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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1,854 | Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2 n \times 2 n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contains two marked cells. | [
"We first construct an example of marking $2 n$ cells satisfying the requirement. Label the rows and columns $1,2, \\ldots, 2 n$ and label the cell in the $i$-th row and the $j$-th column $(i, j)$.\n\nFor $i=1,2, \\ldots, n$, we mark the cells $(i, i)$ and $(i, i+1)$. We claim that the required partition exists and... | [
"$2 n$"
] | 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 |
1,866 | Define $P(n)=n^{2}+n+1$. For any positive integers $a$ and $b$, the set
$$
\{P(a), P(a+1), P(a+2), \ldots, P(a+b)\}
$$
is said to be fragrant if none of its elements is relatively prime to the product of the other elements. Determine the smallest size of a fragrant set. | [
"We have the following observations.\n\n(i) $(P(n), P(n+1))=1$ for any $n$.\n\nWe have $(P(n), P(n+1))=\\left(n^{2}+n+1, n^{2}+3 n+3\\right)=\\left(n^{2}+n+1,2 n+2\\right)$. Noting that $n^{2}+n+1$ is odd and $\\left(n^{2}+n+1, n+1\\right)=(1, n+1)=1$, the claim follows.\n\n(ii) $(P(n), P(n+2))=1$ for $n \\not \\eq... | [
"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 | Number Theory | Math | English |
1,869 | Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-m n$ is nonzero and divides $m f(m)+n f(n)$. | [
"It is given that\n\n$$\nf(m)+f(n)-m n \\mid m f(m)+n f(n) .\n\\tag{1}\n$$\n\nTaking $m=n=1$ in (1), we have $2 f(1)-1 \\mid 2 f(1)$. Then $2 f(1)-1 \\mid 2 f(1)-(2 f(1)-1)=1$ and hence $f(1)=1$.\n\nLet $p \\geqslant 7$ be a prime. Taking $m=p$ and $n=1$ in (1), we have $f(p)-p+1 \\mid p f(p)+1$ and hence\n\n$$\nf(... | [
"$f(n)=n^{2}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Expression | null | Open-ended | Number Theory | Math | English |
1,872 | Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
$$
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x
$$ | [
"First of all, assume that $a_{n}<N / 2$ satisfies the condition. Take $x=1+t$ for $t>0$, we should have\n\n$$\n\\frac{(1+t)^{2 N}+1}{2} \\leqslant\\left(1+t+a_{n} t^{2}\\right)^{N}\n$$\n\nExpanding the brackets we get\n\n$$\n\\left(1+t+a_{n} t^{2}\\right)^{N}-\\frac{(1+t)^{2 N}+1}{2}=\\left(N a_{n}-\\frac{N^{2}}{2... | [
"$\\frac{N}{2}$"
] | 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 |
1,873 | Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
$$
(x+y+z) P(x, y, z)+(x y+y z+z x) Q(x, y, z)+x y z R(x, y, z)
$$
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^{i} y^{j} z^{k} \in \mathcal{B}$ for all nonnegative integers $i, j, k$ satisfying $i+j+k \geqslant n$. | [
"We start by showing that $n \\leqslant 4$, i.e., any monomial $f=x^{i} y^{j} z^{k}$ with $i+j+k \\geqslant 4$ belongs to $\\mathcal{B}$. Assume that $i \\geqslant j \\geqslant k$, the other cases are analogous.\n\nLet $x+y+z=p, x y+y z+z x=q$ and $x y z=r$. Then\n\n$$\n0=(x-x)(x-y)(x-z)=x^{3}-p x^{2}+q x-r\n$$\n\n... | [
"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 | Algebra | Math | English |
1,874 | Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d)=a c+b d$. Find the smallest possible value of
$$
S=\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}
$$ | [
"To show that $S \\geqslant 8$, apply the AM-GM inequality twice as follows:\n\n$$\n\\left(\\frac{a}{b}+\\frac{c}{d}\\right)+\\left(\\frac{b}{c}+\\frac{d}{a}\\right) \\geqslant 2 \\sqrt{\\frac{a c}{b d}}+2 \\sqrt{\\frac{b d}{a c}}=\\frac{2(a c+b d)}{\\sqrt{a b c d}}=\\frac{2(a+c)(b+d)}{\\sqrt{a b c d}} \\geqslant 2... | [
"8"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
1,879 | Let $\mathbb{R}^{+}$be the set of positive real numbers. Determine all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that, for all positive real numbers $x$ and $y$,
$$
f(x+f(x y))+y=f(x) f(y)+1
\tag{*}
$$ | [
"A straightforward check shows that $f(x)=x+1$ satisfies (*). We divide the proof of the converse statement into a sequence of steps.\n\nStep 1: $f$ is injective.\n\nPut $x=1$ in (*) and rearrange the terms to get\n\n$$\ny=f(1) f(y)+1-f(1+f(y))\n$$\n\nTherefore, if $f\\left(y_{1}\\right)=f\\left(y_{2}\\right)$, the... | [
"$f(x)=x+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 | Expression | null | Open-ended | Algebra | Math | English |
1,882 | Let $n$ be an integer with $n \geqslant 2$. On a slope of a mountain, $n^{2}$ checkpoints are marked, numbered from 1 to $n^{2}$ from the bottom to the top. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars numbered from 1 to $k$; each cable car provides a transfer from some checkpoint to a higher one. For each company, and for any $i$ and $j$ with $1 \leqslant i<j \leqslant k$, the starting point of car $j$ is higher than the starting point of car $i$; similarly, the finishing point of car $j$ is higher than the finishing point of car $i$. Say that two checkpoints are linked by some company if one can start from the lower checkpoint and reach the higher one by using one or more cars of that company (no movement on foot is allowed).
Determine the smallest $k$ for which one can guarantee that there are two checkpoints that are linked by each of the two companies. | [
"We start with showing that for any $k \\leqslant n^{2}-n$ there may be no pair of checkpoints linked by both companies. Clearly, it suffices to provide such an example for $k=n^{2}-n$.\n\nLet company $A$ connect the pairs of checkpoints of the form $(i, i+1)$, where $n \\nmid i$. Then all pairs of checkpoints $(i,... | [
"$n^{2}-n+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 | Expression | null | Open-ended | Combinatorics | Math | English |
1,883 | The Fibonacci numbers $F_{0}, F_{1}, F_{2}, \ldots$ are defined inductively by $F_{0}=0, F_{1}=1$, and $F_{n+1}=F_{n}+F_{n-1}$ for $n \geqslant 1$. Given an integer $n \geqslant 2$, determine the smallest size of a set $S$ of integers such that for every $k=2,3, \ldots, n$ there exist some $x, y \in S$ such that $x-y=F_{k}$. | [
"First we show that if a set $S \\subset \\mathbb{Z}$ satisfies the conditions then $|S| \\geqslant \\frac{n}{2}+1$.\n\nLet $d=\\lceil n / 2\\rceil$, so $n \\leqslant 2 d \\leqslant n+1$. In order to prove that $|S| \\geqslant d+1$, construct a graph as follows. Let the vertices of the graph be the elements of $S$.... | [
"$\\lceil n / 2\\rceil+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 | Expression | null | Open-ended | Combinatorics | Math | English |
1,887 | Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds:
(1) one of the numbers on the blackboard is larger than the sum of all other numbers;
(2) there are only zeros on the blackboard.
Player $B$ must then give as many cookies to player $A$ as there are numbers on the blackboard. Player $A$ wants to get as many cookies as possible, whereas player $B$ wants to give as few as possible. Determine the number of cookies that $A$ receives if both players play optimally. | [
"For a positive integer $n$, we denote by $S_{2}(n)$ the sum of digits in its binary representation. We prove that, in fact, if a board initially contains an even number $n>1$ of ones, then $A$ can guarantee to obtain $S_{2}(n)$, but not more, cookies. The binary representation of 2020 is $2020=\\overline{111111001... | [
"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 |
1,914 | Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are $k$ coins showing heads and $k>0$, then he flips the $k^{\text {th }}$ coin over; otherwise he stops the process. (For example, the process starting with THT would be THT $\rightarrow H H T \rightarrow H T T \rightarrow T T T$, which takes three steps.)
Letting $C$ denote the initial configuration (a sequence of $n H$ 's and $T$ 's), write $\ell(C)$ for the number of steps needed before all coins show $T$. Show that this number $\ell(C)$ is finite, and determine its average value over all $2^{n}$ possible initial configurations $C$. | [
"We represent the problem using a directed graph $G_{n}$ whose vertices are the length- $n$ strings of $H$ 's and $T$ 's. The graph features an edge from each string to its successor (except for $T T \\cdots T T$, which has no successor). We will also write $\\bar{H}=T$ and $\\bar{T}=H$.\n\nThe graph $G_{0}$ consis... | [
"$\\frac{1}{4} n(n+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 | Expression | null | Open-ended | Combinatorics | Math | English |
1,915 | On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue.
At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet.
After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls.
Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how Merlin paints the labyrinth $\mathfrak{L}$, Morgana can always place at least $k$ knights such that no two of them can ever meet. For each $n$, what are all possible values for $k(\mathfrak{L})$, where $\mathfrak{L}$ is a labyrinth with $n$ walls? | [
"First we show by induction that the $n$ walls divide the plane into $\\left(\\begin{array}{c}n+1 \\\\ 2\\end{array}\\right)+1$ regions. The claim is true for $n=0$ as, when there are no walls, the plane forms a single region. When placing the $n^{\\text {th }}$ wall, it intersects each of the $n-1$ other walls exa... | [
"$k=n+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 | Expression | null | Open-ended | Combinatorics | Math | English |
1,918 | There are 60 empty boxes $B_{1}, \ldots, B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.
In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(a) Bob chooses an integer $k$ with $1 \leqslant k \leqslant 59$ and splits the boxes into the two groups $B_{1}, \ldots, B_{k}$ and $B_{k+1}, \ldots, B_{60}$.
(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group.
Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. | [
"We present solutions for the general case of $N>1$ boxes, and write $M=\\left\\lfloor\\frac{N}{2}+1\\right\\rfloor\\left\\lceil\\frac{N}{2}+1\\right\\rceil-1$ for the claimed answer. For $1 \\leqslant k<N$, say that Bob makes a $k$-move if he splits the boxes into a left group $\\left\\{B_{1}, \\ldots, B_{k}\\righ... | [
"960"
] | 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 |
1,920 | For any two different real numbers $x$ and $y$, we define $D(x, y)$ to be the unique integer $d$ satisfying $2^{d} \leqslant|x-y|<2^{d+1}$. Given a set of reals $\mathcal{F}$, and an element $x \in \mathcal{F}$, we say that the scales of $x$ in $\mathcal{F}$ are the values of $D(x, y)$ for $y \in \mathcal{F}$ with $x \neq y$.
Let $k$ be a given positive integer. Suppose that each member $x$ of $\mathcal{F}$ has at most $k$ different scales in $\mathcal{F}$ (note that these scales may depend on $x$ ). What is the maximum possible size of $\mathcal{F}$ ? | [
"We first construct a set $\\mathcal{F}$ with $2^{k}$ members, each member having at most $k$ different scales in $\\mathcal{F}$. Take $\\mathcal{F}=\\left\\{0,1,2, \\ldots, 2^{k}-1\\right\\}$. The scale between any two members of $\\mathcal{F}$ is in the set $\\{0,1, \\ldots, k-1\\}$.\n\nWe now show that $2^{k}$ i... | [
"$2^{k}$"
] | 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 |
1,929 | Find all pairs $(m, n)$ of positive integers satisfying the equation
$$
\left(2^{n}-1\right)\left(2^{n}-2\right)\left(2^{n}-4\right) \cdots\left(2^{n}-2^{n-1}\right)=m !
\tag{1}
$$ | [
"For any prime $p$ and positive integer $N$, we will denote by $v_{p}(N)$ the exponent of the largest power of $p$ that divides $N$. The left-hand side of (1) will be denoted by $L_{n}$; that is, $L_{n}=\\left(2^{n}-1\\right)\\left(2^{n}-2\\right)\\left(2^{n}-4\\right) \\cdots\\left(2^{n}-2^{n-1}\\right)$.\n\nWe wi... | [
"$(1,1), (3,2)$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Number Theory | Math | English |
1,930 | Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. | [
"Note that the equation is symmetric. We will assume without loss of generality that $a \\geqslant b \\geqslant c$, and prove that the only solution is $(a, b, c)=(3,2,1)$.\n\n\nWe will start by proving that $c=1$. Note that\n\n$$\n3 a^{3} \\geqslant a^{3}+b^{3}+c^{3}>a^{3} .\n$$\n\nSo $3 a^{3} \\geqslant(a b c)^{2... | [
"$(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Number Theory | Math | English |
1,938 | Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ with the property that
$$
f(x-f(y))=f(f(x))-f(y)-1
\tag{1}
$$
holds for all $x, y \in \mathbb{Z}$. | [
"It is immediately checked that both functions mentioned in the answer are as desired.\n\nNow let $f$ denote any function satisfying (1) for all $x, y \\in \\mathbb{Z}$. Substituting $x=0$ and $y=f(0)$ into (1) we learn that the number $z=-f(f(0))$ satisfies $f(z)=-1$. So by plugging $y=z$ into (1) we deduce that\n... | [
"$f(x)=-1$,$f(x)=x+1$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Expression | null | Open-ended | Algebra | Math | English |
1,939 | Let $n$ be a fixed positive integer. Find the maximum possible value of
$$
\sum_{1 \leqslant r<s \leqslant 2 n}(s-r-n) x_{r} x_{s}
$$
where $-1 \leqslant x_{i} \leqslant 1$ for all $i=1,2, \ldots, 2 n$. | [
"Let $Z$ be the expression to be maximized. Since this expression is linear in every variable $x_{i}$ and $-1 \\leqslant x_{i} \\leqslant 1$, the maximum of $Z$ will be achieved when $x_{i}=-1$ or 1 . Therefore, it suffices to consider only the case when $x_{i} \\in\\{-1,1\\}$ for all $i=1,2, \\ldots, 2 n$.\n\nFor ... | [
"$n(n-1)$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Expression | null | Open-ended | Algebra | Math | English |
1,940 | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the equation
$$
f(x+f(x+y))+f(x y)=x+f(x+y)+y f(x)\tag{1}
$$
for all real numbers $x$ and $y$. | [
"Clearly, each of the functions $x \\mapsto x$ and $x \\mapsto 2-x$ satisfies (1). It suffices now to show that they are the only solutions to the problem.\n\nSuppose that $f$ is any function satisfying (1). Then setting $y=1$ in (1), we obtain\n\n$$\nf(x+f(x+1))=x+f(x+1)\\tag{2}\n$$\n\nin other words, $x+f(x+1)$ i... | [
"$f(x)=x$,$f(x)=2-x$"
] | 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 |
1,945 | For a finite set $A$ of positive integers, we call a partition of $A$ into two disjoint nonempty subsets $A_{1}$ and $A_{2}$ good if the least common multiple of the elements in $A_{1}$ is equal to the greatest common divisor of the elements in $A_{2}$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly 2015 good partitions. | [
"Let $A=\\left\\{a_{1}, a_{2}, \\ldots, a_{n}\\right\\}$, where $a_{1}<a_{2}<\\cdots<a_{n}$. For a finite nonempty set $B$ of positive integers, denote by $\\operatorname{lcm} B$ and $\\operatorname{gcd} B$ the least common multiple and the greatest common divisor of the elements in $B$, respectively.\n\nConsider a... | [
"3024"
] | 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 |
1,953 | Let $A B C$ be an acute triangle, and let $M$ be the midpoint of $A C$. A circle $\omega$ passing through $B$ and $M$ meets the sides $A B$ and $B C$ again at $P$ and $Q$, respectively. Let $T$ be the point such that the quadrilateral $B P T Q$ is a parallelogram. Suppose that $T$ lies on the circumcircle of the triangle $A B C$. Determine all possible values of $B T / B M$. | [
"Let $S$ be the center of the parallelogram $B P T Q$, and let $B^{\\prime} \\neq B$ be the point on the ray $B M$ such that $B M=M B^{\\prime}$ (see Figure 1). It follows that $A B C B^{\\prime}$ is a parallelogram. Then, $\\angle A B B^{\\prime}=\\angle P Q M$ and $\\angle B B^{\\prime} A=\\angle B^{\\prime} B C=... | [
"$\\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 | false | null | Numerical | null | Open-ended | Geometry | Math | English |
1,962 | Determine all triples $(a, b, c)$ of positive integers for which $a b-c, b c-a$, and $c a-b$ are powers of 2 .
Explanation: A power of 2 is an integer of the form $2^{n}$, where $n$ denotes some nonnegative integer. | [
"It can easily be verified that these sixteen triples are as required. Now let $(a, b, c)$ be any triple with the desired property. If we would have $a=1$, then both $b-c$ and $c-b$ were powers of 2 , which is impossible since their sum is zero; because of symmetry, this argument shows $a, b, c \\geqslant 2$.\n\nCa... | [
"$(2,2,2),(2,2,3),(2,3,2),(3,2,2),(2,6,11),(2,11,6),(6,2,11),(6,11,2),(11,2,6),(11,6,2),(3,5,7),(3,7,5),(5,3,7),(5,7,3),(7,3,5),(7,5,3)$"
] | 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 | Number Theory | Math | English |
1,964 | Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ is called $k$-good if $\operatorname{gcd}(f(m)+n, f(n)+m) \leqslant k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function. | [
"For any function $f: \\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z}_{>0}$, let $G_{f}(m, n)=\\operatorname{gcd}(f(m)+n, f(n)+m)$. Note that a $k$-good function is also $(k+1)$-good for any positive integer $k$. Hence, it suffices to show that there does not exist a 1-good function and that there exists a 2-good functio... | [
"$k \\geqslant 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 | Number Theory | Math | English |
1,968 | For a sequence $x_{1}, x_{2}, \ldots, x_{n}$ of real numbers, we define its price as
$$
\max _{1 \leqslant i \leqslant n}\left|x_{1}+\cdots+x_{i}\right|
$$
Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_{1}$ such that $\left|x_{1}\right|$ is as small as possible; among the remaining numbers, he chooses $x_{2}$ such that $\left|x_{1}+x_{2}\right|$ is as small as possible, and so on. Thus, in the $i^{\text {th }}$ step he chooses $x_{i}$ among the remaining numbers so as to minimise the value of $\left|x_{1}+x_{2}+\cdots+x_{i}\right|$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$.
Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G \leqslant c D$. | [
"If the initial numbers are $1,-1,2$, and -2 , then Dave may arrange them as $1,-2,2,-1$, while George may get the sequence $1,-1,2,-2$, resulting in $D=1$ and $G=2$. So we obtain $c \\geqslant 2$.\n\nTherefore, it remains to prove that $G \\leqslant 2 D$. Let $x_{1}, x_{2}, \\ldots, x_{n}$ be the numbers Dave and ... | [
"2"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
1,969 | Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying
$$
f(f(m)+n)+f(m)=f(n)+f(3 m)+2014
\tag{1}
$$
for all integers $m$ and $n$. | [
"Let $f$ be a function satisfying (1). Set $C=1007$ and define the function $g: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ by $g(m)=f(3 m)-f(m)+2 C$ for all $m \\in \\mathbb{Z}$; in particular, $g(0)=2 C$. Now (1) rewrites as\n\n$$\nf(f(m)+n)=g(m)+f(n)\n$$\n\nfor all $m, n \\in \\mathbb{Z}$. By induction in both directi... | [
"$f(n) = 2 n+1007$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Expression | null | Open-ended | Algebra | Math | English |
1,970 | Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has
$$
\left|y^{2}-P(x)\right| \leqslant 2|x| \text { if and only if }\left|x^{2}-P(y)\right| \leqslant 2|y|
\tag{1}
$$
Determine all possible values of $P(0)$. | [
"Part I. We begin by verifying that these numbers are indeed possible values of $P(0)$. To see that each negative real number $-C$ can be $P(0)$, it suffices to check that for every $C>0$ the polynomial $P(x)=-\\left(\\frac{2 x^{2}}{C}+C\\right)$ has the property described in the statement of the problem. Due to sy... | [
"$(-\\infty, 0) \\cup\\{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 | Interval | null | Open-ended | Algebra | Math | English |
1,974 | Let $n \geqslant 2$ be an integer. Consider an $n \times n$ chessboard divided into $n^{2}$ unit squares. We call a configuration of $n$ rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for every happy configuration of rooks, we can find a $k \times k$ square without a rook on any of its $k^{2}$ unit squares. | [
"Let $\\ell$ be a positive integer. We will show that (i) if $n>\\ell^{2}$ then each happy configuration contains an empty $\\ell \\times \\ell$ square, but (ii) if $n \\leqslant \\ell^{2}$ then there exists a happy configuration not containing such a square. These two statements together yield the answer.\n\n(i). ... | [
"$\\lfloor\\sqrt{n-1}\\rfloor$"
] | 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 |
1,977 | We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of 100 cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions:
1. The winner only depends on the relative order of the 200 cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner.
2. If we write the elements of both sets in increasing order as $A=\left\{a_{1}, a_{2}, \ldots, a_{100}\right\}$ and $B=\left\{b_{1}, b_{2}, \ldots, b_{100}\right\}$, and $a_{i}>b_{i}$ for all $i$, then $A$ beats $B$.
3. If three players draw three disjoint sets $A, B, C$ from the deck, $A$ beats $B$ and $B$ beats $C$, then $A$ also beats $C$.
How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets $A$ and $B$ such that $A$ beats $B$ according to one rule, but $B$ beats $A$ according to the other. | [
"We prove a more general statement for sets of cardinality $n$ (the problem being the special case $n=100$, then the answer is $n$ ). In the following, we write $A>B$ or $B<A$ for \" $A$ beats $B$ \".\n\nPart I. Let us first define $n$ different rules that satisfy the conditions. To this end, fix an index $k \\in\\... | [
"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 | Combinatorics | Math | English |
1,988 | Let $n \geqslant 2$ be an integer, and let $A_{n}$ be the set
$$
A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbb{Z}, 0 \leqslant k<n\right\} .
$$
Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_{n}$. | [
"Part I. First we show that every integer greater than $(n-2) 2^{n}+1$ can be represented as such a sum. This is achieved by induction on $n$.\n\nFor $n=2$, the set $A_{n}$ consists of the two elements 2 and 3 . Every positive integer $m$ except for 1 can be represented as the sum of elements of $A_{n}$ in this cas... | [
"$(n-2) 2^{n}+1$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Expression | null | Open-ended | Number Theory | Math | English |
1,997 | Let $k \geqslant 2$ be an integer. Find the smallest integer $n \geqslant k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set. | [
"First we show that $n \\geqslant k+4$. Suppose that there exists such a set with $n$ numbers and denote them by $a_{1}<a_{2}<\\cdots<a_{n}$.\n\nNote that in order to express $a_{1}$ as a sum of $k$ distinct elements of the set, we must have $a_{1} \\geqslant a_{2}+\\cdots+a_{k+1}$ and, similarly for $a_{n}$, we mu... | [
"$n=k+4$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Expression | null | Open-ended | Algebra | Math | English |
1,998 | Let $\mathbb{R}_{>0}$ be the set of positive real numbers. Find all functions $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that, for every $x \in \mathbb{R}_{>0}$, there exists a unique $y \in \mathbb{R}_{>0}$ satisfying
$$
x f(y)+y f(x) \leqslant 2 .
$$ | [
"First we prove that the function $f(x)=1 / x$ satisfies the condition of the problem statement. The AM-GM inequality gives\n\n$$\n\\frac{x}{y}+\\frac{y}{x} \\geqslant 2\n$$\n\nfor every $x, y>0$, with equality if and only if $x=y$. This means that, for every $x>0$, there exists a unique $y>0$ such that\n\n$$\n\\fr... | [
"$f(x)=\\frac{1}{x}$"
] | 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 |
2,000 | Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_{1}<\cdots<a_{n}$ and a real number $r>0$ such that the $\frac{1}{2} n(n-1)$ differences $a_{j}-a_{i}$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^{1}, r^{2}, \ldots, r^{\frac{1}{2} n(n-1)}$. | [
"We first show a solution for each $n \\in\\{2,3,4\\}$. We will later show the impossibility of finding such a solution for $n \\geqslant 5$.\n\nFor $n=2$, take for example $\\left(a_{1}, a_{2}\\right)=(1,3)$ and $r=2$.\n\nFor $n=3$, take the root $r>1$ of $x^{2}-x-1=0$ (the golden ratio) and set $\\left(a_{1}, a_{... | [
"2,3,4"
] | 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 | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,004 | $A \pm 1 \text{-}sequence$ is a sequence of 2022 numbers $a_{1}, \ldots, a_{2022}$, each equal to either +1 or -1 . Determine the largest $C$ so that, for any $\pm 1 -sequence$, there exists an integer $k$ and indices $1 \leqslant t_{1}<\ldots<t_{k} \leqslant 2022$ so that $t_{i+1}-t_{i} \leqslant 2$ for all $i$, and
$$
\left|\sum_{i=1}^{k} a_{t_{i}}\right| \geqslant C
$$ | [
"First, we prove that this can always be achieved. Without loss of generality, suppose at least $\\frac{2022}{2}=1011$ terms of the \\pm 1 -sequence are +1 . Define a subsequence as follows: starting at $t=0$, if $a_{t}=+1$ we always include $a_{t}$ in the subsequence. Otherwise, we skip $a_{t}$ if we can (i.e. if ... | [
"506"
] | 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 |
2,006 | In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height 0 . A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:
- The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller.
- The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter.
We say that a tree is majestic if its height is at least $10^{6}$. Determine the largest number $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays. | [
"We solve the problem for a general $3 N \\times 3 N$ board. First, we prove that the lumberjack has a strategy to ensure there are never more than $5 N^{2}$ majestic trees. Giving the squares of the board coordinates in the natural manner, colour each square where at least one of its coordinates are divisible by 3... | [
"2271380"
] | 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 |
2,010 | Lucy starts by writing $s$ integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=\left(v_{1}, \ldots, v_{2022}\right)$ and $\mathbf{w}=\left(w_{1}, \ldots, w_{2022}\right)$ that she has already written, and apply one of the following operations to obtain a new tuple:
$$
\begin{aligned}
& \mathbf{v}+\mathbf{w}=\left(v_{1}+w_{1}, \ldots, v_{2022}+w_{2022}\right) \\
& \mathbf{v} \vee \mathbf{w}=\left(\max \left(v_{1}, w_{1}\right), \ldots, \max \left(v_{2022}, w_{2022}\right)\right)
\end{aligned}
$$
and then write this tuple on the blackboard.
It turns out that, in this way, Lucy can write any integer-valued 2022-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote? | [
"We solve the problem for $n$-tuples for any $n \\geqslant 3$ : we will show that the answer is $s=3$, regardless of the value of $n$.\n\nFirst, let us briefly introduce some notation. For an $n$-tuple $\\mathbf{v}$, we will write $\\mathbf{v}_{i}$ for its $i$-th coordinate (where $1 \\leqslant i \\leqslant n$ ). F... | [
"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 |
2,011 | Alice fills the fields of an $n \times n$ board with numbers from 1 to $n^{2}$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such that:
(i) The first field in the sequence is one that is only adjacent to fields with larger numbers,
(ii) Each subsequent field in the sequence is adjacent to the previous field,
(iii) The numbers written on the fields in the sequence are in increasing order.
Two fields are considered adjacent if they share a common side. Find the smallest possible number of good paths Alice can obtain, as a function of $n$. | [
"We will call any field that is only adjacent to fields with larger numbers a well. Other fields will be called non-wells. Let us make a second $n \\times n$ board $B$ where in each field we will write the number of good sequences which end on the corresponding field in the original board $A$. We will thus look for... | [
"$2 n^{2}-2 n+1$"
] | 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 | Expression | null | Open-ended | Combinatorics | Math | English |
2,012 | Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right)$, we have $f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, y_{2}\right)$ and $f\left(x_{1}, y_{1}+1\right)>f\left(x_{2}, y_{2}+1\right)$.
Let $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y<100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$. | [
"We defer the constructions to the end of the solution. Instead, we begin by characterizing all such functions $f$, prove a formula and key property for such functions, and then solve the problem, providing constructions.\n\n**Characterization** Suppose $f$ satisfies the given relation. The condition can be written... | [
"2500,7500"
] | 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 |
2,021 | A number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number.
(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.) | [
"Observe that 1344 is a Norwegian number as 6, 672 and 1344 are three distinct divisors of 1344 and $6+672+1344=2022$. It remains to show that this is the smallest such number.\n\nAssume for contradiction that $N<1344$ is Norwegian and let $N / a, N / b$ and $N / c$ be the three distinct divisors of $N$, with $a<b<... | [
"1344"
] | 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 |
2,022 | Find all positive integers $n>2$ such that
$$
n ! \mid \prod_{\substack{p<q \leqslant n \\ p, q \text { primes }}}(p+q)
$$ | [
"Assume that $n$ satisfies $n ! \\mid \\prod_{p<q \\leqslant n}(p+q)$ and let $2=p_{1}<p_{2}<\\cdots<p_{m} \\leqslant n$ be the primes in $\\{1,2, \\ldots, n\\}$. Each such prime divides $n$ !. In particular, $p_{m} \\mid p_{i}+p_{j}$ for some $p_{i}<p_{j} \\leqslant n$. But\n\n$$\n0<\\frac{p_{i}+p_{j}}{p_{m}}<\\fr... | [
"7"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,024 | Find all triples of positive integers $(a, b, p)$ with $p$ prime and
$$
a^{p}=b !+p
$$ | [
"Clearly, $a>1$. We consider three cases.\n\nCase 1: We have $a<p$. Then we either have $a \\leqslant b$ which implies $a \\mid a^{p}-b$ ! $=p$ leading to a contradiction, or $a>b$ which is also impossible since in this case we have $b ! \\leqslant a !<a^{p}-p$, where the last inequality is true for any $p>a>1$.\n\... | [
"$(2,2,2),(3,4,3)$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Tuple | null | Open-ended | Number Theory | Math | English |
2,029 | Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}_{>0} \rightarrow \mathbb{Q}_{>0}$ satisfying
$$
f\left(x^{2} f(y)^{2}\right)=f(x)^{2} f(y)
\tag{*}
$$
for all $x, y \in \mathbb{Q}_{>0}$. | [
"Take any $a, b \\in \\mathbb{Q}_{>0}$. By substituting $x=f(a), y=b$ and $x=f(b), y=a$ into $(*)$ we get\n\n$$\nf(f(a))^{2} f(b)=f\\left(f(a)^{2} f(b)^{2}\\right)=f(f(b))^{2} f(a)\n$$\n\nwhich yields\n\n$$\n\\frac{f(f(a))^{2}}{f(a)}=\\frac{f(f(b))^{2}}{f(b)} \\quad \\text { for all } a, b \\in \\mathbb{Q}>0\n$$\n\... | [
"$f(x)=1$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Expression | null | Open-ended | Algebra | Math | English |
2,032 | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying
$$
a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k}
$$
Find the maximal possible value of $a_{2018}-a_{2017}$. | [
"The claimed maximal value is achieved at\n\n$$\n\\begin{gathered}\na_{1}=a_{2}=\\cdots=a_{2016}=1, \\quad a_{2017}=\\frac{a_{2016}+\\cdots+a_{0}}{2017}=1-\\frac{1}{2017}, \\\\\na_{2018}=\\frac{a_{2017}+\\cdots+a_{1}}{2017}=1-\\frac{1}{2017^{2}} .\n\\end{gathered}\n$$\n\nNow we need to show that this value is optim... | [
"$\\frac{2016}{2017^{2}}$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,035 | Find the maximal value of
$$
S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}}
$$
where $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$. | [
"Since the value $8 / \\sqrt[3]{7}$ is reached, it suffices to prove that $S \\leqslant 8 / \\sqrt[3]{7}$.\n\nAssume that $x, y, z, t$ is a permutation of the variables, with $x \\leqslant y \\leqslant z \\leqslant t$. Then, by the rearrangement inequality,\n\n$$\nS \\leqslant\\left(\\sqrt[3]{\\frac{x}{t+7}}+\\sqrt... | [
"$\\frac{8}{\\sqrt[3]{7}}$"
] | 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 | Algebra | Math | English |
2,037 | Queenie and Horst play a game on a $20 \times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when somebody cannot move.
Find the maximal positive $K$ such that, regardless of the strategy of Queenie, Horst can put at least $K$ knights on the board. | [
"We show two strategies, one for Horst to place at least 100 knights, and another strategy for Queenie that prevents Horst from putting more than 100 knights on the board.\n\nA strategy for Horst: Put knights only on black squares, until all black squares get occupied.\n\nColour the squares of the board black and w... | [
"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 | Combinatorics | Math | English |
2,040 | Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay 1 coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost. | [
"Enumerate the days of the tournament $1,2, \\ldots,\\left(\\begin{array}{c}2 k \\\\ 2\\end{array}\\right)$. Let $b_{1} \\leqslant b_{2} \\leqslant \\cdots \\leqslant b_{2 k}$ be the days the players arrive to the tournament, arranged in nondecreasing order; similarly, let $e_{1} \\geqslant \\cdots \\geqslant e_{2 ... | [
"$k\\left(4 k^{2}+k-1\\right) / 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 | Combinatorics | Math | English |
2,045 | A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold:
(i) each triangle from $T$ is inscribed in $\omega$;
(ii) no two triangles from $T$ have a common interior point.
Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$. | [
"First, we show how to construct a good collection of $n$ triangles, each of perimeter greater than 4 . This will show that all $t \\leqslant 4$ satisfy the required conditions.\n\nConstruct inductively an $(n+2)$-gon $B A_{1} A_{2} \\ldots A_{n} C$ inscribed in $\\omega$ such that $B C$ is a diameter, and $B A_{1}... | [
"$t(0,4]$"
] | 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 | Interval | null | Open-ended | Geometry | Math | English |
2,063 | Let $n$ be a positive integer. Find the smallest integer $k$ with the following property: Given any real numbers $a_{1}, \ldots, a_{d}$ such that $a_{1}+a_{2}+\cdots+a_{d}=n$ and $0 \leqslant a_{i} \leqslant 1$ for $i=1,2, \ldots, d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most 1 . | [
"If $d=2 n-1$ and $a_{1}=\\cdots=a_{2 n-1}=n /(2 n-1)$, then each group in such a partition can contain at most one number, since $2 n /(2 n-1)>1$. Therefore $k \\geqslant 2 n-1$. It remains to show that a suitable partition into $2 n-1$ groups always exists.\n\nWe proceed by induction on $d$. For $d \\leqslant 2 n... | [
"$k=2 n-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 | Expression | null | Open-ended | Algebra | Math | English |
2,064 | In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors.
Find the minimal value of $k$ such that the goal is attainable for every possible configuration of 4027 points. | [
"Firstly, let us present an example showing that $k \\geqslant 2013$. Mark 2013 red and 2013 blue points on some circle alternately, and mark one more blue point somewhere in the plane. The circle is thus split into 4026 arcs, each arc having endpoints of different colors. Thus, if the goal is reached, then each ar... | [
"2013"
] | 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 |
2,075 | Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$
m^{2}+f(n) \mid m f(m)+n
$$
for all positive integers $m$ and $n$. | [
"Setting $m=n=2$ tells us that $4+f(2) \\mid 2 f(2)+2$. Since $2 f(2)+2<2(4+f(2))$, we must have $2 f(2)+2=4+f(2)$, so $f(2)=2$. Plugging in $m=2$ then tells us that $4+f(n) \\mid 4+n$, which implies that $f(n) \\leqslant n$ for all $n$.\n\nSetting $m=n$ gives $n^{2}+f(n) \\mid n f(n)+n$, so $n f(n)+n \\geqslant n^... | [
"$f(n)=n$"
] | 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 |
2,082 | 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
$$
\begin{aligned}
b_{1} \leq b_{2} \leq \ldots \leq b_{2009} & \text { the lengths of the blue sides } \\
r_{1} \leq r_{2} \leq \ldots \leq r_{2009} & \text { the lengths of the red sides, } \\
\text { and } \quad & w_{1} \leq w_{2} \leq \ldots \leq w_{2009} \quad \text { the lengths of the white sides. }
\end{aligned}
$$
Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_{j}, r_{j}, w_{j}$. | [
"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... | [
"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 |
2,084 | 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
$$
x, \quad f(y) \text { and } f(y+f(x)-1) .
$$ | [
"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... | [
"$f(z)=z$"
] | 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 |
2,091 | 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 $i \neq j$, then $a_{i} \neq a_{j}, b_{i} \neq b_{j}$ and $c_{i} \neq c_{j}$.
Determine $N(n)$ for all $n \geq 2$. | [
"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,... | [
"$N(n)=\\left\\lfloor\\frac{2 n}{3}\\right\\rfloor+1$"
] | 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 | Expression | null | Open-ended | Combinatorics | Math | English |
2,095 | 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 consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called cyclic, if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over.
How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? | [
"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... | [
"996000"
] | 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 |
2,099 | 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 $\angle B A C$. | [
"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... | [
"$90^{\\circ}$,$60^{\\circ}$"
] | 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 | Numerical | null | Open-ended | Geometry | Math | English |
2,111 | 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 $2 \leq k \leq n-1$. | [
"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... | [
"1,2,3,4"
] | 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 | true | null | Numerical | null | Open-ended | Number Theory | Math | English |
2,124 | 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 that $B_{i}$ and $B_{j}$ intersect if and only if $i \not \equiv j \pm 1(\bmod n)$. | [
"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 \\... | [
"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 |
2,126 | 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 two points in $T$ are $k$-friends. Find the least positive integer $k$ for which there exists a $k$-clique with more than 200 elements. | [
"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... | [
"180180"
] | 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 |
2,127 | 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 the number of $k$-step sequences ending in the state: lamps $1, \ldots, n$ on, lamps $n+1, \ldots, 2 n$ off.
Let $M$ be the number of $k$-step sequences leading to the same state and not touching lamps $n+1, \ldots, 2 n$ at all.
Find the ratio $N / M$. | [
"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... | [
"$2^{k-n}$"
] | 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 |
2,147 | 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 $f(-1) \neq 0$. | [
"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 $C=g(-1)-1 \\n... | [
"$f(x)=x-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 | Expression | null | Open-ended | Algebra | Math | English |
2,151 | 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 $n$ ? | [
"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... | [
"$\\lfloor\\frac{2 n-1}{5}\\rfloor$"
] | 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 |
2,152 | 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. | [
"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}, ... | [
"$\\frac{4 \\cdot 999^{4}}{27}$"
] | 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 |
2,153 | 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 1 coin from every box to an adjacent box.
- On every move of hers $A$ chooses several coins that were not involved in $B$ 's previous move and are in different boxes. She passes every chosen coin to an adjacent box.
Player $A$ 's goal is to ensure at least 1 coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed. | [
"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 ... | [
"4022"
] | 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 |
2,165 | 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 {. }
$$ | [
"First note that $x$ divides $2012 \\cdot 2=2^{3} \\cdot 503$. If $503 \\mid x$ then the right-hand side of the equation is divisible by $503^{3}$, and it follows that $503^{2} \\mid x y z+2$. This is false as $503 \\mid x$. Hence $x=2^{m}$ with $m \\in\\{0,1,2,3\\}$. If $m \\geq 2$ then $2^{6} \\mid 2012(x y z+2)$... | [
"$(2,251,252)$"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Tuple | null | Open-ended | Number Theory | Math | English |
2,171 | Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that the equation
holds for all rational numbers $x$ and $y$.
$$
f(x f(x)+y)=f(y)+x^{2}
$$
Here, $\mathbb{Q}$ denotes the set of rational numbers. | [
"Denote the equation from the statement by (1). Let $x f(x)=A$ and $x^{2}=B$. The equation (1) is of the form\n\n$$\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... | [
"$f(x)=x,f(x)=-x$"
] | 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 |
2,174 | 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 number of fat triangles. | [
"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... | [
"$2021 \\cdot 505 \\cdot 337$"
] | 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 |
2,177 | 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) There are positive integers $m, n$ such that $f(m+n) \neq f(m)+f(n)$.
In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both (i) and (ii) the positive integers $m, n$ are not necessarily different. | [
"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 $n \... | [
"$k=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 |
2,185 | 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 the minimum number of blue cells. | [
"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... | [
"$6m$"
] | 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 |
2,193 | 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. | [
"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... | [
"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 |
2,198 | 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 contestant $C_{i}$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions.
- If contestant $C_{i}$ has fewer than $i$ other contestants in front of her, the restaurant opens and the process ends.
Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves. | [
"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... | [
"$2^{n}-n-1$"
] | 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 | Expression | null | Open-ended | Combinatorics | Math | English |
2,203 | Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b \text {. }
$$ | [
"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 $(a, b, c)=(1,1... | [
"$(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)$"
] | 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 |
2,204 | 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 $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.) | [
"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 $M \\... | [
"$\\frac{n(n+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 | false | null | Expression | null | Open-ended | Combinatorics | Math | English |
2,212 | 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 $1 \leq k \leq N-1$. | [
"$\\left(a_{k}+a_{k-1}\\right)\\left(a_{k}+a_{k+1}\\right)=a_{k-1}-a_{k+1}$ 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 $\\left(b_{k-1}+1\\right)\\left(b_{k}-1\\right)... | [
"$N=n$"
] | 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 |
2,216 | 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. | [
"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 $\... | [
"11,13"
] | 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 |
2,230 | 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 top-to-bottom and each row contains a string from $\mathcal{D}$ when read from left-to-right.
What is the smallest integer $m$ such that if $\mathcal{D}$ contains at least $m$ different strings, then Lexi can fill her grid in this manner, no matter what strings are in $\mathcal{D}$ ? | [
"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 \\... | [
"$2^{k-1}$"
] | 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 | Expression | null | Open-ended | Combinatorics | Math | English |
2,234 | 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 $d$ ? | [
"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... | [
"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 |
2,235 | 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. | [
"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$$\n\\begin{array}{r}\nn^{2}-8 n-20=0 \\\\\n(n-10)... | [
"10,11,12,13,14,-2,-1,0,1,2"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | true | null | Numerical | null | Open-ended | Algebra | Math | English |
2,236 | If $f(t)=\sin \left(\pi t-\frac{\pi}{2}\right)$, what is the smallest positive value of $t$ at which $f(t)$ attains its minimum value? | [
"Since $t>0, \\pi t-\\frac{\\pi}{2}>-\\frac{\\pi}{2}$. So $\\sin \\left(\\pi t-\\frac{\\pi}{2}\\right)$ first attains its minimum value when\n\n$$\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}\... | [
"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 |
2,238 | Determine all integer values of $x$ such that $\left(x^{2}-3\right)\left(x^{2}+5\right)<0$. | [
"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$."
] | [
"-1,0,1"
] | 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 |
2,239 | At present, the sum of the ages of a husband and wife, $P$, is six times the sum of the ages of their children, $C$. Two years ago, the sum of the ages of the husband and wife was ten times the sum of the ages of the same children. Six years from now, it will be three times the sum of the ages of the same children. Determine the number of children. | [
"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... | [
"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 |
2,241 | What is the value of $x$ such that $\log _{2}\left(\log _{2}(2 x-2)\right)=2$ ? | [
"$$\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$$"
] | [
"9"
] | null | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Not supported with pagination yet | Text-only | Competition | false | null | Numerical | null | Open-ended | Algebra | Math | English |
2,242 | Let $f(x)=2^{k x}+9$, where $k$ is a real number. If $f(3): f(6)=1: 3$, determine the value of $f(9)-f(3)$. | [
"From the given condition,\n\n$$\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... | [
"210"
] | 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 |
2,243 | Determine, with justification, all values of $k$ for which $y=x^{2}-4$ and $y=2|x|+k$ do not intersect. | [
"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\... | [
"(-\\infty,-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 | Interval | null | Open-ended | Algebra | Math | English |
2,247 | If $2 \leq x \leq 5$ and $10 \leq y \leq 20$, what is the maximum value of $15-\frac{y}{x}$ ? | [
"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... | [
"13"
] | 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 |
2,248 | 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 $2 f(2) g(2)$. | [
"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... | [
"-84"
] | 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 |
2,249 | Three different numbers are chosen at random from the set $\{1,2,3,4,5\}$.
The numbers are arranged in increasing order.
What is the probability that the resulting sequence is an arithmetic sequence?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3,5,7,9 is an arithmetic sequence with four terms.) | [
"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 ... | [
"$\\frac{2}{5}$"
] | 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 |
2,251 | What is the largest two-digit number that becomes $75 \%$ greater when its digits are reversed? | [
"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... | [
"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 | Combinatorics | Math | English |
2,253 | Serge likes to paddle his raft down the Speed River from point $A$ to point $B$. The speed of the current in the river is always the same. When Serge paddles, he always paddles at the same constant speed. On days when he paddles with the current, it takes him 18 minutes to get from $A$ to $B$. When he does not paddle, the current carries him from $A$ to $B$ in 30 minutes. If there were no current, how long would it take him to paddle from $A$ to $B$ ? | [
"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... | [
"45"
] | 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 | minute | Numerical | null | Open-ended | Geometry | Math | English |
2,254 | 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 . | [
"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 $(2,0)$ ... | [
"$\\frac{32}{9}$,$\\frac{1}{2}$"
] | 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 | Geometry | Math | English |
2,255 | A 75 year old person has a $50 \%$ chance of living at least another 10 years.
A 75 year old person has a $20 \%$ chance of living at least another 15 years. An 80 year old person has a $25 \%$ chance of living at least another 10 years. What is the probability that an 80 year old person will live at least another 5 years? | [
"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 ... | [
"62.5%"
] | 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 |
2,256 | Determine all values of $x$ for which $2^{\log _{10}\left(x^{2}\right)}=3\left(2^{1+\log _{10} x}\right)+16$. | [
"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... | [
"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 |
2,257 | The Sieve of Sundaram uses the following infinite table of positive integers:
| 4 | 7 | 10 | 13 | $\cdots$ |
| :---: | :---: | :---: | :---: | :---: |
| 7 | 12 | 17 | 22 | $\cdots$ |
| 10 | 17 | 24 | 31 | $\cdots$ |
| 13 | 22 | 31 | 40 | $\cdots$ |
| $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | |
The numbers in each row in the table form an arithmetic sequence. The numbers in each column in the table form an arithmetic sequence. The first four entries in each of the first four rows and columns are shown.
Determine the number in the 50th row and 40th column. | [
"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... | [
"4090"
] | 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 |
2,258 | The Sieve of Sundaram uses the following infinite table of positive integers:
| 4 | 7 | 10 | 13 | $\cdots$ |
| :---: | :---: | :---: | :---: | :---: |
| 7 | 12 | 17 | 22 | $\cdots$ |
| 10 | 17 | 24 | 31 | $\cdots$ |
| 13 | 22 | 31 | 40 | $\cdots$ |
| $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | |
The numbers in each row in the table form an arithmetic sequence. The numbers in each column in the table form an arithmetic sequence. The first four entries in each of the first four rows and columns are shown.
Determine a formula for the number in the $R$ th row and $C$ th column. | [
"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... | [
"$2RC+R+C$"
] | 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 |
2,260 | Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$.
Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$.
Determine the value of $g(2011)$. | [
"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, $g(2011)=2(2011)+\\left\\lfloor\\frac{1+\\sqrt{8(2011)-7}}{2}\\right\\rfloor=4022+\\lfloor 63.9\\rfloor=4022+63=4085$."
] | [
"4085"
] | 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 |
2,261 | Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3.1\rfloor=3$ and $\lfloor-1.4\rfloor=-2$.
Suppose that $f(n)=2 n-\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ and $g(n)=2 n+\left\lfloor\frac{1+\sqrt{8 n-7}}{2}\right\rfloor$ for each positive integer $n$.
Determine a value of $n$ for which $f(n)=100$. | [
"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... | [
"55"
] | 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 |
2,263 | 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 ? | [
"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... | [
"$\\frac{14}{15}$"
] | 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 | false | null | Numerical | null | Open-ended | Combinatorics | Math | English |
2,265 | A goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it travels 1 unit up to $(0,1)$. On move 2 , it travels 2 units right to $(2,1)$. On move 3 , it travels 3 units down to $(2,-2)$. On move 4 , it travels 4 units to $(-2,-2)$. It continues in this fashion, so that on move $n$, it turns $90^{\circ}$ in a clockwise direction from its previous heading and travels $n$ units in this new direction. After $n$ moves, the goat has travelled a total of 55 units. Determine the coordinates of its position at this time. | [
"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... | [
"(6,5)"
] | 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 | Tuple | null | Open-ended | Geometry | Math | English |
2,266 | 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.) | [
"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 ... | [
"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 |
2,268 | If $f(x)=\sin ^{2} x-2 \sin x+2$, what are the minimum and maximum values of $f(x)$ ? | [
"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... | [
"5,1"
] | 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 |
2,275 | What is the sum of the digits of the integer equal to $\left(10^{3}+1\right)^{2}$ ? | [
"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} \\\\... | [
"1002001"
] | 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 |
2,276 | A bakery sells small and large cookies. Before a price increase, the price of each small cookie is $\$ 1.50$ and the price of each large cookie is $\$ 2.00$. The price of each small cookie is increased by $10 \%$ and the price of each large cookie is increased by $5 \%$. What is the percentage increase in the total cost of a purchase of 2 small cookies and 1 large cookie? | [
"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... | [
"$8 \\%$"
] | 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 | Algebra | Math | English |
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