talzoomanzoo/olympiadbench · Datasets at Hugging Face

Question stringlengths 49 4.42k	answer listlengths 1 1
If $f(t)=\sin \left(\pi t-\frac{\pi}{2}\right)$, what is the smallest positive value of $t$ at which $f(t)$ attains its minimum value?	[ "2" ]
Determine all integer values of $x$ such that $\left(x^{2}-3\right)\left(x^{2}+5\right)<0$.	[ "-1,0,1" ]
At present, the sum of the ages of a husband and wife, $P$, is six times the sum of the ages of their children, $C$. Two years ago, the sum of the ages of the husband and wife was ten times the sum of the ages of the same children. Six years from now, it will be three times the sum of the ages of the same children. Determine the number of children.	[ "3" ]
What is the value of $x$ such that $\log _{2}\left(\log _{2}(2 x-2)\right)=2$ ?	[ "9" ]
Let $f(x)=2^{k x}+9$, where $k$ is a real number. If $f(3): f(6)=1: 3$, determine the value of $f(9)-f(3)$.	[ "210" ]
Determine, with justification, all values of $k$ for which $y=x^{2}-4$ and $y=2\|x\|+k$ do not intersect.	[ "(-\\infty,-5)" ]
If $2 \leq x \leq 5$ and $10 \leq y \leq 20$, what is the maximum value of $15-\frac{y}{x}$ ?	[ "13" ]
The functions $f$ and $g$ satisfy $$ \begin{aligned} & f(x)+g(x)=3 x+5 \\ & f(x)-g(x)=5 x+7 \end{aligned} $$ for all values of $x$. Determine the value of $2 f(2) g(2)$.	[ "-84" ]
Three different numbers are chosen at random from the set $\{1,2,3,4,5\}$. The numbers are arranged in increasing order. What is the probability that the resulting sequence is an arithmetic sequence? (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 is an arithmetic sequence with four terms.)	[ "$\\frac{2}{5}$" ]
What is the largest two-digit number that becomes $75 \%$ greater when its digits are reversed?	[ "48" ]
Serge likes to paddle his raft down the Speed River from point $A$ to point $B$. The speed of the current in the river is always the same. When Serge paddles, he always paddles at the same constant speed. On days when he paddles with the current, it takes him 18 minutes to get from $A$ to $B$. When he does not paddle, the current carries him from $A$ to $B$ in 30 minutes. If there were no current, how long would it take him to paddle from $A$ to $B$ ?	[ "45" ]
Square $O P Q R$ has vertices $O(0,0), P(0,8), Q(8,8)$, and $R(8,0)$. The parabola with equation $y=a(x-2)(x-6)$ intersects the sides of the square $O P Q R$ at points $K, L, M$, and $N$. Determine all the values of $a$ for which the area of the trapezoid $K L M N$ is 36 .	[ "$\\frac{32}{9}$,$\\frac{1}{2}$" ]
A 75 year old person has a $50 \%$ chance of living at least another 10 years. A 75 year old person has a $20 \%$ chance of living at least another 15 years. An 80 year old person has a $25 \%$ chance of living at least another 10 years. What is the probability that an 80 year old person will live at least another 5 years?	[ "62.5%" ]
Determine all values of $x$ for which $2^{\log _{10}\left(x^{2}\right)}=3\left(2^{1+\log _{10} x}\right)+16$.	[ "1000" ]
The Sieve of Sundaram uses the following infinite table of positive integers: \| 4 \| 7 \| 10 \| 13 \| $\cdots$ \| \| :---: \| :---: \| :---: \| :---: \| :---: \| \| 7 \| 12 \| 17 \| 22 \| $\cdots$ \| \| 10 \| 17 \| 24 \| 31 \| $\cdots$ \| \| 13 \| 22 \| 31 \| 40 \| $\cdots$ \| \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| \| The numbers in each row in the table form an arithmetic sequence. The numbers in each column in the table form an arithmetic sequence. The first four entries in each of the first four rows and columns are shown. Determine the number in the 50th row and 40th column.	[ "4090" ]
The Sieve of Sundaram uses the following infinite table of positive integers: \| 4 \| 7 \| 10 \| 13 \| $\cdots$ \| \| :---: \| :---: \| :---: \| :---: \| :---: \| \| 7 \| 12 \| 17 \| 22 \| $\cdots$ \| \| 10 \| 17 \| 24 \| 31 \| $\cdots$ \| \| 13 \| 22 \| 31 \| 40 \| $\cdots$ \| \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| \| The numbers in each row in the table form an arithmetic sequence. The numbers in each column in the table form an arithmetic sequence. The first four entries in each of the first four rows and columns are shown. Determine a formula for the number in the $R$ th row and $C$ th column.	[ "$2RC+R+C$" ]
Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$. Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$. Determine the value of $g(2011)$.	[ "4085" ]
Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$. Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$. Determine a value of $n$ for which $f(n)=100$.	[ "55" ]
Six tickets numbered 1 through 6 are placed in a box. Two tickets are randomly selected and removed together. What is the probability that the smaller of the two numbers on the tickets selected is less than or equal to 4 ?	[ "$\\frac{14}{15}$" ]
A goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it travels 1 unit up to $(0,1)$. On move 2 , it travels 2 units right to $(2,1)$. On move 3 , it travels 3 units down to $(2,-2)$. On move 4 , it travels 4 units to $(-2,-2)$. It continues in this fashion, so that on move $n$, it turns $90^{\circ}$ in a clockwise direction from its previous heading and travels $n$ units in this new direction. After $n$ moves, the goat has travelled a total of 55 units. Determine the coordinates of its position at this time.	[ "(6,5)" ]
Determine all possible values of $r$ such that the three term geometric sequence 4, $4 r, 4 r^{2}$ is also an arithmetic sequence. (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.)	[ "1" ]
If $f(x)=\sin ^{2} x-2 \sin x+2$, what are the minimum and maximum values of $f(x)$ ?	[ "5,1" ]
What is the sum of the digits of the integer equal to $\left(10^{3}+1\right)^{2}$ ?	[ "1002001" ]
A bakery sells small and large cookies. Before a price increase, the price of each small cookie is $\$ 1.50$ and the price of each large cookie is $\$ 2.00$. The price of each small cookie is increased by $10 \%$ and the price of each large cookie is increased by $5 \%$. What is the percentage increase in the total cost of a purchase of 2 small cookies and 1 large cookie?	[ "$8 \\%$" ]
Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing and Rayna is 13. Determine their ages.	[ "7,14,18" ]
The parabola with equation $y=-2 x^{2}+4 x+c$ has vertex $V(1,18)$. The parabola intersects the $y$-axis at $D$ and the $x$-axis at $E$ and $F$. Determine the area of $\triangle D E F$.	[ "48" ]
If $3\left(8^{x}\right)+5\left(8^{x}\right)=2^{61}$, what is the value of the real number $x$ ?	[ "$\\frac{58}{3}$" ]
For some real numbers $m$ and $n$, the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order. Determine all possible values of $m$.	[ "1,-1,7,-7" ]
Chinara starts with the point $(3,5)$, and applies the following three-step process, which we call $\mathcal{P}$ : Step 1: Reflect the point in the $x$-axis. Step 2: Translate the resulting point 2 units upwards. Step 3: Reflect the resulting point in the $y$-axis. As she does this, the point $(3,5)$ moves to $(3,-5)$, then to $(3,-3)$, and then to $(-3,-3)$. Chinara then starts with a different point $S_{0}$. She applies the three-step process $\mathcal{P}$ to the point $S_{0}$ and obtains the point $S_{1}$. She then applies $\mathcal{P}$ to $S_{1}$ to obtain the point $S_{2}$. She applies $\mathcal{P}$ four more times, each time using the previous output of $\mathcal{P}$ to be the new input, and eventually obtains the point $S_{6}(-7,-1)$. What are the coordinates of the point $S_{0}$ ?	[ "(-7,-1)" ]
Suppose that $n>5$ and that the numbers $t_{1}, t_{2}, t_{3}, \ldots, t_{n-2}, t_{n-1}, t_{n}$ form an arithmetic sequence with $n$ terms. If $t_{3}=5, t_{n-2}=95$, and the sum of all $n$ terms is 1000 , what is the value of $n$ ? (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant, called the common difference. For example, $3,5,7,9$ are the first four terms of an arithmetic sequence.)	[ "20" ]
Suppose that $a$ and $r$ are real numbers. A geometric sequence with first term $a$ and common ratio $r$ has 4 terms. The sum of this geometric sequence is $6+6 \sqrt{2}$. A second geometric sequence has the same first term $a$ and the same common ratio $r$, but has 8 terms. The sum of this second geometric sequence is $30+30 \sqrt{2}$. Determine all possible values for $a$. (A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant, called the common ratio. For example, $3,-6,12,-24$ are the first four terms of a geometric sequence.)	[ "$a=2$, $a=-6-4 \\sqrt{2}$" ]
A bag contains 3 green balls, 4 red balls, and no other balls. Victor removes balls randomly from the bag, one at a time, and places them on a table. Each ball in the bag is equally likely to be chosen each time that he removes a ball. He stops removing balls when there are two balls of the same colour on the table. What is the probability that, when he stops, there is at least 1 red ball and at least 1 green ball on the table?	[ "$\\frac{4}{7}$" ]
Suppose that $f(a)=2 a^{2}-3 a+1$ for all real numbers $a$ and $g(b)=\log _{\frac{1}{2}} b$ for all $b>0$. Determine all $\theta$ with $0 \leq \theta \leq 2 \pi$ for which $f(g(\sin \theta))=0$.	[ "$\\frac{1}{6} \\pi, \\frac{5}{6} \\pi, \\frac{1}{4} \\pi, \\frac{3}{4} \\pi$" ]
Suppose that $a=5$ and $b=4$. Determine all pairs of integers $(K, L)$ for which $K^{2}+3 L^{2}=a^{2}+b^{2}-a b$.	[ "$(3,2),(-3,2),(3,-2),(-3,-2)$" ]
Determine all values of $x$ for which $0<\frac{x^{2}-11}{x+1}<7$.	[ "$(-\\sqrt{11},-2)\\cup (\\sqrt{11},9)$" ]
The numbers $a_{1}, a_{2}, a_{3}, \ldots$ form an arithmetic sequence with $a_{1} \neq a_{2}$. The three numbers $a_{1}, a_{2}, a_{6}$ form a geometric sequence in that order. Determine all possible positive integers $k$ for which the three numbers $a_{1}, a_{4}, a_{k}$ also form a geometric sequence in that order. (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 are the first four terms of an arithmetic sequence. A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant. For example, $3,6,12$ is a geometric sequence with three terms.)	[ "34" ]
For some positive integers $k$, the parabola with equation $y=\frac{x^{2}}{k}-5$ intersects the circle with equation $x^{2}+y^{2}=25$ at exactly three distinct points $A, B$ and $C$. Determine all such positive integers $k$ for which the area of $\triangle A B C$ is an integer.	[ "1,2,5,8,9" ]
Consider the following system of equations in which all logarithms have base 10: $$ \begin{aligned} (\log x)(\log y)-3 \log 5 y-\log 8 x & =a \\ (\log y)(\log z)-4 \log 5 y-\log 16 z & =b \\ (\log z)(\log x)-4 \log 8 x-3 \log 625 z & =c \end{aligned} $$ If $a=-4, b=4$, and $c=-18$, solve the system of equations.	[ "$(10^{4}, 10^{3}, 10^{10}),(10^{2}, 10^{-1}, 10^{-2})$" ]
Two fair dice, each having six faces numbered 1 to 6 , are thrown. What is the probability that the product of the two numbers on the top faces is divisible by 5 ?	[ "$\\frac{11}{36}$" ]
If $f(x)=x^{2}-x+2, g(x)=a x+b$, and $f(g(x))=9 x^{2}-3 x+2$, determine all possible ordered pairs $(a, b)$ which satisfy this relationship.	[ "$(3,0),(-3,1)$" ]
Digital images consist of a very large number of equally spaced dots called pixels The resolution of an image is the number of pixels/cm in each of the horizontal and vertical directions. Thus, an image with dimensions $10 \mathrm{~cm}$ by $15 \mathrm{~cm}$ and a resolution of 75 pixels/cm has a total of $(10 \times 75) \times(15 \times 75)=843750$ pixels. If each of these dimensions was increased by $n \%$ and the resolution was decreased by $n \%$, the image would have 345600 pixels. Determine the value of $n$.	[ "60" ]
If $T=x^{2}+\frac{1}{x^{2}}$, determine the values of $b$ and $c$ so that $x^{6}+\frac{1}{x^{6}}=T^{3}+b T+c$ for all non-zero real numbers $x$.	[ "-3,0" ]
A Skolem sequence of order $n$ is a sequence $\left(s_{1}, s_{2}, \ldots, s_{2 n}\right)$ of $2 n$ integers satisfying the conditions: i) for every $k$ in $\{1,2,3, \ldots, n\}$, there exist exactly two elements $s_{i}$ and $s_{j}$ with $s_{i}=s_{j}=k$, and ii) if $s_{i}=s_{j}=k$ with $i<j$, then $j-i=k$. For example, $(4,2,3,2,4,3,1,1)$ is a Skolem sequence of order 4. List all Skolem sequences of order 4.	[ "(4,2,3,2,4,3,1,1),(1,1,3,4,2,3,2,4),(4,1,1,3,4,2,3,2),(2,3,2,4,3,1,1,4),(3,4,2,3,2,4,1,1),(1,1,4,2,3,2,4,3)" ]
A Skolem sequence of order $n$ is a sequence $\left(s_{1}, s_{2}, \ldots, s_{2 n}\right)$ of $2 n$ integers satisfying the conditions: i) for every $k$ in $\{1,2,3, \ldots, n\}$, there exist exactly two elements $s_{i}$ and $s_{j}$ with $s_{i}=s_{j}=k$, and ii) if $s_{i}=s_{j}=k$ with $i<j$, then $j-i=k$. For example, $(4,2,3,2,4,3,1,1)$ is a Skolem sequence of order 4. Determine, with justification, all Skolem sequences of order 9 which satisfy all of the following three conditions: I) $s_{3}=1$, II) $s_{18}=8$, and III) between any two equal even integers, there is exactly one odd integer.	[ "(7,5,1,1,9,3,5,7,3,8,6,4,2,9,2,4,6,8)" ]
The three-digit positive integer $m$ is odd and has three distinct digits. If the hundreds digit of $m$ equals the product of the tens digit and ones (units) digit of $m$, what is $m$ ?	[ "623" ]
Eleanor has 100 marbles, each of which is black or gold. The ratio of the number of black marbles to the number of gold marbles is $1: 4$. How many gold marbles should she add to change this ratio to $1: 6$ ?	[ "40" ]
Suppose that $n$ is a positive integer and that the value of $\frac{n^{2}+n+15}{n}$ is an integer. Determine all possible values of $n$.	[ "1, 3, 5, 15" ]
Ada starts with $x=10$ and $y=2$, and applies the following process: Step 1: Add $x$ and $y$. Let $x$ equal the result. The value of $y$ does not change. Step 2: Multiply $x$ and $y$. Let $x$ equal the result. The value of $y$ does not change. Step 3: Add $y$ and 1. Let $y$ equal the result. The value of $x$ does not change. Ada keeps track of the values of $x$ and $y$ : \| \| $x$ \| $y$ \| \| :---: \| :---: \| :---: \| \| Before Step 1 \| 10 \| 2 \| \| After Step 1 \| 12 \| 2 \| \| After Step 2 \| 24 \| 2 \| \| After Step 3 \| 24 \| 3 \| Continuing now with $x=24$ and $y=3$, Ada applies the process two more times. What is the final value of $x$ ?	[ "340" ]
Determine all integers $k$, with $k \neq 0$, for which the parabola with equation $y=k x^{2}+6 x+k$ has two distinct $x$-intercepts.	[ "-2,-1,1,2" ]
The positive integers $a$ and $b$ have no common divisor larger than 1 . If the difference between $b$ and $a$ is 15 and $\frac{5}{9}<\frac{a}{b}<\frac{4}{7}$, what is the value of $\frac{a}{b}$ ?	[ "$\\frac{19}{34}$" ]
A geometric sequence has first term 10 and common ratio $\frac{1}{2}$. An arithmetic sequence has first term 10 and common difference $d$. The ratio of the 6th term in the geometric sequence to the 4th term in the geometric sequence equals the ratio of the 6th term in the arithmetic sequence to the 4 th term in the arithmetic sequence. Determine all possible values of $d$. (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant, called the common difference. For example, 3, 5, 7, 9 are the first four terms of an arithmetic sequence. A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant, called the common ratio. For example, $3,6,12$ is a geometric sequence with three terms.)	[ "$-\\frac{30}{17}$" ]
For each positive real number $x$, define $f(x)$ to be the number of prime numbers $p$ that satisfy $x \leq p \leq x+10$. What is the value of $f(f(20))$ ?	[ "5" ]
Determine all triples $(x, y, z)$ of real numbers that satisfy the following system of equations: $$ \begin{aligned} (x-1)(y-2) & =0 \\ (x-3)(z+2) & =0 \\ x+y z & =9 \end{aligned} $$	[ "(1,-4,-2),(3,2,3),(13,2,-2)" ]
Suppose that the function $g$ satisfies $g(x)=2 x-4$ for all real numbers $x$ and that $g^{-1}$ is the inverse function of $g$. Suppose that the function $f$ satisfies $g\left(f\left(g^{-1}(x)\right)\right)=2 x^{2}+16 x+26$ for all real numbers $x$. What is the value of $f(\pi)$ ?	[ "$4 \\pi^{2}-1$" ]
Determine all pairs of angles $(x, y)$ with $0^{\circ} \leq x<180^{\circ}$ and $0^{\circ} \leq y<180^{\circ}$ that satisfy the following system of equations: $$ \begin{aligned} \log _{2}(\sin x \cos y) & =-\frac{3}{2} \\ \log _{2}\left(\frac{\sin x}{\cos y}\right) & =\frac{1}{2} \end{aligned} $$	[ "$(45^{\\circ}, 60^{\\circ}),(135^{\\circ}, 60^{\\circ})$" ]
Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in which a total of three matches are played. First, two players are chosen randomly to play each other. The other two players also play each other. The winners of the two matches then play to decide the tournament champion. Alain, Bianca and Chen are equally matched (that is, when a match is played between any two of them, the probability that each player wins is $\frac{1}{2}$ ). When Dave plays each of Alain, Bianca and Chen, the probability that Dave wins is $p$, for some real number $p$. Determine the probability that Bianca wins the tournament, expressing your answer in the form $\frac{a p^{2}+b p+c}{d}$ where $a, b, c$, and $d$ are integers.	[ "$\\frac{1-p^{2}}{3}$" ]
Three microphones $A, B$ and $C$ are placed on a line such that $A$ is $1 \mathrm{~km}$ west of $B$ and $C$ is $2 \mathrm{~km}$ east of $B$. A large explosion occurs at a point $P$ not on this line. Each of the three microphones receives the sound. The sound travels at $\frac{1}{3} \mathrm{~km} / \mathrm{s}$. Microphone $B$ receives the sound first, microphone $A$ receives the sound $\frac{1}{2}$ s later, and microphone $C$ receives it $1 \mathrm{~s}$ after microphone $A$. Determine the distance from microphone $B$ to the explosion at $P$.	[ "$\\frac{41}{12}$" ]
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.	[ "(1,7,103, 105), (3, 5, 101, 107)" ]
Determine all values of $x$ for which $\frac{x^{2}+x+4}{2 x+1}=\frac{4}{x}$.	[ "$-1$,$2$,$-2$" ]
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.)	[ "8" ]
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 $k$.	[ "$8$,$4$" ]
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?	[ "17.5%" ]
Suppose that $x$ and $y$ are real numbers with $3 x+4 y=10$. Determine the minimum possible value of $x^{2}+16 y^{2}$.	[ "10" ]
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?	[ "26" ]
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.)	[ "20" ]
If $\frac{(x-2013)(y-2014)}{(x-2013)^{2}+(y-2014)^{2}}=-\frac{1}{2}$, what is the value of $x+y$ ?	[ "4027" ]
Determine all real numbers $x$ for which $$ \left(\log _{10} x\right)^{\log _{10}\left(\log _{10} x\right)}=10000 $$	[ "$10^{100}$,$10^{1 / 100}$" ]
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.)	[ "$221,$8$" ]
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 .	[ "202" ]
If $\log _{10} x=3+\log _{10} y$, what is the value of $\frac{x}{y}$ ?	[ "1000" ]
If $x+\frac{1}{x}=\frac{13}{6}$, determine all values of $x^{2}+\frac{1}{x^{2}}$.	[ "$\\frac{97}{36}$" ]
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?	[ "$\\frac{2}{9}$" ]
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 \|	[ "16,3,1,1" ]
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 $x$ ?	[ "4.25" ]
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 $n$.	[ "6" ]
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?	[ "7,4" ]
Determine all real values of $x$ for which $(x+8)^{4}=(2 x+16)^{2}$.	[ "-6,-8,-10" ]
If $f(x)=2 x+1$ and $g(f(x))=4 x^{2}+1$, determine an expression for $g(x)$.	[ "$g(x)=x^2-2x+2$" ]
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.)	[ "5" ]
Determine all real values of $x$ for which $3^{(x-1)} 9^{\frac{3}{2 x^{2}}}=27$.	[ "$1$,$\\frac{3 + \\sqrt{21}}{2}$,$\\frac{3 - \\sqrt{21}}{2}$" ]
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.	[ "$(1,0),(\\frac{1}{100},-8),(100,8)$" ]
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.	[ "$\\frac{27}{64}$" ]
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 $S=\frac{1}{\sin \alpha}$.	[ "$4^{\\circ}$" ]
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 $T(12)$.	[ "2,4,3" ]
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 $T(n)>2010$.	[ "309" ]
Suppose $0^{\circ}<x<90^{\circ}$ and $2 \sin ^{2} x+\cos ^{2} x=\frac{25}{16}$. What is the value of $\sin x$ ?	[ "$\\frac{3}{4}$" ]
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?	[ "153" ]
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.	[ "5,15" ]
Determine all values of $x$ for which $2+\sqrt{x-2}=x-2$.	[ "6" ]
Determine all values of $x$ for which $(\sqrt{x})^{\log _{10} x}=100$.	[ "$100,\\frac{1}{100}$" ]
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?	[ "23" ]
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?	[ "$\\frac{10}{21}$" ]
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: $$ \begin{aligned} a c+a d+b c+b d & =2023 \\ a+b+c+d & =296 \end{aligned} $$	[ "417" ]
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.	[ "222" ]
Determine all real values of $x$ for which $$ \sqrt{\log _{2} x \cdot \log _{2}(4 x)+1}+\sqrt{\log _{2} x \cdot \log _{2}\left(\frac{x}{64}\right)+9}=4 $$	[ "$[\\frac{1}{2}, 8]$" ]
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.)	[ "590" ]
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 $a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}$.	[ "$p(x)=\\frac{(x-1)(x-2)}{6}$" ]
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 $\mathrm{cm}^{2}$.	[ "$166$" ]
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 $k$ ?	[ "$2,-1,-18$" ]
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.)	[ "-5,7" ]

If $f(t)=\sin \left(\pi t-\frac{\pi}{2}\right)$, what is the smallest positive value of $t$ at which $f(t)$ attains its minimum value?

[ "2" ]

Determine all integer values of $x$ such that $\left(x^{2}-3\right)\left(x^{2}+5\right)<0$.

[ "-1,0,1" ]

At present, the sum of the ages of a husband and wife, $P$, is six times the sum of the ages of their children, $C$. Two years ago, the sum of the ages of the husband and wife was ten times the sum of the ages of the same children. Six years from now, it will be three times the sum of the ages of the same children. Determine the number of children.

[ "3" ]

What is the value of $x$ such that $\log _{2}\left(\log _{2}(2 x-2)\right)=2$ ?

[ "9" ]

Let $f(x)=2^{k x}+9$, where $k$ is a real number. If $f(3): f(6)=1: 3$, determine the value of $f(9)-f(3)$.

[ "210" ]

Determine, with justification, all values of $k$ for which $y=x^{2}-4$ and $y=2|x|+k$ do not intersect.

[ "(-\\infty,-5)" ]

If $2 \leq x \leq 5$ and $10 \leq y \leq 20$, what is the maximum value of $15-\frac{y}{x}$ ?

[ "13" ]

The functions $f$ and $g$ satisfy $$ \begin{aligned} & f(x)+g(x)=3 x+5 \\ & f(x)-g(x)=5 x+7 \end{aligned} $$ for all values of $x$. Determine the value of $2 f(2) g(2)$.

[ "-84" ]

Three different numbers are chosen at random from the set $\{1,2,3,4,5\}$. The numbers are arranged in increasing order. What is the probability that the resulting sequence is an arithmetic sequence? (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 is an arithmetic sequence with four terms.)

[ "$\\frac{2}{5}$" ]

What is the largest two-digit number that becomes $75 \%$ greater when its digits are reversed?

[ "48" ]

Serge likes to paddle his raft down the Speed River from point $A$ to point $B$. The speed of the current in the river is always the same. When Serge paddles, he always paddles at the same constant speed. On days when he paddles with the current, it takes him 18 minutes to get from $A$ to $B$. When he does not paddle, the current carries him from $A$ to $B$ in 30 minutes. If there were no current, how long would it take him to paddle from $A$ to $B$ ?

[ "45" ]

Square $O P Q R$ has vertices $O(0,0), P(0,8), Q(8,8)$, and $R(8,0)$. The parabola with equation $y=a(x-2)(x-6)$ intersects the sides of the square $O P Q R$ at points $K, L, M$, and $N$. Determine all the values of $a$ for which the area of the trapezoid $K L M N$ is 36 .

[ "$\\frac{32}{9}$,$\\frac{1}{2}$" ]

A 75 year old person has a $50 \%$ chance of living at least another 10 years. A 75 year old person has a $20 \%$ chance of living at least another 15 years. An 80 year old person has a $25 \%$ chance of living at least another 10 years. What is the probability that an 80 year old person will live at least another 5 years?

[ "62.5%" ]

Determine all values of $x$ for which $2^{\log _{10}\left(x^{2}\right)}=3\left(2^{1+\log _{10} x}\right)+16$.

[ "1000" ]

The Sieve of Sundaram uses the following infinite table of positive integers: | 4 | 7 | 10 | 13 | $\cdots$ | | :---: | :---: | :---: | :---: | :---: | | 7 | 12 | 17 | 22 | $\cdots$ | | 10 | 17 | 24 | 31 | $\cdots$ | | 13 | 22 | 31 | 40 | $\cdots$ | | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | | The numbers in each row in the table form an arithmetic sequence. The numbers in each column in the table form an arithmetic sequence. The first four entries in each of the first four rows and columns are shown. Determine the number in the 50th row and 40th column.

[ "4090" ]

The Sieve of Sundaram uses the following infinite table of positive integers: | 4 | 7 | 10 | 13 | $\cdots$ | | :---: | :---: | :---: | :---: | :---: | | 7 | 12 | 17 | 22 | $\cdots$ | | 10 | 17 | 24 | 31 | $\cdots$ | | 13 | 22 | 31 | 40 | $\cdots$ | | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | | The numbers in each row in the table form an arithmetic sequence. The numbers in each column in the table form an arithmetic sequence. The first four entries in each of the first four rows and columns are shown. Determine a formula for the number in the $R$ th row and $C$ th column.

[ "$2RC+R+C$" ]

Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$. Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$. Determine the value of $g(2011)$.

[ "4085" ]

Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$. Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$. Determine a value of $n$ for which $f(n)=100$.

[ "55" ]

Six tickets numbered 1 through 6 are placed in a box. Two tickets are randomly selected and removed together. What is the probability that the smaller of the two numbers on the tickets selected is less than or equal to 4 ?

[ "$\\frac{14}{15}$" ]

A goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it travels 1 unit up to $(0,1)$. On move 2 , it travels 2 units right to $(2,1)$. On move 3 , it travels 3 units down to $(2,-2)$. On move 4 , it travels 4 units to $(-2,-2)$. It continues in this fashion, so that on move $n$, it turns $90^{\circ}$ in a clockwise direction from its previous heading and travels $n$ units in this new direction. After $n$ moves, the goat has travelled a total of 55 units. Determine the coordinates of its position at this time.

[ "(6,5)" ]

Determine all possible values of $r$ such that the three term geometric sequence 4, $4 r, 4 r^{2}$ is also an arithmetic sequence. (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.)

[ "1" ]

If $f(x)=\sin ^{2} x-2 \sin x+2$, what are the minimum and maximum values of $f(x)$ ?

[ "5,1" ]

What is the sum of the digits of the integer equal to $\left(10^{3}+1\right)^{2}$ ?

[ "1002001" ]

A bakery sells small and large cookies. Before a price increase, the price of each small cookie is $\$ 1.50$ and the price of each large cookie is $\$ 2.00$. The price of each small cookie is increased by $10 \%$ and the price of each large cookie is increased by $5 \%$. What is the percentage increase in the total cost of a purchase of 2 small cookies and 1 large cookie?

[ "$8 \\%$" ]

Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing and Rayna is 13. Determine their ages.

[ "7,14,18" ]

The parabola with equation $y=-2 x^{2}+4 x+c$ has vertex $V(1,18)$. The parabola intersects the $y$-axis at $D$ and the $x$-axis at $E$ and $F$. Determine the area of $\triangle D E F$.

[ "48" ]

If $3\left(8^{x}\right)+5\left(8^{x}\right)=2^{61}$, what is the value of the real number $x$ ?

[ "$\\frac{58}{3}$" ]

For some real numbers $m$ and $n$, the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order. Determine all possible values of $m$.

[ "1,-1,7,-7" ]

Chinara starts with the point $(3,5)$, and applies the following three-step process, which we call $\mathcal{P}$ : Step 1: Reflect the point in the $x$-axis. Step 2: Translate the resulting point 2 units upwards. Step 3: Reflect the resulting point in the $y$-axis. As she does this, the point $(3,5)$ moves to $(3,-5)$, then to $(3,-3)$, and then to $(-3,-3)$. Chinara then starts with a different point $S_{0}$. She applies the three-step process $\mathcal{P}$ to the point $S_{0}$ and obtains the point $S_{1}$. She then applies $\mathcal{P}$ to $S_{1}$ to obtain the point $S_{2}$. She applies $\mathcal{P}$ four more times, each time using the previous output of $\mathcal{P}$ to be the new input, and eventually obtains the point $S_{6}(-7,-1)$. What are the coordinates of the point $S_{0}$ ?

[ "(-7,-1)" ]

Suppose that $n>5$ and that the numbers $t_{1}, t_{2}, t_{3}, \ldots, t_{n-2}, t_{n-1}, t_{n}$ form an arithmetic sequence with $n$ terms. If $t_{3}=5, t_{n-2}=95$, and the sum of all $n$ terms is 1000 , what is the value of $n$ ? (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant, called the common difference. For example, $3,5,7,9$ are the first four terms of an arithmetic sequence.)

[ "20" ]

Suppose that $a$ and $r$ are real numbers. A geometric sequence with first term $a$ and common ratio $r$ has 4 terms. The sum of this geometric sequence is $6+6 \sqrt{2}$. A second geometric sequence has the same first term $a$ and the same common ratio $r$, but has 8 terms. The sum of this second geometric sequence is $30+30 \sqrt{2}$. Determine all possible values for $a$. (A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant, called the common ratio. For example, $3,-6,12,-24$ are the first four terms of a geometric sequence.)

[ "$a=2$, $a=-6-4 \\sqrt{2}$" ]

A bag contains 3 green balls, 4 red balls, and no other balls. Victor removes balls randomly from the bag, one at a time, and places them on a table. Each ball in the bag is equally likely to be chosen each time that he removes a ball. He stops removing balls when there are two balls of the same colour on the table. What is the probability that, when he stops, there is at least 1 red ball and at least 1 green ball on the table?

[ "$\\frac{4}{7}$" ]

Suppose that $f(a)=2 a^{2}-3 a+1$ for all real numbers $a$ and $g(b)=\log _{\frac{1}{2}} b$ for all $b>0$. Determine all $\theta$ with $0 \leq \theta \leq 2 \pi$ for which $f(g(\sin \theta))=0$.

[ "$\\frac{1}{6} \\pi, \\frac{5}{6} \\pi, \\frac{1}{4} \\pi, \\frac{3}{4} \\pi$" ]

Suppose that $a=5$ and $b=4$. Determine all pairs of integers $(K, L)$ for which $K^{2}+3 L^{2}=a^{2}+b^{2}-a b$.

[ "$(3,2),(-3,2),(3,-2),(-3,-2)$" ]

Determine all values of $x$ for which $0<\frac{x^{2}-11}{x+1}<7$.

[ "$(-\\sqrt{11},-2)\\cup (\\sqrt{11},9)$" ]

The numbers $a_{1}, a_{2}, a_{3}, \ldots$ form an arithmetic sequence with $a_{1} \neq a_{2}$. The three numbers $a_{1}, a_{2}, a_{6}$ form a geometric sequence in that order. Determine all possible positive integers $k$ for which the three numbers $a_{1}, a_{4}, a_{k}$ also form a geometric sequence in that order. (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 are the first four terms of an arithmetic sequence. A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant. For example, $3,6,12$ is a geometric sequence with three terms.)

[ "34" ]

For some positive integers $k$, the parabola with equation $y=\frac{x^{2}}{k}-5$ intersects the circle with equation $x^{2}+y^{2}=25$ at exactly three distinct points $A, B$ and $C$. Determine all such positive integers $k$ for which the area of $\triangle A B C$ is an integer.

[ "1,2,5,8,9" ]

Consider the following system of equations in which all logarithms have base 10: $$ \begin{aligned} (\log x)(\log y)-3 \log 5 y-\log 8 x & =a \\ (\log y)(\log z)-4 \log 5 y-\log 16 z & =b \\ (\log z)(\log x)-4 \log 8 x-3 \log 625 z & =c \end{aligned} $$ If $a=-4, b=4$, and $c=-18$, solve the system of equations.

[ "$(10^{4}, 10^{3}, 10^{10}),(10^{2}, 10^{-1}, 10^{-2})$" ]

Two fair dice, each having six faces numbered 1 to 6 , are thrown. What is the probability that the product of the two numbers on the top faces is divisible by 5 ?

[ "$\\frac{11}{36}$" ]

If $f(x)=x^{2}-x+2, g(x)=a x+b$, and $f(g(x))=9 x^{2}-3 x+2$, determine all possible ordered pairs $(a, b)$ which satisfy this relationship.

[ "$(3,0),(-3,1)$" ]

Digital images consist of a very large number of equally spaced dots called pixels The resolution of an image is the number of pixels/cm in each of the horizontal and vertical directions. Thus, an image with dimensions $10 \mathrm{~cm}$ by $15 \mathrm{~cm}$ and a resolution of 75 pixels/cm has a total of $(10 \times 75) \times(15 \times 75)=843750$ pixels. If each of these dimensions was increased by $n \%$ and the resolution was decreased by $n \%$, the image would have 345600 pixels. Determine the value of $n$.

[ "60" ]

If $T=x^{2}+\frac{1}{x^{2}}$, determine the values of $b$ and $c$ so that $x^{6}+\frac{1}{x^{6}}=T^{3}+b T+c$ for all non-zero real numbers $x$.

[ "-3,0" ]

A Skolem sequence of order $n$ is a sequence $\left(s_{1}, s_{2}, \ldots, s_{2 n}\right)$ of $2 n$ integers satisfying the conditions: i) for every $k$ in $\{1,2,3, \ldots, n\}$, there exist exactly two elements $s_{i}$ and $s_{j}$ with $s_{i}=s_{j}=k$, and ii) if $s_{i}=s_{j}=k$ with $i<j$, then $j-i=k$. For example, $(4,2,3,2,4,3,1,1)$ is a Skolem sequence of order 4. List all Skolem sequences of order 4.

[ "(4,2,3,2,4,3,1,1),(1,1,3,4,2,3,2,4),(4,1,1,3,4,2,3,2),(2,3,2,4,3,1,1,4),(3,4,2,3,2,4,1,1),(1,1,4,2,3,2,4,3)" ]

A Skolem sequence of order $n$ is a sequence $\left(s_{1}, s_{2}, \ldots, s_{2 n}\right)$ of $2 n$ integers satisfying the conditions: i) for every $k$ in $\{1,2,3, \ldots, n\}$, there exist exactly two elements $s_{i}$ and $s_{j}$ with $s_{i}=s_{j}=k$, and ii) if $s_{i}=s_{j}=k$ with $i<j$, then $j-i=k$. For example, $(4,2,3,2,4,3,1,1)$ is a Skolem sequence of order 4. Determine, with justification, all Skolem sequences of order 9 which satisfy all of the following three conditions: I) $s_{3}=1$, II) $s_{18}=8$, and III) between any two equal even integers, there is exactly one odd integer.

[ "(7,5,1,1,9,3,5,7,3,8,6,4,2,9,2,4,6,8)" ]

The three-digit positive integer $m$ is odd and has three distinct digits. If the hundreds digit of $m$ equals the product of the tens digit and ones (units) digit of $m$, what is $m$ ?

[ "623" ]

Eleanor has 100 marbles, each of which is black or gold. The ratio of the number of black marbles to the number of gold marbles is $1: 4$. How many gold marbles should she add to change this ratio to $1: 6$ ?

[ "40" ]

Suppose that $n$ is a positive integer and that the value of $\frac{n^{2}+n+15}{n}$ is an integer. Determine all possible values of $n$.

[ "1, 3, 5, 15" ]

Ada starts with $x=10$ and $y=2$, and applies the following process: Step 1: Add $x$ and $y$. Let $x$ equal the result. The value of $y$ does not change. Step 2: Multiply $x$ and $y$. Let $x$ equal the result. The value of $y$ does not change. Step 3: Add $y$ and 1. Let $y$ equal the result. The value of $x$ does not change. Ada keeps track of the values of $x$ and $y$ : | | $x$ | $y$ | | :---: | :---: | :---: | | Before Step 1 | 10 | 2 | | After Step 1 | 12 | 2 | | After Step 2 | 24 | 2 | | After Step 3 | 24 | 3 | Continuing now with $x=24$ and $y=3$, Ada applies the process two more times. What is the final value of $x$ ?

[ "340" ]

Determine all integers $k$, with $k \neq 0$, for which the parabola with equation $y=k x^{2}+6 x+k$ has two distinct $x$-intercepts.

[ "-2,-1,1,2" ]

The positive integers $a$ and $b$ have no common divisor larger than 1 . If the difference between $b$ and $a$ is 15 and $\frac{5}{9}<\frac{a}{b}<\frac{4}{7}$, what is the value of $\frac{a}{b}$ ?

[ "$\\frac{19}{34}$" ]

A geometric sequence has first term 10 and common ratio $\frac{1}{2}$. An arithmetic sequence has first term 10 and common difference $d$. The ratio of the 6th term in the geometric sequence to the 4th term in the geometric sequence equals the ratio of the 6th term in the arithmetic sequence to the 4 th term in the arithmetic sequence. Determine all possible values of $d$. (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant, called the common difference. For example, 3, 5, 7, 9 are the first four terms of an arithmetic sequence. A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant, called the common ratio. For example, $3,6,12$ is a geometric sequence with three terms.)

[ "$-\\frac{30}{17}$" ]

For each positive real number $x$, define $f(x)$ to be the number of prime numbers $p$ that satisfy $x \leq p \leq x+10$. What is the value of $f(f(20))$ ?

[ "5" ]

Determine all triples $(x, y, z)$ of real numbers that satisfy the following system of equations: $$ \begin{aligned} (x-1)(y-2) & =0 \\ (x-3)(z+2) & =0 \\ x+y z & =9 \end{aligned} $$

[ "(1,-4,-2),(3,2,3),(13,2,-2)" ]

Suppose that the function $g$ satisfies $g(x)=2 x-4$ for all real numbers $x$ and that $g^{-1}$ is the inverse function of $g$. Suppose that the function $f$ satisfies $g\left(f\left(g^{-1}(x)\right)\right)=2 x^{2}+16 x+26$ for all real numbers $x$. What is the value of $f(\pi)$ ?

[ "$4 \\pi^{2}-1$" ]

Determine all pairs of angles $(x, y)$ with $0^{\circ} \leq x<180^{\circ}$ and $0^{\circ} \leq y<180^{\circ}$ that satisfy the following system of equations: $$ \begin{aligned} \log _{2}(\sin x \cos y) & =-\frac{3}{2} \\ \log _{2}\left(\frac{\sin x}{\cos y}\right) & =\frac{1}{2} \end{aligned} $$

[ "$(45^{\\circ}, 60^{\\circ}),(135^{\\circ}, 60^{\\circ})$" ]

Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in which a total of three matches are played. First, two players are chosen randomly to play each other. The other two players also play each other. The winners of the two matches then play to decide the tournament champion. Alain, Bianca and Chen are equally matched (that is, when a match is played between any two of them, the probability that each player wins is $\frac{1}{2}$ ). When Dave plays each of Alain, Bianca and Chen, the probability that Dave wins is $p$, for some real number $p$. Determine the probability that Bianca wins the tournament, expressing your answer in the form $\frac{a p^{2}+b p+c}{d}$ where $a, b, c$, and $d$ are integers.

[ "$\\frac{1-p^{2}}{3}$" ]

Three microphones $A, B$ and $C$ are placed on a line such that $A$ is $1 \mathrm{~km}$ west of $B$ and $C$ is $2 \mathrm{~km}$ east of $B$. A large explosion occurs at a point $P$ not on this line. Each of the three microphones receives the sound. The sound travels at $\frac{1}{3} \mathrm{~km} / \mathrm{s}$. Microphone $B$ receives the sound first, microphone $A$ receives the sound $\frac{1}{2}$ s later, and microphone $C$ receives it $1 \mathrm{~s}$ after microphone $A$. Determine the distance from microphone $B$ to the explosion at $P$.

[ "$\\frac{41}{12}$" ]

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.

[ "(1,7,103, 105), (3, 5, 101, 107)" ]

Determine all values of $x$ for which $\frac{x^{2}+x+4}{2 x+1}=\frac{4}{x}$.

[ "$-1$,$2$,$-2$" ]

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.)

[ "8" ]

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 $k$.

[ "$8$,$4$" ]

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?

[ "17.5%" ]

Suppose that $x$ and $y$ are real numbers with $3 x+4 y=10$. Determine the minimum possible value of $x^{2}+16 y^{2}$.

[ "10" ]

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?

[ "26" ]

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.)

[ "20" ]

If $\frac{(x-2013)(y-2014)}{(x-2013)^{2}+(y-2014)^{2}}=-\frac{1}{2}$, what is the value of $x+y$ ?

[ "4027" ]

Determine all real numbers $x$ for which $$ \left(\log _{10} x\right)^{\log _{10}\left(\log _{10} x\right)}=10000 $$

[ "$10^{100}$,$10^{1 / 100}$" ]

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.)

[ "$221,$8$" ]

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 .

[ "202" ]

If $\log _{10} x=3+\log _{10} y$, what is the value of $\frac{x}{y}$ ?

[ "1000" ]

If $x+\frac{1}{x}=\frac{13}{6}$, determine all values of $x^{2}+\frac{1}{x^{2}}$.

[ "$\\frac{97}{36}$" ]

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?

[ "$\\frac{2}{9}$" ]

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 |

[ "16,3,1,1" ]

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 $x$ ?

[ "4.25" ]

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 $n$.

[ "6" ]

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?

[ "7,4" ]

Determine all real values of $x$ for which $(x+8)^{4}=(2 x+16)^{2}$.

[ "-6,-8,-10" ]

If $f(x)=2 x+1$ and $g(f(x))=4 x^{2}+1$, determine an expression for $g(x)$.

[ "$g(x)=x^2-2x+2$" ]

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.)

[ "5" ]

Determine all real values of $x$ for which $3^{(x-1)} 9^{\frac{3}{2 x^{2}}}=27$.

[ "$1$,$\\frac{3 + \\sqrt{21}}{2}$,$\\frac{3 - \\sqrt{21}}{2}$" ]

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.

[ "$(1,0),(\\frac{1}{100},-8),(100,8)$" ]

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.

[ "$\\frac{27}{64}$" ]

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 $S=\frac{1}{\sin \alpha}$.

[ "$4^{\\circ}$" ]

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 $T(12)$.

[ "2,4,3" ]

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 $T(n)>2010$.

[ "309" ]

Suppose $0^{\circ}<x<90^{\circ}$ and $2 \sin ^{2} x+\cos ^{2} x=\frac{25}{16}$. What is the value of $\sin x$ ?

[ "$\\frac{3}{4}$" ]

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?

[ "153" ]

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.

[ "5,15" ]

Determine all values of $x$ for which $2+\sqrt{x-2}=x-2$.

[ "6" ]

Determine all values of $x$ for which $(\sqrt{x})^{\log _{10} x}=100$.

[ "$100,\\frac{1}{100}$" ]

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?

[ "23" ]

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?

[ "$\\frac{10}{21}$" ]

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: $$ \begin{aligned} a c+a d+b c+b d & =2023 \\ a+b+c+d & =296 \end{aligned} $$

[ "417" ]

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.

[ "222" ]

Determine all real values of $x$ for which $$ \sqrt{\log _{2} x \cdot \log _{2}(4 x)+1}+\sqrt{\log _{2} x \cdot \log _{2}\left(\frac{x}{64}\right)+9}=4 $$

[ "$[\\frac{1}{2}, 8]$" ]

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.)

[ "590" ]

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 $a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}$.

[ "$p(x)=\\frac{(x-1)(x-2)}{6}$" ]

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 $\mathrm{cm}^{2}$.

[ "$166$" ]

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 $k$ ?

[ "$2,-1,-18$" ]

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.)

[ "-5,7" ]