Observe the characteristics of the numbers given and write the numbers underlined below.~(adapted from $$2006$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$8$$)  $$3$$~ ~ $$4$$~ ~ $$7$$~ ~ $$11$$~ ~$18$~\uline{~~~~~~~~~~}~$47$~ ~$76$~ ~$123$~ $$\cdots$$

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

The sum of the first digit and the second digit is the third digit, and so on. The horizontal line is $11+18=29$.


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 $11: 6$, and the the ratio of $8^{\text {th }}$-graders to $7^{\text {th }}$ graders is $8: 13$. 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 4 on both sides, and the second ratio by 3 to get the same number for 8 th graders, in order that we can put the two ratios together: $$ \begin{aligned} \&11: 6=11(4): 6(4)=44: 24 \textbackslash\textbackslash{} \&8: 13=8(3): 13(3)=24: 39 \end{aligned} $$ Therefore, the ratio of 8th graders to 7th graders to 6th graders is $44: 24: 39$. Since the ratio is in lowest terms, the smallest number of students participating in the project is $$ 44+24+39=\text { (C) } 107 $$


Let $f$ be a linear function for which $f(5)-f(2)=10$. What is $f(8)-f(2)$? (  Adapted From 2003 AMC 12B Problems, Question \#9)

Overseas Competition->Knowledge Point->Calculation Modules

Let $f$ be a linear function with slope $m$.  $$ \begin{gathered} m=\frac{f(5)-f(2)}{\Delta x}=\frac{10}{5-2}=\frac{10}{3} \textbackslash\textbackslash{} f(8)-f(2)=m \Delta x=\frac{10}{3}(8-2)=20 \Rightarrow(C) \end{gathered}$$


How many odd numbers are there?  1, 3, 4, 6, 7, 9, 5, 8.

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


For how many integers $x$ is the number $x^{4}-51 x^{2}+50$ negative?

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

First, note that $50+1=51$, which motivates us to factor the polynomial as $\left(x^{2}-50\right)\left(x^{2}-1\right)$. 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 $x^{2}-50\textless0\textless x^{2}-1$. Solving this inequality, we find $1\textless x^{2}\textless50$. There are exactly 12 integers $x$ that satisfy this inequality, $\pm\textbackslash{2,3,4,5,6,7\textbackslash}$. Thus our answer is $(\mathbf{C}) 12$.


The following are the heights (in cm) of eight people: $157$, $175$, $165$, $184$, $180$, $197$, $160$, $177$. Find the $42$-th percentile.


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


\textbf{Scientists estimate that the distribution of the life span of the Galápagos Islands giant tortoise is approximately normal with mean 100 years and standard deviation 15 years. Based on the estimate, which of the following is closest to the age of a Galápagos Islands giant tortoise at the 90th percentile of the distribution?}

\textbf{80 years}

\textbf{115 years}

\textbf{120 years}

\textbf{125 years}

\textbf{130 years}


The sum of three numbers is $96$. The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers? (2022 AMC 10A Problems, Question \#3)

Let $x$ be the third number. It follows that the first number is $6 x$, and the second number is $x+40$.  We have $$ 6 x+(x+40)+x=8 x+40=96, $$ from which $x=7$.  Therefore, the first number is $42$ , and the second number is $47$ . Their absolute value of the difference is $\textbar42-47\textbar=(\mathbf{E}) 5$.


Which of the following expression is written correctly?

$3 \times x+4$

$x \times y+z$

$3a+b \times c$

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

Only $ \text C$ is in accordance with the rules of writing expression.


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


Which of the following is the largest fraction?  $$\dfrac{2}{15}$$,$$\dfrac{11}{15}$$,$$\dfrac{7}{15}$$,$$\dfrac{4}{15}$$

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

Same denominator, so larger numerator means larger fraction


\textbf{One of the values in a normal distribution is 43, and its z-score is 1.65. If the mean of the distribution is 40, what is the standard deviation of the distribution?}


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


Joe writes an expression $\frac59\times\frac9{13}\times\frac{13}{17}\cdots $ Following the pattern, he writes the expression with $\frac{45}{49}$ as the multiplier in the middle. What is the result of the expression?

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

The last fraction should be $\frac{85}{89}$, so the answer is $\frac5{89}.$


None of the above


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


The most likely height of a single-decker bus is．

$$0.25\rm cm$$

$$2.5\rm cm$$

$$25\rm cm$$

$$250\rm cm$$

$$2500\rm cm$$

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


Round $$398^{}\circ \rm C$$ to the nearest $$10^{}\circ $$.

$$380^{}\circ \rm C$$

$$390^{}\circ \rm C$$

$$399^{}\circ \rm C$$

$$400^{}\circ \rm C$$

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

Rounding, $$398^{}\circ \rm C$$ is closer to $$400^{}\circ \rm C$$ than to $$390^{}\circ \rm C$$.


What is the median of the following list of 4040 numbers: $$ 1,2,3, \ldots, 2020,1^{2}, 2^{2}, 3^{2}, \ldots, 2020^{2}$$? (2020 AMC 10A Problems, Question \#11)

We want to know the $2020$-th term and the $2021$-st term to get the median.  We know that $44^{2}=1936$. So, numbers $1^{2}, 2^{2}, \ldots, 44^{2}$ are in between $1$ and $1936$.  So, the sum of $44$ and $1936$ will result in $1980$ , which means that $1936$ is the $1980$-th number.  Also, notice that $45^{2}=2025$, which is larger than $2021$.  Then the $2020$-th term will be $1936+40=1976$, and similarly the $2021$-th term will be $1977$.  Solving for the median of the two numbers, we get (C) $1976.5$.


Big Al, the ape, ate $100$ bananas from May $1$ through May $5$ . Each day he ate six more bananas than on the previous day. How many bananas did Big Al eat on May $5$ ? (2005 AMC 8 Problems, Question \#12)

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

There are $5$ days from May $1$ to May $5$ . If we set the first day as $n$, the second day can be expressed as $n+6$, the third as $n+12$, and so on, for five days. The sum $n+(n+6)+(n+12)+(n+18)+(n+24)$ is equal to $100$ , as stated in the problem. We can write a very simple equation, that is: $5 n+60=100$. Now all we do is just solve. $5 n=40$, so Big Al eats 8 bananas on the first day. The fifth day, $n+24$, is then 32 , which is your answer.


For $\triangle ABC$, all its side lengths are integer. The primeter of $\triangle ABC$ with a side of length $2$ and a side length of $5$ is at least ．

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

We know from the triangle inequality that the last side, $s$, fulfills $s+2\textgreater5$. Therefore, $P\textgreater5+5$. The least integer value of $P$ is $11$


One basket can contain no more than $5$ eggs. What is the smallest number of baskets needed to contain $37$ eggs?


Let $f(x)=a x^{2}+b x+c$, where $a, b$, and $c$ are integers. Suppose that $f(1)=0,50\textless f(7)\textless60,70\textless f(8)\textless80$, $5000 k\textless f(100)\textless5000(k+1)$ for some integer $k$. What is $k$? (2011 AMC 12A Problems, Question 20)

From $f(1)=0$, we know that $a+b+c=0$.  From the first inequality, we get $50\textless49 a+7 b+c\textless60$.  Subtracting $a+b+c=0$ from this gives us $50\textless48 a+6 b\textless60$, and thus $\frac{25}{3}\textless8 a+b\textless10$. Since $8 a+b$ must be an integer, it follows that $8 a+b=9$.  Similarly, from the second inequality, we get $70\textless64 a+8 b+c\textless80$. Again subtracting $a+b+c=0$ from this gives us $70\textless63 a+7 b\textless80$, or $10\textless9 a+b\textless\frac{80}{7}$.  It follows from this that $9 a+b=11$. We now have a system of three equations: $a+b+c=0,8 a+b=9$, and $9 a+b=11$. Solving gives us $(a, b, c)=(2,-7,5)$ and from this we find that $f(100)=2(100)^{2}-7(100)+5=19305$.  Since $15000\textless19305\textless20000 \rightarrow 5000(3)\textless19305\textless5000(4)$, we find that $k=3 \rightarrow(\mathbf{C}) 3$.


What is the sum of the tenths and the hundredths digits in the num-ber $$12345.6789$$?

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

The tenths digit is $$6$$ and the hundredths digit is $$7$$. Their sum is $$13$$.


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


Find the difference between $$\frac{5}{9}$$ and $$\frac{1}{3}$$


In the arithmetic sequence $5$, $7$, $9$, $11$ $\cdots$ , the $9$\textsuperscript{th} term is~\uline{~~~~~~~~~~}~.

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

Observe that each number is the sum of the number of previous term and the difference between adjacent numbers.


A 95\% confidence interval for the difference between two population proportions is found to be (0.07, 0.19). Which of the following statement is (are) true?  I. It is unlikely that the two populations have the same proportions.  II. We are 95\% confidence that the true difference between population proportions is between 0.07 and 0.19.  III. The probability is 0.95 that the true difference between the population proportions is between 0.07 and 0.19.

I

II

I, II

I, III

II, III

Overseas Competition->Knowledge Point->Calculation Modules

Because 0 is not in the interval (0.07, 0.19). It is unlikely to be the true difference between the proportions. III is just plain wrong! We cannot make a probability statement about an interval we have already constructed. All we can say is that the process used to generate this interval has a 0.95 chance of producing an interval that does contain the true population proportion.


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


If $x\textgreater0$, which of the following is equivalent to $\sqrt[3]{x^{4}}$ ?  $ $ $I. x+x^{\frac{1}{3}}$  $ $ $II. \left(x^{\frac{1}{3}}\right)^{4}$  $ $ $III. x^{2}\left(x^{-\frac{2}{3}}\right)$

$I$ and $II$ only

$II$ and $III$ only

$I$, $II$, and $III$

$\sqrt[3]{x^{4}}$ can be written as $x^{}\frac{4}{3}$, which is equivalent to II and III.


What is the $$30$$th term of the following arithmetic sequence $$1, 4, 7, 10, \cdots$$?


How many digits are there in the correct answer to the calculation $$123123123123\div 123$$?

The correct answer to the calculation $$123123123123\div123=1001001001$$.  This has $$10$$ digits.


A computer makes $$4\times {{10}^{9}}$$ operations per second. How many operations does it make in $$5\times {{10}^{2}}$$ seconds?

$$4\times {{10}^{11}}$$

$$2\times {{10}^{11}}$$

$$2\times {{10}^{12}}$$

$$20\times {{10}^{18}}$$

$$2\times {{10}^{19}}$$

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


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." Her sequence is ---, ---, ---, ~$$18$$, $$29$$.  What is her first term? ．

$$0 $$

$$ 3 $$

$$ 4 $$

$$ 5 $$

$$ 7$$

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 $$a=4$$.


Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean\textquotesingle s average speed, in miles per hour, until they meet? (2021 AMC 10A Problems, Question \#6)

Let $2 d$ miles be the distance from the trailhead to the fire tower, where $d\textgreater0$. When Chantal meets Jean, the two have traveled for $$ \frac{d}{4}+\frac{d}{2}+\frac{d}{3}=d\left(\frac{1}{4}+\frac{1}{2}+\frac{1}{3}\right)=d\left(\frac{3}{12}+\frac{6}{12}+\frac{4}{12}\right)=\frac{13}{12} d $$ hours.  At that point, Jean has traveled for $d$ miles, so his average speed is $\frac{d}{\frac{13}{12} d}=(\mathbf{A}) \frac{12}{13}$ miles per hour.


After simplying the following expressions, which one is different from others?

$(x+y)-(x-y)$

$(x-y)+(-x+y)$


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


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


\textbf{From a random sample of 1,005 adults in the United States, it was found that 32 percent own an e-reader. Which of the following is the appropriate 90 percent confidence interval to estimate the proportion of all adults in the United States who own an e-reader?~}

\textbf{$0.32 \pm 1.960(\frac{(0.32)(0.68)}{\sqrt{1,005}})$}

\textbf{$0.32 \pm 1.645(\frac{(0.32)(0.68)}{\sqrt{1,005}})$}

\textbf{$0.32 \pm 2.575\sqrt{(\frac{(0.32)(0.68)}{1,005})}$}

\textbf{$0.32 \pm 1.960\sqrt{(\frac{(0.32)(0.68)}{1,005})}$}

\textbf{$0.32 \pm 1.645\sqrt{(\frac{(0.32)(0.68)}{1,005})}$}


None of the above

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


It is known that the number $2017$ is placed at $M^{\text{th}}$ row, $N^{\text{th}}$ entry. Find the value of $M+N$.

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


Definite @ as the operation of choosing the larger number between two numbers. Definite \&~as the operation of choosing the smaller number between two numbers. Find the result of $(37$\&$23)\times(25$@$45)\div(23$@$15)$.

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


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->Word Problem Modules->Equation Word Problems

In weight, $$45$$ mice $$=3\times (15$$ mice$$)=3\times (2$$ cats$$)= 2\times (3$$ cats$$)=2\times (2$$ dogs$$)= 4$$ dogs.  On Planet Pythagoras, the people use a different money system to us.  two pog is worth six pings.


$$10\times 99$$

$$10\div 99$$

$$9\times 99$$

$$99\times 99$$

The sum of ten $$99$$\textquotesingle s is the same as $$10\times 99$$.


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


When three positive integers $a$, $b$, and $c$ are multiplied together, their product is 100. Suppose $a \textless{} b \textless{} c$. In how many ways can the numbers be chosen?


If $*abcd*=a\times d+b\times c$ ,then $*2543*=$ ．

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


\textbf{According to a report for veterinarians in the United States, 36.5 percent of households in the United States own dogs and 30.4 percent of households in the United States own cats. If one household in the United States is selected at random, what is the probability that the selected household will own a dog or a cat?}

\textbf{Not enough information is given to determine the probability}

\textbf{The sample in the report is about veterinarians. The population cannot be a household in the United States.}


An amusement park has a collection of scale models, with ratio $1: 35$, of buildings and other sights from around the country. If the height of a signal tower is $2250$ 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 $1: 35$. If the height of the tower is $2250$ feet, to find the height of the model, we divide by $35$ . That gives us $64.28$ which rounds to $64$ . Therefore, to the nearest whole number, the duplicate is (C) $64$ feet.


The total value of $$25$$ dimes is $$125$$ times to total value of．

$$1$$ penny

$$1$$ nickel

$$2$$ pennies

$$2$$ nickels


If the 9-digit number $$2017122\square2$$ can be divisible by $$4$$, then the number in $$\square $$ can be ．

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

We check if it is divisible by $$4$$ by looking at the last two digits.  $$72$$ is divisible by $$4$$.


The last four digits of Linda\textquotesingle s phone number are $2022$. If the numbers $2,0,2,2$ are summed, what is the final result?~(adapted from $$2011$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$7$$)

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


Amelia cuts a $75$ cm by $30$ cm rectangle into identical squares without any leftovers. If the length of the squares is a whole number of cm, what is the least number of squares can she get?

None of the above

Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Extracting Common Factors from Whole Numbers

75 is divisible by 1, 3, 5,15, 25 and 75.  30 is divisible by 1, 2, 3, 5\textsubscript{Z} 6,10,15 and 30.  15 is the highest common divisor for both 75 and 30.  $\textasciitilde$  To get the least number of squares, the length of the squares must be the largest possible. Hence the length must be 15 and there must be 5 x 2 = \textbf{10} squares.


Seven numbers add up to $2022$ and one of the numbers is $522$. Replace $522$ with $321$. What is the new sum of the seven numbers?

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


There is a rule that the symbol "\&" represents an operation of producing the larger one of the two numbers (for example, $7$ \&~$15 = 15$). Calculate: ($13$ \&~$22$ ) $\times$ ($3$ \&~$6$)=~\uline{~~~~~~~~~~}~.


If $3^{}p+3^{4}=90$, and $2^{}r+44=76$, what is the product of $p$ and $r$? (Adapted from 2013 AMC 8 Problem, Question \#15)

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

First, solve for $p$. Start with $3^{}p+3^{4}=90$. Then, change $3^{4}$ to $81$. Subtract $81$ from both sides to get $3^{}p=9$ and see that $p$ is $2$. Now, solve for $r$. Since $2^{}r+44=76$, $2^{}r$ must equal $32$, $r=5$. $pr$ equals $2\times5$ which equals $10$. So, the answer is $10$.


What is the biggest digit in ones place?


What is the least possible value of $$ (x+1)(x+2)(x+3)(x+4)+2019 $$ where $x$ is a real number?

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

Grouping the first and last terms and two middle terms gives $\left(x^{2}+5 x+4\right)\left(x^{2}+5 x+6\right)+2019$, which can be simplified to $\left(x^{2}+5 x+5\right)^{2}-1+2019$. Noting that squares are nonnegative, and verifying that $x^{2}+5 x+5=0$ for some real $x$, the answer is 2018.


Avril\textquotesingle s father bought 14 books. Avril takes away 6 books. How many books does Avril\textquotesingle s father have left?

Overseas Competition->Knowledge Point->Calculation Modules


Before I began snacking, there were $$60\div 4+1\times 3$$ gumballs here. If I ate all of them, how many gumballs did I eat?

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

To evaluate $$60\div 4+1\times 3$$, we first do the $$\times $$ and $$\div $$ in the order in which they appear. Do the addition last. We get $$15+3=18$$.


$$199\times 99$$

$$198\times 99$$

$$100\times 100$$

$$100\times 99$$

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


Gary sold $5$ times as many tickets as Louis.They sold a total of $54$ tickets. How many tickets did Louis sell?


What is the value of $\dfrac{21w}{10}-w+1$ when $w=2$?

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


What is the correct ordering of the three numbers, $$10^{8}$$, $$5^{12}$$, and $$2^{24}$$? ($$2010$$ AMC $$8$$ Problem, Question \#$$24$$)

$$2^{24}$$\textless{} $$10^{8}$$\textless$$5^{12}$$

$$2^{24}$$\textless$$5^{12}$$\textless$$10^{8}$$

$$5^{12}$$\textless$$2^{24}$$\textless$$10^{8}$$

$$10^{8}$$\textless$$5^{12}$$\textless$$2^{24}$$

$$10^{8}$$\textless$$2^{24}$$\textless$$5^{12}$$

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

$$\rm Method$$ $$1$$:  Since all of the exponents are multiples of $4$, we can simplify the problem by taking the fourth root of each number. Evaluating we get $$10^{2}=100$$, $$5^{3}=125$$, and $$2^{6}=64$$. $$64\textless100\textless125$$. So, $$2^{24} \textless$$ $$10^{8}$$ $$\textless{} 5^{12}$$.  $$\rm Method$$ $$2$$:  First, let us make all exponents equal to $$8$$. Then, it will be easy to order the  numbers without doing any computations. $$10^{8}$$ is fine as it is. We can rewrite $$2^{24}$$ as $$(2^{3})^{8}=8^{8}$$. We can rewrite $$5^{12}$$ as $$\left( 5^{\frac{3}{2}}\right)^{8}=\left( \sqrt{125}\right)^{8}$$. We take the eighth root of all of these to get $$10$$, $$8$$,~$\sqrt{125}$. Obviously, $$8\textless10\textless\sqrt{ 125}$$, so $$2^{24}\textless10^{8}\textless5^{12}$$.


$3+\dfrac{1}{1+\dfrac{1}{5+\dfrac{1}{16}}}=\left( \textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{} \right).$

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


Which of the following is an algebraic equation with variable(s)? ．


$$\frac{2}{3}; \frac{1}{2}$$

$$\frac{2}{3}; \frac{1}{9}$$

$$\frac{1}{12}; \frac{1}{9}$$


If $n$ is an even positive integer, the double factorial notation $n!!$ represents the product of all the even integers from $2$ to $n$. For example, $8!! = 2·4·6·8$. What is the units digit of the following sum?  $2!! + 4!! + 6!! + · · · + 2018!! + 2020!! + 2022!!$


Happy Hotel is offering $$40 \textbackslash\%$$ off discount for any bookings. David booked a room, the new price is $$80$$ dollars cheaper than the original price, the original price is~\uline{~~~~~~~~~~}~.

$$80\div40\textbackslash\%=200$$．  so choose $$\text{B}$$．


Among all the whole numbers from $$1$$ to $$30$$, how many numbers are multiples of $$3$$?


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


Starting with some gold coins and some empty treasure chests, I tried to put $15$ gold coins in each treasure chest, but that left $$1$$ treasure chests empty. So instead I put $12$ gold coins in each treasure chest, but then I had $6$ gold coins left over. How many gold coins did I have? ( adapted from 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: $$ \begin{gathered} 15 c-15=g \textbackslash\textbackslash{} 12 c+6=g \end{gathered} $$  Therefore, $15 c-15=12 c+6$. This implies that $c=7$. We therefore have $g=90$. So, our answer is (E) $90$


From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon? (2018 AMC 8 Problems, Question \#23)

We will use constructive counting to solve this. There are $2$ cases: Either all $3$ points are adjacent, or exactly $2$ points are adjacent. If all $3$ points are adjacent, then we have $8$ choices. If we have exactly $2$ adjacent points, then we will have $8$ places to put the adjacent points and $4$ places to put the remaining point, so we have $8 \cdot 4$ choices. The total amount of choices is $\left(\begin{array}{l}8 \textbackslash\textbackslash{} 3\end{array}\right)=8 \cdot 7$. Thus, our answer is $\frac{8+8 \cdot 4}{8 \cdot 7}=\frac{1+4}{7}=$ (D) $\frac{5}{7}$.


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 $20.$


Calculate: $$3.75\times 20\textbackslash\%\times \frac{3}{23}-4\times \frac{7}{23}+\frac{9}{23}\times 3\frac{1}{4}=$$．


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?


\textbf{If P(A) = 0.34 and P(A or B) = 0.71, which of the following is false?}

\textbf{P(B) = 0.37 , if A and B are mutually exclusive.}

\textbf{P(B) = 0.561, if A and B are independent.}

\textbf{P(B) cannot be determined if A and B are neither mutually exclusive nor independent.}

\textbf{P(A and B) = 0.191 , if A and B are independent.}

\textbf{P(A\textbar B) = 0.34 , if A and B are mutually exclusive.}

\textbf{P(A) = 0.34, P(A ∪ B) = 0.71 =~ P(A) + P(B) -- P(A ∩ B)}  \textbf{(A) If A and B are mutually exclusive, P(A ∩ B) = 0. Then, P(A ∪ B) = P(A) + P(B) which is 0.71= 0.34+P(B). So P(B) = 0.37.}  \textbf{(B) If A and B is independent, P(A ∩ B) = P(A)*P(B). Then, P(A ∪ B) =~ P(A) + P(B) --~ P(A)*P(B) which 0.71 = 0.34 + P(B) - 0.34*P(B). So P(B) = 0.561}  \textbf{(C) P(A ∪ B) = P(A) + P(B) -- P(A ∩ B). We have to know it is mutually exclusive or independent in order to know P(A ∩ B).}  \textbf{(D) If A and B is independent, P(A ∩ B) = P(A)*P(B) =0.34*0.561=0.191}  \textbf{(E) P(A \textbar{} B)=P(A∩B)P(B) = 0}


After the trainer\textquotesingle s first whistle, the monkeys at the circus formed $$4$$ rows. There were $$4$$ monkeys in each row. After the second whistle, they rearranged themselves into $$8$$ rows. How many monkeys were there in each row after the second whistle? (2005 Math Kangaroo Problem, Level 3-4, Question \#4)

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

There were $$4 \times 4 = 16$$ monkeys in total. After the second whistle, there were $$16 \div 8 = 2$$ monkeys in each row.


Let $X$ be the smallest positive integer, consisting of only the digit $4$ and $9$ (at least one of each), that is divisible by both $4$ and $9$. What is the last four digits of $X$?


Like terms are terms that have the same variables. For each variable, the number of times it is multiplied by itself is also the same.


Calculate: $$\frac{1}{3}\times \frac{3}{5}\times \frac{5}{7}\times \cdots \times \frac{2019}{2021}=$$．

$$\textasciitilde\textasciitilde\textasciitilde\textasciitilde\frac{1}{3}\times \frac{3}{5}\times \frac{5}{7}\times \cdots \cdots \times \frac{2019}{2021}$$  $$=\frac{1}{2021}$$．


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 fraciton is exactly the same as the other\textquotesingle s numerator. What is the product of the two fractions?

It cannot be determined.

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


What is the area of the triangle formed by the lines $y=-2$, $y=3+x$, and $y=3-x$?


\textbf{A basketball player makes 160 out of 200 free throws. We would estimate the probability that the player makes his next free throw to be heads the fourth time.}

\textbf{0.50; either he makes it or he doesn't}


None of the answer above


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 $$\text{D}$$.


The pages of a book are numbered $1, 2, 3, 4, 5$, and so on. The digit $5$ appears exactly $16$ times. What is the maximum number of pages this book could have? (2019 Math Kangaroo Problem, Level 3-4, Question \#19)

Before page $50$, the digit $5$ appears $5$ times.  From page $50$ to $59$, the digit $5$ appears $11$ times.  Then, the digit $5$ appears the $17$\textsuperscript{th}~time in $65$. Thus, the maximum page number is $64$.


\textbf{Refer to the previous problem. On the final exam Carla scored 98. What is the value of her residual?}

None of the above


The base of a triangle increases by $30\textbackslash\%$ and its height decreases $20\textbackslash\%$ at the same time. The area of the new triangle is~\uline{~~~~~~~~~~}~$\textbackslash\%$ of the original triangle.

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


None of the above


two

three

four

five

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


How many different isosceles triangles have integer side lengths and perimeter $23$ ? (2005 AMC 8 Problem, Question \#15)

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


Robin had a great meal in a local Thai restaurant and he spent $34$ dollars on the dishes. The tax is $10\textbackslash\%$ of the dishes and Robin also needed to pay for the tip after paying the food and tax. Robin brought $60$ dollars with him and he\textquotesingle s thinking about how many tips should he give.  Which of the following is an inequality that represents the situation?

$x=60-34\times 10\textbackslash\%-34$, let $x$ be the amount of tips

$x=60-34$, let $x$ be the amount of tips

$x \leq 60-34\times 10\textbackslash\%-34$, let $x$ be the amount of tips

$x\leq 60-34$, let $x$ be the amount of tips

Robin needs to pay the food and taxes, the rest he can decide how much he wants to pay for tips. Therefore, the tip should be represented using inequality.


The $2021^{}\text{st}$ digit to the right of the decimal point in the decimal expansion of $\dfrac{5}{37}$ is~\uline{~~~~~~~~~~}~.

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

$$\frac{5}{37}=0.\overline{135}$$, it is a decimal which repeats in cycles of $3$ digits. $2021\div 3=673$$R2$, so the $2021$$^{st}$ digit is $3$.


Martin and Dai divide some sweets between them. There are $65$ sweets, and Martin takes $7$ more than Dai. How many does Martin take?


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


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? (2022 AMC 8 Problems, Question \#6)

Let the smallest number be $x$. It follows that the largest number is $4 x$.  Since $x, 15$, and $4 x$ are equally spaced on a number line, we have  $$ \begin{aligned} 4 x-15 \& =15-x \textbackslash\textbackslash{} 5 x \& =30 \textbackslash\textbackslash{} x \& =(C) 6. \end{aligned} $$


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


What is $$40\textbackslash\%$$ of $$28.5$$ plus $$28.5\textbackslash\%$$ of $$60$$? ．

$$14.25 $$

$$28.5 $$

$$42.75 $$

$$57 $$

Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Extracting Common Factors from Decimals

It is a general rule that $$x\textbackslash\%$$ of $$y$$ equals $$y\textbackslash\%$$ of $$x$$. This is because $$x\textbackslash\%$$ of $$y= \frac{x}{100} \times y= \frac{xy}{100}$$ and $$y \textbackslash\% $$ of $$x= \frac{y}{100} \times x= \frac{yx}{100}$$.  So $$40\textbackslash\%$$ of $$28.5$$ plus $$28.5\textbackslash\%$$ of $$60 = 40\textbackslash\%$$ of $$28.5$$ plus $$60\textbackslash\%$$ of $$28.5 = 100\textbackslash\%$$ of $$28.5 = 28.5$$.


In an arithmetic sequence, the $1$\textsuperscript{st} number is $5$, the $2$\textsuperscript{nd} number is $8$, the $3$\textsuperscript{rd} number is $11$, and so on. What is the $25$\textsuperscript{th} term of this sequence?

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

The $25$\textsuperscript{th} number: $5+(25-1)\times3=77$


In the arithmetic sequence: $77$, $86$, $95$, $104$, $\cdots$, the $14$\textsuperscript{th}~term is~\uline{~~~~~~~~~~}~.


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?

Overseas Competition->Knowledge Point->Calculation Modules


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


What is $$0.9949$$ when rounded to the nearest hundredth?

Since the thousandth\textquotesingle s digit is $$4$$, round $$0.9949$$ down to $$0.99$$.


There are four types of number machines: $$A$$, $$B$$, $$C$$, $$D$$.  Device A: Add $$3$$ to the input number;  Device B: Divide the input number by $$3$$;  Device C: Subtract $$3$$ from the input;  Device D: Multiply the input number by $$3$$.  These devices can be connected. If the device $$A$$ is followed by the device $$B$$, it is written as $$A-B$$. For example, input $$6$$, $$6 + 3 = 9, 9 \div3 = 3 $$. Thus the output would be $$3$$. If you input a number in the device $$B$$-$$A$$-$$D$$-$$C$$, and get $$60$$, the input number was ．

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

$$B$$-$$A$$-$$D$$-$$C$$ has the rule: divide the input number by $$3$$; add $$3$$ to the input number;~multiply the input number by $$3$$; subtract $$3$$ from the input. So the inverse operation gives $$((60+3)\div3-3)\times3=54$$.


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


There are some identical candies on a electronic scale. The little bear wants to eat some of them. After eating $3$ candies, the scale shows $105$ grams. After eating a total of $5$ candies, the scale shows $75$ grams. How many candies are on the scale at first?

The quality of each candy: $(105 - 75) \div 2 = 15$ g  The amount of the apples at first: $75~ \div 15 + 5 = 10$


Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas, would like to cover the package with a large number of stamps. Suppose he has a collection of $5$-cent, $10$-cent, and $25$-cent stamps, with exactly $20$ of each type. What is the greatest number of stamps Nicolas can use to make exactly $\textbackslash$7.10$ in postage?


What is the second to last digit when $$5^{7}$$ is calculated?

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

The second to last digit is always $$2$$. For example $$5^{3}$$ is $$125$$. When multiplying this by $$5$$, we see that is it inevitable that the second to last digit remains $$2$$. Try repeated multiplication of $$5$$ on a calculator.


Overseas Competition->Knowledge Point->Calculation Modules->Operations through Formulas->Perfect Square Factorization


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


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

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

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


Starting with some gold coins and some empty treasure chests, I tried to put $8$ gold coins in each treasure chest, but that makes $1$ gold short to fill all the chests. So instead I put $6$ gold coins in each treasure chest, but then I had $15$ gold coins left over. How many gold coins did I have? ( adapted from 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: $$ \begin{gathered} 8 c-1=g \textbackslash\textbackslash{} 6 c+15=g \end{gathered} $$  Therefore, $8 c-1=6 c+15$. This implies that $c=8$. We therefore have $g=63$. So, our answer is (D) $63$


Which does NOT have $-1$ as a solution?~\uline{~~~~~~~~~~}~

$2 x-4\textless-5$

$1-x \geq 2$

$\frac{2}{x}\textgreater x$

$A$ gives $-6\textless-5$. True.  $B$ gives $2 \geq 2$. True.  $C$ gives 1=1. True.  $D$ gives $-2\textgreater-1$. False.


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 $\frac{\frac{6 s}{5}+\frac{2 s}{5}}{\frac{6 s}{5}+s}=\frac{\frac{6 s+6 s}{15}}{\frac{11 s}{5}}=\frac{\frac{4 s}{5}}{\frac{11 s}{5}}=\left(\right.$ B) $\frac{4}{11}$.


Twelve friends met for dinner at Oscar\textquotesingle s Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for $18$ people. If they shared, how many meals should they have ordered to have just enough food for the $12$ of them? (2004 AMC 8 Problems, Question \#3)

Set up the proportion $\frac{12 \text { meals }}{18 \text { people }}=\frac{x \text { meals }}{12 \text { people }}$. Solving for $x$ gives us $x=8$.


Which of the following is the largest fraction?  $$\dfrac{2}{6}$$,$$\dfrac{2}{7}$$,$$\dfrac{2}{8}$$,$$\dfrac{2}{9}$$

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

Same numerator, so smaller denominator means larger fraction.


David measured the length of his garden.  It was $$15$$ metres to the nearest tenth of a metre.  Between what limits was the actual length?

$$14.995\leqslant $$ the actual length $$\leqslant 15.005$$

$$14.9\leqslant $$ the actual length $$\textless~15.1$$

$$14.95\leqslant $$ the actual length $$\textless{} 15.05$$

$$14.99\leqslant $$ the actual length $$\textless15.01$$

$$14.5\textless$$ the actual length $$\textless15.5$$

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


There were three piles of plums and a camel wanted to eat some. Each pile had $30$ plums. The camel ate a few plums from the first pile and then ate as many strawberries from the third pile as were left in the first pile. Then it ate some plums in the second plie, and finally there were $5$ plums left in the second pile. How many plums in total did the camel eat?


In a group of $40$ students, $40\textbackslash\%$ of them can swim. In this group of students, how many of them can swim?

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


$$33$$ hours and $$36$$ minutes $$=$$  minutes.

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


Polly has more than $90$ candies. The candies cabe divided evenly between $2$, $3$ or $4$, children. However, they cannot be divided evenly between $9$ children because $3$ more candies would be needed. How many candies does she have at least?


Avril\textquotesingle s father bought 15 books. Avril takes away 8 books. How many books does Avril\textquotesingle s father have left?


Find the value of $$\left\textbar{} -19 \right\textbar$$.

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

By the definition, we can remove the mimus sign before $$-19$$. We get $$19$$ and choose $$\text{A}$$.


The ones digit of $$1!\times2!\times3!\times ···\times50!$$ is~\uline{~~~~~~~~~~}~.

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

As long as we can find the factor $2$ and $5$, the ones digit is $0$.


Evaluate the following expression:  $$2^{7}+2^{8}+\cdots +2^{19}+2^{20}$$

None of the above


Fill iin the missing number.  $$-7\textless{}11+9$$

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


Some numbers add up to $4077$ and one of the number is $93$. Double this number. What is the new sum of the these numbers?


If the repeating decimal $2.0232323\cdots $ can be written as $\frac{m}{n}$, where the fraction is in its simplest form. Find $m+n$

None of the above

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


Think Academy will hold a concert. The stage and seats have been set up. There are $$40$$ benches that can only seat one people each, and there are $$7$$ benches that can seat three people each ,How many people can all the chairs and benches hold?


There are $20$ balls of the same size in a box.  Lucas says: "$\frac12$ of them are red."  Peter says: "The probability of drawing a green ball without observing them is $\frac15$."  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?


In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x\textgreater y\textgreater0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? (2012 AMC 10B Problem, Question \#6)

Her estimate is larger than $x-y$

Her estimate is smaller than $x-y$

Her estimate equals $x-y$

Her estimate equals $y-x$

Her estimate is $0$

Overseas Competition->Knowledge Point->Calculation Modules

Let\textquotesingle s define $z$ as the amount rounded up by and down by.  The problem statement tells us that Xiaoli performed the following computation: $(x+z)-(y-z)=x+z-y+z=x-y+2 z$  We can see that $x-y+2 z$ is greater than $x-y$, and so the answer is $A$.


What is the correct ordering of the three numbers $$\dfrac{5}{6}$$, $$\dfrac{7}{8}$$, and $$\dfrac{9}{10}$$, in increasing order? (Adapted from$$2012$$ AMC $$8$$ Problem, Question \#$$4$$)

$$\dfrac{9}{10}\textless{} \dfrac{7}{8}\textless\dfrac{5}{6}$$

$$\dfrac{5}{6}\textless{} \dfrac{7}{8}\textless{} \dfrac{9}{10}$$

$$\dfrac{9}{10}\textless{} \dfrac{5}{6}\textless{} \dfrac{7}{8}$$

$$\dfrac{5}{6}\textless{} \dfrac{9}{10}\textless{} \dfrac{7}{8}$$

$$\dfrac{7}{8}\textless{} \dfrac{5}{6}\textless{} \dfrac{9}{10}$$

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

Instead of finding the LCD, we can subtract each fraction from $$1$$ to get a common numerator. Thus,  $$1- \dfrac{5}{6}= \dfrac{1}{6}$$,  $$1- \dfrac{7}{8}= \dfrac{1}{8}$$,  $$1- \dfrac{9}{10}= \dfrac{1}{10}$$.  All three fractions have the common numerator $$1$$. Now the order of the fractions is obvious. $$\dfrac{1}{6}\textgreater\dfrac{1}{8}\textgreater\dfrac{1}{10}\Rightarrow\dfrac{5}{6}\textless\dfrac{7}{8}\textless\dfrac{9}{10}$$. Therefore, $$\dfrac{5}{6}\textless\dfrac{7}{8}\textless\dfrac{9}{10}$$.


Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3 x+y}{x-3 y}=-2$. What is the value of $\frac{x+3 y}{3 x-y}$?

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

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 $\frac{x+3 x}{3 x-x}=\frac{4 x}{2 x}= 2$ More step-by-step explanation:  $$ \begin{aligned} \&\frac{3 x+y}{x-3 y}=-2 \textbackslash\textbackslash{} \&3 x+y=-2(x-3 y) \textbackslash\textbackslash{} \&3 x+y=-2 x+6 y \textbackslash\textbackslash{} \&5 x=5 y \textbackslash\textbackslash{} \&x=y \textbackslash\textbackslash{} \&\frac{x+3 y}{3 x-y}=\frac{1+3(1)}{3(1)-1}=\frac{4}{2}=2 \end{aligned} $$


A binomial event has n = 60 trials. The probability of success on each trial is 0.4. Let X be the count of successes of the vent during the 60 trials. Whart are the $$\mu\_x$$ and $$\sigma\_x$$?

24, 3.49

24, 14.4

4.90, 3.79

4.90, 14.4

2.4, 3.79


Observe the sequence below and fill in the blank.  $$2, 1, 3, 4, 7, $$~\uline{~~~~~~~~~~}~$$,18, 29, 47$$

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


Which of the following is not an algebraic expression?

$x=y$


Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the front of the ship to the back. In terms of Emily\textquotesingle s equal steps, what is the length of the ship? (2021 Fall AMC 10A Problems, Question \#11)

Let $x$ be the length of the ship. Then, in the time that Emily walks $210$ steps, the ship moves $210-x$ steps. Also, in the time that Emily walks $42$ steps, the ship moves $x-42$ steps. Since the ship and Emily both travel at some constant rate, $\frac{210}{210-x}=\frac{42}{x-42}$. Dividing both sides by $42$ and cross multiplying, we get $5(x-42)=210-x$, so $6 x=420$, and $x=70$.


Fill in the operation sign "$$+$$" or~"$$-$$" in the circles below to make the number statement true.  $$5$$ 　$$5$$ 　$$5$$ 　$$5$$　$$5$$ 　$$5=0$$

I don\textquotesingle t know o(╥﹏╥)o


\textbf{At a certain restaurant, the distribution of wait times between ordering a meal and receiving the meal has mean 11.4 minutes and standard deviation 2.6 minutes. The restaurant manager wants to find the probability that the mean wait time will be greater than 12.0 minutes for a random sample of 84 customers. Assuming the wait times among customers are independent, which of the following describes the sampling distribution of the sample mean wait time for random samples of size 84 ?}

\textbf{~Approximately normal with mean 11.4 minutes and standard deviation 2.6 minutes}

\textbf{~Approximately normal with mean 11.4 minutes and standard deviation $\frac{2.6}{\sqrt{84}}$ minute~}

\textbf{Approximately normal with mean 12.0 minutes and standard deviation 2.6 minutes}

\textbf{Binomial with mean 84(0.41) minutes and standard deviation 84(0.41)(0.59) minutes~}

\textbf{Binomial with mean 84(0.5) minutes and standard deviation 84(0.5)(0.5) minutes}

\textbf{For a sufficiently large n, the sampling distribution of $$\bar\_{X}$$ is approximately normal, with mean $\mu\_{\bar\_{X}}=\mu$ and standard deviation $\sigma\_\bar\_{X}=\sqrt{\frac{\sigma}{\sqrt{n}}}$}


Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned $6$ years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is $30$ years. How many years older than Bella is Anna?


Which calculation has the greatest value?

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

$$2\times 0\times 2\times 2$$

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


The result of $\frac12\times \frac23\times \frac34$ is~\uline{~~~~~~~~~~}~.

$$\frac 1{3}$$

$$\frac 14$$

$$\frac 12$$

$$\frac {3}{4}$$

$$\frac {1}{8}$$

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

We notice that a lot of terms can be canceled. In fact, every term in the numerator except for the $$1$$ and every term in the denominator except for the $$4$$ will be canceled out, so the answer is $$\frac 1{4}$$.


Dave bought $3$ packets of chips at $2.40$ dollars each and $3$ cans of soft drink at $70$ cents each. How much did he spend altogether?

$3.10$ dollars

$10.70$ dollars

$9.30$ dollars

$8.70$ dollars

$10.50$ dollars


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


The bookstore has the "Exchanging Sales". One school bag exchanges 3 books, 2 books exchange 6 pens, and 3 pens exchange 6 erasers. One school bag exchangeserasers．

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


For how many integers $x$ is the number $x^{4}-51 x^{2}+50$ negative? (2014 AMC 10B Problems, Question \#20)

First, note that $50+1=51$, which motivates us to factor the polynomial as $\left(x^{2}-50\right)\left(x^{2}-1\right)$. 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 $x^{2}-50\textless0\textless x^{2}-1$. Solving this inequality, we find $1\textless x^{2}\textless50$. There are exactly $12$ integers $x$ that satisfy this inequality, $\pm\textbackslash{2,3,4,5,6,7\textbackslash}$. Thus our answer is $(\mathbf{C}) 12$.


$$2015$$ Rosyth School, First Continual Assessment, Primary $$5$$, Question \#$$11$$  What is the value of $$\left( 84\div 7 \right)+7\times 6-3$$?


Which of the following fractions is equivalent to $$0.3\dot{8}$$？

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


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


Bruce is a talented writer and graphic artist who enjoys both types of work equally. Instead of earning 45,000 dollars as a writer, Bruce now earns 25,000 dollars in accounting profits as a graphic artist using the same computer equipment he would have used as a writer. What is Bruce\textquotesingle s economic profit from choosing to work as a graphic artist?

-45,000 dollars

-20,000 dollars

20,000 dollars

45,000 dollars

70,000 dollars

Economic profit = total revenue - economic costs = total revenue - (explicit costs + implicit costs)


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

Separate into two equations~$\dfrac{3}{5}=\dfrac{M}{45}$~and~$\dfrac{3}{5}=\dfrac{60}{N}$~and solve for the unknowns.~$M=27$~and $N=100$, therefore $M+N=\boxed{\left( E\right)127}.$.


How many digits are there in the correct answer to the calculation $$1234123412340\div 1234$$?

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

The correct answer to the calculation $$1234123412340\div1234=1000100010$$.  This has $$10$$ digits.


Four numbers are written in a row. The average of the first two is $21$, the average of the middle two is $26$, and the average of the last two is $30$. What is the average of the first and last of the numbers?


$$\textgreater$$, $$\textgreater$$

$$\textgreater$$, $$\textless$$

$$\textless$$, $$\textgreater$$

$$\textless$$, $$\textless$$


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


Find the next number in the following sequence  $$21, 24, 19, 26, 17, \cdots $$

None of the above


$$\textasciitilde\textasciitilde\textasciitilde\textasciitilde1+1\times 2+1\times 2\times 3+1\times 2\times 3\times 4$$  $$=1+2+6+24$$  $$=33$$


There are two girls and two cats in the room. How many legs are there in this room? (Adapted from 2011 Math Kangaroo Problem, Level 1-2, Question \#4)


Peter Rabbit eats cabbages and carrots. Each day he eats either $$10$$ carrots or $$2$$ cabbages. Last week Peter ate $$6$$ cabbages. How many carrots did he eat last week? (2014 Math Kangaroo Problem, Level 1-2, Question \#18)

Peter ate $6$ cabbages in $6\div2=3$ days. In $7-3=4$ days, Peter ate $4\times10=10+10+10+10=40$ carrots.


An amusement park has a collection of scale models, with ratio $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number? (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 $1: 20$. We also know that the U.S Capitol is 289 feet in real life, so to find the height of the model, we divide by 20 . That gives us $14.45$ which rounds to 14 . Therefore, to the nearest whole number, the duplicate is (A) 14 feet.


Seventy $$-$$ seven hundred is equal to．

$$70\times 70$$

$$7\times 7000$$

$$7\times 10$$

$$11 \times 700$$

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

Seventy $$-$$ seven hundred $$= 7700 = 11\times700$$.


Alen and Austin share $60$ apples between them.  Alen has twice as much as Austin.  How many apples does Alen have?


Fifteen integers $a\_1, a\_2, a\_3, \cdots , a\_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that  $$ 1 \leq a\_1 \leq 10$$,~~$$13 \leq a\_2 \leq 20$$, and $$241 \leq a\_{15} \leq 250$$.  What is the sum of the digits of $a\_{14}$?


Given a twelve-digit number $$370592318436$$. Brandon wants to delete $4$ odd digits from the number to create an eight-digit number. What is the smallest eight-digit number that Brandon can form?


How many inequalities are there in the following options?  1. $x = 1$  2. $y \textless{} 2$  3. $1 + a \geq g + p$  4. $1 \textgreater{} 2$

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


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


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


Devi is $$9$$ years old and Valli is $$13$$ years old now. What is the ratio of Devi\textquotesingle s age to Valli\textquotesingle s age in $$5$$ years\textquotesingle{} time?

$5$ years later,  Devi is $9+5=14$ years old and Valli is $13+5=18$ years old.  $D:V$ $=$ $14:18$ $\to$ $7:9$


Four fifths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 10 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 $\frac{4}{5}$ full, i.e. $80 \textbackslash\%$ full. Therefore each cup receives $\frac{80}{10}=(\mathbf{C}) 8$ percent of the total capacity.


If Anna takes one step forward, it means she walks $"+1"$ step; If Anna takes one step backward, it means she walks $"-1"$ step. Now, Anna walks $"-5"$ steps, then she walks $"+4"$ steps, then she walks another $"-1"$ step. Anna\textquotesingle s position is from her original location.

$1$ step forward

$1$ step backward

$2$ step forward

$2$ step backward

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

Since walking one step forward and one step backward gets you back to your original location. Anna takes $4$ steps foward and $1+5=6$ steps backward.  $6-4=2$(steps)  Therefore, Anna is now $2$ steps backward from her original location.


The last five digits of Zoe\textquotesingle s phone number are $3,4,8,5,7.$ 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


When $$115$$ is divided by~ \uline{?~} , the quotient is 4 and remainder is $$3$$.

Overseas Competition->Knowledge Point->Calculation Modules


$$\left( {} \right.$$The number of seconds in a week$$\left. {} \right)$$$$\div$$$$\left( {} \right.$$the number of minutes in a week$$\left. {} \right)=$$．

There are $$60$$ seconds in each minute, so the quotient of the two quantities is $$60$$.


There are $12$ boys in the class. The ratio of boys to girls is $1:2$. How many students are there in the class?


\textbf{A news organization conducted a survey about preferred methods for obtaining the news. A random sample of 1,605 adults living in a certain state was selected, and 16.2 percent of the adults in the sample reported that television was their preferred method. Which of the following is an appropriate margin of error for a 90 percent confidence interval to estimate the population proportion of all adults living in the state who would report that television is their preferred method for obtaining the news?}

\textbf{$1.645 \sqrt{\frac{(0.162)(1-0.162)}{1605}}$}

\textbf{$1.645 \sqrt{\frac{(0.5)(1-0.5)}{1605}}$}

\textbf{$1.96 \sqrt{\frac{(0.162)(1-0.162)}{1605}}$}

\textbf{$1.96 \sqrt{\frac{(0.5)(1-0.5)}{1605}}$}

\textbf{$1.83 \sqrt{\frac{(0.162)(1-0.162)}{1605}}$}


Cassandra sets her watch to the correct time at noon. At the actual time of $1: 00$ PM, she notices that her watch reads $12: 57$ and $36$ seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads $$10:00$$ PM? (2003 AMC 12B Problems, Question \#11)

$10:22$ PM and $24$ seconds

$10:24$ PM

$10:25$ PM

$10:27$ PM

$10:30$ PM

For every $60$ minutes that pass by in actual time, $57+\frac{36}{60}=57.6$ minutes pass by on Cassandra\textquotesingle s watch. When her watch first reads, $10: 00$ PM, $10(60)=600$ minutes have passed by on her watch. Setting up a proportion, $$ \frac{57.6}{60}=\frac{600}{x} $$ where $x$ is the number of minutes that have passed by in actual time. Solve for $x$ to get $625$ minutes, or $10$ hours and $25$ minutes $$ \Rightarrow \text { (C) 10:25} $$ PM.


Yesterday is Saturday, what day is the day after tomorrow? (Adapted from 2019 Math Kangaroo Problem, Level 3-4, Question \#3)

Monday

Tuesday

Wednesday

Thursday

Sunday

Yesterday: Saturday  Today: Sunday  Tomorrow: Monday  The day after tomorrow: Tuesday


\textbf{Free-response questions on the AP Statistics Exam are graded 4, 3, 2, 1, or~ 0. Question 2 on the exam was of moderate difficulty. The average score on question 2 was 2.05 with a standard deviation of 1. To the nearest tenth, what score was achieved by a student who was at the 90th percentile of all students on the test? You may assume that the scores on the question were approximately normally distributed.}


What is the simplest form of $\frac{15}{20}$?

Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Basic Understanding of Fractions->Using Common Factors to Simplify Fractions

Factors of numerator $15$ are $1, 3, 5, 15$ and factors of denominator $20$ are $1, 2, 4, 5, 10, 20$.  The highest common factor(HCF) is $5$.  Divide $15$ and $20$ by the HCF:  $\frac{15 \div 5}{20 \div 5}$=$\frac{3}{4}$.


Which of the following sequences is different from the rest?

$$1$$，$$1$$，$$2$$，$$3$$，$$5$$，$$8$$，$$13$$，$$\cdots $$

$$0$$，$$2$$，$$2$$，$$4$$，$$6$$，$$10$$，$$16$$，$$\cdots $$

$$1$$，$$3$$，$$4$$，$$7$$，$$11$$，$$18$$，$$29$$，$$\cdots $$

$$1$$，$$2$$，$$3$$，$$6$$，$$11$$，$$20$$，$$37$$，$$\cdots $$

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

All sequences are Fibonacci numbers except for ($$\text{D}$$).  （Maths Olympiad 《Looking for a Pattern》 Pr$$3$$\&$$4$$ Question \#$$18$$）


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


There is a total of $3567$ dollars in Thompson\textquotesingle s wallet. What is the sum of the digits of his money? (adapted from $$2011$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$7$$)


Sreshtha needs to estimate the quantity $\frac{a}{b}-c$, where $a, b$, and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\frac{a}{b}-c ?$ ．(2015 AMC 12A Problem, Question \#5)

She rounds all three numbers up.

She rounds $a$ and $b$ up, and she rounds $c$ down.

She rounds $a$ and $c$ up, and she rounds $b$ down.

She rounds $a$ up, and she rounds $b$ and $c$ down.

She rounds $c$ up, and she rounds $a$ and $b$ down.

To maximize our estimate, we want to maximize $\frac{a}{b}$ and minimize $c$, because both terms are positive values. Therefore we round $c$ down. To maximize $\frac{a}{b}$, round $a$ up and $b$ down. $\Rightarrow(\mathbf{D})$


Let $a, b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$?

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

Expanding the equation and combining like terms results in $2 x^{2}-(a+2 b+c) x+(a b+b c)=0$. By Vieta\textquotesingle s formula the sum of the roots is $\frac{-[-(a+2 b+c)]}{2}=\frac{a+2 b+c}{2}$. To maximize this expression we want $b$ to be the largest, and from there we can assign the next highest values to $a$ and $c$. So let $b=9, a=8$, and $c=7$. Then the answer is $\frac{8+18+7}{2}=16.5$.


$$3\times 20$$

$$5\times 20$$


Which of the following is the correct order of the fractions $$\dfrac{15}{11}$$, $$\dfrac{19}{15}$$ and $$\dfrac{17}{13}$$, from least to greatest? ($$2019$$ AMC $$8$$ Problem, Question \#$$3$$)

$$\dfrac{15}{11}\textless{} \dfrac{17}{13}\textless{} \dfrac{19}{15}$$

$$\dfrac{15}{11}\textless{} \dfrac{19}{15}\textless{} \dfrac{17}{13}$$

$$\dfrac{17}{13}\textless{} \dfrac{19}{15}\textless{} \dfrac{15}{11}$$

$$\dfrac{19}{15}\textless{} \dfrac{15}{11}\textless{} \dfrac{17}{13}$$

$$\dfrac{19}{15}\textless{} \dfrac{17}{13}\textless{} \dfrac{15}{11}$$

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


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


For real numbers $w$ and $z$, $$ \frac{\frac{1}{w}+\frac{1}{z}}{\frac{1}{w}-\frac{1}{z}}=2014 $$ What is $\frac{w+z}{w-z} ?$

- 2014

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

Muliply both sides by $\left(\frac{1}{w}-\frac{1}{z}\right)$ to get $\frac{1}{w}+\frac{1}{z}=2014\left(\frac{1}{w}-\frac{1}{z}\right)$. Then, add $2014 \cdot \frac{1}{z}$ to both sides and subtract $\frac{1}{w}$ from both sides to get $2015 \cdot \frac{1}{z}=2013 \cdot \frac{1}{w}$. Then, we can plug in the most simple values for $z$ and $w(2015$ and 2013 , respectively), and find $\frac{2013+2015}{2013-2015}=\frac{2(2014)}{-2}=-2014$.


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

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


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 $7: 6$, and the the ratio of $8^{\text {th }}$-graders to $7^{\text {th }}$ graders is $4: 7$. What is the smallest number of students that could be participating in the project? (2013 AMC 8, Question 16)

We multiply the first ratio by 2 on both sides, and the second ratio by 3 to get the same number for 8 th graders, in order that we can put the two ratios together: $$ \begin{aligned} \&7: 6=7(2): 6(2)=14: 12 \textbackslash\textbackslash{} \&4: 7=4(3): 7(3)=12: 21\end{aligned} $$ Therefore, the ratio of 8th graders to 7th graders to 6th graders is $14: 12: 21$. Since the ratio is in lowest terms, the smallest number of students participating in the project is $$ 14+12+21=\text { (B) } 47 $$  Notice if you get $$28:24:42$$, you have to simplfy to $14: 12: 21$ for the smallest number of students.


There is a tournament at the pool. First, $$13$$ children signed up and then another $$19$$ children signed up. Six teams with an equal number of members each are needed for the tournament. At least how many more children need to sign up so that the six teams can be formed?~

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

$$13 + 19 = 32$$; $$32 \div 6 = 5R2$$. There are $$2$$ students left, which means that $$6 - 2 = 4$$ more students are needed to form $6$ teams.


A data set consists of $6$ (not distinct) positive integers: $1,7,5,2,5$, and $X$. The average (arithmetic mean) of the 6 numbers equals a value in the data set. What is the sum of all positive values of $X$? (2022 AMC 10A Problems, Question \#8)

First, note that $1+7+5+2+5=20$. The mean cannot be $1$ or $2$.  There are $3$ possible cases:  Case 1: the mean is $5$ . $$ X=5 \cdot 6-20=10 \text {. } $$  Case 2: the mean is $7$. $$ X=7 \cdot 6-20=22 \text {. } $$  Case 3: the mean is $X$. $$ \frac{20+X}{6}=X \Rightarrow X=4 \text {. } $$  Hence, the answer is $10+22+4=$ (D) $36$ .


Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability


Which of the following number is different from the others．

Eight percents

Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Basic Understanding of Fractions->Converting Between Fractions, Percentage and Decimals

$$8\textbackslash\%=0.08$$，  But $$\frac{8}{10}=0.8$$．  So, the answer is $$\text{C}$$．


An operator $$\star$$ acts on two numbers to give the following outcomes:  $$3\star2=51$$  $$5\star3=82$$  $$6\star1=75$$  $$9\star4=135$$  What is $$7\star5$$ equal to?

None of the above

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


What value does the hundredth place represent in the number $12.345$?

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

The hundredth place of a number is the second digit after the decimal point, which is $$4$$. Since the number $$4$$ is on the hundredth place, it represents $$0.04$$.  Check Lesson 4 Concept 1 on textbook


\textbf{Suppose a certain scale is not calibrated correctly, and as a result, the mass of any object is displayed as 0.75 kilogram less than its actual mass. What is the correlation between the actual masses of a set of objects and the respective masses of the same set of objects displayed by the scale?}

\textbf{Since the actual mass is always 0.75kg more than the displayed mass, they are perfectly correlated.~}


\textbf{High school students from track teams in the state participated in a training program to improve running times. Before the training, the mean running time for the students to run a mile was 402 seconds with standard deviation 40 seconds. After completing the program, the mean running time for the students to run a mile was 368 seconds with standard deviation 30 seconds. Let X represent the running time of a randomly selected student before training, and let Y represent the running time of the same student after training. Which of the following is true about the distribution of X - Y ?}

\textbf{The variables X and Y are independent; therefore, the mean is 34 seconds and the standard deviation is 10 seconds.}

\textbf{~The variables X and Y are independent; therefore, the mean is 34 seconds and the standard deviation is 50 seconds.}

\textbf{The variables X and Y are not independent; therefore, the standard deviation is 50 seconds and the mean cannot be determined with the information given.}

\textbf{The variables X and Y are not independent; therefore, the mean is 34 seconds and the standard deviation cannot be determined with the information given.}

\textbf{The variables X and Y are not independent; therefore, neither the mean nor the standard deviation can be determined with the information given.}

\textbf{Mean: 402 - 368 = 34}  \textbf{SD: X and Y are not independent, correlation is unknown → SD unknown}


There is a number $X.$ Three times the sum of $4$ and number $X$ is $36$. What is the value of number $X$?

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


insert pic  Which number is hidden behind the square?

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


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


Vivian creates a new operation: $m$@$n=m\times m-n\times n$. What is the value of $11$@$9$?


How many two-digits positive integers has at least one $8$ as its digit?


When Koko and Koala does not sleep, he eats 50 grams of leaves per hour. Yesterday, he slept 20 hours. How many grams of leaves did he eat yesterday?

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


Two non-zero real numbers, $a$ and $b$, satisfy $a b=a-b$. Which of the following is a possible value of $\frac{a}{b}+\frac{b}{a}-a b$? (2000 AMC 12 Problems, Question \#11)

Overseas Competition->Knowledge Point->Calculation Modules


$$4.07 $$

$$ 3.925 $$

$$3.979 $$

$$4.12 $$


Nancy bought 17 cones of ice-cream for her three children. Misha ate twice as many cones as Ana. Dan ate more ice-cream than Ana but less than Misha. How many cones of ice-cream did Dan eat?

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

Let x be the number of ice-cream cones that Ana ate.  Then, 2x is the number of cones that Misha ate, and the number of cones that Dan ate is:  17 - x - 2x = 17 - 3x.  It is given that Dan ate more than Misha but less than Ana; therefore,  x \textless{} 17 - 3x \textless{} 2x  4x \textless{} 17 \textless{} 5x  only x=4 satisfies this inequality.  The number of cones that Dan ate is thus 17- 3 x 4 = 5.


For $\triangle ABC$, all its side lengths are integer. The primeter of $\triangle ABC$ with a side of length $14$ and a side length of $8$ is at least ．

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

We know from the triangle inequality that the last side, $s$, fulfills $s+8\textgreater14$. Therefore, $P\textgreater14+14$. The least integer value of $P$ is $29$


Gilda has a bag of marbles. She gives $30 \textbackslash\%$ of them to her friend Pedro. Then Gilda gives $25 \textbackslash\%$ of what is left to another friend, Ebony. Finally, Gilda gives $10 \textbackslash\%$ 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? (adapted 2019 AMC 8, Question \#8)

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

After Gilda gives $30 \textbackslash\%$ of the marbles to Pedro, she has $70 \textbackslash\%$ of the marbles left. If she then gives $25 \textbackslash\%$ of what\textquotesingle s left to Ebony, she has $(0.75 * 0.70)=52.5 \textbackslash\%$ of what she had at the beginning. Finally, she gives $10 \textbackslash\%$ of what\textquotesingle s left to her brother, so she has $(0.9 * 0.525) =$ (A) $47.25\textbackslash\%$ of what she had in the beginning left.


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


$$-1$$ or $$6$$

$$1$$ or $$-6$$

$$1$$ or $$6$$


A circle is centered at $O$, $AB$ is a diameter and $C$ is a point on the circle with $\angle COB=50^{\circ}$. What is the degree measure of $\angle CAB$?


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 $\left(1, a^{2}-4\right)$ will be on the line $y=x+a^{2}-6$, $a^{2} -4=1+a^{2}-6$. Solve for $a$, there is no solution.


Tiara\textquotesingle s toll pass account has a value of $$$24$$. Each time she uses the toll road, $$$1.4$$ is deducted from the account. When the value drops below $$$10$$, she must add value to the toll pass. Assume there are $$x$$ times Tiara can use the toll road without having to add value to the toll pass. Which inequality can represent the situation?


How many $0$s are there in the end of the result of the factorial of $25$?

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

We can get $1+1+1+1+2=6$ factor $5$s from $5, 10, 15, 20, 25$ in total, which can make six $10$s after multiplying $2$. Six $10$s means that there should be $6$ $0$s in the end.


From a starting number, Olivia was supposed to subtract $3$, and then divide by $9$, but instead, Olivia subtracted $9$, then divided by $3$, getting $43$. If the correct instructions were followed, what would the result be?


What is the product of $$5$$ and $$830$$?

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


In how many ways can the letters in \textbf{BEEKEEPER} be rearranged so that two or more \textbf{E}s do not appear together?


There is some water in a bowl, which is on a electronic scales. Mike pours water into the bowl. After pouring $2$ glasses of water, the sacle shows $300$ grams. After pouring a total of $6$ glasses of water, the scale shows $700$ grams. What is the quality of the water at the start?

$$100$$ g

$$150$$ g

$$200$$ g

$$250$$ g

$$275$$ g

The quality of each glass of water: $(700 - 300) \div (6 - 2) = 100$ g  At first: $300 - 100 \times 2 = 100$ g


$\left (876\times543\right ) -\left (543\times876\right )=$．

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


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


Ace and Sam were painting a house. Ace painted $$\frac{3}{10}$$ of the house. Sam painted $$\frac{2}{10}$$ more than Ace. What fraction of the house still needs to be painted?

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


Which of the following is not an expression?

$$a$$

$$a+b=a+b$$


Which of these pairs of fractions are equivalent?

$$3\frac{1}{2}$$ and $$2\frac{1}{4}$$

$$1\frac{1}{3}$$ and $$\frac{8}{6}$$

$$1\frac{1}{4}$$ and $$\frac{3}{4}$$

$$3\frac{3}{9}$$ and $$\frac{9}{9}$$

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


Cassandra is helping her mother to pack $$75$$ cupcakes. The boxes that her mother prepare can only fit $$7$$ cupcakes. She must ensure the box is full before she can use the next box. How many boxes she can fill up?

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


Judy has less than $40$ game coins. The game coins can be divided evenly between $3, 4$ or $6$ students. However, they cannot be divided evenly between $7$ students because $4$ more game coins would be needed. How many game coins does Judy have?

The game coins can be divided evenly between $3, 4$ or $6$ students, which means the number of game coins should be divisible by $3, 4,$ or $6$, so we can eliminate the options $E$ .  The game coins cannot be divided evenly between $7$ students because $4$ more game coins would be needed, so after the number of game coins plus $4$, the result can be divisible by $7$. Thus, the answer is $24$.


How many odd numbers are there?  5, 12, 3, 6, 7, 9, 10, 4

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


Which of the followings is not an algebraic expression?


For $\triangle ABC$, all its side lengths are integers. The perimeter of $\triangle ABC$ with a side of length $25$ and a side length of $18$ is at least ．

We know from the triangle inequality that the last side, $s$, fulfills $s+18\textgreater25$. Therefore, $P\textgreater25+25$. The least integer value of $P$ is $51$


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?


Find the least dividend and divisor in each division equation below.  $\textasciitilde$  ~\uline{~~~~~~~~~~}~$$\div$$5$$=4R4$$  $\textasciitilde$  $$22\div$$~\uline{~~~~~~~~~~}~$$=7R1$$

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

The remainder should be less than the divisor.


At equilibrium in the market of apartments, the consumer surplus is $\textbackslash$ 120,000$ and the producer surplus is $\textbackslash$ 180,000$. The government decide to set a price ceiling. As a result, the consumer surplus becomes $ \textbackslash$ 140,000$ and the producer surplus becomes $\textbackslash$ 140,000$. What is the deadweight loss from the price ceiling?

At the equilibrium, the total surplus is the sum of consumer and producer surplus, which is $ \textbackslash$120,000 + \textbackslash$180,000 = \textbackslash$300,000$.  After the price ceiling is imposed, the consumer surplus increases to $\textbackslash$140,000$, while the producer surplus decreases to $\textbackslash$140,000$. The new total surplus is $ \textbackslash$140,000 + \textbackslash$140,000 = \textbackslash$280,000$.  Therefore, the deadweight loss is the difference between the total surplus before and after the price ceiling, which is $\textbackslash$300,000 - \textbackslash$280,000 = \textbackslash$20,000$ .


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


Simplify: $$b\left( a+1 \right)-a\left( b-1 \right)$$．

$$b+a$$

$$b-a$$

$$b\left( a+1 \right)-a\left( b-1 \right)=ab+b-ab+a=b+a$$, so the answer is$\text{A}$$.


Stefani is ordering fertilizer for her garden. A local garden supply store sells fertilizer by the yard (that really means cubic yard, but that is what they call it).For every yard she orders there is a $\textbackslash$$15charge and ,in addition ,there is a $\textbackslash$$40 delivery charge. ( Total Cost =~$\textbackslash$$15(yard)+~$\textbackslash$$40.)  $\textasciitilde$  If the average order is 4.3 yards with standard deviation of 2.9 yards, what is the mean and standard deviation of the Total Cost of a typical order?

mean = $\textbackslash$$4.30; standard deviation =~$\textbackslash$$2.90

mean = $\textbackslash$$64.50; standard deviation = $\textbackslash$$43.50

mean = $\textbackslash$$105.50; standard deviation =~$\textbackslash$$43.50

mean = $\textbackslash$$105.50; standard deviation = $\textbackslash$$83.50

Cannot be determined because each order may not be independent~

The correct answer is (c). When using a linear transformation (Total cost 15 (yard)+40), the mean is changed by both the multiplication and addition. But the standard deviation is only changed by the multiplication. Mean = 15(4.3) +40=$\textbackslash$$ 104.50; =15(2.9) =~$\textbackslash$$43.50.


$$40 \times 50 \times 60 = 4 \times 5\times 6 \times$$．


The average of $$4000$$ fours is four times the average of $$2000$$．

ones

twos

fours

eights

The average of $$4000$$ fours is $$4$$; this is $$4$$ times the average of $$2000$$ ones.


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


The mean, median, and mode of the $7$ data values $60,100, x, 40,50,200,90$ are all equal to $x$. What is the value of $x$? (2016 AMC 10A Problems, Question \#7)

Since $x$ is the mean, $$ \begin{aligned} x \& =\frac{60+100+x+40+50+200+90}{7} \textbackslash\textbackslash{} \& =\frac{540+x}{7} . \end{aligned} $$  Therefore, $7 x=540+x$, so $x=$ (D) $90$.


What is the simplest form of $5$ minutes $: 30$ seconds?

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

We need to make units same first. $5$ minutes equal to $300$ seconds. Now we could remove the same unit, second. We get $300:30$ and simplify it to $10:1$.


Given the symbol $$\otimes $$ defines a new operation and $$3\otimes 3=3\times 4\times 5$$, $$7\otimes 2=7\times 8$$, and $$2\otimes 4=2\times 3\times 4\times 5$$, then $$5\otimes 3=$$~\uline{~~~~~~~~~~}~.

Non of the above

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


Observe the sequence $10$, $15$, $20$, $25$, $\cdots$ , the $9$\textsuperscript{th} term is~\uline{~~~~~~~~~~}~.

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


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


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->Fractions->Basic Understanding of Fractions


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

Yes

No

Overseas Competition->Knowledge Point->Calculation Modules


None of the above


tens

ones

we don\textquotesingle t know


Evaluate the expression shown below:  $$36 \left( \frac{1}{1\times 6} + \frac{1}{6\times 11} +\frac{1}{11\times 16} + \frac{1}{16\times 21} + \frac{1}{21\times 26} + \frac{1}{26\times 31} + \frac{1}{31\times 36} \right)$$


A container had $27$ ℓ of longan drink. The drink is made up of three $2$-ℓ\textbf{~}bottles of longan syrup and some water. What was the volume of water used to make the drink?

6ℓ

Overseas Competition->Knowledge Point->Calculation Modules->Unit Conversion->Converting between Units of Volume and Capacity


Given that $$a \Omega b=a\times b-3$$, find $$4\Omega 5$$.

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


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


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


The following are the weights (in pounds) of ten people: $100, 115, 135, 140, 180, 197, 230, 250, 260, 270$. Find the $80$-th percentile.

None of the above


What is the sum of all numbers x for which~$\left\textbar{} x^{2}-12x+34\right\textbar=2$

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


17, 85

18, 40

9,~ 64

17, 40

20, 50


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?

The correct answer is A. $\textbackslash$2$, as 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->Basic Concepts of Equation->Linear Equations with Multiple Variables


What is the \uline{\textbf{average}} amount of sleep adults get each night?

5 hours

6 hours

10 hours

8 hours

12 hours

The average amount of sleep an adult gets is 8 hours. Everyone is different, however; where some adults function off of 6 hours, others need at least 9 hours or more to feel healthy and awake.


(2016) A test has a mean of 80 with a standard deviation of 4. Which of the following scores is within one standard deviation of the mean?


Cristi has to sell 10 glass bells that vary in price: 1 euro, 2 euro, 3 euro, 4 euro, 5 euro, 6 euro, 7 euro, 8 euro, 9 euro, 10 euro. In how many ways can Cristi divide all the grass bells in three packages so that all the packages have the same price?

Such a division is not possible

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


What is the solution of this equation: $$2^{2007}=4^{1003}\cdot x$$?

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


Which of the following is a solution of $$\begin{cases}x-3=0 \textbackslash\textbackslash{} 3x-2y=7 \end{cases}$$?


$$16\times 2$$

$$12\times 3$$

$$7\times 5$$

$$38\times 1$$


How many different three-digit numbers can we make using $6,7,8,$ and $9$? The digits can be repeated.

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


What is the cost of eight mugs at £$$2.99$$ each?

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


Given that $n!$ represent $n$ factorial, with $n!=1\times 2\times 3 \times \cdots \times n$, then how many positive integer divisors of $12!$ are perfect squares?


Happy Hotel is offering $$40 \textbackslash\%$$ off discount for any bookings. David booked a room, the new price is $$80$$ dollars cheaper than the original price, what was the original price．

$$80\div40\textbackslash\%=200$$．  so choose $$\text{B}$$．


Which of the following is equivalent to $\sqrt{16a^{16}}$?


Calculate:  $$\left(1+\frac 12+\frac 13+\cdots +\frac 1{149}\right)\times \left(\frac 12+\frac 13+\cdots +\frac 1{149}+\frac 1{150}\right)$$$$-\left(1+\frac 12+\frac 13+\cdots +\frac 1{149}+\frac 1{150}\right)\times \left(\frac 12+\frac 13+\cdots +\frac 1{149}\right)=$$~\uline{~~~~~~~~~~}~.

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

Suppose $$\left(\frac 12+\frac 13+\cdots +\frac 1{149}\right)$$ as $$A$$, $$\left(\frac 12+\frac 13+\cdots +\frac 1{150}\right)$$ as $$B$$.  $$(1+A)\times B-(1+B)\times A=B+AB-A-AB=B-A=\frac 1{150}$$.


Which of the following is equal to the product shown below?  $$\frac{8}{4} \cdot \frac{12}{8} \cdot \frac{16}{12} \cdot \frac{20}{16} \cdot \cdot \cdot \frac{2024}{2020}$$

None of the above


If $$6$$ pens cost as much as $$5$$ pencils, then $$36$$ pens cost as much as pencils.

Overseas Competition->Knowledge Point->Heuristics Skills-> Equivalent Substitution

$$6$$ pens = $$5$$ pencils, we use $\times6$ which give us $36$ pens = $30$ pencils


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 $s\textless12+13=25$. Adding $12+13$ to both sides of the inequality, we get $s+12+13\textless25$, and because $s+12+13$ is the perimeter of our triangle, (B) 50 is our answer.


is a factor of $$1\times2\times3\times4\times5\times6\times7\times8\times9\times10$$.

$$1\times2\times3\times4\times5\times6\times7\times8\times9\times10$$ is divisible by $$3$$ and $$25$$.


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


Teacher Nicole bought some badges and divided it equally among $8$ children. If everyone got $9$ badges, there would still be some badges remaining. What is the biggest possible and smallest possible number of badges Teacher Nicole could have bought?

~\uline{~~~~~~~~~~}~$\div 8=9$ $\text{R}$~\uline{~~~~~~~~~~}~  Biggest possible remainder is $7$ while smallest possible remainder is $1$.  Biggest possible number of badges is $$8\times 9+7=79$$, while the least possible number of sweets is $$8\times 9+1=73$$.


In a rectangle, what is the ratio of one angle to the sum of all inner angles?

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


Real numbers $x$ and $y$ satisfy $x^{3}+y^{3}=4$ and $xy = 2$. What is the value of $2xy+\frac{x^{4}}{y^{2}}+\frac{y^{4}}{x^{2}}$? (Adapted From 2020 AMC 10A Problem, Question \#14)


Real numbers $x$ and $y$ satisfy $x+y=4$ and $x\cdot y=-2$. What is the value of $x+\frac{x^{3}}{y^{2}}+\frac{y^{3}}{x^{2}}+y$? (2020 AMC 10A Problem, Question \#14)


Given that $$a\Phi b=2\times a-b$$, for example, $$2\Phi 1 = 2\times2 -1$$, what is $$3\Phi4$$?

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


For how many integers $x$ is the number $x^{4}-51 x^{2}+50$ negative? (  2014 AMC 10B Problems, Question \#20)

Overseas Competition->Knowledge Point->Calculation Modules

First, note that $50+1=51$, which motivates us to factor the polynomial as $\left(x^{2}-50\right)\left(x^{2}-1\right)$. 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 $x^{2}-50\textless0\textless x^{2}-1$. Solving this inequality, we find $1\textless x^{2}\textless50$. There are exactly $12$ integers $x$ that satisfy this inequality, $\pm\textbackslash{2,3,4,5,6,7\textbackslash}$. Thus our answer is $(\mathbf{C}) 12$.


How many of the following options are equations?  1. $x = y$  2. $x \textgreater{} 1$  3. $x \geq x-1$  4. $ 1 = 2$

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


How many sections will Linda get if she cuts a piece of wood $4$ times?

It depends on how long the piece of wood is.


What is the product of $$628$$ and $$6$$?（)

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 2 in 628 stands for 2 tens(20) ,the 6 in the 628 stands for 6 hundreds(600)  First, multiply the ones~ $6\times8=48$~, regroup the 4 tens to the tens column  Write 8 in the ones place.  Then,~ Multiply and add the tens .~$2\times6+4=16$  Write 6 in the tens place and regroup the 1 hundred.  Last, multiply and add the hundreds.~$6\times6+1=37$  Write 7 in the hundreds place and write 3 in the thousands place.


In the animal school, some sheep are taking lessons. The teacher cow finds out that the sheep have $$24$$ legs altogether. How many sheep are there? (Adapted from 2012 Math Kangaroo Problem, Level 3-4, Question \#14)


The $1985^{}\text{th}$ digit at the right of the decimal point in the decimal expression of $\dfrac{1}{7}$ is~\uline{~~~~~~~~~~}~.

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

$$\frac{1}{7}=0.\overline{142857}$$, it is a decimal which repeats in cycles of $6$ digits. Every $6$\textsuperscript{th}~digit is $7$. The $1986$\textsuperscript{th} digit is $7$, so the $1985$\textsuperscript{th} digit is $5$.


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


How many different three-digit numbers can we make using $6,7,8,$ and $9$? The digits can be repeated.


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

infinitely many

We see that the vertex of the quadratic function $y=x^{2}-4x+a^{2}$ is $\left(2, a^{2}-4\right)$. If $\left(2, a^{2}-4\right)$ will be on the line $y=x+a^{2}-6$, $a^{2} -4=2+a^{2}-6$. Solve for $a$, any value of $a$ will satisfy this equation.


Line $l\_1$ has equation $2x-y=3$ and goes through $A=(1,-1)$. Line $l\_2$ has equation $y=1$ and meets line $l\_1$ at point $B$. Line $l\_3$ has negative 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 $l\_3$? (Adapted From 2013 AMC 12B Problems, Question \#8)

Line $l\_1$ has the equation $y=2x-3$ when rearranged. Substituting $1$ for $y$, we find that line $l\_2$ will meet this line at point $(2,1)$, 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 $\textbar-1-1\textbar=2$. The area of $\triangle A B C=3$. Since $A=\frac{bh}{2}, b=3$. Points that are on the line $y= 1$ and has a distance of $3$ from $B$ are $(5,1)$ and $(-1,1)$. Since $l\_3$ has negative slope, point $C$ is $(-1,1)$. $l\_3$ passes through $(-1,1)$ and $(1,-1)$, and thus has slope $\frac{1-(-1)}{-1-1}=(\mathbf{C}) -1$.


What is the median of the following distribution: 6, 2, 9, 4, 7, 3?

The median of the distribution is 5. The problem is easier if vou put the scores in order: 2, 3, 4, 6, 7, 9. Since the distribution has an even number of scores, there is no middle score and you must average the two middle scores, 4 and 6.


Overseas In-curriculum->Knowledge Point->Operations of Numbers ->Addition and Subtraction of Whole Numbers->Adding and Subtracting within 10000->Subtraction of 3-digit Numbers

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


Find the missing number: $$512\times2 = 32\times $$．

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


Alicia and Emily agreed to meet at the cinema at $3.55\rm{pm}$. Emily left her house at $1.47\rm{pm}$ 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.


Calculate~ ~ ~$\dfrac{2014}{2013-\dfrac{2012}{2011-\dfrac{2010}{5-\dfrac{4}{3-\dfrac{2}{1}}}}}$  After calculating, Chuan says that answer is D. Is he right or wrong? If not, choose the right option.

None of the above

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


Which number has to be subtracted from $-17$ in order to obtain $-33$? (2017 Math Kangaroo Problem, Level 7-8, Question \#3)

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


$$-1$$ or $$\frac{11}{5}$$

$$1$$ or $$\frac{1}{5}$$

$$1$$ or $$-\frac{11}{5}$$

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


The operator $$\bigtriangleup$$ acts on two numbers to give the following outcomes:  $$3 \bigtriangleup 2 = 12$$  $$4 \bigtriangleup 5 = 40$$  $$5 \bigtriangleup 9 = 90$$  $$6 \bigtriangleup 1 = 12$$  What is $$2 \bigtriangleup 7$$ equal to?

None of the above

The pattern of the operation is $$a \bigtriangleup b = a\times b\times2$$


Find the integer part of the following fractional expression:  $$ \frac{1}{\frac{1}{50}+\frac{1}{51} +\frac{1}{52} +\frac{1}{53} +\frac{1}{54}} $$

None of the above

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


Three football teams participate in a sport tournament. Each team plays the other two teams exactly once. In each game, the winner gets three points and the loser doesn\textquotesingle t get any points. If the game ends in a tie, each team gets $1$ point. At the end of tournament, Which number of points is it impossible for any team to have? (2022 Math Kangaroo Problem, Level 3-4, Question \#18)

Each team plays $2$ games.  $1$ points: a lost and a tie.  $2$ points: two ties.  $3$ points: a lost and a win.  $4$ points: a win and a tie.  $6$ points: two wins.


Suppose $5\times8=\triangle$, $4\times7=\angle$, $3\times9=\square$. Arrange the three shapes according to their values from smallest to largest.

$\angle$ $\triangle$ $\square$

$\square$ $\angle$ $\triangle$

$\triangle$ $\angle$ $\square$

$\triangle$ $\square$ $\angle$

$\square$ $\triangle$ $\angle$

$\square=27$  $\angle=28$  $\triangle=40$


The students in Mr. Neatkin\textquotesingle s class took a penmaship test. Two-thirds of the boys and~$\dfrac{3}{4}$~of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class? (2008 AMC 8, 20)

Let~$b$~be the number of boys and~$g$~be the number of girls.  $\dfrac{2}{3}b=\dfrac{3}{4}g\textbackslash{} \Rightarrow\textbackslash{} b=\dfrac{9}{8}g$  For~$g$~and~$b$~to be integers,~~must cancel out with the denominator, and the smallest possible value is . This yields~~boys. The minimum number of students is~$8+9=\boxed{\left( B\right)17}$  Solution 2  We know that~$\dfrac{2}{3}B=\dfrac{3}{4}G\textbackslash{} or\textbackslash{} \dfrac{6}{9}B=\dfrac{6}{8}G.$~So, the ratio of the number of boys to girls is~$9:8$. The So, the ratio of the number of boys to girls is~$8+9=\boxed{\left( B\right)17}$.


In how many different ways can Chloe select two digits out of $$ 0$$, $$1$$, $$ 2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, and $$9$$ and put into the two boxes to make the below equality correct?  $$ 20- \square =22-\square $$

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


I multiply a whole number by itself, then multiply that product by itself. The ones digit of my final product cannot be．

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