id int64 1.61k 3.1k | subfield stringclasses 4
values | context null | solution listlengths 1 5 | is_multiple_answer bool 2
classes | unit stringclasses 8
values | answer_type stringclasses 4
values | error stringclasses 1
value | problem stringlengths 49 4.42k | answer listlengths 1 1 |
|---|---|---|---|---|---|---|---|---|---|
2,082 | Algebra | null | [
"We will prove that the largest possible number $k$ of indices satisfying the given condition is one.\n\nFirstly we prove that $b_{2009}, r_{2009}, w_{2009}$ are always lengths of the sides of a triangle. Without loss of generality we may assume that $w_{2009} \\geq r_{2009} \\geq b_{2009}$. We show that the inequa... | false | null | Numerical | null | Find the largest possible integer $k$, such that the following statement is true:
Let 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort the lengths of the sides. We obtain
$... | [
"1"
] |
2,084 | Algebra | null | [
"The identity function $f(x)=x$ is the only solution of the problem.\n\nIf $f(x)=x$ for all positive integers $x$, the given three lengths are $x, y=f(y)$ and $z=$ $f(y+f(x)-1)=x+y-1$. Because of $x \\geq 1, y \\geq 1$ we have $z \\geq \\max \\{x, y\\}>|x-y|$ and $z<x+y$. From this it follows that a triangle with t... | false | null | Expression | null | Determine all functions $f$ from the set of positive integers into the set of positive integers such that for all $x$ and $y$ there exists a non degenerated triangle with sides of lengths
$$
x, \quad f(y) \text { and } f(y+f(x)-1) .
$$ | [
"$f(z)=z$"
] |
2,091 | Combinatorics | null | [
"Let $n \\geq 2$ be an integer and let $\\left\\{T_{1}, \\ldots, T_{N}\\right\\}$ be any set of triples of nonnegative integers satisfying the conditions (1) and (2). Since the $a$-coordinates are pairwise distinct we have\n\n$$\n\\sum_{i=1}^{N} a_{i} \\geq \\sum_{i=1}^{N}(i-1)=\\frac{N(N-1)}{2}\n$$\n\nAnalogously,... | false | null | Expression | null | For any integer $n \geq 2$, let $N(n)$ be the maximal number of triples $\left(a_{i}, b_{i}, c_{i}\right), i=1, \ldots, N(n)$, consisting of nonnegative integers $a_{i}, b_{i}$ and $c_{i}$ such that the following two conditions are satisfied:
(1) $a_{i}+b_{i}+c_{i}=n$ for all $i=1, \ldots, N(n)$,
(2) If $i \neq j$, t... | [
"$N(n)=\\left\\lfloor\\frac{2 n}{3}\\right\\rfloor+1$"
] |
2,095 | Combinatorics | null | [
"First we show that this number is an upper bound for the number of cells a limp rook can visit. To do this we color the cells with four colors $A, B, C$ and $D$ in the following way: for $(i, j) \\equiv(0,0) \\bmod 2$ use $A$, for $(i, j) \\equiv(0,1) \\bmod 2$ use $B$, for $(i, j) \\equiv(1,0) \\bmod 2$ use $C$ a... | false | null | Numerical | null | On a $999 \times 999$ board a limp rook can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A nonintersecting route of the limp rook c... | [
"996000"
] |
2,099 | Geometry | null | [
"Let $I$ be the incenter of triangle $A B C$, then $K$ lies on the line $C I$. Let $F$ be the point, where the incircle of triangle $A B C$ touches the side $A C$; then the segments $I F$ and $I D$ have the same length and are perpendicular to $A C$ and $B C$, respectively.\n\n<img_3148>\n\nFigure 1\n\n<img_3229>\n... | true | null | Numerical | null | Let $A B C$ be a triangle with $A B=A C$. The angle bisectors of $A$ and $B$ meet the sides $B C$ and $A C$ in $D$ and $E$, respectively. Let $K$ be the incenter of triangle $A D C$. Suppose that $\angle B E K=45^{\circ}$. Find all possible values of $\angle B A C$. | [
"$90^{\\circ}$,$60^{\\circ}$"
] |
2,111 | Number Theory | null | [
"Such a sequence exists for $n=1,2,3,4$ and no other $n$. Since the existence of such a sequence for some $n$ implies the existence of such a sequence for all smaller $n$, it suffices to prove that $n=5$ is not possible and $n=4$ is possible.\n\nAssume first that for $n=5$ there exists a sequence of positive intege... | true | null | Numerical | null | Find all positive integers $n$ such that there exists a sequence of positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying
$$
a_{k+1}=\frac{a_{k}^{2}+1}{a_{k-1}+1}-1
$$
for every $k$ with $2 \leq k \leq n-1$. | [
"1,2,3,4"
] |
2,124 | Combinatorics | null | [
"The maximum number of such boxes is 6 . One example is shown in the figure.\n\n<img_3912>\n\nNow we show that 6 is the maximum. Suppose that boxes $B_{1}, \\ldots, B_{n}$ satisfy the condition. Let the closed intervals $I_{k}$ and $J_{k}$ be the projections of $B_{k}$ onto the $x$ - and $y$-axis, for $1 \\leq k \\... | false | null | Numerical | null | In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary.
Find the largest $n$ for which there exist $n$ boxes $B_{1}, \ldots, B_{n}$ such t... | [
"6"
] |
2,126 | Combinatorics | null | [
"To begin, let us describe those points $B \\in S$ which are $k$-friends of the point $(0,0)$. By definition, $B=(u, v)$ satisfies this condition if and only if there is a point $C=(x, y) \\in S$ such that $\\frac{1}{2}|u y-v x|=k$. (This is a well-known formula expressing the area of triangle $A B C$ when $A$ is t... | false | null | Numerical | null | In the coordinate plane consider the set $S$ of all points with integer coordinates. For a positive integer $k$, two distinct points $A, B \in S$ will be called $k$-friends if there is a point $C \in S$ such that the area of the triangle $A B C$ is equal to $k$. A set $T \subset S$ will be called a $k$-clique if every ... | [
"180180"
] |
2,127 | Combinatorics | null | [
"A sequence of $k$ switches ending in the state as described in the problem statement (lamps $1, \\ldots, n$ on, lamps $n+1, \\ldots, 2 n$ off) will be called an admissible process. If, moreover, the process does not touch the lamps $n+1, \\ldots, 2 n$, it will be called restricted. So there are $N$ admissible proc... | false | null | Expression | null | Let $n$ and $k$ be fixed positive integers of the same parity, $k \geq n$. We are given $2 n$ lamps numbered 1 through $2 n$; each of them can be on or off. At the beginning all lamps are off. We consider sequences of $k$ steps. At each step one of the lamps is switched (from off to on or from on to off).
Let $N$ be t... | [
"$2^{k-n}$"
] |
2,147 | Algebra | null | [
"The only solution is the function $f(x)=x-1, x \\in \\mathbb{R}$.\n\nWe set $g(x)=f(x)+1$ and show that $g(x)=x$ for all real $x$. The conditions take the form\n\n$$\ng(1+x y)-g(x+y)=(g(x)-1)(g(y)-1) \\quad \\text { for all } x, y \\in \\mathbb{R} \\text { and } g(-1) \\neq 1\n\\tag{1}\n$$\n\nDenote $C=g(-1)-1 \\n... | false | null | Expression | null | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
$$
f(1+x y)-f(x+y)=f(x) f(y) \text { for all } x, y \in \mathbb{R}
$$
and $f(-1) \neq 0$. | [
"$f(x)=x-1$"
] |
2,151 | Combinatorics | null | [
"Consider $x$ such pairs in $\\{1,2, \\ldots, n\\}$. The sum $S$ of the $2 x$ numbers in them is at least $1+2+\\cdots+2 x$ since the pairs are disjoint. On the other hand $S \\leq n+(n-1)+\\cdots+(n-x+1)$ because the sums of the pairs are different and do not exceed $n$. This gives the inequality\n\n$$\n\\frac{2 x... | false | null | Expression | null | Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{1,2, \ldots, n\}$ such that the sums of the different pairs are different integers not exceeding $n$ ? | [
"$\\lfloor\\frac{2 n-1}{5}\\rfloor$"
] |
2,152 | Combinatorics | null | [
"We prove that in an $n \\times n$ square table there are at most $\\frac{4 n^{4}}{27}$ such triples.\n\nLet row $i$ and column $j$ contain $a_{i}$ and $b_{j}$ white cells respectively, and let $R$ be the set of red cells. For every red cell $(i, j)$ there are $a_{i} b_{j}$ admissible triples $\\left(C_{1}, C_{2}, ... | false | null | Expression | null | In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $\left(C_{1}, C_{2}, C_{3}\right)$ of cells, the first two in the same row and the last two in the same column, with $C_{1}$ and $C_{3}$ white and $C_{2}$ red. Find the maximum value $T$ can attain. | [
"$\\frac{4 \\cdot 999^{4}}{27}$"
] |
2,153 | Combinatorics | null | [
"We argue for a general $n \\geq 7$ instead of 2012 and prove that the required minimum $N$ is $2 n-2$. For $n=2012$ this gives $N_{\\min }=4022$.\n\na) If $N=2 n-2$ player $A$ can achieve her goal. Let her start the game with a regular distribution: $n-2$ boxes with 2 coins and 2 boxes with 1 coin. Call the boxes ... | false | null | Numerical | null | Players $A$ and $B$ play a game with $N \geq 2012$ coins and 2012 boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order $B, A, B, A, \ldots$ by the following rules:
- On every move of his $B$ passes... | [
"4022"
] |
2,165 | Number Theory | null | [
"First note that $x$ divides $2012 \\cdot 2=2^{3} \\cdot 503$. If $503 \\mid x$ then the right-hand side of the equation is divisible by $503^{3}$, and it follows that $503^{2} \\mid x y z+2$. This is false as $503 \\mid x$. Hence $x=2^{m}$ with $m \\in\\{0,1,2,3\\}$. If $m \\geq 2$ then $2^{6} \\mid 2012(x y z+2)$... | false | null | Tuple | null | Find all triples $(x, y, z)$ of positive integers such that $x \leq y \leq z$ and
$$
x^{3}\left(y^{3}+z^{3}\right)=2012(x y z+2) \text {. }
$$ | [
"$(2,251,252)$"
] |
2,171 | Algebra | null | [
"Denote the equation from the statement by (1). Let $x f(x)=A$ and $x^{2}=B$. The equation (1) is of the form\n\n$$\nf(A+y)=f(y)+B\n$$\n\nAlso, if we put $y \\rightarrow-A+y$, we have $f(A-A+y)=f(-A+y)+B$. Therefore\n\n$$\nf(-A+y)=f(y)-B\n$$\n\nWe can easily show that for any integer $n$ we even have\n\n$$\nf(n A+y... | true | null | Expression | null | Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that the equation
holds for all rational numbers $x$ and $y$.
$$
f(x f(x)+y)=f(y)+x^{2}
$$
Here, $\mathbb{Q}$ denotes the set of rational numbers. | [
"$f(x)=x,f(x)=-x$"
] |
2,174 | Geometry | null | [
"We will count minimal number of triangles that are not fat. Let $F$ set of fat triangles, and $\\mathrm{S}$ set of triangles that are not fat. If triangle $X Y Z \\in S$, we call $X$ and $Z$ good vertices if $O Y$ is located between $O X$ and $O Z$. For $A \\in P$ let $S_{A} \\subseteq S$ be set of triangles in $S... | false | null | Numerical | null | A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane, such that
(i) no three points in $P$ lie on a line and
(ii) no two points in $P$ lie on a line through the origin.
A triangle with vertices in $P$ is $f a t$, if $O$ is strictly inside the triangle. Find the maximum numb... | [
"$2021 \\cdot 505 \\cdot 337$"
] |
2,177 | Combinatorics | null | [
"The answer is $k=3$.\n\nFirst we show that there is such a function and coloring for $k=3$. Consider $f: \\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z}_{>0}$ given by $f(n)=n$ for all $n \\equiv 1$ or 2 modulo 3 , and $f(n)=2 n$ for $n \\equiv 0$ modulo 3 . Moreover, give a positive integer $n$ the $i$-th color if $n \... | false | null | Numerical | null | Find the smallest positive integer $k$ for which there exist a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ with the following two properties:
(i) For all positive integers $m, n$ of the same colour, $f(m+n)=f(m)+f(n)$.
(ii) Ther... | [
"$k=3$"
] |
2,185 | Combinatorics | null | [
"The required minimum is $6 m$ and is achieved by a diagonal string of $m$ $4 \\times 4$ blocks of the form below (bullets mark centres of blue cells):\n\n<img_3402>\n\nIn particular, this configuration shows that the required minimum does not exceed $6 m$.\n\nWe now show that any configuration of blue cells satisf... | false | null | Expression | null | Let $m$ be a positive integer. Consider a $4 m \times 4 m$ array of square unit cells. Two different cells are related to each other if they are in either the same row or in the same column. No cell is related to itself. Some cells are coloured blue, such that every cell is related to at least two blue cells. Determine... | [
"$6m$"
] |
2,193 | Number Theory | null | [
"Consider an integer $m>1$ for which the sequence defined in the problem statement contains only perfect squares. We shall first show that $m-1$ is a power of 3 .\n\nSuppose that $m-1$ is even. Then $a_{4}=5 m-1$ should be divisible by 4 and hence $m \\equiv 1(\\bmod 4)$. But then $a_{5}=5 m^{2}+3 m-1 \\equiv 3(\\b... | true | null | Numerical | null | Let $m>1$ be an integer. A sequence $a_{1}, a_{2}, a_{3}, \ldots$ is defined by $a_{1}=a_{2}=1$, $a_{3}=4$, and for all $n \geq 4$,
$$
a_{n}=m\left(a_{n-1}+a_{n-2}\right)-a_{n-3} .
$$
Determine all integers $m$ such that every term of the sequence is a square. | [
"1,2"
] |
2,198 | Combinatorics | null | [
"The maximal number of euros is $2^{n}-n-1$.\n\nTo begin with, we show that it is possible for the Jury to collect this number of euros. We argue by induction. Let us assume that the Jury can collect $M_{n}$ euros in a configuration with $n$ contestants. Then we show that the Jury can collect at least $2 M_{n}+n$ m... | false | null | Expression | null | The $n$ contestants of an EGMO are named $C_{1}, \ldots, C_{n}$. After the competition they queue in front of the restaurant according to the following rules.
- The Jury chooses the initial order of the contestants in the queue.
- Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$.
- If contestan... | [
"$2^{n}-n-1$"
] |
2,203 | Algebra | null | [
"First suppose that $a=0$. Then we have $b c=1$ and $c=b^{2} c=b$. So $b=c$, which implies $b^{2}=1$ and hence $b= \\pm 1$. This leads to the solutions $(a, b, c)=(0,1,1)$ and $(a, b, c)=(0,-1,-1)$. Similarly, $b=0$ gives the solutions $(a, b, c)=(1,0,1)$ and $(a, b, c)=(-1,0,-1)$, while $c=0$ gives $(a, b, c)=(1,1... | true | null | Tuple | null | Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b \text {. }
$$ | [
"$(0,1,1),(0,-1,-1),(1,0,1),(-1,0,-1),(1,1,0)$,$(-1,-1,0),\\left(\\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}\\right)$,$\\left(-\\frac{1}{\\sqrt{3}},-\\frac{1}{\\sqrt{3}},-\\frac{1}{\\sqrt{3}}\\right)$"
] |
2,204 | Combinatorics | null | [
"Let $M$ denote the maximum number of dominoes which satisfy the condition of the problem. We claim that $M=n(n+1) / 2$. The proof naturally splits into two parts: we first prove that $n(n+1) / 2$ dominoes can be placed on the board, and then show that $M \\leq n(n+1) / 2$ to complete the proof. To prove that $M \\... | false | null | Expression | null | Let $n$ be a positive integer. Dominoes are placed on a $2 n \times 2 n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way.
(A domino is a tile of size $2 \times 1$ or $1 \times 2$.... | [
"$\\frac{n(n+1)}{2}$"
] |
2,212 | Algebra | null | [
"$\\left(a_{k}+a_{k-1}\\right)\\left(a_{k}+a_{k+1}\\right)=a_{k-1}-a_{k+1}$ is equivalent to $\\left(a_{k}+a_{k-1}+1\\right)\\left(a_{k}+a_{k+1}-1\\right)=-1$. Let $b_{k}=a_{k}+a_{k+1}$. Thus we need $b_{0}, b_{1}, \\ldots$ the following way: $b_{0}=-\\frac{1}{n}$ and $\\left(b_{k-1}+1\\right)\\left(b_{k}-1\\right)... | false | null | Expression | null | Given a positive integer $n \geq 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_{0}, a_{1}, \ldots, a_{N}$ such that
(1) $a_{0}+a_{1}=-\frac{1}{n}$, and
(2) $\left(a_{k}+a_{k-1}\right)\left(a_{k}+a_{k+1}\right)=a_{k-1}-a_{k+1}$ for $1 \leq k \leq N-1$. | [
"$N=n$"
] |
2,216 | Combinatorics | null | [
"The solution naturally divides into three different parts: we first obtain some bounds on $m$. We then describe the structure of possible dissections, and finally, we deal with the few remaining cases.\n\nIn the first part of the solution, we get rid of the cases with $m \\leqslant 10$ or $m \\geqslant 14$. Let $\... | true | null | Numerical | null | Determine all integers $m$ for which the $m \times m$ square can be dissected into five rectangles, the side lengths of which are the integers $1,2,3, \ldots, 10$ in some order. | [
"11,13"
] |
2,230 | Combinatorics | null | [
"We claim the minimum value of $m$ is $2^{k-1}$.\n\nFirstly, we provide a set $\\mathcal{S}$ of size $2^{k-1}-1$ for which Lexi cannot fill her grid. Consider the set of all length- $k$ strings containing only $A \\mathrm{~s}$ and $B \\mathrm{~s}$ which end with a $B$, and remove the string consisting of $k$ $B \\... | false | null | Expression | null | Let $k$ be a positive integer. Lexi has a dictionary $\mathcal{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathcal{D}$ when read from t... | [
"$2^{k-1}$"
] |
2,234 | Algebra | null | [
"Let the number of terms in the sequence be $2 k+1$.\n\nWe label the terms $a_{1}, a_{2}, \\ldots, a_{2 k+1}$.\n\nThe middle term here is $a_{k+1}=302$.\n\nSince the difference between any two consecutive terms in this increasing sequence is $d$, $a_{m+1}-a_{m}=d$ for $m=1,2, \\ldots, 2 k$.\n\nWhen the last 4 terms... | false | null | Numerical | null | In an increasing sequence of numbers with an odd number of terms, the difference between any two consecutive terms is a constant $d$, and the middle term is 302 . When the last 4 terms are removed from the sequence, the middle term of the resulting sequence is 296. What is the value of $d$ ? | [
"3"
] |
2,235 | Algebra | null | [
"Let $n$ be the smallest integer in one of these sequences.\n\nSo we want to solve the equation $n^{2}+(n+1)^{2}+(n+2)^{2}=(n+3)^{2}+(n+4)^{2}$ (translating the given problem into an equation).\n\nThus $n^{2}+n^{2}+2 n+1+n^{2}+4 n+4=n^{2}+6 n+9+n^{2}+8 n+16$\n\n\n\n$$\n\\begin{array}{r}\nn^{2}-8 n-20=0 \\\\\n(n-10)... | true | null | Numerical | null | There are two increasing sequences of five consecutive integers, each of which have the property that the sum of the squares of the first three integers in the sequence equals the sum of the squares of the last two. Determine these two sequences. | [
"10,11,12,13,14,-2,-1,0,1,2"
] |
2,236 | Algebra | null | [
"Since $t>0, \\pi t-\\frac{\\pi}{2}>-\\frac{\\pi}{2}$. So $\\sin \\left(\\pi t-\\frac{\\pi}{2}\\right)$ first attains its minimum value when\n\n$$\n\\begin{aligned}\n\\pi t-\\frac{\\pi}{2} & =\\frac{3 \\pi}{2} \\\\\nt & =2 .\n\\end{aligned}\n$$",
"Rewriting $f(t)$ as, $f(t)=\\sin \\left[\\pi\\left(t-\\frac{1}{2}\... | false | null | Numerical | null | If $f(t)=\sin \left(\pi t-\frac{\pi}{2}\right)$, what is the smallest positive value of $t$ at which $f(t)$ attains its minimum value? | [
"2"
] |
2,238 | Algebra | null | [
"Since $x^{2} \\geq 0$ for all $x, x^{2}+5>0$. Since $\\left(x^{2}-3\\right)\\left(x^{2}+5\\right)<0, x^{2}-3<0$, so $x^{2}<3$ or $-\\sqrt{3}<x<\\sqrt{3}$. Thus $x=-1,0,1$."
] | true | null | Numerical | null | Determine all integer values of $x$ such that $\left(x^{2}-3\right)\left(x^{2}+5\right)<0$. | [
"-1,0,1"
] |
2,239 | Algebra | null | [
"Let $n$ be the number of children.\n\nAt the present, $P=6 C$, where $P$ and $C$ are as given. (1)\n\nTwo years ago, the sum of the ages of the husband and wife was $P-4$, since they were each two years younger.\n\nSimilarly, the sum of the ages of the children was $C-n(2)$ ( $n$ is the number of children).\n\nSo... | false | null | Numerical | null | At present, the sum of the ages of a husband and wife, $P$, is six times the sum of the ages of their children, $C$. Two years ago, the sum of the ages of the husband and wife was ten times the sum of the ages of the same children. Six years from now, it will be three times the sum of the ages of the same children. Det... | [
"3"
] |
2,241 | Algebra | null | [
"$$\n\\begin{aligned}\n\\log _{2}\\left(\\log _{2}(2 x-2)\\right) & =2 \\\\\n\\log _{2}(2 x-2) & =2^{2} \\\\\n2 x-2 & =2^{\\left(2^{2}\\right)} \\\\\n2 x-2 & =2^{4} \\\\\n2 x-2 & =16 \\\\\n2 x & =18 \\\\\nx & =9\n\\end{aligned}\n$$"
] | false | null | Numerical | null | What is the value of $x$ such that $\log _{2}\left(\log _{2}(2 x-2)\right)=2$ ? | [
"9"
] |
2,242 | Algebra | null | [
"From the given condition,\n\n$$\n\\begin{aligned}\n\\frac{f(3)}{f(6)}=\\frac{2^{3 k}+9}{2^{6 k}+9} & =\\frac{1}{3} \\\\\n3\\left(2^{3 k}+9\\right) & =2^{6 k}+9 \\\\\n0 & =2^{6 k}-3\\left(2^{3 k}\\right)-18 .\n\\end{aligned}\n$$\n\nWe treat this as a quadratic equation in the variable $x=2^{3 k}$, so\n\n$$\n\\begin... | false | null | Numerical | null | Let $f(x)=2^{k x}+9$, where $k$ is a real number. If $f(3): f(6)=1: 3$, determine the value of $f(9)-f(3)$. | [
"210"
] |
2,243 | Algebra | null | [
"Since each of these two graphs is symmetric about the $y$-axis (i.e. both are even functions), then we only need to find $k$ so that there are no points of intersection with $x \\geq 0$.\n\nSo let $x \\geq 0$ and consider the intersection between $y=2 x+k$ and $y=x^{2}-4$.\n\nEquating, we have, $2 x+k=x^{2}-4$.\n\... | false | null | Interval | null | Determine, with justification, all values of $k$ for which $y=x^{2}-4$ and $y=2|x|+k$ do not intersect. | [
"(-\\infty,-5)"
] |
2,247 | Algebra | null | [
"Since we want to make $15-\\frac{y}{x}$ as large as possible, then we want to subtract as little as possible from 15.\n\nIn other words, we want to make $\\frac{y}{x}$ as small as possible.\n\nTo make a fraction with positive numerator and denominator as small as possible, we make the numerator as small as possibl... | false | null | Numerical | null | If $2 \leq x \leq 5$ and $10 \leq y \leq 20$, what is the maximum value of $15-\frac{y}{x}$ ? | [
"13"
] |
2,248 | Algebra | null | [
"First, we add the two given equations to obtain\n\n$$\n(f(x)+g(x))+(f(x)-g(x))=(3 x+5)+(5 x+7)\n$$\n\nor $2 f(x)=8 x+12$ which gives $f(x)=4 x+6$.\n\nSince $f(x)+g(x)=3 x+5$, then $g(x)=3 x+5-f(x)=3 x+5-(4 x+6)=-x-1$.\n\n(We could also find $g(x)$ by subtracting the two given equations or by using the second of th... | false | null | Numerical | null | The functions $f$ and $g$ satisfy
$$
\begin{aligned}
& f(x)+g(x)=3 x+5 \\
& f(x)-g(x)=5 x+7
\end{aligned}
$$
for all values of $x$. Determine the value of $2 f(2) g(2)$. | [
"-84"
] |
2,249 | Combinatorics | null | [
"We consider choosing the three numbers all at once.\n\nWe list the possible sets of three numbers that can be chosen:\n\n$$\n\\{1,2,3\\}\\{1,2,4\\}\\{1,2,5\\} \\quad\\{1,3,4\\} \\quad\\{1,3,5\\} \\quad\\{1,4,5\\} \\quad\\{2,3,4\\} \\quad\\{2,3,5\\} \\quad\\{2,4,5\\} \\quad\\{3,4,5\\}\n$$\n\nWe have listed each in ... | false | null | Numerical | null | Three different numbers are chosen at random from the set $\{1,2,3,4,5\}$.
The numbers are arranged in increasing order.
What is the probability that the resulting sequence is an arithmetic sequence?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding... | [
"$\\frac{2}{5}$"
] |
2,251 | Combinatorics | null | [
"Let $n$ be the original number and $N$ be the number when the digits are reversed. Since we are looking for the largest value of $n$, we assume that $n>0$.\n\nSince we want $N$ to be $75 \\%$ larger than $n$, then $N$ should be $175 \\%$ of $n$, or $N=\\frac{7}{4} n$.\n\nSuppose that the tens digit of $n$ is $a$ a... | false | null | Numerical | null | What is the largest two-digit number that becomes $75 \%$ greater when its digits are reversed? | [
"48"
] |
2,253 | Geometry | null | [
"Suppose that the distance from point $A$ to point $B$ is $d \\mathrm{~km}$.\n\nSuppose also that $r_{c}$ is the speed at which Serge travels while not paddling (i.e. being carried by just the current), that $r_{p}$ is the speed at which Serge travels with no current (i.e. just from his paddling), and $r_{p+c}$ his... | false | minute | Numerical | null | Serge likes to paddle his raft down the Speed River from point $A$ to point $B$. The speed of the current in the river is always the same. When Serge paddles, he always paddles at the same constant speed. On days when he paddles with the current, it takes him 18 minutes to get from $A$ to $B$. When he does not paddle, ... | [
"45"
] |
2,254 | Geometry | null | [
"First, we note that $a \\neq 0$. (If $a=0$, then the \"parabola\" $y=a(x-2)(x-6)$ is actually the horizontal line $y=0$ which intersects the square all along $O R$.)\n\nSecond, we note that, regardless of the value of $a \\neq 0$, the parabola has $x$-intercepts 2 and 6 , and so intersects the $x$-axis at $(2,0)$ ... | true | null | Numerical | null | Square $O P Q R$ has vertices $O(0,0), P(0,8), Q(8,8)$, and $R(8,0)$. The parabola with equation $y=a(x-2)(x-6)$ intersects the sides of the square $O P Q R$ at points $K, L, M$, and $N$. Determine all the values of $a$ for which the area of the trapezoid $K L M N$ is 36 . | [
"$\\frac{32}{9}$,$\\frac{1}{2}$"
] |
2,255 | Algebra | null | [
"Consider a population of 100 people, each of whom is 75 years old and who behave according to the probabilities given in the question.\n\nEach of the original 100 people has a $50 \\%$ chance of living at least another 10 years, so there will be $50 \\% \\times 100=50$ of these people alive at age 85 .\n\nEach of ... | false | null | Numerical | null | A 75 year old person has a $50 \%$ chance of living at least another 10 years.
A 75 year old person has a $20 \%$ chance of living at least another 15 years. An 80 year old person has a $25 \%$ chance of living at least another 10 years. What is the probability that an 80 year old person will live at least another 5 y... | [
"62.5%"
] |
2,256 | Algebra | null | [
"Using logarithm rules, the given equation is equivalent to $2^{2 \\log _{10} x}=3\\left(2 \\cdot 2^{\\log _{10} x}\\right)+16$ or $\\left(2^{\\log _{10} x}\\right)^{2}=6 \\cdot 2^{\\log _{10} x}+16$.\n\nSet $u=2^{\\log _{10} x}$. Then the equation becomes $u^{2}=6 u+16$ or $u^{2}-6 u-16=0$.\n\nFactoring, we obtain... | false | null | Numerical | null | Determine all values of $x$ for which $2^{\log _{10}\left(x^{2}\right)}=3\left(2^{1+\log _{10} x}\right)+16$. | [
"1000"
] |
2,257 | Algebra | null | [
"First, we determine the first entry in the 50th row.\n\nSince the first column is an arithmetic sequence with common difference 3, then the 50th entry in the first column (the first entry in the 50th row) is $4+49(3)=4+147=151$.\n\nSecond, we determine the common difference in the 50th row by determining the secon... | false | null | Numerical | null | The Sieve of Sundaram uses the following infinite table of positive integers:
| 4 | 7 | 10 | 13 | $\cdots$ |
| :---: | :---: | :---: | :---: | :---: |
| 7 | 12 | 17 | 22 | $\cdots$ |
| 10 | 17 | 24 | 31 | $\cdots$ |
| 13 | 22 | 31 | 40 | $\cdots$ |
| $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | |
The numbers in each ... | [
"4090"
] |
2,258 | Algebra | null | [
"First, we determine the first entry in the $R$ th row.\n\nSince the first column is an arithmetic sequence with common difference 3 , then the $R$ th entry in the first column (that is, the first entry in the $R$ th row) is $4+(R-1)(3)$ or $4+3 R-3=3 R+1$.\n\nSecond, we determine the common difference in the $R$ t... | false | null | Expression | null | The Sieve of Sundaram uses the following infinite table of positive integers:
| 4 | 7 | 10 | 13 | $\cdots$ |
| :---: | :---: | :---: | :---: | :---: |
| 7 | 12 | 17 | 22 | $\cdots$ |
| 10 | 17 | 24 | 31 | $\cdots$ |
| 13 | 22 | 31 | 40 | $\cdots$ |
| $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | |
The numbers in each ... | [
"$2RC+R+C$"
] |
2,260 | Number Theory | null | [
"If $n=2011$, then $8 n-7=16081$ and so $\\sqrt{8 n-7} \\approx 126.81$.\n\nThus, $\\frac{1+\\sqrt{8 n-7}}{2} \\approx \\frac{1+126.81}{2} \\approx 63.9$.\n\nTherefore, $g(2011)=2(2011)+\\left\\lfloor\\frac{1+\\sqrt{8(2011)-7}}{2}\\right\\rfloor=4022+\\lfloor 63.9\\rfloor=4022+63=4085$."
] | false | null | Numerical | null | Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$.
Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$.
Determ... | [
"4085"
] |
2,261 | Number Theory | null | [
"To determine a value of $n$ for which $f(n)=100$, we need to solve the equation\n\n$$\n2 n-\\left\\lfloor\\frac{1+\\sqrt{8 n-7}}{2}\\right\\rfloor=100\n$$\n\nWe first solve the equation\n\n$$\n2 x-\\frac{1+\\sqrt{8 x-7}}{2}=100 \\quad(* *)\n$$\n\nbecause the left sides of $(*)$ and $(* *)$ do not differ by much an... | false | null | Numerical | null | Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$.
Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$.
Determ... | [
"55"
] |
2,263 | Combinatorics | null | [
"The possible pairs of numbers on the tickets are (listed as ordered pairs): (1,2), (1,3), $(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)$, and $(5,6)$.\n\nThere are fifteen such pairs. (We treat the pair of tickets numbered 2 and 4 as being the same as the pair numbered 4 and 2.)\n\nThe p... | false | null | Numerical | null | Six tickets numbered 1 through 6 are placed in a box. Two tickets are randomly selected and removed together. What is the probability that the smaller of the two numbers on the tickets selected is less than or equal to 4 ? | [
"$\\frac{14}{15}$"
] |
2,265 | Geometry | null | [
"After 2 moves, the goat has travelled $1+2=3$ units.\n\nAfter 3 moves, the goat has travelled $1+2+3=6$ units.\n\nSimilarly, after $n$ moves, the goat has travelled a total of $1+2+3+\\cdots+n$ units.\n\nFor what value of $n$ is $1+2+3+\\cdots+n$ equal to 55 ?\n\nThe fastest way to determine the value of $n$ is by... | false | null | Tuple | null | A goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it travels 1 unit up to $(0,1)$. On move 2 , it travels 2 units right to $(2,1)$. On move 3 , it travels 3 units down to $(2,-2)$. On move 4 , it travels 4 units to $(-2,-2)$. It continues in this fashion, so that on move $n$, it turns $90^{\... | [
"(6,5)"
] |
2,266 | Algebra | null | [
"Since the sequence $4,4 r, 4 r^{2}$ is also arithmetic, then the difference between $4 r^{2}$ and $4 r$ equals the difference between $4 r$ and 4 , or\n\n$$\n\\begin{aligned}\n4 r^{2}-4 r & =4 r-4 \\\\\n4 r^{2}-8 r+4 & =0 \\\\\nr^{2}-2 r+1 & =0 \\\\\n(r-1)^{2} & =0\n\\end{aligned}\n$$\n\nTherefore, the only value ... | false | null | Numerical | null | Determine all possible values of $r$ such that the three term geometric sequence 4, $4 r, 4 r^{2}$ is also an arithmetic sequence.
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9, 11 is an arithmetic sequence.) | [
"1"
] |
2,268 | Algebra | null | [
"We rewrite by completing the square as $f(x)=\\sin ^{2} x-2 \\sin x+2=(\\sin x-1)^{2}+1$.\n\nTherefore, since $(\\sin x-1)^{2} \\geq 0$, then $f(x) \\geq 1$, and in fact $f(x)=1$ when $\\sin x=1$ (which occurs for instance when $x=90^{\\circ}$ ).\n\nThus, the minimum value of $f(x)$ is 1 .\n\nTo maximize $f(x)$, w... | true | null | Numerical | null | If $f(x)=\sin ^{2} x-2 \sin x+2$, what are the minimum and maximum values of $f(x)$ ? | [
"5,1"
] |
2,275 | Algebra | null | [
"Using a calculator, we see that\n\n$$\n\\left(10^{3}+1\\right)^{2}=1001^{2}=1002001\n$$\n\nThe sum of the digits of this integer is $1+2+1$ which equals 4 .\n\nTo determine this integer without using a calculator, we can let $x=10^{3}$.\n\nThen\n\n$$\n\\begin{aligned}\n\\left(10^{3}+1\\right)^{2} & =(x+1)^{2} \\\\... | false | null | Numerical | null | What is the sum of the digits of the integer equal to $\left(10^{3}+1\right)^{2}$ ? | [
"1002001"
] |
2,276 | Algebra | null | [
"Before the price increase, the total cost of 2 small cookies and 1 large cookie is $2 \\cdot \\$ 1.50+\\$ 2.00=\\$ 5.00$.\n\n$10 \\%$ of $\\$ 1.50$ is $0.1 \\cdot \\$ 1.50=\\$ 0.15$. After the price increase, 1 small cookie costs $\\$ 1.50+\\$ 0.15=\\$ 1.65$.\n\n$5 \\%$ of $\\$ 2.00$ is $0.05 \\cdot \\$ 2.00=\\$ 0... | false | null | Numerical | null | A bakery sells small and large cookies. Before a price increase, the price of each small cookie is $\$ 1.50$ and the price of each large cookie is $\$ 2.00$. The price of each small cookie is increased by $10 \%$ and the price of each large cookie is increased by $5 \%$. What is the percentage increase in the total cos... | [
"$8 \\%$"
] |
2,277 | Algebra | null | [
"Suppose that Rayna's age is $x$ years.\n\nSince Qing is twice as old as Rayna, Qing's age is $2 x$ years.\n\nSince Qing is 4 years younger than Paolo, Paolo's age is $2 x+4$ years.\n\nSince the average of their ages is 13 years, we obtain\n\n$$\n\\frac{x+(2 x)+(2 x+4)}{3}=13\n$$\n\nThis gives $5 x+4=39$ and so $5 ... | true | null | Numerical | null | Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing and Rayna is 13. Determine their ages. | [
"7,14,18"
] |
2,280 | Geometry | null | [
"Since $V(1,18)$ is on the parabola, then $18=-2\\left(1^{2}\\right)+4(1)+c$ and so $c=18+2-4=16$.\n\nThus, the equation of the parabola is $y=-2 x^{2}+4 x+16$.\n\nThe $y$-intercept occurs when $x=0$, and so $y=16$. Thus, $D$ has coordinates $(0,16)$.\n\nThe $x$-intercepts occur when $y=0$. Here,\n\n$$\n\\begin{arr... | false | null | Numerical | null | The parabola with equation $y=-2 x^{2}+4 x+c$ has vertex $V(1,18)$. The parabola intersects the $y$-axis at $D$ and the $x$-axis at $E$ and $F$. Determine the area of $\triangle D E F$. | [
"48"
] |
2,281 | Algebra | null | [
"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... | false | null | Numerical | null | If $3\left(8^{x}\right)+5\left(8^{x}\right)=2^{61}$, what is the value of the real number $x$ ? | [
"$\\frac{58}{3}$"
] |
2,282 | Number Theory | null | [
"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... | true | null | Numerical | null | For some real numbers $m$ and $n$, the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order. Determine all possible values of $m$. | [
"1,-1,7,-7"
] |
2,283 | Combinatorics | null | [
"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... | false | null | Tuple | null | 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,-... | [
"(-7,-1)"
] |
2,286 | Algebra | null | [
"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... | false | null | Numerical | null | 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 fro... | [
"20"
] |
2,287 | Algebra | null | [
"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$$\na+a r+a r^{2}+a r^{3}+a r^{4}+... | true | null | Expression | null | Suppose that $a$ and $r$ are real numbers. A geometric sequence with first term $a$ and common ratio $r$ has 4 terms. The sum of this geometric sequence is $6+6 \sqrt{2}$. A second geometric sequence has the same first term $a$ and the same common ratio $r$, but has 8 terms. The sum of this second geometric sequence is... | [
"$a=2$, $a=-6-4 \\sqrt{2}$"
] |
2,288 | Combinatorics | null | [
"Victor stops when there are either 2 green balls on the table or 2 red balls on the table. If the first 2 balls that Victor removes are the same colour, Victor will stop.\n\nIf the first 2 balls that Victor removes are different colours, Victor does not yet stop, but when he removes a third ball, its colour must m... | false | null | Numerical | null | A bag contains 3 green balls, 4 red balls, and no other balls. Victor removes balls randomly from the bag, one at a time, and places them on a table. Each ball in the bag is equally likely to be chosen each time that he removes a ball. He stops removing balls when there are two balls of the same colour on the table. Wh... | [
"$\\frac{4}{7}$"
] |
2,289 | Algebra | null | [
"Using the definition of $f$, the following equations are equivalent:\n\n$$\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 ... | true | null | Numerical | null | Suppose that $f(a)=2 a^{2}-3 a+1$ for all real numbers $a$ and $g(b)=\log _{\frac{1}{2}} b$ for all $b>0$. Determine all $\theta$ with $0 \leq \theta \leq 2 \pi$ for which $f(g(\sin \theta))=0$. | [
"$\\frac{1}{6} \\pi, \\frac{5}{6} \\pi, \\frac{1}{4} \\pi, \\frac{3}{4} \\pi$"
] |
2,292 | Number Theory | null | [
"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 ... | true | null | Tuple | null | Suppose that $a=5$ and $b=4$. Determine all pairs of integers $(K, L)$ for which $K^{2}+3 L^{2}=a^{2}+b^{2}-a b$. | [
"$(3,2),(-3,2),(3,-2),(-3,-2)$"
] |
2,298 | Algebra | null | [
"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 ... | false | null | Interval | null | Determine all values of $x$ for which $0<\frac{x^{2}-11}{x+1}<7$. | [
"$(-\\sqrt{11},-2)\\cup (\\sqrt{11},9)$"
] |
2,300 | Algebra | null | [
"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... | false | null | Numerical | null | 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.
(A... | [
"34"
] |
2,301 | Number Theory | null | [
"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... | true | null | Numerical | null | For some positive integers $k$, the parabola with equation $y=\frac{x^{2}}{k}-5$ intersects the circle with equation $x^{2}+y^{2}=25$ at exactly three distinct points $A, B$ and $C$. Determine all such positive integers $k$ for which the area of $\triangle A B C$ is an integer. | [
"1,2,5,8,9"
] |
2,303 | Algebra | null | [
"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... | true | null | Tuple | null | Consider the following system of equations in which all logarithms have base 10:
$$
\begin{aligned}
(\log x)(\log y)-3 \log 5 y-\log 8 x & =a \\
(\log y)(\log z)-4 \log 5 y-\log 16 z & =b \\
(\log z)(\log x)-4 \log 8 x-3 \log 625 z & =c
\end{aligned}
$$
If $a=-4, b=4$, and $c=-18$, solve the system of equations. | [
"$(10^{4}, 10^{3}, 10^{10}),(10^{2}, 10^{-1}, 10^{-2})$"
] |
2,312 | Combinatorics | null | [
"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$$\n(1,5),(2,5),(3,5),(4,5),(5,5)... | false | null | Numerical | null | Two fair dice, each having six faces numbered 1 to 6 , are thrown. What is the probability that the product of the two numbers on the top faces is divisible by 5 ? | [
"$\\frac{11}{36}$"
] |
2,313 | Algebra | null | [
"First, we compute an expression for the composition of the two given functions:\n\n$$\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$$\n\nBut we already know that $f(g(x))=9... | true | null | Tuple | null | If $f(x)=x^{2}-x+2, g(x)=a x+b$, and $f(g(x))=9 x^{2}-3 x+2$, determine all possible ordered pairs $(a, b)$ which satisfy this relationship. | [
"$(3,0),(-3,1)$"
] |
2,315 | Algebra | null | [
"When the dimensions were increased by $n \\%$ from 10 by 15 , the new dimensions were $10\\left(1+\\frac{n}{100}\\right)$ by $15\\left(1+\\frac{n}{100}\\right)$.\n\nWhen the resolution was decreased by $n$ percent, the new resolution was $75\\left(1-\\frac{n}{100}\\right)$.\n\n(Note that $n$ cannot be larger than ... | false | null | Numerical | null | Digital images consist of a very large number of equally spaced dots called pixels The resolution of an image is the number of pixels/cm in each of the horizontal and vertical directions.
Thus, an image with dimensions $10 \mathrm{~cm}$ by $15 \mathrm{~cm}$ and a resolution of 75 pixels/cm has a total of $(10 \times 7... | [
"60"
] |
2,317 | Algebra | null | [
"Consider the right side of the given equation:\n\n$$\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... | true | null | Numerical | null | If $T=x^{2}+\frac{1}{x^{2}}$, determine the values of $b$ and $c$ so that $x^{6}+\frac{1}{x^{6}}=T^{3}+b T+c$ for all non-zero real numbers $x$. | [
"-3,0"
] |
2,318 | Combinatorics | null | [
"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... | true | null | Tuple | null | 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 examp... | [
"(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)"
] |
2,319 | Combinatorics | null | [
"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... | false | null | Tuple | null | 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 examp... | [
"(7,5,1,1,9,3,5,7,3,8,6,4,2,9,2,4,6,8)"
] |
2,321 | Number Theory | null | [
"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... | false | null | Numerical | null | The three-digit positive integer $m$ is odd and has three distinct digits. If the hundreds digit of $m$ equals the product of the tens digit and ones (units) digit of $m$, what is $m$ ? | [
"623"
] |
2,322 | Combinatorics | null | [
"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... | false | null | Numerical | null | Eleanor has 100 marbles, each of which is black or gold. The ratio of the number of black marbles to the number of gold marbles is $1: 4$. How many gold marbles should she add to change this ratio to $1: 6$ ? | [
"40"
] |
2,323 | Number Theory | null | [
"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... | true | null | Numerical | null | Suppose that $n$ is a positive integer and that the value of $\frac{n^{2}+n+15}{n}$ is an integer. Determine all possible values of $n$. | [
"1, 3, 5, 15"
] |
2,325 | Combinatorics | null | [
"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 | ... | false | null | Numerical | null | 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... | [
"340"
] |
2,326 | Number Theory | null | [
"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 $... | true | null | Numerical | null | Determine all integers $k$, with $k \neq 0$, for which the parabola with equation $y=k x^{2}+6 x+k$ has two distinct $x$-intercepts. | [
"-2,-1,1,2"
] |
2,327 | Number Theory | null | [
"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... | false | null | Numerical | null | The positive integers $a$ and $b$ have no common divisor larger than 1 . If the difference between $b$ and $a$ is 15 and $\frac{5}{9}<\frac{a}{b}<\frac{4}{7}$, what is the value of $\frac{a}{b}$ ? | [
"$\\frac{19}{34}$"
] |
2,328 | Algebra | null | [
"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... | false | null | Numerical | null | 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 a... | [
"$-\\frac{30}{17}$"
] |
2,329 | Algebra | null | [
"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,... | false | null | Numerical | null | For each positive real number $x$, define $f(x)$ to be the number of prime numbers $p$ that satisfy $x \leq p \leq x+10$. What is the value of $f(f(20))$ ? | [
"5"
] |
2,330 | Algebra | null | [
"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... | true | null | Tuple | null | Determine all triples $(x, y, z)$ of real numbers that satisfy the following system of equations:
$$
\begin{aligned}
(x-1)(y-2) & =0 \\
(x-3)(z+2) & =0 \\
x+y z & =9
\end{aligned}
$$ | [
"(1,-4,-2),(3,2,3),(13,2,-2)"
] |
2,331 | Algebra | null | [
"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... | false | null | Numerical | null | Suppose that the function $g$ satisfies $g(x)=2 x-4$ for all real numbers $x$ and that $g^{-1}$ is the inverse function of $g$. Suppose that the function $f$ satisfies $g\left(f\left(g^{-1}(x)\right)\right)=2 x^{2}+16 x+26$ for all real numbers $x$. What is the value of $f(\pi)$ ? | [
"$4 \\pi^{2}-1$"
] |
2,332 | Algebra | null | [
"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}(\... | true | null | Tuple | null | Determine all pairs of angles $(x, y)$ with $0^{\circ} \leq x<180^{\circ}$ and $0^{\circ} \leq y<180^{\circ}$ that satisfy the following system of equations:
$$
\begin{aligned}
\log _{2}(\sin x \cos y) & =-\frac{3}{2} \\
\log _{2}\left(\frac{\sin x}{\cos y}\right) & =\frac{1}{2}
\end{aligned}
$$ | [
"$(45^{\\circ}, 60^{\\circ}),(135^{\\circ}, 60^{\\circ})$"
] |
2,333 | Combinatorics | null | [
"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... | false | null | Expression | null | 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 C... | [
"$\\frac{1-p^{2}}{3}$"
] |
2,334 | Geometry | null | [
"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\... | false | km | Numerical | null | 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 $\frac{1}{3} \mathrm{~km} / \mathrm{s}$. Microphon... | [
"$\\frac{41}{12}$"
] |
2,336 | Number Theory | null | [
"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\... | true | null | Tuple | null | 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'... | [
"(1,7,103, 105), (3, 5, 101, 107)"
] |
2,338 | Algebra | null | [
"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$$\n\\begin{aligned}\n\\frac{x^{2}+x+4}{2 x+1} & =\\frac{4}{x} \\\\\nx\\left(x^{2}+x+4\\right) & =4(2 x+1) \\\\\nx^{3}+x^{2}+4 x & =8 x+4 \\\\\nx^{3}+x^{2}-4 x-4 & =0 \\\\\nx^... | true | null | Numerical | null | Determine all values of $x$ for which $\frac{x^{2}+x+4}{2 x+1}=\frac{4}{x}$. | [
"$-1$,$2$,$-2$"
] |
2,339 | Number Theory | null | [
"Since $900=30^{2}$ and $30=2 \\times 3 \\times 5$, then $900=2^{2} 3^{2} 5^{2}$.\n\nThe positive divisors of 900 are those integers of the form $d=2^{a} 3^{b} 5^{c}$, where each of $a, b, c$ is 0,1 or 2 .\n\nFor $d$ to be a perfect square, the exponent on each prime factor in the prime factorization of $d$ must be... | false | null | Numerical | null | Determine the number of positive divisors of 900, including 1 and 900, that are perfect squares. (A positive divisor of 900 is a positive integer that divides exactly into 900.) | [
"8"
] |
2,340 | Geometry | null | [
"In isosceles triangle $A B C, \\angle A B C=\\angle A C B$, so the sides opposite these angles $(A C$ and $A B$, respectively) are equal in length.\n\nSince the vertices of the triangle are $A(k, 3), B(3,1)$ and $C(6, k)$, then we obtain\n\n$$\n\\begin{aligned}\nA C & =A B \\\\\n\\sqrt{(k-6)^{2}+(3-k)^{2}} & =\\sq... | true | null | Numerical | null | Points $A(k, 3), B(3,1)$ and $C(6, k)$ form an isosceles triangle. If $\angle A B C=\angle A C B$, determine all possible values of $k$. | [
"$8$,$4$"
] |
2,341 | Combinatorics | null | [
"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}... | false | null | Numerical | null | 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 bottl... | [
"17.5%"
] |
2,342 | Algebra | null | [
"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) \\\... | false | null | Numerical | null | Suppose that $x$ and $y$ are real numbers with $3 x+4 y=10$. Determine the minimum possible value of $x^{2}+16 y^{2}$. | [
"10"
] |
2,343 | Combinatorics | null | [
"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... | false | null | Numerical | null | A bag contains 40 balls, each of which is black or gold. Feridun reaches into the bag and randomly removes two balls. Each ball in the bag is equally likely to be removed. If the probability that two gold balls are removed is $\frac{5}{12}$, how many of the 40 balls are gold? | [
"26"
] |
2,344 | Algebra | null | [
"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... | false | null | Numerical | null | 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... | [
"20"
] |
2,345 | Algebra | null | [
"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... | false | null | Numerical | null | If $\frac{(x-2013)(y-2014)}{(x-2013)^{2}+(y-2014)^{2}}=-\frac{1}{2}$, what is the value of $x+y$ ? | [
"4027"
] |
2,346 | Algebra | null | [
"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(\\... | true | null | Numerical | null | Determine all real numbers $x$ for which
$$
\left(\log _{10} x\right)^{\log _{10}\left(\log _{10} x\right)}=10000
$$ | [
"$10^{100}$,$10^{1 / 100}$"
] |
2,349 | Algebra | null | [
"Let $S=\\sin ^{6} 1^{\\circ}+\\sin ^{6} 2^{\\circ}+\\sin ^{6} 3^{\\circ}+\\cdots+\\sin ^{6} 87^{\\circ}+\\sin ^{6} 88^{\\circ}+\\sin ^{6} 89^{\\circ}$.\n\nSince $\\sin \\theta=\\cos \\left(90^{\\circ}-\\theta\\right)$, then $\\sin ^{6} \\theta=\\cos ^{6}\\left(90^{\\circ}-\\theta\\right)$, and so\n\n$$\n\\begin{al... | true | null | Numerical | null | 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^{\ci... | [
"$221,$8$"
] |
2,350 | Number Theory | null | [
"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$$\n5 \\quad 4,1 \\quad 3,2 \\quad 3,1,1 \\quad 2,2,1 \\quad 2,1,1,1 \\quad 1,1,1,... | false | null | Numerical | null | Let $f(n)$ be the number of positive integers that have exactly $n$ digits and whose digits have a sum of 5. Determine, with proof, how many of the 2014 integers $f(1), f(2), \ldots, f(2014)$ have a units digit of 1 . | [
"202"
] |
2,351 | Algebra | null | [
"$$\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$$"
] | false | null | Numerical | null | If $\log _{10} x=3+\log _{10} y$, what is the value of $\frac{x}{y}$ ? | [
"1000"
] |
2,352 | Algebra | null | [
"$\\left(x+\\frac{1}{x}\\right)^{2}=\\left(\\frac{13}{6}\\right)^{2}$; 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)... | false | null | Numerical | null | If $x+\frac{1}{x}=\frac{13}{6}$, determine all values of $x^{2}+\frac{1}{x^{2}}$. | [
"$\\frac{97}{36}$"
] |
2,355 | Combinatorics | null | [
"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... | false | null | Numerical | null | 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 wi... | [
"$\\frac{2}{9}$"
] |
2,356 | Algebra | null | [
"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 $... | true | null | Numerical | null | 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... | [
"16,3,1,1"
] |
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