ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Kerry has a list of $n$ integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$. Kerry calculates the pairwise sums of all $m=\frac{1}{2} n(n-1)$ possible pairs of integers in her list and orders these pairwise sums as $s_{1} \leq s_{2} \leq \ldots \leq s_{m}$. For example, if Kerry's list consists of the three integers $1,2,4$, the three pairwise sums are $3,5,6$.


Suppose that $n=4$ and that the 6 pairwise sums are $s_{1}=8, s_{2}=104, s_{3}=106$, $s_{4}=110, s_{5}=112$, and $s_{6}=208$. Determine two possible lists $(a_{1}, a_{2}, a_{3}, a_{4})$ that Kerry could have.
Answer: (1,7,103, 105), (3, 5, 101, 107)
Problem: Determine all values of $x$ for which $\frac{x^{2}+x+4}{2 x+1}=\frac{4}{x}$.
Answer: -1$,$2$,$-2
Problem: Determine the number of positive divisors of 900, including 1 and 900, that are perfect squares. (A positive divisor of 900 is a positive integer that divides exactly into 900.)
Answer: 8
Problem: Points $A(k, 3), B(3,1)$ and $C(6, k)$ form an isosceles triangle. If $\angle A B C=\angle A C B$, determine all possible values of $k$.
Answer: 8$,$4
Problem: A chemist has three bottles, each containing a mixture of acid and water:

- bottle A contains $40 \mathrm{~g}$ of which $10 \%$ is acid,
- bottle B contains $50 \mathrm{~g}$ of which $20 \%$ is acid, and
- bottle C contains $50 \mathrm{~g}$ of which $30 \%$ is acid.

She uses some of the mixture from each of the bottles to create a mixture with mass $60 \mathrm{~g}$ of which $25 \%$ is acid. Then she mixes the remaining contents of the bottles to create a new mixture. What percentage of the new mixture is acid?
Answer: 17.5%
Problem: Suppose that $x$ and $y$ are real numbers with $3 x+4 y=10$. Determine the minimum possible value of $x^{2}+16 y^{2}$.
Answer: 10
Problem: A bag contains 40 balls, each of which is black or gold. Feridun reaches into the bag and randomly removes two balls. Each ball in the bag is equally likely to be removed. If the probability that two gold balls are removed is $\frac{5}{12}$, how many of the 40 balls are gold?
Answer: 26
Problem: The geometric sequence with $n$ terms $t_{1}, t_{2}, \ldots, t_{n-1}, t_{n}$ has $t_{1} t_{n}=3$. Also, the product of all $n$ terms equals 59049 (that is, $t_{1} t_{2} \cdots t_{n-1} t_{n}=59049$ ). Determine the value of $n$.

(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a constant. For example, $3,6,12$ is a geometric sequence with three terms.)
Answer: 20
Problem: If $\frac{(x-2013)(y-2014)}{(x-2013)^{2}+(y-2014)^{2}}=-\frac{1}{2}$, what is the value of $x+y$ ?
Answer: 4027
Problem: Determine all real numbers $x$ for which

$$
\left(\log _{10} x\right)^{\log _{10}\left(\log _{10} x\right)}=10000
$$
Answer: 10^{100}$,$10^{1 / 100}
Problem: Without using a calculator, determine positive integers $m$ and $n$ for which

$$
\sin ^{6} 1^{\circ}+\sin ^{6} 2^{\circ}+\sin ^{6} 3^{\circ}+\cdots+\sin ^{6} 87^{\circ}+\sin ^{6} 88^{\circ}+\sin ^{6} 89^{\circ}=\frac{m}{n}
$$

(The sum on the left side of the equation consists of 89 terms of the form $\sin ^{6} x^{\circ}$, where $x$ takes each positive integer value from 1 to 89.)
Answer: 221,$8
Problem: Let $f(n)$ be the number of positive integers that have exactly $n$ digits and whose digits have a sum of 5. Determine, with proof, how many of the 2014 integers $f(1), f(2), \ldots, f(2014)$ have a units digit of 1 .
Answer: 202
Problem: If $\log _{10} x=3+\log _{10} y$, what is the value of $\frac{x}{y}$ ?
Answer: 1000
Problem: If $x+\frac{1}{x}=\frac{13}{6}$, determine all values of $x^{2}+\frac{1}{x^{2}}$.
Answer: \frac{97}{36}
Problem: A die, with the numbers $1,2,3,4,6$, and 8 on its six faces, is rolled. After this roll, if an odd number appears on the top face, all odd numbers on the die are doubled. If an even number appears on the top face, all the even numbers are halved. If the given die changes in this way, what is the probability that a 2 will appear on the second roll of the die?
Answer: \frac{2}{9}
Problem: The table below gives the final standings for seven of the teams in the English Cricket League in 1998. At the end of the year, each team had played 17 matches and had obtained the total number of points shown in the last column. Each win $W$, each draw $D$, each bonus bowling point $A$, and each bonus batting point $B$ received $w, d, a$ and $b$ points respectively, where $w, d, a$ and $b$ are positive integers. No points are given for a loss. Determine the values of $w, d, a$ and $b$ if total points awarded are given by the formula: Points $=w \times W+d \times D+a \times A+b \times B$.



Final Standings

|  | $W$ | Losses | $D$ | $A$ | $B$ | Points |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| Sussex | 6 | 7 | 4 | 30 | 63 | 201 |
| Warks | 6 | 8 | 3 | 35 | 60 | 200 |
| Som | 6 | 7 | 4 | 30 | 54 | 192 |
| Derbys | 6 | 7 | 4 | 28 | 55 | 191 |
| Kent | 5 | 5 | 7 | 18 | 59 | 178 |
| Worcs | 4 | 6 | 7 | 32 | 59 | 176 |
| Glam | 4 | 6 | 7 | 36 | 55 | 176 |
Answer: 16,3,1,1
Problem: Let $\lfloor x\rfloor$ represent the greatest integer which is less than or equal to $x$. For example, $\lfloor 3\rfloor=3,\lfloor 2.6\rfloor=2$. If $x$ is positive and $x\lfloor x\rfloor=17$, what is the value of $x$ ?
Answer: 4.25
Problem: A cube has edges of length $n$, where $n$ is an integer. Three faces, meeting at a corner, are painted red. The cube is then cut into $n^{3}$ smaller cubes of unit length. If exactly 125 of these cubes have no faces painted red, determine the value of $n$.
Answer: 6
Problem: Thurka bought some stuffed goats and some toy helicopters. She paid a total of $\$ 201$. She did not buy partial goats or partial helicopters. Each stuffed goat cost $\$ 19$ and each toy helicopter cost $\$ 17$. How many of each did she buy?
Answer: 7,4
Problem: Determine all real values of $x$ for which $(x+8)^{4}=(2 x+16)^{2}$.
Answer: -6,-8,-10
Problem: If $f(x)=2 x+1$ and $g(f(x))=4 x^{2}+1$, determine an expression for $g(x)$.
Answer: g(x)=x^2-2x+2
Problem: A geometric sequence has 20 terms.

The sum of its first two terms is 40 .

The sum of its first three terms is 76 .

The sum of its first four terms is 130 .

Determine how many of the terms in the sequence are integers.

(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a constant. For example, $3,6,12$ is a geometric sequence with three terms.)
Answer: 5
Problem: Determine all real values of $x$ for which $3^{(x-1)} 9^{\frac{3}{2 x^{2}}}=27$.
Answer: 1$,$\frac{3 + \sqrt{21}}{2}$,$\frac{3 - \sqrt{21}}{2}
Problem: Determine all points $(x, y)$ where the two curves $y=\log _{10}\left(x^{4}\right)$ and $y=\left(\log _{10} x\right)^{3}$ intersect.
Answer: (1,0),(\frac{1}{100},-8),(100,8)
Problem: Oi-Lam tosses three fair coins and removes all of the coins that come up heads. George then tosses the coins that remain, if any. Determine the probability that George tosses exactly one head.
Answer: \frac{27}{64}
Problem: Ross starts with an angle of measure $8^{\circ}$ and doubles it 10 times until he obtains $8192^{\circ}$. He then adds up the reciprocals of the sines of these 11 angles. That is, he calculates

$$
S=\frac{1}{\sin 8^{\circ}}+\frac{1}{\sin 16^{\circ}}+\frac{1}{\sin 32^{\circ}}+\cdots+\frac{1}{\sin 4096^{\circ}}+\frac{1}{\sin 8192^{\circ}}
$$

Determine, without using a calculator, the measure of the acute angle $\alpha$ so that $S=\frac{1}{\sin \alpha}$.
Answer: 4^{\circ}
Problem: For each positive integer $n$, let $T(n)$ be the number of triangles with integer side lengths, positive area, and perimeter $n$. For example, $T(6)=1$ since the only such triangle with a perimeter of 6 has side lengths 2,2 and 2 .
Determine the values of $T(10), T(11)$ and $T(12)$.
Answer: 2,4,3
Problem: For each positive integer $n$, let $T(n)$ be the number of triangles with integer side lengths, positive area, and perimeter $n$. For example, $T(6)=1$ since the only such triangle with a perimeter of 6 has side lengths 2,2 and 2 .
Determine the smallest positive integer $n$ such that $T(n)>2010$.
Answer: 309
Problem: Suppose $0^{\circ}<x<90^{\circ}$ and $2 \sin ^{2} x+\cos ^{2} x=\frac{25}{16}$. What is the value of $\sin x$ ?
Answer: \frac{3}{4}
Problem: The first term of a sequence is 2007. Each term, starting with the second, is the sum of the cubes of the digits of the previous term. What is the 2007th term?
Answer: 153
Problem: Sequence A has $n$th term $n^{2}-10 n+70$.

(The first three terms of sequence $\mathrm{A}$ are $61,54,49$. )

Sequence B is an arithmetic sequence with first term 5 and common difference 10. (The first three terms of sequence $\mathrm{B}$ are $5,15,25$.)

Determine all $n$ for which the $n$th term of sequence $\mathrm{A}$ is equal to the $n$th term of sequence B. Explain how you got your answer.
Answer: 5,15
Problem: Determine all values of $x$ for which $2+\sqrt{x-2}=x-2$.
Answer: 6
Problem: Determine all values of $x$ for which $(\sqrt{x})^{\log _{10} x}=100$.
Answer: 100,\frac{1}{100}
Problem: Suppose that $f(x)=x^{2}+(2 n-1) x+\left(n^{2}-22\right)$ for some integer $n$. What is the smallest positive integer $n$ for which $f(x)$ has no real roots?
Answer: 23
Problem: A bag contains 3 red marbles and 6 blue marbles. Akshan removes one marble at a time until the bag is empty. Each marble that they remove is chosen randomly from the remaining marbles. Given that the first marble that Akshan removes is red and the third marble that they remove is blue, what is the probability that the last two marbles that Akshan removes are both blue?
Answer: \frac{10}{21}
Problem: Determine the number of quadruples of positive integers $(a, b, c, d)$ with $a<b<c<d$ that satisfy both of the following system of equations:

$$
\begin{aligned}
a c+a d+b c+b d & =2023 \\
a+b+c+d & =296
\end{aligned}
$$
Answer: 417
Problem: Suppose that $\triangle A B C$ is right-angled at $B$ and has $A B=n(n+1)$ and $A C=(n+1)(n+4)$, where $n$ is a positive integer. Determine the number of positive integers $n<100000$ for which the length of side $B C$ is also an integer.
Answer: 222
Problem: Determine all real values of $x$ for which

$$
\sqrt{\log _{2} x \cdot \log _{2}(4 x)+1}+\sqrt{\log _{2} x \cdot \log _{2}\left(\frac{x}{64}\right)+9}=4
$$
Answer: [\frac{1}{2}, 8]
Problem: For every real number $x$, define $\lfloor x\rfloor$ to be equal to the greatest integer less than or equal to $x$. (We call this the "floor" of $x$.) For example, $\lfloor 4.2\rfloor=4,\lfloor 5.7\rfloor=5$, $\lfloor-3.4\rfloor=-4,\lfloor 0.4\rfloor=0$, and $\lfloor 2\rfloor=2$.
Determine the integer equal to $\left\lfloor\frac{1}{3}\right\rfloor+\left\lfloor\frac{2}{3}\right\rfloor+\left\lfloor\frac{3}{3}\right\rfloor+\ldots+\left\lfloor\frac{59}{3}\right\rfloor+\left\lfloor\frac{60}{3}\right\rfloor$. (The sum has 60 terms.)
Answer: 590
Problem: For every real number $x$, define $\lfloor x\rfloor$ to be equal to the greatest integer less than or equal to $x$. (We call this the "floor" of $x$.) For example, $\lfloor 4.2\rfloor=4,\lfloor 5.7\rfloor=5$, $\lfloor-3.4\rfloor=-4,\lfloor 0.4\rfloor=0$, and $\lfloor 2\rfloor=2$.
Determine a polynomial $p(x)$ so that for every positive integer $m>4$,

$$
\lfloor p(m)\rfloor=\left\lfloor\frac{1}{3}\right\rfloor+\left\lfloor\frac{2}{3}\right\rfloor+\left\lfloor\frac{3}{3}\right\rfloor+\ldots+\left\lfloor\frac{m-2}{3}\right\rfloor+\left\lfloor\frac{m-1}{3}\right\rfloor
$$

(The sum has $m-1$ terms.)

A polynomial $f(x)$ is an algebraic expression of the form $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$ for some integer $n \geq 0$ and for some real numbers $a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}$.
Answer: p(x)=\frac{(x-1)(x-2)}{6}
Problem: One of the faces of a rectangular prism has area $27 \mathrm{~cm}^{2}$. Another face has area $32 \mathrm{~cm}^{2}$. If the volume of the prism is $144 \mathrm{~cm}^{3}$, determine the surface area of the prism in $\mathrm{cm}^{2}$.
Answer: 166
Problem: The equations $y=a(x-2)(x+4)$ and $y=2(x-h)^{2}+k$ represent the same parabola. What are the values of $a, h$ and $k$ ?
Answer: 2,-1,-18
Problem: In an arithmetic sequence with 5 terms, the sum of the squares of the first 3 terms equals the sum of the squares of the last 2 terms. If the first term is 5 , determine all possible values of the fifth term.

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3,5,7,9,11 is an arithmetic sequence with five terms.)
Answer: -5,7
Problem: Dan was born in a year between 1300 and 1400. Steve was born in a year between 1400 and 1500. Each was born on April 6 in a year that is a perfect square. Each lived for 110 years. In what year while they were both alive were their ages both perfect squares on April 7?
Answer: 1469
Problem: Determine all values of $k$ for which the points $A(1,2), B(11,2)$ and $C(k, 6)$ form a right-angled triangle.
Answer: 1,3,9,11
Problem: If $\cos \theta=\tan \theta$, determine all possible values of $\sin \theta$, giving your answer(s) as simplified exact numbers.
Answer: \frac{-1+\sqrt{5}}{2}
Problem: Linh is driving at $60 \mathrm{~km} / \mathrm{h}$ on a long straight highway parallel to a train track. Every 10 minutes, she is passed by a train travelling in the same direction as she is. These trains depart from the station behind her every 3 minutes and all travel at the same constant speed. What is the constant speed of the trains, in $\mathrm{km} / \mathrm{h}$ ?
Answer: \frac{600}{7}
Problem: Determine all pairs $(a, b)$ of real numbers that satisfy the following system of equations:

$$
\begin{aligned}
\sqrt{a}+\sqrt{b} & =8 \\
\log _{10} a+\log _{10} b & =2
\end{aligned}
$$

Give your answer(s) as pairs of simplified exact numbers.
Answer: (22+8 \sqrt{6}, 22-8 \sqrt{6})$,$(22-8 \sqrt{6}, 22+8 \sqrt{6})
Problem: A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$.
Determine the average value of

$$
\left|a_{1}-a_{2}\right|+\left|a_{3}-a_{4}\right|
$$

over all permutations $a_{1}, a_{2}, a_{3}, a_{4}$ of $1,2,3,4$.
Answer: \frac{10}{3}
Problem: A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$.
Determine the average value of

$$
a_{1}-a_{2}+a_{3}-a_{4}+a_{5}-a_{6}+a_{7}
$$

over all permutations $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}$ of $1,2,3,4,5,6,7$.
Answer: 4
Problem: A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$.
Determine the average value of

$$
\left|a_{1}-a_{2}\right|+\left|a_{3}-a_{4}\right|+\cdots+\left|a_{197}-a_{198}\right|+\left|a_{199}-a_{200}\right|
$$

over all permutations $a_{1}, a_{2}, a_{3}, \ldots, a_{199}, a_{200}$ of $1,2,3,4, \ldots, 199,200$. (The sum labelled (*) contains 100 terms of the form $\left|a_{2 k-1}-a_{2 k}\right|$.)
Answer: 6700
Problem: If $0^{\circ}<x<90^{\circ}$ and $3 \sin (x)-\cos \left(15^{\circ}\right)=0$, what is the value of $x$ to the nearest tenth of a degree?
Answer: 18.8^{\circ}
Problem: The function $f(x)$ has the property that $f(2 x+3)=2 f(x)+3$ for all $x$. If $f(0)=6$, what is the value of $f(9)$ ?
Answer: 33
Problem: Suppose that the functions $f(x)$ and $g(x)$ satisfy the system of equations

$$
\begin{aligned}
f(x)+3 g(x) & =x^{2}+x+6 \\
2 f(x)+4 g(x) & =2 x^{2}+4
\end{aligned}
$$

for all $x$. Determine the values of $x$ for which $f(x)=g(x)$.
Answer: 5,-2
Problem: In a short-track speed skating event, there are five finalists including two Canadians. The first three skaters to finish the race win a medal. If all finalists have the same chance of finishing in any position, what is the probability that neither Canadian wins a medal?
Answer: \frac{1}{10}
Problem: Determine the number of positive integers less than or equal to 300 that are multiples of 3 or 5 , but are not multiples of 10 or 15 .
Answer: 100
Problem: In the series of odd numbers $1+3+5-7-9-11+13+15+17-19-21-23 \ldots$ the signs alternate every three terms, as shown. What is the sum of the first 300 terms of the series?
Answer: -900
Problem: A two-digit number has the property that the square of its tens digit plus ten times its units digit equals the square of its units digit plus ten times its tens digit. Determine all two-digit numbers which have this property, and are prime numbers.
Answer: 11,19,37,73
Problem: A lead box contains samples of two radioactive isotopes of iron. Isotope A decays so that after every 6 minutes, the number of atoms remaining is halved. Initially, there are twice as many atoms of isotope $\mathrm{A}$ as of isotope $\mathrm{B}$, and after 24 minutes there are the same number of atoms of each isotope. How long does it take the number of atoms of isotope B to halve?
Answer: 8
Problem: Solve the system of equations:

$$
\begin{aligned}
& \log _{10}\left(x^{3}\right)+\log _{10}\left(y^{2}\right)=11 \\
& \log _{10}\left(x^{2}\right)-\log _{10}\left(y^{3}\right)=3
\end{aligned}
$$
Answer: 10^{3},10
Problem: A positive integer $n$ is called "savage" if the integers $\{1,2,\dots,n\}$ can be partitioned into three sets $A, B$ and $C$ such that

i) the sum of the elements in each of $A, B$, and $C$ is the same,

ii) $A$ contains only odd numbers,

iii) $B$ contains only even numbers, and

iv) C contains every multiple of 3 (and possibly other numbers).
Determine all even savage integers less than 100.
Answer: 8,32,44,68,80
Problem: Tanner has two identical dice. Each die has six faces which are numbered 2, 3, 5, $7,11,13$. When Tanner rolls the two dice, what is the probability that the sum of the numbers on the top faces is a prime number?
Answer: \frac{1}{6}
Problem: If $\frac{1}{\cos x}-\tan x=3$, what is the numerical value of $\sin x$ ?
Answer: -\frac{4}{5}
Problem: Determine all linear functions $f(x)=a x+b$ such that if $g(x)=f^{-1}(x)$ for all values of $x$, then $f(x)-g(x)=44$ for all values of $x$. (Note: $f^{-1}$ is the inverse function of $f$.)
Answer: f(x)=x+22
Problem: Determine all pairs $(a, b)$ of positive integers for which $a^{3}+2 a b=2013$.
Answer: (1,1006),(3,331),(11,31)
Problem: Determine all real values of $x$ for which $\log _{2}\left(2^{x-1}+3^{x+1}\right)=2 x-\log _{2}\left(3^{x}\right)$.
Answer: \frac{\log 2}{\log 2-\log 3}
Problem: A multiplicative partition of a positive integer $n \geq 2$ is a way of writing $n$ as a product of one or more integers, each greater than 1. Note that we consider a positive integer to be a multiplicative partition of itself. Also, the order of the factors in a partition does not matter; for example, $2 \times 3 \times 5$ and $2 \times 5 \times 3$ are considered to be the same partition of 30 . For each positive integer $n \geq 2$, define $P(n)$ to be the number of multiplicative partitions of $n$. We also define $P(1)=1$. Note that $P(40)=7$, since the multiplicative partitions of 40 are $40,2 \times 20,4 \times 10$, $5 \times 8,2 \times 2 \times 10,2 \times 4 \times 5$, and $2 \times 2 \times 2 \times 5$.

(In each part, we use "partition" to mean "multiplicative partition". We also call the numbers being multiplied together in a given partition the "parts" of the partition.)
Determine the value of $P(64)$.
Answer: 11
Problem: A multiplicative partition of a positive integer $n \geq 2$ is a way of writing $n$ as a product of one or more integers, each greater than 1. Note that we consider a positive integer to be a multiplicative partition of itself. Also, the order of the factors in a partition does not matter; for example, $2 \times 3 \times 5$ and $2 \times 5 \times 3$ are considered to be the same partition of 30 . For each positive integer $n \geq 2$, define $P(n)$ to be the number of multiplicative partitions of $n$. We also define $P(1)=1$. Note that $P(40)=7$, since the multiplicative partitions of 40 are $40,2 \times 20,4 \times 10$, $5 \times 8,2 \times 2 \times 10,2 \times 4 \times 5$, and $2 \times 2 \times 2 \times 5$.

(In each part, we use "partition" to mean "multiplicative partition". We also call the numbers being multiplied together in a given partition the "parts" of the partition.)
Determine the value of $P(1000)$.
Answer: 31
Problem: What are all values of $x$ such that

$$
\log _{5}(x+3)+\log _{5}(x-1)=1 ?
$$
Answer: 2
Problem: A chef aboard a luxury liner wants to cook a goose. The time $t$ in hours to cook a goose at $180^{\circ} \mathrm{C}$ depends on the mass of the goose $m$ in kilograms according to the formula

$$
t=a m^{b}
$$

where $a$ and $b$ are constants. The table below gives the times observed to cook a goose at $180^{\circ} \mathrm{C}$.

| Mass, $m(\mathrm{~kg})$ | Time, $t(\mathrm{~h})$ |
| :---: | :---: |
| 3.00 | 2.75 |
| 6.00 | 3.75 |
Using the data in the table, determine both $a$ and $b$ to two decimal places.
Answer: 1.68,0.45
Problem: A circle passes through the origin and the points of intersection of the parabolas $y=x^{2}-3$ and $y=-x^{2}-2 x+9$. Determine the coordinates of the centre of this circle.
Answer: (-\frac{1}{2}, \frac{7}{2})
Problem: In a soccer league with 5 teams, each team plays 20 games(that is, 5 games with each of the other 4 teams). For each team, every game ends in a win (W), a loss (L), or a tie (T). The numbers of wins, losses and ties for each team at the end of the season are shown in the table. Determine the values of $x, y$ and $z$.

| Team | W | L | T |
| :---: | ---: | ---: | ---: |
| A | 2 | 15 | 3 |
| B | 7 | 9 | 4 |
| C | 6 | 12 | 2 |
| D | 10 | 8 | 2 |
| E | $x$ | $y$ | $z$ |
Answer: 19,0,1
Problem: Three thin metal rods of lengths 9,12 and 15 are welded together to form a right-angled triangle, which is held in a horizontal position. A solid sphere of radius 5 rests in the triangle so that it is tangent to each of the three sides. Assuming that the thickness of the rods can be neglected, how high above the plane of the triangle is the top of the sphere?
Answer: 5
Problem: Triangle $A B C$ has vertices $A(0,5), B(3,0)$ and $C(8,3)$. Determine the measure of $\angle A C B$.
Answer: 45^{\circ}
Problem: Blaise and Pierre will play 6 games of squash. Since they are equally skilled, each is equally likely to win any given game. (In squash, there are no ties.) The probability that each of them will win 3 of the 6 games is $\frac{5}{16}$. What is the probability that Blaise will win more games than Pierre?
Answer: \frac{11}{32}
Problem: Determine all real values of $x$ for which

$$
3^{x+2}+2^{x+2}+2^{x}=2^{x+5}+3^{x}
$$
Answer: 3
Problem: Determine all real values of $x$ such that

$$
\log _{5 x+9}\left(x^{2}+6 x+9\right)+\log _{x+3}\left(5 x^{2}+24 x+27\right)=4
$$
Answer: 0,-1,-\frac{3}{2}
Problem: For each positive integer $N$, an Eden sequence from $\{1,2,3, \ldots, N\}$ is defined to be a sequence that satisfies the following conditions:

(i) each of its terms is an element of the set of consecutive integers $\{1,2,3, \ldots, N\}$,

(ii) the sequence is increasing, and

(iii) the terms in odd numbered positions are odd and the terms in even numbered positions are even.

For example, the four Eden sequences from $\{1,2,3\}$ are

$$
\begin{array}{llll}
1 & 3 & 1,2 & 1,2,3
\end{array}
$$
Determine the number of Eden sequences from $\{1,2,3,4,5\}$.
Answer: 12