id int64 | question string | solution list | final_answer list | context string | image_1 image | image_2 image | image_3 image | image_4 image | image_5 image | modality string | difficulty string | is_multiple_answer bool | unit string | answer_type string | error string | question_type string | subfield string | subject string | language string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2,277 | 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. | [
"Suppose that Rayna's age is $x$ years.\n\nSince Qing is twice as old as Rayna, Qing's age is $2 x$ years.\n\nSince Qing is 4 years younger than Paolo, Paolo's age is $2 x+4$ years.\n\nSince the average of their ages is 13 years, we obtain\n\n$$\n\\frac{x+(2 x)+(2 x+4)}{3}=13\n$$\n\nThis gives $5 x+4=39$ and so $5 ... | [
"7,14,18"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,280 | 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 $\triangle D E F$. | [
"Since $V(1,18)$ is on the parabola, then $18=-2\\left(1^{2}\\right)+4(1)+c$ and so $c=18+2-4=16$.\n\nThus, the equation of the parabola is $y=-2 x^{2}+4 x+16$.\n\nThe $y$-intercept occurs when $x=0$, and so $y=16$. Thus, $D$ has coordinates $(0,16)$.\n\nThe $x$-intercepts occur when $y=0$. Here,\n\n$$\n\\begin{arr... | [
"48"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Geometry | Math | English |
2,281 | If $3\left(8^{x}\right)+5\left(8^{x}\right)=2^{61}$, what is the value of the real number $x$ ? | [
"We obtain successively\n\n$$\n\\begin{aligned}\n3\\left(8^{x}\\right)+5\\left(8^{x}\\right) & =2^{61} \\\\\n8\\left(8^{x}\\right) & =2^{61} \\\\\n8^{x+1} & =2^{61} \\\\\n\\left(2^{3}\\right)^{x+1} & =2^{61} \\\\\n2^{3(x+1)} & =2^{61}\n\\end{aligned}\n$$\n\nThus, $3(x+1)=61$ and so $3 x+3=61$ which gives $3 x=58$ o... | [
"$\\frac{58}{3}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,282 | For some real numbers $m$ and $n$, the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order. Determine all possible values of $m$. | [
"Since the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order, then\n\n$$\n\\begin{aligned}\n2(n+1)^{2}-3 n^{2} & =2 \\\\\n2 n^{2}+4 n+2-3 n^{2} & =2 \\\\\n-n^{2}+4 n & =0 \\\\\n-n(n-4) & =0\n\\end{aligned}\n$$\n\nand so $n=0$ or $n=4$.\n\nIf $n=0$, the list becomes... | [
"1,-1,7,-7"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,283 | 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}$ ? | [
"Suppose that $S_{0}$ has coordinates $(a, b)$.\n\nStep 1 moves $(a, b)$ to $(a,-b)$.\n\nStep 2 moves $(a,-b)$ to $(a,-b+2)$.\n\nStep 3 moves $(a,-b+2)$ to $(-a,-b+2)$.\n\nThus, $S_{1}$ has coordinates $(-a,-b+2)$.\n\nStep 1 moves $(-a,-b+2)$ to $(-a, b-2)$.\n\nStep 2 moves $(-a, b-2)$ to $(-a, b)$.\n\nStep 3 moves... | [
"(-7,-1)"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Tuple | null | Open-ended | Combinatorics | Math | English |
2,286 | 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.) | [
"Since the sequence $t_{1}, t_{2}, t_{3}, \\ldots, t_{n-2}, t_{n-1}, t_{n}$ is arithmetic, then\n\n$$\nt_{1}+t_{n}=t_{2}+t_{n-1}=t_{3}+t_{n-2}\n$$\n\nThis is because, if $d$ is the common difference, we have $t_{2}=t_{1}+d$ and $t_{n-1}=t_{n}-d$, as well as having $t_{3}=t_{1}+2 d$ and $t_{n-2}=t_{n}-2 d$.\n\nSince... | [
"20"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,287 | Suppose that $a$ and $r$ are real numbers. A geometric sequence with first term $a$ and common ratio $r$ has 4 terms. The sum of this geometric sequence is $6+6 \sqrt{2}$. A second geometric sequence has the same first term $a$ and the same common ratio $r$, but has 8 terms. The sum of this second geometric sequence is $30+30 \sqrt{2}$. Determine all possible values for $a$.
(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant, called the common ratio. For example, $3,-6,12,-24$ are the first four terms of a geometric sequence.) | [
"Since the sum of a geometric sequence with first term $a$, common ratio $r$ and 4 terms is $6+6 \\sqrt{2}$, then\n\n$$\na+a r+a r^{2}+a r^{3}=6+6 \\sqrt{2}\n$$\n\nSince the sum of a geometric sequence with first term $a$, common ratio $r$ and 8 terms is $30+30 \\sqrt{2}$, then\n\n$$\na+a r+a r^{2}+a r^{3}+a r^{4}+... | [
"$a=2$, $a=-6-4 \\sqrt{2}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Expression | null | Open-ended | Algebra | Math | English |
2,288 | 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? | [
"Victor stops when there are either 2 green balls on the table or 2 red balls on the table. If the first 2 balls that Victor removes are the same colour, Victor will stop.\n\nIf the first 2 balls that Victor removes are different colours, Victor does not yet stop, but when he removes a third ball, its colour must m... | [
"$\\frac{4}{7}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,289 | 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 $f(g(\sin \theta))=0$. | [
"Using the definition of $f$, the following equations are equivalent:\n\n$$\n\\begin{aligned}\nf(a) & =0 \\\\\n2 a^{2}-3 a+1 & =0 \\\\\n(a-1)(2 a-1) & =0\n\\end{aligned}\n$$\n\nTherefore, $f(a)=0$ exactly when $a=1$ or $a=\\frac{1}{2}$.\n\nThus, $f(g(\\sin \\theta))=0$ exactly when $g(\\sin \\theta)=1$ or $g(\\sin ... | [
"$\\frac{1}{6} \\pi, \\frac{5}{6} \\pi, \\frac{1}{4} \\pi, \\frac{3}{4} \\pi$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,292 | Suppose that $a=5$ and $b=4$. Determine all pairs of integers $(K, L)$ for which $K^{2}+3 L^{2}=a^{2}+b^{2}-a b$. | [
"When $a=5$ and $b=4$, we obtain $a^{2}+b^{2}-a b=5^{2}+4^{2}-5 \\cdot 4=21$.\n\nTherefore, we want to find all pairs of integers $(K, L)$ with $K^{2}+3 L^{2}=21$.\n\nIf $L=0$, then $L^{2}=0$, which gives $K^{2}=21$ which has no integer solutions.\n\nIf $L= \\pm 1$, then $L^{2}=1$, which gives $K^{2}=18$ which has ... | [
"$(3,2),(-3,2),(3,-2),(-3,-2)$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Number Theory | Math | English |
2,298 | Determine all values of $x$ for which $0<\frac{x^{2}-11}{x+1}<7$. | [
"We consider two cases: $x>-1$ (that is, $x+1>0$ ) and $x<-1$ (that is, $x+1<0$ ). Note that $x \\neq-1$.\n\nCase 1: $x>-1$\n\nWe take the given inequality $0<\\frac{x^{2}-11}{x+1}<7$ and multiply through by $x+1$, which is positive, to obtain $0<x^{2}-11<7 x+7$.\n\nThus, $x^{2}-11>0$ and $x^{2}-11<7 x+7$.\n\nFrom ... | [
"$(-\\sqrt{11},-2)\\cup (\\sqrt{11},9)$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Interval | null | Open-ended | Algebra | Math | English |
2,300 | 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.) | [
"Suppose that the arithmetic sequence $a_{1}, a_{2}, a_{3}, \\ldots$ has first term $a$ and common difference $d$.\n\nThen, for each positive integer $n, a_{n}=a+(n-1) d$.\n\nSince $a_{1}=a$ and $a_{2}=a+d$ and $a_{1} \\neq a_{2}$, then $d \\neq 0$.\n\nSince $a_{1}, a_{2}, a_{6}$ form a geometric sequence in that o... | [
"34"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,301 | 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. | [
"First, we note that since $k$ is a positive integer, then $k \\geq 1$.\n\nNext, we note that the given parabola passes through the point $(0,-5)$ as does the given circle. (This is because if $x=0$, then $y=\\frac{0^{2}}{k}-5=-5$ and if $(x, y)=(0,-5)$, then $x^{2}+y^{2}=0^{2}+(-5)^{2}=25$, so $(0,-5)$ satisfies e... | [
"1,2,5,8,9"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,303 | 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. | [
"Using $\\log$ arithm rules $\\log (u v)=\\log u+\\log v$ and $\\log \\left(s^{t}\\right)=t \\log s$ for all $u, v, s>0$, the first equation becomes\n\n$$\n\\begin{aligned}\n(\\log x)(\\log y)-3 \\log 5-3 \\log y-\\log 8-\\log x & =a \\\\\n(\\log x)(\\log y)-\\log x-3 \\log y-\\log 8-\\log 5^{3} & =a \\\\\n(\\log x... | [
"$(10^{4}, 10^{3}, 10^{10}),(10^{2}, 10^{-1}, 10^{-2})$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Algebra | Math | English |
2,312 | Two fair dice, each having six faces numbered 1 to 6 , are thrown. What is the probability that the product of the two numbers on the top faces is divisible by 5 ? | [
"There are 36 possibilities for the pair of numbers on the faces when the dice are thrown. For the product of the two numbers, each of which is between 1 and 6 , to be divisible by 5 , one of the two numbers must be equal to 5 .\n\nTherefore, the possible pairs for the faces are\n\n$$\n(1,5),(2,5),(3,5),(4,5),(5,5)... | [
"$\\frac{11}{36}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,313 | 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. | [
"First, we compute an expression for the composition of the two given functions:\n\n$$\n\\begin{aligned}\nf(g(x)) & =f(a x+b) \\\\\n& =(a x+b)^{2}-(a x+b)+2 \\\\\n& =a^{2} x^{2}+2 a b x+b^{2}-a x-b+2 \\\\\n& =a^{2} x^{2}+(2 a b-a) x+\\left(b^{2}-b+2\\right)\n\\end{aligned}\n$$\n\nBut we already know that $f(g(x))=9... | [
"$(3,0),(-3,1)$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Algebra | Math | English |
2,315 | 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 $n$. | [
"When the dimensions were increased by $n \\%$ from 10 by 15 , the new dimensions were $10\\left(1+\\frac{n}{100}\\right)$ by $15\\left(1+\\frac{n}{100}\\right)$.\n\nWhen the resolution was decreased by $n$ percent, the new resolution was $75\\left(1-\\frac{n}{100}\\right)$.\n\n(Note that $n$ cannot be larger than ... | [
"60"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,317 | 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 $x$. | [
"Consider the right side of the given equation:\n\n$$\n\\begin{aligned}\nT^{3}+b T+c & =\\left(x^{2}+\\frac{1}{x^{2}}\\right)^{3}+b\\left(x^{2}+\\frac{1}{x^{2}}\\right)+c \\\\\n& =\\left(x^{4}+2+\\frac{1}{x^{4}}\\right)\\left(x^{2}+\\frac{1}{x^{2}}\\right)+b\\left(x^{2}+\\frac{1}{x^{2}}\\right)+c \\\\\n& =x^{6}+3 x... | [
"-3,0"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,318 | 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. | [
"We start by placing the two 4's. We systematically try each pair of possible positions from positions 1 and 5 to positions 4 and 8 . For each of these positions, we try placing\n\n\nthe two 3's in each pair of possible positions, and then see if the two 2's and two 1's will fit.\n\n(We can reduce our work by notic... | [
"(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)"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Combinatorics | Math | English |
2,319 | 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. | [
"Since we are trying to create a Skolem sequence of order 9 , then there are 18 positions to fill with 10 odd numbers and 8 even numbers.\n\nWe are told that $s_{18}=8$, so we must have $s_{10}=8$, since the 8 's must be 8 positions apart. By condition III, between the two 8's, there can be only one odd integer. Bu... | [
"(7,5,1,1,9,3,5,7,3,8,6,4,2,9,2,4,6,8)"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Tuple | null | Open-ended | Combinatorics | Math | English |
2,321 | 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$ ? | [
"Suppose that $m$ has hundreds digit $a$, tens digit $b$, and ones (units) digit $c$.\n\nFrom the given information, $a, b$ and $c$ are distinct, each of $a, b$ and $c$ is less than 10, $a=b c$, and $c$ is odd (since $m$ is odd).\n\nThe integer $m=623$ satisfies all of these conditions. Since we are told there is o... | [
"623"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,322 | 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$ ? | [
"Since Eleanor has 100 marbles which are black and gold in the ratio $1: 4$, then $\\frac{1}{5}$ of her marbles are black, which means that she has $\\frac{1}{5} \\cdot 100=20$ black marbles.\n\nWhen more gold marbles are added, the ratio of black to gold is $1: 6$, which means that she has $6 \\cdot 20=120$ gold m... | [
"40"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,323 | 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$. | [
"First, we see that $\\frac{n^{2}+n+15}{n}=\\frac{n^{2}}{n}+\\frac{n}{n}+\\frac{15}{n}=n+1+\\frac{15}{n}$.\n\nThis means that $\\frac{n^{2}+n+15}{n}$ is an integer exactly when $n+1+\\frac{15}{n}$ is an integer.\n\nSince $n+1$ is an integer, then $\\frac{n^{2}+n+15}{n}$ is an integer exactly when $\\frac{15}{n}$ is... | [
"1, 3, 5, 15"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,325 | 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$ ? | [
"We apply the process two more times:\n\n| | $x$ | $y$ |\n| :---: | :---: | :---: |\n| Before Step 1 | 24 | 3 |\n| After Step 1 | 27 | 3 |\n| After Step 2 | 81 | 3 |\n| After Step 3 | 81 | 4 |\n\n\n| | $x$ | $y$ |\n| :---: | :---: | :---: |\n| Before Step 1 | 81 | 4 |\n| After Step 1 | 85 | 4 |\n| After Step 2 | ... | [
"340"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,326 | 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. | [
"The parabola with equation $y=k x^{2}+6 x+k$ has two distinct $x$-intercepts exactly when the discriminant of the quadratic equation $k x^{2}+6 x+k=0$ is positive.\n\nHere, the disciminant equals $\\Delta=6^{2}-4 \\cdot k \\cdot k=36-4 k^{2}$.\n\nThe inequality $36-4 k^{2}>0$ is equivalent to $k^{2}<9$.\n\nSince $... | [
"-2,-1,1,2"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,327 | 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}$ ? | [
"Since $\\frac{a}{b}<\\frac{4}{7}$ and $\\frac{4}{7}<1$, then $\\frac{a}{b}<1$.\n\nSince $a$ and $b$ are positive integers, then $a<b$.\n\nSince the difference between $a$ and $b$ is 15 and $a<b$, then $b=a+15$.\n\nTherefore, we have $\\frac{5}{9}<\\frac{a}{a+15}<\\frac{4}{7}$.\n\nWe multiply both sides of the left... | [
"$\\frac{19}{34}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,328 | 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.) | [
"The first 6 terms of a geometric sequence with first term 10 and common ratio $\\frac{1}{2}$ are $10,5, \\frac{5}{2}, \\frac{5}{4}, \\frac{5}{8}, \\frac{5}{16}$.\n\nHere, the ratio of its 6 th term to its 4 th term is $\\frac{5 / 16}{5 / 4}$ which equals $\\frac{1}{4}$. (We could have determined this without writi... | [
"$-\\frac{30}{17}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,329 | 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))$ ? | [
"Let $a=f(20)$. Then $f(f(20))=f(a)$.\n\nTo calculate $f(f(20))$, we determine the value of $a$ and then the value of $f(a)$.\n\nBy definition, $a=f(20)$ is the number of prime numbers $p$ that satisfy $20 \\leq p \\leq 30$.\n\nThe prime numbers between 20 and 30, inclusive, are 23 and 29 , so $a=f(20)=2$.\n\nThus,... | [
"5"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,330 | 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}
$$ | [
"Since $(x-1)(y-2)=0$, then $x=1$ or $y=2$.\n\nSuppose that $x=1$. In this case, the remaining equations become:\n\n$$\n\\begin{aligned}\n(1-3)(z+2) & =0 \\\\\n1+y z & =9\n\\end{aligned}\n$$\n\nor\n\n$$\n\\begin{array}{r}\n-2(z+2)=0 \\\\\ny z=8\n\\end{array}\n$$\n\nFrom the first of these equations, $z=-2$.\n\nFrom... | [
"(1,-4,-2),(3,2,3),(13,2,-2)"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Algebra | Math | English |
2,331 | Suppose that the function $g$ satisfies $g(x)=2 x-4$ for all real numbers $x$ and that $g^{-1}$ is the inverse function of $g$. Suppose that the function $f$ satisfies $g\left(f\left(g^{-1}(x)\right)\right)=2 x^{2}+16 x+26$ for all real numbers $x$. What is the value of $f(\pi)$ ? | [
"Since the function $g$ is linear and has positive slope, then it is one-to-one and so invertible. This means that $g^{-1}(g(a))=a$ for every real number $a$ and $g\\left(g^{-1}(b)\\right)=b$ for every real number $b$.\n\nTherefore, $g\\left(f\\left(g^{-1}(g(a))\\right)\\right)=g(f(a))$ for every real number $a$.\n... | [
"$4 \\pi^{2}-1$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,332 | Determine all pairs of angles $(x, y)$ with $0^{\circ} \leq x<180^{\circ}$ and $0^{\circ} \leq y<180^{\circ}$ that satisfy the following system of equations:
$$
\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}
$$ | [
"Using logarithm laws, the given equations are equivalent to\n\n$$\n\\begin{aligned}\n& \\log _{2}(\\sin x)+\\log _{2}(\\cos y)=-\\frac{3}{2} \\\\\n& \\log _{2}(\\sin x)-\\log _{2}(\\cos y)=\\frac{1}{2}\n\\end{aligned}\n$$\n\nAdding these two equations, we obtain $2 \\log _{2}(\\sin x)=-1$ which gives $\\log _{2}(\... | [
"$(45^{\\circ}, 60^{\\circ}),(135^{\\circ}, 60^{\\circ})$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Algebra | Math | English |
2,333 | 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. | [
"Let $x$ be the probability that Bianca wins the tournament.\n\nBecause Alain, Bianca and Chen are equally matched and because their roles in the tournament are identical, then the probability that each of them wins will be the same.\n\nThus, the probability that Alain wins the tournament is $x$ and the probability... | [
"$\\frac{1-p^{2}}{3}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Expression | null | Open-ended | Combinatorics | Math | English |
2,334 | 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$. | [
"Throughout this solution, we will mostly not include units, but will assume that all lengths are in kilometres, all times are in seconds, and all speeds are in kilometres per second.\n\nWe place the points in the coordinate plane with $B$ at $(0,0), A$ on the negative $x$-axis, and $C$ on the positive $x$-axis.\n\... | [
"$\\frac{41}{12}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | km | Numerical | null | Open-ended | Geometry | Math | English |
2,336 | 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. | [
"Here, the pairwise sums of the numbers $a_{1} \\leq a_{2} \\leq a_{3} \\leq a_{4}$ are $s_{1} \\leq s_{2} \\leq s_{3} \\leq s_{4} \\leq s_{5} \\leq s_{6}$. The six pairwise sums of the numbers in the list can be expressed as\n\n$$\na_{1}+a_{2}, a_{1}+a_{3}, a_{1}+a_{4}, a_{2}+a_{3}, a_{2}+a_{4}, a_{3}+a_{4}\n$$\n\... | [
"(1,7,103, 105), (3, 5, 101, 107)"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Number Theory | Math | English |
2,338 | Determine all values of $x$ for which $\frac{x^{2}+x+4}{2 x+1}=\frac{4}{x}$. | [
"Manipulating the given equation and noting that $x \\neq 0$ and $x \\neq-\\frac{1}{2}$ since neither denominator can equal 0 , we obtain\n\n$$\n\\begin{aligned}\n\\frac{x^{2}+x+4}{2 x+1} & =\\frac{4}{x} \\\\\nx\\left(x^{2}+x+4\\right) & =4(2 x+1) \\\\\nx^{3}+x^{2}+4 x & =8 x+4 \\\\\nx^{3}+x^{2}-4 x-4 & =0 \\\\\nx^... | [
"$-1$,$2$,$-2$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,339 | 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.) | [
"Since $900=30^{2}$ and $30=2 \\times 3 \\times 5$, then $900=2^{2} 3^{2} 5^{2}$.\n\nThe positive divisors of 900 are those integers of the form $d=2^{a} 3^{b} 5^{c}$, where each of $a, b, c$ is 0,1 or 2 .\n\nFor $d$ to be a perfect square, the exponent on each prime factor in the prime factorization of $d$ must be... | [
"8"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,340 | 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 $k$. | [
"In isosceles triangle $A B C, \\angle A B C=\\angle A C B$, so the sides opposite these angles $(A C$ and $A B$, respectively) are equal in length.\n\nSince the vertices of the triangle are $A(k, 3), B(3,1)$ and $C(6, k)$, then we obtain\n\n$$\n\\begin{aligned}\nA C & =A B \\\\\n\\sqrt{(k-6)^{2}+(3-k)^{2}} & =\\sq... | [
"$8$,$4$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Geometry | Math | English |
2,341 | A chemist has three bottles, each containing a mixture of acid and water:
- bottle A contains $40 \mathrm{~g}$ of which $10 \%$ is acid,
- bottle B contains $50 \mathrm{~g}$ of which $20 \%$ is acid, and
- bottle C contains $50 \mathrm{~g}$ of which $30 \%$ is acid.
She uses some of the mixture from each of the bottles to create a mixture with mass $60 \mathrm{~g}$ of which $25 \%$ is acid. Then she mixes the remaining contents of the bottles to create a new mixture. What percentage of the new mixture is acid? | [
"Bottle A contains $40 \\mathrm{~g}$ of which $10 \\%$ is acid.\n\nThus, it contains $0.1 \\times 40=4 \\mathrm{~g}$ of acid and $40-4=36 \\mathrm{~g}$ of water.\n\nBottle B contains $50 \\mathrm{~g}$ of which $20 \\%$ is acid.\n\nThus, it contains $0.2 \\times 50=10 \\mathrm{~g}$ of acid and $50-10=40 \\mathrm{~g}... | [
"17.5%"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,342 | Suppose that $x$ and $y$ are real numbers with $3 x+4 y=10$. Determine the minimum possible value of $x^{2}+16 y^{2}$. | [
"Since $3 x+4 y=10$, then $4 y=10-3 x$.\n\nTherefore, when $3 x+4 y=10$,\n\n$$\n\\begin{aligned}\nx^{2}+16 y^{2} & =x^{2}+(4 y)^{2} \\\\\n& =x^{2}+(10-3 x)^{2} \\\\\n& =x^{2}+\\left(9 x^{2}-60 x+100\\right) \\\\\n& =10 x^{2}-60 x+100 \\\\\n& =10\\left(x^{2}-6 x+10\\right) \\\\\n& =10\\left(x^{2}-6 x+9+1\\right) \\\... | [
"10"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,343 | 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? | [
"Suppose that the bag contains $g$ gold balls.\n\nWe assume that Feridun reaches into the bag and removes the two balls one after the other.\n\nThere are 40 possible balls that he could remove first and then 39 balls that he could remove second. In total, there are 40(39) pairs of balls that he could choose in this... | [
"26"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,344 | 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.) | [
"Suppose that the first term in the geometric sequence is $t_{1}=a$ and the common ratio in the sequence is $r$.\n\nThen the sequence, which has $n$ terms, is $a, a r, a r^{2}, a r^{3}, \\ldots, a r^{n-1}$.\n\nIn general, the $k$ th term is $t_{k}=a r^{k-1}$; in particular, the $n$th term is $t_{n}=a r^{n-1}$.\n\nS... | [
"20"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,345 | If $\frac{(x-2013)(y-2014)}{(x-2013)^{2}+(y-2014)^{2}}=-\frac{1}{2}$, what is the value of $x+y$ ? | [
"Let $a=x-2013$ and let $b=y-2014$.\n\nThe given equation becomes $\\frac{a b}{a^{2}+b^{2}}=-\\frac{1}{2}$, which is equivalent to $2 a b=-a^{2}-b^{2}$ and $a^{2}+2 a b+b^{2}=0$.\n\nThis is equivalent to $(a+b)^{2}=0$ which is equivalent to $a+b=0$.\n\nSince $a=x-2013$ and $b=y-2014$, then $x-2013+y-2014=0$ or $x+y... | [
"4027"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,346 | Determine all real numbers $x$ for which
$$
\left(\log _{10} x\right)^{\log _{10}\left(\log _{10} x\right)}=10000
$$ | [
"Let $a=\\log _{10} x$.\n\nThen $\\left(\\log _{10} x\\right)^{\\log _{10}\\left(\\log _{10} x\\right)}=10000$ becomes $a^{\\log _{10} a}=10^{4}$.\n\nTaking the base 10 logarithm of both sides and using the fact that $\\log _{10}\\left(a^{b}\\right)=b \\log _{10} a$, we obtain $\\left(\\log _{10} a\\right)\\left(\\... | [
"$10^{100}$,$10^{1 / 100}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,349 | 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.) | [
"Let $S=\\sin ^{6} 1^{\\circ}+\\sin ^{6} 2^{\\circ}+\\sin ^{6} 3^{\\circ}+\\cdots+\\sin ^{6} 87^{\\circ}+\\sin ^{6} 88^{\\circ}+\\sin ^{6} 89^{\\circ}$.\n\nSince $\\sin \\theta=\\cos \\left(90^{\\circ}-\\theta\\right)$, then $\\sin ^{6} \\theta=\\cos ^{6}\\left(90^{\\circ}-\\theta\\right)$, and so\n\n$$\n\\begin{al... | [
"$221,$8$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,350 | Let $f(n)$ be the number of positive integers that have exactly $n$ digits and whose digits have a sum of 5. Determine, with proof, how many of the 2014 integers $f(1), f(2), \ldots, f(2014)$ have a units digit of 1 . | [
"First, we prove that $f(n)=\\frac{n(n+1)(n+2)(n+3)}{24}$ in two different ways.\n\nMethod 1\n\nIf an $n$-digit integer has digits with a sum of 5 , then there are several possibilities for the combination of non-zero digits used:\n\n$$\n5 \\quad 4,1 \\quad 3,2 \\quad 3,1,1 \\quad 2,2,1 \\quad 2,1,1,1 \\quad 1,1,1,... | [
"202"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,351 | If $\log _{10} x=3+\log _{10} y$, what is the value of $\frac{x}{y}$ ? | [
"$$\n\\begin{gathered}\n\\log _{10} x-\\log _{10} y=3 \\\\\n\\Leftrightarrow \\log _{10}\\left(\\frac{x}{y}\\right)=3 \\\\\n\\Leftrightarrow \\frac{x}{y}=10^{3}=1000\n\\end{gathered}\n$$"
] | [
"1000"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,352 | If $x+\frac{1}{x}=\frac{13}{6}$, determine all values of $x^{2}+\frac{1}{x^{2}}$. | [
"$\\left(x+\\frac{1}{x}\\right)^{2}=\\left(\\frac{13}{6}\\right)^{2}$; squaring\n\n$x^{2}+2+\\frac{1}{x^{2}}=\\frac{169}{36}$\n\n$x^{2}+\\frac{1}{x^{2}}=\\frac{169}{32}-2$\n\n$x^{2}+\\frac{1}{x^{2}}=\\frac{169}{36}-\\frac{72}{36}=\\frac{97}{36}$",
"$6 x\\left(x+\\frac{1}{x}\\right)=6 x\\left(\\frac{13}{6}\\right)... | [
"$\\frac{97}{36}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,355 | 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? | [
"There are only two possibilities on the first roll - it can either be even or odd.\n\nPossibility 1 'The first roll is odd'\n\nThe probability of an odd outcome on the first roll is $\\frac{1}{3}$.\n\nAfter doubling all the numbers, the possible outcomes on the second roll would now be 2, 2, 6, $4,6,8$ with the pr... | [
"$\\frac{2}{9}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,356 | 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 | | [
"There are a variety of ways to find the unknowns.\n\nThe most efficient way is to choose equations that have like coefficients. Here is one way to solve the problem using this method.\n\nFor Sussex: $\\quad 6 w+4 d+30 a+63 b=201$\n\nFor Som: $\\quad 6 w+4 d+30 a+54 b=192$\n\nSubtracting, $\\quad 9 b=9 b=1$\n\nIf $... | [
"16,3,1,1"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,359 | 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 $x$ ? | [
"We deduce that $4<x<5$.\n\nOtherwise, if $x \\leq 4, x\\lfloor x\\rfloor \\leq 16$, and if $x \\geq 5, x\\lfloor x\\rfloor \\geq 25$.\n\nTherefore $\\lfloor x\\rfloor=4$\n\nSince $x\\lfloor x\\rfloor=17$\n\n$$\n\\begin{aligned}\n4 x & =17 \\\\\nx & =4.25\n\\end{aligned}\n$$"
] | [
"4.25"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,361 | 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 $n$. | [
"If we remove the cubes which have red paint, we are left with a smaller cube with measurements, $(n-1) \\times(n-1) \\times(n-1)$\n\nThus, $(n-1)^{3}=125$\n\n$$\nn=6 \\text {. }\n$$"
] | [
"6"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Geometry | Math | English |
2,369 | 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? | [
"Suppose that Thurka bought $x$ goats and $y$ helicopters.\n\nThen $19 x+17 y=201$.\n\nSince $x$ and $y$ are non-negative integers, then $19 x \\leq 201$ so $x \\leq 10$.\n\nIf $x=10$, then $17 y=201-19 x=11$, which does not have an integer solution because 11 is not divisible by 17 .\n\nIf $x=9$, then $17 y=201-19... | [
"7,4"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,370 | Determine all real values of $x$ for which $(x+8)^{4}=(2 x+16)^{2}$. | [
"Manipulating algebraically,\n\n$$\n\\begin{aligned}\n(x+8)^{4} & =(2 x+16)^{2} \\\\\n(x+8)^{4}-2^{2}(x+8)^{2} & =0 \\\\\n(x+8)^{2}\\left((x+8)^{2}-2^{2}\\right) & =0 \\\\\n(x+8)^{2}((x+8)+2)((x+8)-2) & =0 \\\\\n(x+8)^{2}(x+10)(x+6) & =0\n\\end{aligned}\n$$\n\nTherefore, $x=-8$ or $x=-10$ or $x=-6$.",
"Manipulati... | [
"-6,-8,-10"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,371 | If $f(x)=2 x+1$ and $g(f(x))=4 x^{2}+1$, determine an expression for $g(x)$. | [
"We use the fact that $g(x)=g\\left(f\\left(f^{-1}(x)\\right)\\right)$.\n\nSince $f(x)=2 x+1$, then to determine $f^{-1}(x)$ we solve $x=2 y+1$ for $y$ to get $2 y=x-1$ or $y=\\frac{1}{2}(x-1)$. Thus, $f^{-1}(x)=\\frac{1}{2}(x-1)$.\n\nSince $g(f(x))=4 x^{2}+1$, then\n\n$$\n\\begin{aligned}\ng(x) & =g\\left(f\\left(... | [
"$g(x)=x^2-2x+2$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Expression | null | Open-ended | Algebra | Math | English |
2,372 | A geometric sequence has 20 terms.
The sum of its first two terms is 40 .
The sum of its first three terms is 76 .
The sum of its first four terms is 130 .
Determine how many of the terms in the sequence are integers.
(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a constant. For example, $3,6,12$ is a geometric sequence with three terms.) | [
"Since the sum of the first two terms is 40 and the sum of the first three terms is 76, then the third term is $76-40=36$.\n\nSince the sum of the first three terms is 76 and the sum of the first four terms is 130, then the fourth term is $130-76=54$.\n\nSince the third term is 36 and the fourth term is 54 , then t... | [
"5"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,375 | Determine all real values of $x$ for which $3^{(x-1)} 9^{\frac{3}{2 x^{2}}}=27$. | [
"Using the facts that $9=3^{2}$ and $27=3^{3}$, and the laws for manipulating exponents, we have\n\n$$\n\\begin{aligned}\n3^{x-1} 9^{\\frac{3}{2 x^{2}}} & =27 \\\\\n3^{x-1}\\left(3^{2}\\right)^{\\frac{3}{2 x^{2}}} & =3^{3} \\\\\n3^{x-1} 3^{\\frac{3}{x^{2}}} & =3^{3} \\\\\n3^{x-1+\\frac{3}{x^{2}}} & =3^{3}\n\\end{al... | [
"$1$,$\\frac{3 + \\sqrt{21}}{2}$,$\\frac{3 - \\sqrt{21}}{2}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,376 | 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. | [
"To determine the points of intersection, we equate $y$ values of the two curves and obtain $\\log _{10}\\left(x^{4}\\right)=\\left(\\log _{10} x\\right)^{3}$.\n\nSince $\\log _{10}\\left(a^{b}\\right)=b \\log _{10} a$, the equation becomes $4 \\log _{10} x=\\left(\\log _{10} x\\right)^{3}$.\n\nWe set $u=\\log _{10... | [
"$(1,0),(\\frac{1}{100},-8),(100,8)$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Algebra | Math | English |
2,377 | Oi-Lam tosses three fair coins and removes all of the coins that come up heads. George then tosses the coins that remain, if any. Determine the probability that George tosses exactly one head. | [
"If Oi-Lam tosses 3 heads, then George has no coins to toss, so cannot toss exactly 1 head. If Oi-Lam tosses 2, 1 or 0 heads, then George has at least one coin to toss, so can toss exactly 1 head.\n\nTherefore, the following possibilities exist:\n\n* Oi-Lam tosses 2 heads out of 3 coins and George tosses 1 head out... | [
"$\\frac{27}{64}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,380 | 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 $S=\frac{1}{\sin \alpha}$. | [
"We first prove Lemma(i): If $\\theta$ is an angle whose measure is not an integer multiple of $90^{\\circ}$, then\n$$\n\\cot \\theta-\\cot 2 \\theta=\\frac{1}{\\sin 2 \\theta}\n$$\n\nProof. \n$$\n\\begin{aligned}\n\\mathrm{LS} & =\\cot \\theta-\\cot 2 \\theta \\\\\n& =\\frac{\\cos \\theta}{\\sin \\theta}-\\frac{\\... | [
"$4^{\\circ}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,382 | 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)$. | [
"Denote the side lengths of a triangle by $a, b$ and $c$, with $0<a \\leq b \\leq c$.\n\nIn order for these lengths to form a triangle, we need $c<a+b$ and $b<a+c$ and $a<b+c$. Since $0<a \\leq b \\leq c$, then $b<a+c$ and $a<b+c$ follow automatically, so only $c<a+b$ ever needs to be checked.\n\nInstead of directl... | [
"2,4,3"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,384 | 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$. | [
"Denote the side lengths of a triangle by $a, b$ and $c$, with $0<a \\leq b \\leq c$.\n\nIn order for these lengths to form a triangle, we need $c<a+b$ and $b<a+c$ and $a<b+c$. Since $0<a \\leq b \\leq c$, then $b<a+c$ and $a<b+c$ follow automatically, so only $c<a+b$ ever needs to be checked.\n\nInstead of directl... | [
"309"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,388 | Suppose $0^{\circ}<x<90^{\circ}$ and $2 \sin ^{2} x+\cos ^{2} x=\frac{25}{16}$. What is the value of $\sin x$ ? | [
"Since $2 \\sin ^{2} x+\\cos ^{2} x=\\frac{25}{16}$ and $\\sin ^{2} x+\\cos ^{2} x=1\\left(\\right.$ so $\\left.\\cos ^{2} x=1-\\sin ^{2} x\\right)$, then we get\n\n$$\n\\begin{aligned}\n2 \\sin ^{2} x+\\left(1-\\sin ^{2} x\\right) & =\\frac{25}{16} \\\\\n\\sin ^{2} x & =\\frac{25}{16}-1 \\\\\n\\sin ^{2} x & =\\fra... | [
"$\\frac{3}{4}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Geometry | Math | English |
2,390 | 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? | [
"From the given information, the first term in the sequence is 2007 and each term starting with the second can be determined from the previous term.\n\nThe second term is $2^{3}+0^{3}+0^{3}+7^{3}=8+0+0+343=351$.\n\nThe third term is $3^{3}+5^{3}+1^{3}=27+125+1=153$.\n\nThe fourth term is $1^{3}+5^{3}+3^{3}=27+125+1... | [
"153"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,391 | 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. | [
"The $n$th term of sequence $\\mathrm{A}$ is $n^{2}-10 n+70$.\n\nSince sequence B is arithmetic with first term 5 and common difference 10 , then the $n$th term of sequence $\\mathrm{B}$ is equal to $5+10(n-1)=10 n-5$. (Note that this formula agrees with the first few terms.)\n\nFor the $n$th term of sequence $\\ma... | [
"5,15"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,392 | Determine all values of $x$ for which $2+\sqrt{x-2}=x-2$. | [
"Rearranging and then squaring both sides,\n\n$$\n\\begin{aligned}\n2+\\sqrt{x-2} & =x-2 \\\\\n\\sqrt{x-2} & =x-4 \\\\\nx-2 & =(x-4)^{2} \\\\\nx-2 & =x^{2}-8 x+16 \\\\\n0 & =x^{2}-9 x+18 \\\\\n0 & =(x-3)(x-6)\n\\end{aligned}\n$$\n\nso $x=3$ or $x=6$.\n\nWe should check both solutions, because we may have introduced... | [
"6"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,397 | Determine all values of $x$ for which $(\sqrt{x})^{\log _{10} x}=100$. | [
"Using rules for manipulating logarithms,\n\n$$\n\\begin{aligned}\n(\\sqrt{x})^{\\log _{10} x} & =100 \\\\\n\\log _{10}\\left((\\sqrt{x})^{\\log _{10} x}\\right) & =\\log _{10} 100 \\\\\n\\left(\\log _{10} x\\right)\\left(\\log _{10} \\sqrt{x}\\right) & =2 \\\\\n\\left(\\log _{10} x\\right)\\left(\\log _{10} x^{\\f... | [
"$100,\\frac{1}{100}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Geometry | Math | English |
2,403 | Suppose that $f(x)=x^{2}+(2 n-1) x+\left(n^{2}-22\right)$ for some integer $n$. What is the smallest positive integer $n$ for which $f(x)$ has no real roots? | [
"The quadratic function $f(x)=x^{2}+(2 n-1) x+\\left(n^{2}-22\\right)$ has no real roots exactly when its discriminant, $\\Delta$, is negative.\n\nThe discriminant of this function is\n\n$$\n\\begin{aligned}\n\\Delta & =(2 n-1)^{2}-4(1)\\left(n^{2}-22\\right) \\\\\n& =\\left(4 n^{2}-4 n+1\\right)-\\left(4 n^{2}-88\... | [
"23"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,407 | 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? | [
"Each possible order in which Akshan removes the marbles corresponds to a sequence of 9 colours, 3 of which are red and 6 of which are blue.\n\nWe write these as sequences of 3 R's and 6 B's.\n\nSince are told that the first marble is red and the third is blue, we would like to consider all sequences of the form\n\... | [
"$\\frac{10}{21}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,408 | 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:
$$
\begin{aligned}
a c+a d+b c+b d & =2023 \\
a+b+c+d & =296
\end{aligned}
$$ | [
"Factoring the first equation, we obtain\n\n$$\na c+a d+b c+b d=a(c+d)+b(c+d)=(a+b)(c+d)\n$$\n\nWe now have the equations\n\n$$\n\\begin{aligned}\n(a+b)(c+d) & =2023 \\\\\n(a+b)+(c+d) & =296\n\\end{aligned}\n$$\n\nIf we let $s=a+b$ and $t=c+d$, we obtain the equations\n\n$$\n\\begin{aligned}\ns t & =2023 \\\\\ns+t ... | [
"417"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,409 | Suppose that $\triangle A B C$ is right-angled at $B$ and has $A B=n(n+1)$ and $A C=(n+1)(n+4)$, where $n$ is a positive integer. Determine the number of positive integers $n<100000$ for which the length of side $B C$ is also an integer. | [
"Since $\\triangle A B C$ is right-angled at $B$, then\n\n$$\n\\begin{aligned}\nB C^{2} & =A C^{2}-A B^{2} \\\\\n& =((n+1)(n+4))^{2}-(n(n+1))^{2} \\\\\n& =(n+1)^{2}(n+4)^{2}-n^{2}(n+1)^{2} \\\\\n& =(n+1)^{2}\\left((n+4)^{2}-n^{2}\\right) \\\\\n& =(n+1)^{2}\\left(n^{2}+8 n+16-n^{2}\\right) \\\\\n& =(n+1)^{2}(8 n+16)... | [
"222"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Geometry | Math | English |
2,410 | Determine all real values of $x$ for which
$$
\sqrt{\log _{2} x \cdot \log _{2}(4 x)+1}+\sqrt{\log _{2} x \cdot \log _{2}\left(\frac{x}{64}\right)+9}=4
$$ | [
"Let $f(x)=\\sqrt{\\log _{2} x \\cdot \\log _{2}(4 x)+1}+\\sqrt{\\log _{2} x \\cdot \\log _{2}\\left(\\frac{x}{64}\\right)+9}$.\n\nUsing logarithm laws,\n\n$$\n\\begin{aligned}\n\\log _{2} x \\cdot \\log _{2}(4 x)+1 & =\\log _{2} x\\left(\\log _{2} 4+\\log _{2} x\\right)+1 \\\\\n& =\\log _{2} x\\left(2+\\log _{2} x... | [
"$[\\frac{1}{2}, 8]$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Interval | null | Open-ended | Algebra | Math | English |
2,414 | 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.) | [
"Since $0<\\frac{1}{3}<\\frac{2}{3}<1$, then $\\left\\lfloor\\frac{1}{3}\\right\\rfloor=\\left\\lfloor\\frac{2}{3}\\right\\rfloor=0$.\n\nSince $1 \\leq \\frac{3}{3}<\\frac{4}{3}<\\frac{5}{3}<2$, then $\\left\\lfloor\\frac{3}{3}\\right\\rfloor=\\left\\lfloor\\frac{4}{3}\\right\\rfloor=\\left\\lfloor\\frac{5}{3}\\rig... | [
"590"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,415 | 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
$$
(The sum has $m-1$ terms.)
A polynomial $f(x)$ is an algebraic expression of the form $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$ for some integer $n \geq 0$ and for some real numbers $a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}$. | [
"For every positive integer $m>4$, let\n\n$$\nq(m)=\\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\n$$\n\nExtending our work from (a), we know... | [
"$p(x)=\\frac{(x-1)(x-2)}{6}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Expression | null | Open-ended | Number Theory | Math | English |
2,419 | One of the faces of a rectangular prism has area $27 \mathrm{~cm}^{2}$. Another face has area $32 \mathrm{~cm}^{2}$. If the volume of the prism is $144 \mathrm{~cm}^{3}$, determine the surface area of the prism in $\mathrm{cm}^{2}$. | [
"Suppose that the rectangular prism has dimensions $a \\mathrm{~cm}$ by $b \\mathrm{~cm}$ by $c \\mathrm{~cm}$.\n\nSuppose further that one of the faces that is $a \\mathrm{~cm}$ by $b \\mathrm{~cm}$ is the face with area $27 \\mathrm{~cm}^{2}$ and that one of the faces that is $a \\mathrm{~cm}$ by $c \\mathrm{~cm}... | [
"$166$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | $cm^2$ | Numerical | null | Open-ended | Geometry | Math | English |
2,420 | The equations $y=a(x-2)(x+4)$ and $y=2(x-h)^{2}+k$ represent the same parabola. What are the values of $a, h$ and $k$ ? | [
"We expand the right sides of the two equations, collecting like terms in each case:\n\n$$\n\\begin{aligned}\n& y=a(x-2)(x+4)=a\\left(x^{2}+2 x-8\\right)=a x^{2}+2 a x-8 a \\\\\n& y=2(x-h)^{2}+k=2\\left(x^{2}-2 h x+h^{2}\\right)+k=2 x^{2}-4 h x+\\left(2 h^{2}+k\\right)\n\\end{aligned}\n$$\n\nSince these two equatio... | [
"$2,-1,-18$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,421 | In an arithmetic sequence with 5 terms, the sum of the squares of the first 3 terms equals the sum of the squares of the last 2 terms. If the first term is 5 , determine all possible values of the fifth term.
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3,5,7,9,11 is an arithmetic sequence with five terms.) | [
"Let the common difference in this arithmetic sequence be $d$.\n\nSince the first term in the sequence is 5 , then the 5 terms are $5,5+d, 5+2 d, 5+3 d, 5+4 d$.\n\nFrom the given information, $5^{2}+(5+d)^{2}+(5+2 d)^{2}=(5+3 d)^{2}+(5+4 d)^{2}$.\n\nManipulating, we obtain the following equivalent equations:\n\n$$\... | [
"-5,7"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,422 | 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? | [
"First, we determine the perfect squares between 1300 and 1400 and between 1400 and 1500.\n\nSince $\\sqrt{1300} \\approx 36.06$, then the first perfect square larger than 1300 is $37^{2}=1369$.\n\nThe next perfect squares are $38^{2}=1444$ and $39^{2}=1521$.\n\nSince Dan was born between 1300 and 1400 in a year th... | [
"1469"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,423 | Determine all values of $k$ for which the points $A(1,2), B(11,2)$ and $C(k, 6)$ form a right-angled triangle. | [
"$\\triangle A B C$ is right-angled exactly when one of the following statements is true:\n\n- $A B$ is perpendicular to $B C$, or\n- $A B$ is perpendicular to $A C$, or\n- $A C$ is perpendicular to $B C$.\n\nSince $A(1,2)$ and $B(11,2)$ share a $y$-coordinate, then $A B$ is horizontal.\n\nFor $A B$ and $B C$ to be... | [
"1,3,9,11"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Geometry | Math | English |
2,425 | If $\cos \theta=\tan \theta$, determine all possible values of $\sin \theta$, giving your answer(s) as simplified exact numbers. | [
"Since $\\tan \\theta=\\frac{\\sin \\theta}{\\cos \\theta}$, then we assume that $\\cos \\theta \\neq 0$.\n\nTherefore, we obtain the following equivalent equations:\n\n$$\n\\begin{aligned}\n\\cos \\theta & =\\tan \\theta \\\\\n\\cos \\theta & =\\frac{\\sin \\theta}{\\cos \\theta} \\\\\n\\cos ^{2} \\theta & =\\sin ... | [
"$\\frac{-1+\\sqrt{5}}{2}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,426 | Linh is driving at $60 \mathrm{~km} / \mathrm{h}$ on a long straight highway parallel to a train track. Every 10 minutes, she is passed by a train travelling in the same direction as she is. These trains depart from the station behind her every 3 minutes and all travel at the same constant speed. What is the constant speed of the trains, in $\mathrm{km} / \mathrm{h}$ ? | [
"Suppose that the trains are travelling at $v \\mathrm{~km} / \\mathrm{h}$.\n\nConsider two consecutive points in time at which the car is passed by a train.\n\nSince these points are 10 minutes apart, and 10 minutes equals $\\frac{1}{6}$ hour, and the car travels at $60 \\mathrm{~km} / \\mathrm{h}$, then the car t... | [
"$\\frac{600}{7}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | km/h | Numerical | null | Open-ended | Algebra | Math | English |
2,427 | Determine all pairs $(a, b)$ of real numbers that satisfy the following system of equations:
$$
\begin{aligned}
\sqrt{a}+\sqrt{b} & =8 \\
\log _{10} a+\log _{10} b & =2
\end{aligned}
$$
Give your answer(s) as pairs of simplified exact numbers. | [
"From the first equation, we note that $a \\geq 0$ and $b \\geq 0$, since the argument of a square root must be non-negative.\n\nFrom the second equation, we note that $a>0$ and $b>0$, since the argument of a logarithm must be positive.\n\nCombining these restrictions, we see that $a>0$ and $b>0$.\n\nFrom the equat... | [
"$(22+8 \\sqrt{6}, 22-8 \\sqrt{6})$,$(22-8 \\sqrt{6}, 22+8 \\sqrt{6})$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Algebra | Math | English |
2,430 | A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$.
Determine the average value of
$$
\left|a_{1}-a_{2}\right|+\left|a_{3}-a_{4}\right|
$$
over all permutations $a_{1}, a_{2}, a_{3}, a_{4}$ of $1,2,3,4$. | [
"There are 4 ! $=4 \\cdot 3 \\cdot 2 \\cdot 1=24$ permutations of $1,2,3,4$.\n\nThis is because there are 4 possible choices for $a_{1}$, and for each of these there are 3 possible choices for $a_{2}$, and for each of these there are 2 possible choices for $a_{3}$, and then 1 possible choice for $a_{4}$.\n\nConside... | [
"$\\frac{10}{3}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,431 | A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$.
Determine the average value of
$$
a_{1}-a_{2}+a_{3}-a_{4}+a_{5}-a_{6}+a_{7}
$$
over all permutations $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}$ of $1,2,3,4,5,6,7$. | [
"There are $7 !=7 \\cdot 6 \\cdot 5 \\cdot 4 \\cdot 3 \\cdot 2 \\cdot 1$ permutations of $1,2,3,4,5,6,7$, because there are 7 choices for $a_{1}$, then 6 choices for $a_{2}$, and so on.\n\nWe determine the average value of $a_{1}-a_{2}+a_{3}-a_{4}+a_{5}-a_{6}+a_{7}$ over all of these permutations by determining the... | [
"4"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,432 | A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$.
Determine the average value of
$$
\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|
$$
over all permutations $a_{1}, a_{2}, a_{3}, \ldots, a_{199}, a_{200}$ of $1,2,3,4, \ldots, 199,200$. (The sum labelled (*) contains 100 terms of the form $\left|a_{2 k-1}-a_{2 k}\right|$.) | [
"There are 200! permutations of $1,2,3, \\ldots, 198,199,200$.\n\nWe determine the average value of\n\n$$\n\\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|\n$$\n\nover all of these permutations by determining the sum of all 200! values of th... | [
"6700"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,435 | If $0^{\circ}<x<90^{\circ}$ and $3 \sin (x)-\cos \left(15^{\circ}\right)=0$, what is the value of $x$ to the nearest tenth of a degree? | [
"Rearranging the equation,\n\n$$\n\\begin{aligned}\n3 \\sin (x) & =\\cos \\left(15^{\\circ}\\right) \\\\\n\\sin (x) & =\\frac{1}{3} \\cos \\left(15^{\\circ}\\right) \\\\\n\\sin (x) & \\approx 0.3220\n\\end{aligned}\n$$\n\nUsing a calculator, $x \\approx 18.78^{\\circ}$. To the nearest tenth of a degree, $x=18.8^{\\... | [
"$18.8^{\\circ}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Geometry | Math | English |
2,438 | The function $f(x)$ has the property that $f(2 x+3)=2 f(x)+3$ for all $x$. If $f(0)=6$, what is the value of $f(9)$ ? | [
"Since we are looking for the value of $f(9)$, then it makes sense to use the given equation and to set $x=3$ in order to obtain $f(9)=2 f(3)+3$.\n\nSo we need to determine the value of $f(3)$. We use the equation again and set $x=0$ since we will then get $f(3)$ on the left side and $f(0)$ (whose value we already ... | [
"33"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,439 | Suppose that the functions $f(x)$ and $g(x)$ satisfy the system of equations
$$
\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)$. | [
"We solve the system of equations for $f(x)$ and $g(x)$.\n\nDividing out the common factor of 2 from the second equation, we get\n\n$f(x)+2 g(x)=x^{2}+2$.\n\nSubtracting from the first equation, we get $g(x)=x+4$.\n\nThus, $f(x)=x^{2}+2-2 g(x)=x^{2}+2-2(x+4)=x^{2}-2 x-6$.\n\nEquating $f(x)$ and $g(x)$, we obtain\n\... | [
"$5,-2$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,440 | 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? | [
"We label the 5 skaters A, B, C, D, and E, where D and E are the two Canadians.\n\nThere are then $5 !=5 \\times 4 \\times 3 \\times 2 \\times 1=120$ ways of arranging these skaters in their order of finish (for example, $\\mathrm{ADBCE}$ indicates that A finished first, $\\mathrm{D}$ second, etc.), because there a... | [
"$\\frac{1}{10}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,441 | 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 . | [
"Since the least common multiple of $3,5,10$ and 15 is 30 , then we can count the number of positive integers less than or equal to 30 satisfying these conditions, and multiply the total by 10 to obtain the number less than 300. (This is because each group of 30 consecutive integers starting with 1 more than a mult... | [
"100"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,442 | 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? | [
"Since the signs alternate every three terms, it makes sense to look at the terms in groups of 6 .\n\nThe sum of the first 6 terms is $1+3+5-7-9-11=-18$.\n\nThe sum of the next 6 terms is $13+15+17-19-21-23=-18$.\n\nIn fact, the sum of each group of 6 terms will be the same, since in each group, 12 has been added t... | [
"-900"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,443 | 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. | [
"Let the two digit integer have tens digit $a$ and units digit $b$. Then the given information tells us\n\n$$\n\\begin{aligned}\na^{2}+10 b & =b^{2}+10 a \\\\\na^{2}-b^{2}-10 a+10 b & =0 \\\\\n(a+b)(a-b)-10(a-b) & =0 \\\\\n(a-b)(a+b-10) & =0\n\\end{aligned}\n$$\n\nand so $a=b$ or $a+b=10$.\n\nSo the possibilities f... | [
"11,19,37,73"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,444 | A lead box contains samples of two radioactive isotopes of iron. Isotope A decays so that after every 6 minutes, the number of atoms remaining is halved. Initially, there are twice as many atoms of isotope $\mathrm{A}$ as of isotope $\mathrm{B}$, and after 24 minutes there are the same number of atoms of each isotope. How long does it take the number of atoms of isotope B to halve? | [
"In 24 minutes, the number of atoms of isotope $\\mathrm{A}$ has halved 4 times, so the initial number of atoms is $2^{4}=16$ times the number of atoms of isotope $\\mathrm{A}$ at time 24 minutes.\n\nBut there were initially half as many atoms of isotope B as of isotope B, so there was 8 times the final number of a... | [
"8"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | min | Numerical | null | Open-ended | Algebra | Math | English |
2,445 | Solve the system of equations:
$$
\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}
$$ | [
"Using the facts that $\\log _{10} A+\\log _{10} B=\\log _{10} A B$ and that $\\log _{10} A-\\log _{10} B=\\log _{10} \\frac{A}{B}$, then we can convert the two equations to\n\n$$\n\\begin{aligned}\n\\log _{10}\\left(x^{3} y^{2}\\right) & =11 \\\\\n\\log _{10}\\left(\\frac{x^{2}}{y^{3}}\\right) & =3\n\\end{aligned}... | [
"$10^{3},10$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,453 | 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. | [
"First, we prove lemma (b): if $n$ is an even savage integer, then $\\frac{n+4}{12}$ is an integer.\n\nProof of lemma (b):\nWe use the strategy of putting all of the multiples of 3 between 1 and $n$ in the set $C$, all of the remaining even numbers in the set $B$, and all of the remaining numbers in the set $A$. Th... | [
"8,32,44,68,80"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,454 | 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? | [
"We make a table of the 36 possible combinations of rolls and the resulting sums:\n\n| | 2 | 3 | 5 | 7 | 11 | 13 |\n| :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| 2 | 4 | 5 | 7 | 9 | 13 | 15 |\n| 3 | 5 | 6 | 8 | 10 | 14 | 16 |\n| 5 | 7 | 8 | 10 | 12 | 16 | 18 |\n| 7 | 9 | 10 | 12 | 14 | 18 | 20 |\n| ... | [
"$\\frac{1}{6}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,458 | If $\frac{1}{\cos x}-\tan x=3$, what is the numerical value of $\sin x$ ? | [
"Beginning with the given equation, we have\n\n$$\n\\begin{aligned}\n\\frac{1}{\\cos x}-\\tan x & =3 \\\\\n\\frac{1}{\\cos x}-\\frac{\\sin x}{\\cos x} & =3 \\\\\n1-\\sin x & =3 \\cos x \\quad(\\text { since } \\cos x \\neq 0) \\\\\n(1-\\sin x)^{2} & =9 \\cos ^{2} x \\quad \\text { (squaring both sides) } \\\\\n1-2 ... | [
"$-\\frac{4}{5}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,459 | 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$.) | [
"Since $f(x)=a x+b$, we can determine an expression for $g(x)=f^{-1}(x)$ by letting $y=f(x)$ to obtain $y=a x+b$. We then interchange $x$ and $y$ to obtain $x=a y+b$ which we solve for $y$ to obtain $a y=x-b$ or $y=\\frac{x}{a}-\\frac{b}{a}$.\n\nTherefore, $f^{-1}(x)=\\frac{x}{a}-\\frac{b}{a}$.\n\nNote that $a \\ne... | [
"$f(x)=x+22$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Expression | null | Open-ended | Algebra | Math | English |
2,460 | Determine all pairs $(a, b)$ of positive integers for which $a^{3}+2 a b=2013$. | [
"First, we factor the left side of the given equation to obtain $a\\left(a^{2}+2 b\\right)=2013$.\n\nNext, we factor the integer 2013 as $2013=3 \\times 671=3 \\times 11 \\times 61$. Note that each of 3,11 and 61 is prime, so we can factor 2013 no further. (We can find the factors of 3 and 11 using tests for divisi... | [
"$(1,1006),(3,331),(11,31)$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Number Theory | Math | English |
2,461 | 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)$. | [
"We successively manipulate the given equation to produce equivalent equations:\n\n$$\n\\begin{aligned}\n\\log _{2}\\left(2^{x-1}+3^{x+1}\\right) & =2 x-\\log _{2}\\left(3^{x}\\right) \\\\\n\\log _{2}\\left(2^{x-1}+3^{x+1}\\right)+\\log _{2}\\left(3^{x}\\right) & =2 x \\\\\n\\log _{2}\\left(\\left(2^{x-1}+3^{x+1}\\... | [
"$\\frac{\\log 2}{\\log 2-\\log 3}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,463 | 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 $P(64)$. | [
"We determine the multiplicative partitions of 64 by considering the number of parts in the various partitions. Note that 64 is a power of 2 so any divisor of 64 is also a power of 2 . In each partition, since the order of parts is not important, we list the parts in increasing order to make it easier to systematic... | [
"11"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
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