you are willing to be, in this case, 0.05}  \textbf{$n \geq (\frac{1.96}{2*0.05})^{2} = 384.16$ round up}


Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers


Convert the decimal $$0.65$$ to a fraction in its simplest form.

Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals->Converting Decimals into Fractions

First, write down the decimal "over" the number 1 :~$0.65=\dfrac{0.65}{1}$  Then multiply top and bottom by 100 since there are two numbers after the decimal point :~$\dfrac{0.65}{1}=\dfrac{0.65\times100}{1\times100}=\dfrac{65}{100}$  This makes it a correctly formed fraction. Then simplify the fraction (in this case by dividing top and bottom by 5) :~$\dfrac{65}{100}=\dfrac{13}{20}$


Choose the correct number to make the number statement true.  $$17-9+$$$$\textgreater15+3-8$$

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers

$$15+3-8=10$$，$$17-9=8$$，$$8+\left( 2 \right)=10$$，$$number$$ has to be bigger than $$2$$ so $$5$$ fits the requirement.


A consumer is willing to pay $\textbackslash$ 12$ for a good, but is able to purchase it for $\textbackslash$ 10$. What is the consumer surplus in this scenario?

$\textbackslash$ 2$

$\textbackslash$ 10$

$\textbackslash$ 12$

$\textbackslash$ 22$

Overseas Competition->Knowledge Point->Calculation Modules

Consumer surplus is calculated as the difference between the maximum price a consumer is willing to pay for a good and the actual price they pay, which is $\textbackslash$ 12$- $\textbackslash$10$ = $\textbackslash$ 2$.


Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Operations of Fractions


What is the next number in the sequence below?  $$3, 5, 8, 13, 21, $$~\uline{~~~~~~~~~~}~

None of the above


Which calculation has a result of an odd number．

$$2\times \left( 0+2+2 \right)\times 2021$$

$$2\times 2021-0-2-2$$

$$2021-2\times 0\times 2\times 2$$

$$2+\left( 0\times 2\times 2 \right)\times 2021$$

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations


The $2021^{st}$ digit at the right of the decimal point in the decimal expression of $\dfrac{2}{7}$ is~\uline{~~~~~~~~~~}~.

Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals

$$\frac{2}{7}=0.\overline{285714}$$, it is a decimal which repeats in cycles of $6$ digits. Every $6$$^{th}$ digit is $4$. The $2022$$$^{nd}$$ digit is $4$, so the $2021$$^{st}$ digit is $1$.


Betty has more than $90$ toys. All her toys can be divided evenly between $2$, $3$, or $4$ children. However, they cannot be divided evenly between $9$ children because $3$ more toys would be needed. How many toys does she have at least?

After adding $3$ toys, the number of toys should be divisible by $9$, and it also should be divisible by $2, 3, $ and $4$. Thus, the answer is $B$.


Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Multiplication and Division of Decimals


Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?


None of the above

Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Fast Calculation in Fractions


Express the recurring decimal~$0.2\dot{5}$~as a fraction. ．

Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Recurring Decimals


Ethan is shopping for an Eiffel Tower model online. The description says the scale of the model is $1:1000$. The height of the Eiffel Tower is $1083$ feet, so the model should be~\uline{~~~~~~~~~~}~inches tall (round to the nearest integer, $12$ inches $=1$ foot ).


For $\triangle ABC$, all of its side lengths are integers. The perimeter of $\triangle ABC$ with a side of length $12$ and a side length of $7$ is at least ．

We know from the triangle inequality that the last side, $s$, fulfills $s+7\textgreater12$. $P=s+7+12\textgreater12+12$. Therefore, $P\textgreater24+1=25$


There are seven people in a bus. Four passengers get on and $6$ passengers get off at the first station. Then seven passengers get on and three get off at the second station. How many passengers are there on the bus at this time?

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Addition and Subtraction of Whole Numbers


What is the sum: $$3+7+2+8+1=?$$ (2007 Math Kangaroo Problem, Level 1-2, Question \#17)


Which of the following statements is true?


What is the product of $$ \frac { 3 } { 2 } \times \frac { 4 } { 3 } \times \frac { 5 } { 4 } \times \cdots \times \frac { 2 0 0 6 } { 2 0 0 5 }$$?

Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Fast Calculation in Fractions->Reducing Fractions by cancelling out successively

By telescoping, it\textquotesingle s easy to see the sum becomes $$\frac {2006}2=1003$$.


In a dining room, there are $$15$$ chairs, $$5$$ tables, and $$20$$ cups. What is the ratio of chairs to cups?

Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions->Ratio

There are $$15$$ chairs and $$20$$ cups. So the ratio of chairs to cups is $$15:20$$. The simplest form is $$3:4$$.


Which of the following expression has the maximum value?


Teacher Judy has $71$ stickers in total, and she gives all the stickers to her students Martin and David. Martin gets $7$ more than David. How many stickers does David get?


What is the last number on the $20^{th}$ row?

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns of Number Tables->Triangle Number Table


Pinocchio has a magic nose, which will stretch out $5$ cm if he tells a lie and will shorten itself $1$ cm if he tells a truth. At the beginning his nose was $11$ cm long. The length of his nose changed into $13$ cm after he had said $10$ sentences. If these sentences were not true, they were definitely false. Thus, there were~\uline{~~~~~~~~~~}~truths among these $10$ sentences.

If these $10$ sentences are true, the nose will shorten $10$ cm. There are $(13-11+10)\div(5+1)=2$ sentences which are false. Thus, $10-2=8$ sentences are true.


Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Complex Fractions


Which numbers should be filled in the parentheses to make the equation correct?  $\frac14=\frac{(\textasciitilde\textasciitilde\textasciitilde\textasciitilde)}{16}=\frac8{(\textasciitilde\textasciitilde\textasciitilde\textasciitilde)}$

$4; 20$

$16; 32$

$4; 32$

$4; 40$

Overseas Competition->Knowledge Point->Calculation Modules->Fractions

When the denominator and numerator are multiplied by the same number, the value of the fraction remains equal.


A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^{\text {th }}$-graders to $6^{\text {th }}$-graders is $5: 3$, and the the ratio of $8^{\text {th }}$-graders to $7^{\text {th }}$ graders is $8: 5$. What is the smallest number of students that could be participating in the project? (2013 AMC 8, Question 16)

Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions

We multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8 th graders, in order that we can put the two ratios together: $$ \begin{aligned} \&5: 3=5(8): 3(8)=40: 24 \textbackslash\textbackslash{} \&8: 5=8(5): 5(5)=40: 25 \end{aligned} $$ Therefore, the ratio of 8th graders to 7th graders to 6th graders is $40: 25: 24$. Since the ratio is in lowest terms, the smallest number of students participating in the project is $$ 40+25+24=\text { (E) } 89 $$


What is the largest number which is both less than $$2\times3\times5\times7$$ and also a divisor of $$2 \times 3 \times 5\times 7$$?

$$2 \times5 \times7$$

$$3 \times5 \times7$$

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division

The largest divisor of $$2\times3\times5\times7$$ that is less than~ is $$3\times5\times7$$, which we get by dropping the smallest factor.


Which of the following is not an algebraic expression?

$xyzabc$

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions


$$2^{5}$$ means $$2$$ multipled by itself $$5$$ times, i.e. $$2^{5}=2\times2\times2\times2\times2=32$$. What is $$3^{4}$$ equal to?

None of the above


Given that $$\left[ x+0.19 \right]+\left[ x+0.20 \right]+\left[ x+0.21 \right]+\cdots \cdots +\left[ x+0.91 \right]=546$$. Find $$\left[ 100x \right]$$

None of the above


Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132$. (2016 AMC 8 Problems, Question \#11)

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Calculation of Multi-digit Numbers

We can write the two digit number in the form of $10 a+b$; reverse of $10 a+b$ is $10 b+a$. The sum of those numbers is:  $$ \begin{gathered} (10 a+b)+(10 b+a)=132 \textbackslash\textbackslash{} 11 a+11 b=132 \textbackslash\textbackslash{} a+b=12 \end{gathered} $$  We can use brute force to find order pairs $(a, b)$ such that $a+b=12$. Since $a$ and $b$ are both digits, both $a$ and $b$ have to be integers less than $10$.  Thus our ordered pairs are $(3,9) ;(4,8) ;(5,7) ;(6,6) ;(7,5) ;(8,4) ;(9,3)$ or $(\mathbf{B}) 7$ ordered pairs.


There is an equal number of motorcycles and cars in the parking lot. They have $$48$$ wheels in total. How many cars are there in the parking lot?

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Linear Equations with one Variable


Judy~ is waiting in a line to buy some toys. There are $7$ people in front of her, and $3$ people behind her. In total, how many people are there in the line?

Overseas Competition->Knowledge Point->Calculation Modules


What is the value of $1+3+5+\ldots+2017+2019-2 -4-6-\ldots-2016-2018$? (Adapted from $2018$ AMC 8 Problem, Question \#5)

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Grouping in Fast Addition and Subtraction of Whole Numbers


At an ice cream shop, a sundae is made by selecting two flavors of ice cream and topping them with fudge, whipped cream, nuts, and a cherry. The available flavors of ice cream are chocolate, vanilla, strawberry, rocky road, chocolate chip cookie dough, and mint chip. How many difference sundae combinations are possible?


Six pencils and two erasers cost $$3.20$$ dollars. One pencil costs $$40$$ cents. How much does one eraser cost? (Adapted from 2000 Math Kangaroo Problem, Level 3-4, Question \#5)

$$20$$ cents

$$30$$ cents

$$40$$ cents

$$50$$ cents

$$60$$ cents

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division->Division of Whole Numbers->Division out of the Multiplication Table


Kitty writes down a sequence of five integers. The rule she uses is, "after the first two terms, each term is the sum of the two previous terms." She sequence is~\uline{~~~~~~~~~~}~,~\uline{~~~~~~~~~~}~,~\uline{~~~~~~~~~~}~, $$18$$, $$29$$.  What is her first term? ．

$$0 $$

$$ 3 $$

$$ 4 $$

$$ 5 $$

$$ 7$$

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns in Number Sequences

Let the first six terms of Kitty\textquotesingle s sequence be $$a$$, $$b$$, $$c$$, $$18$$ and $$29$$ respectively.  Then $$c+ 18= 29$$, so $$c= 11$$. Hence $$b+11= 18$$, so $$b=7$$.  Therefore, $$a+7=11$$, so $$a=4$$.


\textbf{A set of 5,000 scores on a college readiness exam are known to be approximately normally distributed with a mean of 72 and a standard deviation of 6. To the nearest integer value, approximately how many scores are between 63 and 75.}


If $$\frac{2\textbar x-2\textbar+1}{3}\textless{}1$$, the range of $x$ is~\uline{~~~~~~~~~~}~.

$$-3\textless x\textless-1$$

$$x\textgreater-1$$ or $$x\textless-3$$

$$1\textless x\textless3$$

$$x\textgreater3$$ or $$x\textless1$$

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities

$$\frac{2\textbar x-2\textbar+1}{3}\textless{}1$$  $$2\left\textbar x-2\right\textbar+1\textless3$$  $$2\left\textbar x-2\right\textbar\textless2$$  $$1\left\textbar x-2\right\textbar\textless1$$  $$-1\textless x-2\textless1$$  $$1\textless x\textless3$$


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction


Alicia and Emily agreed to meet at the cinema at 3.55pm. Emily left her house at 1.47pm but arrived at the cinema 17 minutes late. How long was Emily\textquotesingle s journey from her house to the cinema?

$$189$$ minutes

$$172$$ minutes

$$216$$ minutes

$$206$$ minutes

None of the above.

Overseas Competition->Knowledge Point->Calculation Modules->Unit Conversion

3: 55pm - 1: 47om + 15 minutes = 145 minutes.


The average weight of dogs that come to a certain vet\textquotesingle s office is 55.6 lbs, with a standard deviation of 2.2 lbs. If the weights are normally distributed, what percent of dogs weight more than 60 lbs?


There are four more girls than boys in Ms. Raub\textquotesingle s class of 28 students. What is the ratio of number of girls to the number of boys in her class? (2014 AMC 8, Question 7)

We can set up an equation with $x$ being the number of girls in the class. The number of boys in the class is equal to $x-4$. Since the total number of students is equal to 28 , we get $x+x-4=28$. Solving this equation, we get $x=16$. There are $16-4=12$ boys in our class, and our answer is $16: 12=$ (B) $4: 3$.


A rectangular water tank is filled to a depth of $$70$$cm. It contains $$1050$$ litres of water. Some water is taken out of the tank. The water level drops by $$25$$cm. How much water is left in the tank?

$$625\rm L$$

$$375\rm L$$

$$525\rm L$$

$$270\rm L$$

$$675\rm L$$


There is a sequence of squares of some natural numbers: $1$, $4$, $9$, $16$, $\cdots$ One of the numbers in this sequence is $10^{8}$. What is the next number after $10^{8}$? ($2001$ Math Kangaroo Problem, Level $$11$$-$$12$$, Question \#$14$)

$\left (10^{4}+1\right )^{2}$

$\left (10^{8}+1\right )^{2}$

$\left (10^{5}\right )^{2}$

$\left (10^{8}\right )^{2}$

$\left (10^{4}\right )^{2}+1$

Overseas Competition->Knowledge Point->Calculation Modules->Power->Application of Powers


Which is the solution of the following inequality:  $-11x-10\geq30-x$.


Mary went to the stationery store and she bought three pencils, a notebook, and a stationery box. A pencil costs $1$ dollar, a notebook costs $2$ dollars, a pencil case costs $3$ dollars, and Mary has a $1$ dollar discount coupon. How much did Mary spend in the end?~(adapted from $$2005$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$6$$)

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Addition and Subtraction of Whole Numbers->Questions Involving Addition and Subtraction

Three pencils are three dollars. Notebook is two dollars. A three dollar pencil case and a one dollar free at last. So the last answer is $3+2+3-1=7$


Mike has $7y+1$ cans of coke. He gives his friends $$x$$ cans of coke. And he takes one out. How many cans of coke does he have left?

$$7y + 1$$

$$7y + 1 -x$$


The sum of the whole numbers from $$1$$ through $$100$$ is $$5050$$. What is the sum of the whole numbers from $$1$$ through $$200$$?

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Arithmetic Sequences


Which of the following is the same as 2018 cm?

2 metres and 18 centimetres

2 kilometres and 18 centimetres

20 metres and 18 centimetres

201 metres and 8 centimetres

None of the above


What fraction of the integers from $$1$$ to $$1000$$ inclusive are cubes? ．

Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers

As $$1=1^{3}$$ and $$1000 = 10^{3}$$, there are $$10$$ cubes from $$1$$ to $$1000$$. So the fraction of the integers from $$1$$ to $$1000$$ inclusive which are cubes is $$\frac{10}{1000}= \frac{1}{100}$$.


After the final exam results came out, four students wanted to compare who had done best. Who had the best grades?~(adapted from $$2007$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$4$$)  Alice：$12+8$  Bob：$7+9$  Alan：$14+5$~  Tom：$6+8$  Susan: $6+15$

Alice

Bob

Alan

Tom

Susan

Overseas Competition->Knowledge Point->Calculation Modules->Comparing, Ordering and Estimating->Comparing and Ordering->Comparing and Ordering the Value of Expressions

Alice：$12+8=20$  Bob：$7+9=16$  Alan：$14+5=19$~  Tom：$6+8=14$  Susan: $6+15=21$  $21\textgreater20\textgreater19\textgreater16\textgreater14$


Overseas Competition->Knowledge Point->Calculation Modules->Negative Numbers->Operations on Negative numbers

Subtract a negative number is the same as add a postive number with the same absolute value. $a-(-b)=a+b$


Kit played computer games for $225$ minutes. When he stopped gaming and decided to head to sleep, he checked the time and noticed that it was $11.20\text{p.m.}$ What time did he start playing computer games?

Overseas In-curriculum->Knowledge Point->Measurement->Time->Time Calculation

$225$ minutes $=3$ hours $45$ minutes.  He started playing at $7.35\text{p.m.}$


Daniel had a package of $$36$$ pieces of candy. Without breaking any pieces of candy, he divided all the candy equally among his friends without remaining. Which of the following was definitely not the number of his friends? (2013 Math Kangaroo Problem, Level 3 - 4, Question \#6)

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division->Division of Whole Numbers->Division with Remainders


One tour bus can seat no more than $$50$$ people. What is the smallest number of buses needed to take $$160$$ people? (Adapted from 2000 Math Kangaroo Problem, Level 3-4, Question \#6)


If the degree measures of the angles of a convex quadrilateral are in the ratio~$3:4:5:6$, by how many degrees does the measure of the largest angle exceed the measure of the smallest angle?

Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions


There were five books on the shelf. Later, Jack took away three, and Mike took back four. Jim also took back three. How many books were there on the shelf?~(adapted from $$2005$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$5$$)


How many different four-digit numbers can be formed by rearranging the four digits in $2004$?~(2004 AMC 8 Problem, Question \#2)


The sum of the first $m$ positive odd integers is 212 more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$ ?

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Unary Quadratic Equations

The sum of the first $m$ odd integers is given by $m^{2}$. The sum of the first $n$ even integers is given by $n(n+1)$. Thus, $m^{2}=n^{2}+n+212$. Since we want to solve for $n$, rearrange as a quadratic equation: $n^{2}+n+\left(212-m^{2}\right)=0$. Use the quadratic formula: $n=\frac{-1+\sqrt{1-4\left(212-m^{2}\right)}}{2}$. Since $n$ is clearly an integer, $1-4\left(212-m^{2}\right)=4 m^{2}-847$ must be not only a perfect square, but also an odd perfect square for $n$ to be an integer.  Let $x=\sqrt{4 m^{2}-847}$; note that this means $n=\frac{-1+x}{2}$. It can be rewritten as $x^{2}=4 m^{2}-847$, so $4 m^{2}-x^{2}=847$. Factoring the left side by using the difference of squares, we get $(2 m+x)(2 m-x)=847=7 \cdot 11^{2}$. Our goal is to find possible values for $x$, then use the equation above to find $n$. The difference between the factors is $(2 m+x)-(2 m-x)=2 m+x-2 m+x=2 x$. We have three pairs of factors, $847 \cdot 1,121 \cdot 7$, and $77 \cdot 11$. The differences between these factors are 846,114 , and 66 - those are all possible values for $2 x$. Thus the possibilities for $x$ are 423,57 , and 33 . Now plug in these values into the equation $n=\frac{-1+x}{2}$, so $n$ can equal 211,28 , or 16 , hence the answer is 255 .


1. In the number 98, the digit "9" is in the ones place.~\uline{~~~~~~~~~~}~

Yes

No

Overseas Competition->Knowledge Point->Calculation Modules


What is the area of the triangle formed by the lines $y=5, y=1+x$, and $y=1-x$? (2019 AMC 8 Problems, Question \#21)

$y=x+1$ and $y=-x+1$ have $y$-intercepts at $(0,1)$ and slopes of 1 and $-1$, respectively. Since the product of these slopes is $-1$, the two lines are perpendicular. From $y=5$, we see that $(-4,5)$ and $(4,5)$ are the other two intersection points, and they are 8 units apart. By symmetry, this triangle is a $45-45-90$ triangle, so the legs are $4 \sqrt{2}$ each and the area is $\frac{(4 \sqrt{2})^{2}}{2}=(\mathbf{E}) 16$.


1. In the number 98, the digit "9" is in the ones place.

Yes

No


Teacher Ying bought some sweets and divided them equally among $$9$$ children. If everyone got $$6$$ sweets, there would still be some sweets remaining. What is the most number of sweets Teacher Ying could have bought? What is the least amount of sweets Teacher Ying could have bought?

$$(\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} )\div 9=6 R (\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} )$$  Largest remainder: $$8$$, $$9\times 6+8=62$$  Smallest remainder: $$1$$, $$9\times 6+1=55$$


In the class, the teacher gave $4$ numbers:$15, 27, 36, 8$. Billy added another number to make the sum result of these $5$ digits to $100$. Guess what\textquotesingle s the number Billy added ? ( adapted from $$2009$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$5$$)

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Subtraction of Whole Numbers->Subtraction in Horizontal Form


Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Linear Equations with Multiple Variables


For how many values of $a$ is it true that the line $y=x+a^{2}-6$ passes through the vertex of the parabola $y=4x^{2}-8x+a^{2}$? (Adapted From 2005 AMC 12B Problem, Question \#8)

infinitely many

We see that the vertex of the quadratic function $y=4x^{2}-8x+a^{2}$ is $\left(1, a^{2}-4\right)$. If $\left(2, a^{2}-1\right)$ will be on the line $y=x+a^{2}-6$, $a^{2} -4=1+a^{2}-6$. Solve for $a$, there is no solution.


\textbf{Recently you learned that the probability of getting a critical hit on an attack in your favorite game is 15\%. You've noticed that you do 20 attacks per fight. You're interested in figuring out how likely it is that you would get 5 critical hits during a fight. Which of the following distributions should you use to answer this question?}

\textbf{A binomial distribution with p = 0.15 and n = 20}

\textbf{A binomial distribution with p = 0.2 and n = 15}

\textbf{A geometric distribution with p = 0.15}

\textbf{A geometric distribution with p = 0.2}

\textbf{A cumulative geometric distribution with p = 0.15}

\textbf{Binomial distribution: how likely it is to get x successes in n trials given that your probability of success is p}  \textbf{Geometric distribution: how likely is it that I'll have my first critical strike on the 5th attack}  \textbf{Cumulative geometric distribution: how likely is it that I'll have my first critical strike on or before the 5th attack.}


True or False: It is impossible for the $25$-th percentile to be equal to the mean.

True

False

For example, consider a dataset with the following values: 0, 2, 2, 3, 3. The mean of this dataset is $2$, and the $25$-th percentile is also $2$.


One ticket to a mini concert costs $\textbackslash$20$ at full price. Nicole buys $4$ tickets using a coupon that gives her a $25\textbackslash\%$ discount. Bel buys $5$ tickets using a coupon that gives her a $30\textbackslash\%$ discount. How many more dollars does Bel pay than Nicole?


What is the smallest whole number larger than the perimeter of any triangle with a side of length $7$ and a side of length $15$? (2015 AMC8, Question 8)

We know from the triangle inequality that the last side, $s$, fulfills $s\textless7+15$. Adding $7+15$ to both sides of the inequality, we get $s+7+15\textless44$, and because $s+7+15$ is the perimeter of our triangle, (D) 44 is our answer.


Find the missing number: $$12345 + 123450 = 12345\times $$．

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations


Find the result of $$1\times12\times23\times34\times45\times \cdots \times78\times89$$. What is the sum of its last $2$ digits?

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Calculation of Multi-digit Numbers

$$\textasciitilde\textasciitilde\textasciitilde\textasciitilde1\times12\times23\times34\times45\times \cdots \times78\times89$$  $$=1\times6\times23\times34\times9\times \cdots \times78\times89\times2\times5$$  $$1\times6\times3\times4\times9\times6\times7\times8\times9$$ has the last digit of $2$.  The last two digits are $2$ and $0$.


A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^{\text {th }}$-graders to $6^{\text {th }}$-graders is $5: 3$, and the the ratio of $8^{\text {th }}$-graders to $7^{\text {th }}$ graders is $8: 5$. What is the smallest number of students that could be participating in the project?

We multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8 th graders, in order that we can put the two ratios together: $$ \begin{aligned} \&5: 3=5(8): 3(8)=40: 24 \textbackslash\textbackslash{} \&8: 5=8(5): 5(5)=40: 25 \end{aligned} $$ Therefore, the ratio of 8th graders to 7th graders to 6th graders is $40: 25: 24$. Since the ratio is in lowest terms, the smallest number of students participating in the project is $$ 40+25+24=\text { (E) } 89 $$


A box has fewer than $50$ cookies in it. The cookies can be divided evenly between $2, 3,$ or $4$ children. However, they cannot be divided evenly between $7$ children because $6$ more cookies would be needed. How many cookies are there in the box? (2021 Math Kangaroo Problem, Level 3-4, Question \#21)


Which of the following numbers\textquotesingle{} value does not change after removing all ``$$0$$''s ．

Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals

Only $$3.5260$$\textquotesingle s $$"0"$$ can be removed without causing other digits to change their places from the original number.


Gilda has a bag of marbles. She gives $20 \textbackslash\%$ of them to her friend Pedro. Then Gilda gives $10 \textbackslash\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25 \textbackslash\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself? (2019 AMC 8, Question \#8)

$33 \frac{1}{3}$

Overseas Competition->Knowledge Point->Calculation Modules->Percentage Calculation

After Gilda gives $20 \textbackslash\%$ of the marbles to Pedro, she has $80 \textbackslash\%$ of the marbles left. If she then gives $10 \textbackslash\%$ of what\textquotesingle s left to Ebony, she has $(0.8 * 0.9)=72 \textbackslash\%$ of what she had at the beginning. Finally, she gives $25 \textbackslash\%$ of what\textquotesingle s left to her brother, so she has $(0.75 * 0.72)$ (E) 54 . of what she had in the beginning left.


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction


For what positive value of $k$ does the following system of equations have no solutions?~\uline{~~~~~~~~~~}~  $$ \begin{cases} 8 x+6 k y=17 \textbackslash\textbackslash{} k x+12 y=16 \end{cases}$$

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation

We want the lines to be parallel and not the same line in order to have 0 solutions. Parallel implies the slopes of the lines are equal, so we have $-\frac{8}{6 k}=-\frac{k}{12}$. Cross multiplying, we get $96=6 k^{2}$, so $k^{2}=$ 16 and our positive solution is then $k=4$.


A little rabbit wants to cross the river. He needs to lay ten stones on the river. At this time, there are five stones on the river. The rabbit moves three stones back for the first time and two stones back for the second time. But the river washes away one stone. How many stones the rabbit still need to move?~(adapted from $$2005$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$6$$)

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Addition and Subtraction of Whole Numbers->Questions Involving Addition and Subtraction


What is the $5^{}\text{th}$ number in the $21^{}\text{st}$ row?

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns of Number Tables->Triangle Number Table


Which of the numbers below is the greatest? ($2003$ Math Kangaroo Problem, Level $9-10$, Question \#$11$)

Overseas Competition->Knowledge Point->Calculation Modules->Power->Application of Powers

$2^{2222}\textgreater2^{2220}=\left (2^{10}\right )^{222}=1024^{222}\textgreater22^{222}$. According to this rule, $\text{E}\textgreater\text{D}\textgreater\text{C}\textgreater\text{B}\textgreater\text{A}$.


What is the ones digit of the result of $3$\textsuperscript{50}?

Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers->Exponentiation of Powers

The ones digits of exponents based on $3$ follow the rule: $3, 9, 7, 1, 3, 9, 7, 1\cdots $  $50\div4=12R2$, which means the ones digit is $9$.


Express $$0.\dot{6}\dot{3}$$ as a fraction.

Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Recurring Decimals

Let $$x = 0.636363\cdots$$  Multiply by $$100$$, which means move the decimal point two places to the right: $$100x = 63.636363\cdots$$  $$x$$ and $$100x$$ have exactly the same decimal part, so if we subtract, it will disappear: $$100x - x = 63.636363\cdots - 0.636363\cdots$$  Which is: $$99x = 63$$ ,$x=\dfrac{63}{99}=\dfrac{7}{11}$  So there is our answer： $$0.636363\cdots=$$$\dfrac{7}{11}$


The 2000 Census identified the ethnic breakdown of the state of California to be approximately as follows: White:46％, Latino:32\%, Asian:11\%, Blcak:7\%,and Other:4\%.~Assuming that these are mutually exclusive categories (this is not a realistic assumption), what is the probability that a randomly selected person from the state of California is of Asian or Latino descent?

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables

The correct answer is (c).There are 12 values in the A and E cell out the total of 125. When we are given colwnn E, the total is 63. Of those,28 are C.


Let $$x = 0.555\cdots$$  Multiply by $$10$$ , which means move the decimal point one place to the right: $$10x = 5.555\cdots$$.  $$x$$ and $$10x$$ have exactly the same decimal part, so if we subtract, it will disappear: $$10x - x = 5.555\cdots - 0.555\cdots$$.  Simplify:~ $$9x = 5$$，$x=\dfrac{5}{9}$  So there\textquotesingle s our answer : $$0.555\cdots=$$$\dfrac{5}{9}$


Alina writes the numbers $1,2,\cdots ,9$ on a separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?


The ones digit of the fourth power of an integer cannot be．

The ones digit of a $$4\text{th}$$ power canbe $$0$$, $$1$$, $$5$$, or $$6$$. It can never be $$3$$.


If a study has three factors, each with three levels, how many treatments are there?

The number of values you multiply together is equal to the number of factors in a study. There are three factors, so multiply \_\_x\_\_x\_\_. The values that go into each slot represent the number of levels for each factor. In this case, 3x3x3=27.


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division

$$99\times9=(100-1)\times9$$. This is slightly less than $$100\times9$$, so it\textquotesingle s $$\text{D}$$.


Ten balls numbered from $1$ to $10$ are put into a bag. $3$ balls are taken out and numbers on them are added up. The ball with number $5$ is included in the balls that are taken out, and the sum of numbers on the $3$ balls can be divisible by both $3$ and $4$. Which of the following balls is definitely not taken out?

Overseas Competition->Knowledge Point->Calculation Modules

The sum can be divisible by both $3$ and $4$, so the sum is the common multiple of $3$ and $4$.  If the sum is $12$, $12-5=7$, so the sum of the other numbers should be $7$. The other two numbers can be $1$ and $6$ or $3$ and $4$.  If the sum is $24$, $24-5=19$, the other two numbers can only be $10$ and $9$.  If the sum is $36$, it is impossible, because $36-5=31$ and the largest sum of two numbers from $1$ to $10$ is $19$.


In which of the following expressions, the value of the constant is larger than the value of the coefficient of $x$?

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions

the sign in front is also part of constant or coefficient


There is a rule that the symbol "\#" represents an operation of producing the smaller one of the two numbers (for example, $17$ \# $8 = 8$). Calculate: ($6$ \# $3$ ) $\times$($10$ \# $9$)=~\uline{~~~~~~~~~~}~.

Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Operating Directly


Overseas Competition->Knowledge Point->Calculation Modules->Comparing, Ordering and Estimating

Sugar water theory. 1 gram of sugar added each time, and the sugar water gets sweeter.


\textbf{Which of the following data sets has the largest standard deviation?}

\textbf{{100, 101, 102, 103, 104}}

\textbf{{1000.3, 999.56, 1000.49, 1000, 998.32}}

\textbf{{1, 1, 1, 1, 1}}

\textbf{{5, 10, 15, 20, 25}}

\textbf{{1, 2, 3, 4, 5}}

\textbf{variance is used to measure the spread of data. A, B, C, E are densely distributed.We can verify by calculating the variances out. (A) 2.5, (B) 0.598, (C) 0, (D) 50, (E) 2}


None of the above

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations


\textbf{Events D and E are independent, with P(D) = 0.6 and P(D and E) = 0.18. Which of the following is true?}

\textbf{P(E) = 0.12}

\textbf{P(E) = 0.4}

\textbf{P(D or E) = 0.28}

\textbf{P(D or E) = 0.72}

\textbf{P(D or E) = 0.9}

\textbf{Since D and E are independent, P(D ∩ E) = P(D)*P(E). So P(E) = 0.3.}  \textbf{P(D ∪ E) = P(D) + P(E) -- P(D ∩ E) = 0.6 +0.3 -0.18 = 0.72}


17, 80

18, 70

17, 70

20, 80


Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers


What is the 100\textsuperscript{th}~number in the arithmetic sequence $$1$$, $$5$$, $$9$$, $$13$$, $$17$$, $$21$$, $$25$$, $$\cdots$$．

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Arithmetic Sequences


In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If $\frac{1}{3}$ of all the ninth graders are paired with $\frac{2}{5}$ of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy? (2015 AMC 8 Problems, Question \#16)

Let the number of sixth graders be $s$, and the number of ninth graders be $n$. Thus, $\frac{n}{3}=\frac{2 s}{5}$, which simplifies to $n=\frac{6 s}{5}$. Since we are trying to find the value of $\frac{\frac{n}{3}+\frac{2 s}{5}}{n+s}$, we can just substitute $\frac{6 s}{5}$ for $n$ into the equation. We then get a value of $\frac{\frac{6 s}{5}\cdot\frac13+\frac{2 s}{5}}{\frac{6 s}{5}+s}=\frac{\frac{6 s+6 s}{15}}{\frac{11 s}{5}}=\frac{\frac{4 s}{5}}{\frac{11 s}{5}}=\frac{4}{11}$.


Line $l\_1$ has equation $3 x-2 y=1$ and goes through $A=(-1,-2)$. Line $l\_2$ has equation $y=1$ and meets line $l\_1$ at point $B$. Line $l\_3$ has positive slope, goes through point $A$, and meets $l\_2$ at point $C$. The area of $\triangle A B C$ is $3$ . What is the slope of $l\_3$? (2013 AMC 12B Problems, Question \#8)

Line $l\_1$ has the equation $y=\frac{3x}{2}-\frac12$ when rearranged. Substituting $1$ for $y$, we find that line $l\_2$ will meet this line at point $(1,1)$, which is point $B$. We call $\overline{B C}$ the base and the altitude from $A$ to the line connecting $B$ and $C, y=1$, the height. The altitude has length $\textbar-2-1\textbar=3$, and the area of $\triangle A B C=3$. Since $A=\frac{bh}{2}, b=2$. Because $l\_3$ has positive slope, it will meet $l\_2$ to the right of $B$, and the point that is $2$ to the right of $B$ is $(3,1)$. $l\_3$ passes through $(-1,-2)$ and $(3,1)$, and thus has slope $\frac{\textbar1-(-2)\textbar}{\textbar3-(-1)\textbar}=(\mathbf{B}) \frac{3}{4}$.


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction


Which of the following equations are not equivalent to $x+3=11$

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation

Equations are equivalent when you can obtain one by subtracting, adding, dividing, or multiplying the same number on the other.


There are $18$ bottles of water in each of $10$ boxes. Six customers want to buy some botlles of water. Four of them buy $30$ bottles of water, and two of them buy $20$ bottles of water. How many bottles of water are left in the boxes


Erik and Ivy each writes down a fraction. They are surprised to find that the fractions they write are very similar: the denominator of each fraction is exactly the same as the other\textquotesingle s numerator. What is the product of the two fractions?

It cannot be determined.

Overseas Competition->Knowledge Point->Calculation Modules->Fractions

Their fractions are just reciprocal of the other one. Thus, the product of the two fractions should be $1.$


$$200001\times 10^{6}$$

$$100001\times 10^{6}$$

Overseas Competition->Knowledge Point->Calculation Modules->Operations through Formulas-> Difference of Two Squares Formula


Overseas Competition->Knowledge Point->Calculation Modules->Operations through Formulas->1²+2²+3²+......+n²=1/6n(n+1)(n+2)

Directly substitute $$n$$ with $$18$$ into the formula to get $$1^{2}+2^{2}+3^{2}+\cdots +18^{2}=\frac{18\times 19\times 37}{6}=2109$$. So the answer is $$\text{D}$$.


For how many integers $x$ is the number $x^{4}-25 x^{2}+24$ negative? (  2014 AMC 10B Problems, Question \#20)

First, note that $24+1=25$, which motivates us to factor the polynomial as $\left(x^{2}-24\right)\left(x^{2}-1\right)$. Since this expression is negative, one term must be negative and the other positive. Also, the first term is obviously smaller than the second, so $x^{2}-24\textless0\textless x^{2}-1$. Solving this inequality, we find $1\textless x^{2}\textless24$. There are exactly $6$ integers $x$ that satisfy this inequality, $\pm\textbackslash{2,3,4\textbackslash}$. Thus our answer is $(\mathbf{B}) 6$.


In a class of $$40$$ pupils, there are $$10$$ more boys than girls.  What is the ratio of the number of girls to the number of boys?

Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions->Ratio


What is the largest factor of $$2^{2}\times3^{3} \times5^{5} \times7^{7}\times11^{11}$$ less than $$100$$?

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Calculation of Multi-digit Numbers

The largest two-digit factor is $$3^{2} \times11 = 99$$.


The following are the weights (in pounds) of seven people: $100, 115, 135, 140, 180, 197, 230$. Find the $36$-th percentile.


Which of the following pairs is a solution of $$\begin{cases}2x-4=0 \textbackslash\textbackslash{} 4x-y=7 \end{cases}$$?

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Linear Equations with Multiple Variables


Which one is an equivalent fraction of $$\frac{16}{24}$$?

Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Basic Understanding of Fractions->Properties of Fractions


Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division

Overseas In-curriculum->Knowledge Point->Operations of Numbers ->Multiplication of Whole Numbers->Multiplication of Multi-Digit Numbers and 1-Digit Numbers->Multiplication of 3-Digit and 1-Digit (with regrouping for more than once)

Stack the two numbers as shown below ,~ lining up the unit digits  Remember that the 7 in 272 stands for 7 tens(70) ,the first 2 in the 272 stands for 2 hundreds(200)  First, multiply the ones~ $2\times3=6$~, regroup the 0 tens to the tens column  Write ~6 in the ones place.  Then,~ Multiply and add the tens .~$3\times7+0=21$  Write 1 in the tens place and regroup the 2 hundreds.  Last, multiply and add the hundreds.~$3\times2+2=8$  Write 8 in the hundreds place


Teacher Jason leaves home every day at $8:00\text{AM}$ to go to work. If he drives at an average speed of $40$ km/h, he will be late by $3$ minutes. If he drives at an average speed of $60$ km/h, he will be early by $3$ minutes. How many km/h does Teacher Jason need to drive to get to work exactly on time?

Overseas Competition->Knowledge Point->Calculation Modules


Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals->Comparing Decimals


Fill ``$$+$$'' or ``$$-$$'' between neighbouring numbers to make the number statement correct.  $$5$$ 　$$5$$ 　$$5$$ 　$$5$$　$$5$$ 　$$5=0$$

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers


Find the value of the expression:  $$3- \frac{6}{1 \times \left (1+2\right )}- \frac{9}{\left (1+2\right ) \times \left (1+2+3\right )}-\frac{12}{\left (1+2+3\right ) \times \left (1+2+3+4\right )}- \cdots$$  $$- \frac{60}{\left (1+2+ \cdots +19\right ) \times \left (1+2+ \cdots +20\right )}$$

None of the above


Find the missing number: $$64 \div 2 = 2 \times $$．


How many real numbers $x$ satisfy the following equation: $$ (x^{2}+7)^{2} = 289$$


Aline writes a correct calculation. Then she covers two digits which are the same with a sticker:  [insert pic]  What digit is under the stickers?

Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Finding Patterns->Encryption and Decryption

Two same digits that add up to "4" in the last digit, hence it must be either 2 or 7.  If we try digit 2, 42+52=94, wrong.  If we try digit 7, 27+57=104. correct.


The number that is 8.9 larger than 1.2 is．

Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Addition and Subtraction of Decimals


What is the value of $$\left( 330+22 \right)\div 11$$ ?

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations


Let $p, q$, and $r$ be the distinct roots of the polynomial $x^{3}-22 x^{2}+80 x-67$. It is given that there exist real numbers $A, B$, and $C$ such that $$ \frac{1}{s^{3}-22 s^{2}+80 s-67}=\frac{A}{s-p}+\frac{B}{s-q}+\frac{C}{s-r} $$ for all $s \notin\textbackslash{p, q, r\textbackslash}$. What is $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$?

Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Proportional Equations

Multiplying both sides by $(s-p)(s-q)(s-r)$ yields $$ 1=A(s-q)(s-r)+B(s-p)(s-r)+C(s-p)(s-q) $$ As this is a polynomial identity, and it is true for infinitely many $s$, it must be true for all $s$ (since a polynomial with infinitely many roots must in fact be the constant polynomial 0$)$. This means we can plug in $s=p$ to find that $\frac{1}{A}=(p-q)(p-r)$. Similarly, we can find $\frac{1}{B}=(q-p)(q-r)$ and $\frac{1}{C}=(r-p)(r-q)$. Summing them up, we get that $$ \frac{1}{A}+\frac{1}{B}+\frac{1}{C}=p^{2}+q^{2}+r^{2}-p q-q r-p r $$ By Vieta\textquotesingle s Formulas, we know that $p^{2}+q^{2}+r^{2}=(p+q+r)^{2}-2(p q+q r+p r)=324$ and $p q+q r+p r=80$. Thus the answer is $324-80=$ 244 .


There were four piles of strawberries and a rabbit wanted to eat some. Each pile had $24$ strawberries. The rabbit ate a few strawberries from the first pile and then ate as many strawberries from the third pile as were left in the first pile. After that the rabbit ate a few strawberries from the second pile and then ate as many strawberries from the fourth pile as were left in the second pile. How many strawberries in total did the rabbit eat?


Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10 \textbackslash\%$?

infinitely many

You can create the equation $\frac{x+1}{y+1}=\frac{11 x}{10 y}$.  Cross multiplying and combining like terms gives $x y+11 x-10 y=0$.  This can be factored into $(x-10)(y+11)=-110$.  $x$ and $y$ must be positive, so $x\textgreater0$ and $y\textgreater0$, so $x-10\textgreater-10$ and $y+11\textgreater11$.  Using the factors of 110 , we can get the factor pairs: $(-1,110),(-2,55)$, and $(-5,22)$.  But we can\textquotesingle t stop here because $x$ and $y$ must be relatively prime.  $(-1,110)$ gives $x=9$ and $y=99.9$ and 99 are not relatively prime, so this doesn\textquotesingle t work.  $(-2,55)$ gives $x=8$ and $y=44$. This doesn\textquotesingle t work.  $(-5,22)$ gives $x=5$ and $y=11$. This does work. We found one valid solution so the answer is (B)$1$.


On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had sabing $20$ days? (2015 AMC 8 Problems, Question \#9)

Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables

We can easily find out she makes $2 \cdot 20-1=39$ widgets on Day $20$ . Then, we make the sum of $1,3,5, \ldots \ldots, 35,37,39$ by adding in this way: $(1+39)+(3+37)+(5+35)+\ldots+(19+21)$, which include $10$ pairs of $40$ . So the sum of $1,3,5, \ldots \ldots \ldots 39$ is $(40 \cdot 10)=(\text{D}) 400$


Three fifths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 6 cups. What percent of the total capacity of the pitcher did each cup receive?~ (adapted from 2020 AMC 8, Question \#5)

Overseas Competition->Knowledge Point->Calculation Modules->Percentage Calculation

The pitcher is $\frac{3}{5}$ full, i.e. $60 \textbackslash\%$ full. Therefore each cup receives $\frac{60}{6}=(\mathbf{C}) 10$ percent of the total capacity.