year stringdate 1961-01-01 00:00:00 2025-01-01 00:00:00 ⌀ | tier stringclasses 5
values | problem_label stringclasses 119
values | problem_type stringclasses 13
values | exam stringclasses 28
values | problem stringlengths 87 2.77k | solution stringlengths 834 13k | metadata dict | problem_tokens int64 50 903 | solution_tokens int64 500 3.93k |
|---|---|---|---|---|---|---|---|---|---|
2020 | T1 | 3 | null | APMO | Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n>m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways. | . We give an alternative proof of the first half of the lemma in the Solution 1 above.
Let $s_{1}<s_{2}<\cdots$ be the elements of $S$. For any positive integer $r$, define $A_{r}(x)=\prod_{n=1}^{r}\left(1+x^{s_{n}}\right)$. For each $n$ such that $m \leq n<s_{r+1}$, all $k$ ways of writing $n$ as a sum of elements of ... | {
"problem_match": "\nProblem 3.",
"resource_path": "APMO/segmented/en-apmo2020_sol.jsonl",
"solution_match": "\nSolution 2"
} | 55 | 707 |
2020 | T1 | 4 | null | APMO | Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property:
For any infinite sequence $a_{1}, a_{2}, \ldots$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i<j$ and an integer $k$ such that $a_{... | Part 1: All polynomials with $\operatorname{deg} P=1$ satisfy the given property.
Suppose $P(x)=c x+d$, and assume without loss of generality that $c>d \geq 0$. Denote $s_{i}=a_{1}+a_{2}+$ $\cdots+a_{i}(\bmod c)$. It suffices to show that there exist indices $i$ and $j$ such that $j-i \geq 2$ and $s_{j}-s_{i} \equiv d$... | {
"problem_match": "\nProblem 4.",
"resource_path": "APMO/segmented/en-apmo2020_sol.jsonl",
"solution_match": "# Solution:"
} | 103 | 1,448 |
2020 | T1 | 5 | null | APMO | Let $n \geq 3$ be a fixed integer. The number 1 is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers 1 and $a+b$, then adding one stone to the first bucket and $\operatorname... | The answer is the set of all rational numbers in the interval $[1, n-1)$. First, we show that no other numbers are possible. Clearly the ratio is at least 1, since for every move, at least one stone is added to the second bucket. Note that the number $s$ of stones in the first bucket is always equal to $p-n$, where $p$... | {
"problem_match": "\nProblem 5.",
"resource_path": "APMO/segmented/en-apmo2020_sol.jsonl",
"solution_match": "# Solution:"
} | 151 | 2,136 |
2021 | T1 | 2 | null | APMO | For a polynomial $P$ and a positive integer $n$, define $P_{n}$ as the number of positive integer pairs $(a, b)$ such that $a<b \leq n$ and $|P(a)|-|P(b)|$ is divisible by $n$.
Determine all polynomial $P$ with integer coefficients such that for all positive integers $n, P_{n} \leq 2021$. | There are two possible families of solutions:
- $P(x)=x+d$, for some integer $d \geq-2022$.
- $P(x)=-x+d$, for some integer $d \leq 2022$.
Suppose $P$ satisfies the problem conditions. Clearly $P$ cannot be a constant polynomial. Notice that a polynomial $P$ satifies the conditions if and only if $-P$ also satisfies ... | {
"problem_match": "\nProblem 2.",
"resource_path": "APMO/segmented/en-apmo2021_sol.jsonl",
"solution_match": "\nSolution "
} | 92 | 737 |
2021 | T1 | 3 | null | APMO | Let $A B C D$ be a cyclic convex quadrilateral and $\Gamma$ be its circumcircle. Let $E$ be the intersection of the diagonals $A C$ and $B D$, let $L$ be the center of the circle tangent to sides $A B, B C$, and $C D$, and let $M$ be the midpoint of the arc $B C$ of $\Gamma$ not containing $A$ and $D$. Prove that the e... | Let $L$ be the intersection of the bisectors of $\angle A B C$ and $\angle B C D$. Let $N$ be the $E$-excenter of $\triangle B C E$. Let $\angle B A C=\angle B D C=\alpha, \angle D B C=\beta$ and $\angle A C B=\gamma$.
We have the following:
$$
\begin{array}{r}
\angle C B L=\frac{1}{2} \angle A B C=90^{\circ}-\frac{1}... | {
"problem_match": "\nProblem 3.",
"resource_path": "APMO/segmented/en-apmo2021_sol.jsonl",
"solution_match": "# Solution 1"
} | 120 | 1,171 |
2021 | T1 | 4 | null | APMO | Given a $32 \times 32$ table, we put a mouse (facing up) at the bottom left cell and a piece of cheese at several other cells. The mouse then starts moving. It moves forward except that when it reaches a piece of cheese, it eats a part of it, turns right, and continues moving forward. We say that a subset of cells cont... | (a) For the sake of contradiction, assume a good subset consisting of 888 cells exists. We call those cheese-cells and the other ones gap-cells. Observe that since each cheese-cell is visited once, each gap-cell is visited at most twice (once vertically and once horizontally). Define a finite sequence $s$ whose $i$-th ... | {
"problem_match": "\nProblem 4.",
"resource_path": "APMO/segmented/en-apmo2021_sol.jsonl",
"solution_match": "# Solution."
} | 137 | 798 |
2021 | T1 | 5 | null | APMO | Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(f(a)-b)+b f(2 a)$ is a perfect square for all integers $a$ and $b$. | .
There are two families of functions which satisfy the condition:
(1) $f(n)= \begin{cases}0 & \text { if } n \text { is even, and } \\ \text { any perfect square } & \text { if } n \text { is odd }\end{cases}$
(2) $f(n)=n^{2}$, for every integer $n$.
It is straightforward to verify that the two families of functions... | {
"problem_match": "\nProblem 5.",
"resource_path": "APMO/segmented/en-apmo2021_sol.jsonl",
"solution_match": "# Solution 1"
} | 50 | 1,913 |
2022 | T1 | 1 | null | APMO | Find all pairs $(a, b)$ of positive integers such that $a^{3}$ is a multiple of $b^{2}$ and $b-1$ is a multiple of $a-1$. Note: An integer $n$ is said to be a multiple of an integer $m$ if there is an integer $k$ such that $n=k m$. | .2
We will start by showing that there are positive integers $x, c, d$ such that $a=x^{2} c d$ and $b=x^{3} c$. Let $g=\operatorname{gcd}(a, b)$ so that $a=g d$ and $b=g x$ for some coprime $d$ and $x$. Then, $b^{2} \mid a^{3}$ is equivalent to $g^{2} x^{2} \mid g^{3} d^{3}$, which is equivalent to $x^{2} \mid g d^{3}... | {
"problem_match": "\nProblem 1.",
"resource_path": "APMO/segmented/en-apmo2022_sol.jsonl",
"solution_match": "# Solution 1"
} | 76 | 544 |
2022 | T1 | 3 | null | APMO | Find all positive integers $k<202$ for which there exists a positive integer $n$ such that
$$
\left\{\frac{n}{202}\right\}+\left\{\frac{2 n}{202}\right\}+\cdots+\left\{\frac{k n}{202}\right\}=\frac{k}{2}
$$
where $\{x\}$ denote the fractional part of $x$.
Note: $\{x\}$ denotes the real number $k$ with $0 \leq k<1$ su... | Denote the equation in the problem statement as $\left(^{*}\right)$, and note that it is equivalent to the condition that the average of the remainders when dividing $n, 2 n, \ldots, k n$ by 202 is 101 . Since $\left\{\frac{i n}{202}\right\}$ is invariant in each residue class modulo 202 for each $1 \leq i \leq k$, it ... | {
"problem_match": "\nProblem 3.",
"resource_path": "APMO/segmented/en-apmo2022_sol.jsonl",
"solution_match": "# Solution\n\n"
} | 131 | 812 |
2022 | T1 | 4 | null | APMO | Let $n$ and $k$ be positive integers. Cathy is playing the following game. There are $n$ marbles and $k$ boxes, with the marbles labelled 1 to $n$. Initially, all marbles are placed inside one box. Each turn, Cathy chooses a box and then moves the marbles with the smallest label, say $i$, to either any empty box or the... | We claim Cathy can win if and only if $n \leq 2^{k-1}$.
First, note that each non-empty box always contains a consecutive sequence of labeled marbles. This is true since Cathy is always either removing from or placing in the lowest marble in a box. As a consequence, every move made is reversible.
Next, we prove by in... | {
"problem_match": "\nProblem 4.",
"resource_path": "APMO/segmented/en-apmo2022_sol.jsonl",
"solution_match": "# Solution\n\n"
} | 127 | 758 |
2022 | T1 | 5 | null | APMO | Let $a, b, c, d$ be real numbers such that $a^{2}+b^{2}+c^{2}+d^{2}=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a, b, c, d)$ such that the minimum value is achieved. | .1
Since the expression is cyclic, we could WLOG $a=\max \{a, b, c, d\}$. Let
$$
S(a, b, c, d)=(a-b)(b-c)(c-d)(d-a)
$$
Note that we have given $(a, b, c, d)$ such that $S(a, b, c, d)=-\frac{1}{8}$. Therefore, to prove that $S(a, b, c, d) \geq$ $-\frac{1}{8}$, we just need to consider the case where $S(a, b, c, d)<0$... | {
"problem_match": "\nProblem 5.",
"resource_path": "APMO/segmented/en-apmo2022_sol.jsonl",
"solution_match": "# Solution 5"
} | 77 | 1,191 |
2022 | T1 | 5 | null | APMO | Let $a, b, c, d$ be real numbers such that $a^{2}+b^{2}+c^{2}+d^{2}=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a, b, c, d)$ such that the minimum value is achieved. | .2
The minimum value is $-\frac{1}{8}$. There are eight equality cases in total. The first one is
$$
\left(\frac{1}{4}+\frac{\sqrt{3}}{4},-\frac{1}{4}-\frac{\sqrt{3}}{4}, \frac{1}{4}-\frac{\sqrt{3}}{4},-\frac{1}{4}+\frac{\sqrt{3}}{4}\right) .
$$
Cyclic shifting all the entries give three more quadruples. Moreover, f... | {
"problem_match": "\nProblem 5.",
"resource_path": "APMO/segmented/en-apmo2022_sol.jsonl",
"solution_match": "# Solution 5"
} | 77 | 1,440 |
2023 | T1 | 1 | null | APMO | Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1,2, \ldots, n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices.
Show that it is possible to arrange these squares in a way s... | Set aside the squares with sidelengths $n-3, n-2, n-1$, and $n$ and suppose we can split the remaining squares into two sets $A$ and $B$ such that the sum of the sidelengths of the squares in $A$ is 1 or 2 units larger than the sum of the sidelengths of the squares in $B$.
String the squares of each set $A, B$ along tw... | {
"problem_match": "# Problem 1",
"resource_path": "APMO/segmented/en-apmo2023_sol.jsonl",
"solution_match": "# Solution 1"
} | 91 | 1,266 |
2023 | T1 | 2 | null | APMO | Find all integers $n$ satisfying $n \geq 2$ and $\frac{\sigma(n)}{p(n)-1}=n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.
Answer: $n=6$. | Let $n=p_{1}^{\alpha_{1}} \cdot \ldots \cdot p_{k}^{\alpha_{k}}$ be the prime factorization of $n$ with $p_{1}<\ldots<p_{k}$, so that $p(n)=p_{k}$ and $\sigma(n)=\left(1+p_{1}+\cdots+p_{1}^{\alpha_{1}}\right) \ldots\left(1+p_{k}+\cdots+p_{k}^{\alpha_{k}}\right)$. Hence
$p_{k}-1=\frac{\sigma(n)}{n}=\prod_{i=1}^{k}\left(... | {
"problem_match": "# Problem 2",
"resource_path": "APMO/segmented/en-apmo2023_sol.jsonl",
"solution_match": "# Solution\n\n"
} | 70 | 823 |
2023 | T1 | 3 | null | APMO | Let $A B C D$ be a parallelogram. Let $W, X, Y$, and $Z$ be points on sides $A B, B C, C D$, and $D A$, respectively, such that the incenters of triangles $A W Z, B X W, C Y X$ and $D Z Y$ form a parallelogram. Prove that $W X Y Z$ is a parallelogram. | Let the four incenters be $I_{1}, I_{2}, I_{3}$, and $I_{4}$ with inradii $r_{1}, r_{2}, r_{3}$, and $r_{4}$ respectively (in the order given in the question). Without loss of generality, let $I_{1}$ be closer to $A B$ than $I_{2}$. Let the acute angle between $I_{1} I_{2}$ and $A B$ (and hence also the angle between $... | {
"problem_match": "# Problem 3",
"resource_path": "APMO/segmented/en-apmo2023_sol.jsonl",
"solution_match": "# Solution\n\n"
} | 93 | 1,372 |
2023 | T1 | 4 | null | APMO | Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that
$$
f((c+1) x+f(y))=f(x+2 y)+2 c x \quad \text { for all } x, y \in \mathbb{R}_{>0}
$$
Answer: $f(x)=2 x$ for all $x>0$. | We first prove that $f(x) \geq 2 x$ for all $x>0$. Suppose, for the sake of contradiction, that $f(y)<2 y$ for some positive $y$. Choose $x$ such that $f((c+1) x+f(y))$ and $f(x+2 y)$ cancel out, that is,
$$
(c+1) x+f(y)=x+2 y \Longleftrightarrow x=\frac{2 y-f(y)}{c}
$$
Notice that $x>0$ because $2 y-f(y)>0$. Then $2... | {
"problem_match": "# Problem 4",
"resource_path": "APMO/segmented/en-apmo2023_sol.jsonl",
"solution_match": "# Solution 1"
} | 121 | 951 |
2023 | T1 | 5 | null | APMO | There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2 n-1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows:
First, he chooses an endp... | Draw a circle that encloses all the intersection points between line segments and extend all line segments until they meet the circle, and then move Tony and all his friends to the circle. Number the intersection points with the circle from 1 to $2 n$ anticlockwise, starting from Tony (Tony has number 1). We will prove... | {
"problem_match": "# Problem 5",
"resource_path": "APMO/segmented/en-apmo2023_sol.jsonl",
"solution_match": "# Solution 1"
} | 179 | 1,127 |
2023 | T1 | 5 | null | APMO | There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2 n-1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows:
First, he chooses an endp... | First part: at most n friends can receive a present.
Similarly to the first solution, consider a circle that encompasses all line segments, extend the lines, and use the endpoints of the chords instead of the line segments, and prove that each chord connects vertices $k$ and $k+n$. We also consider, even in the first p... | {
"problem_match": "# Problem 5",
"resource_path": "APMO/segmented/en-apmo2023_sol.jsonl",
"solution_match": "# Solution 2"
} | 179 | 644 |
2024 | T1 | 2 | null | APMO | Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b, 1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves fr... | Cell $(x, y)$ is directly reachable from another cell if and only if $x-k \geq 1$ or $x+k \leq 100$ or $y-k \geq 1$ or $y+k \leq 100$, that is, $x \geq k+1$ or $x \leq 100-k$ or $y \geq k+1$ or $y \leq 100-k(*)$. Therefore the cells $(x, y)$ for which $101-k \leq x \leq k$ and $101-k \leq y \leq k$ are unreachable. Let... | {
"problem_match": "# Problem 2",
"resource_path": "APMO/segmented/en-apmo2024_sol.jsonl",
"solution_match": "# Solution\n\n"
} | 446 | 627 |
2024 | T1 | 3 | null | APMO | Let $n$ be a positive integer and $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers. Prove that
$$
\sum_{i=1}^{n} \frac{1}{2^{i}}\left(\frac{2}{1+a_{i}}\right)^{2^{i}} \geq \frac{2}{1+a_{1} a_{2} \ldots a_{n}}-\frac{1}{2^{n}}
$$ | We first prove the following lemma:
Lemma 1. For $k$ positive integer and $x, y>0$,
$$
\left(\frac{2}{1+x}\right)^{2^{k}}+\left(\frac{2}{1+y}\right)^{2^{k}} \geq 2\left(\frac{2}{1+x y}\right)^{2^{k-1}}
$$
The proof goes by induction. For $k=1$, we have
$$
\left(\frac{2}{1+x}\right)^{2}+\left(\frac{2}{1+y}\right)^{2}... | {
"problem_match": "# Problem 3",
"resource_path": "APMO/segmented/en-apmo2024_sol.jsonl",
"solution_match": "# Solution\n\n"
} | 110 | 1,506 |
2024 | T1 | 4 | null | APMO | Prove that for every positive integer $t$ there is a unique permutation $a_{0}, a_{1}, \ldots, a_{t-1}$ of $0,1, \ldots, t-$ 1 such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2 a_{i}}$ is odd and $2 a_{i} \neq t+i$. | We constantly make use of Kummer's theorem which, in particular, implies that $\binom{n}{k}$ is odd if and only if $k$ and $n-k$ have ones in different positions in binary. In other words, if $S(x)$ is the set of positions of the digits 1 of $x$ in binary (in which the digit multiplied by $2^{i}$ is in position $i),\bi... | {
"problem_match": "# Problem 4",
"resource_path": "APMO/segmented/en-apmo2024_sol.jsonl",
"solution_match": "# Solution\n\n"
} | 98 | 2,402 |
2024 | T1 | 5 | null | APMO | Line $\ell$ intersects sides $B C$ and $A D$ of cyclic quadrilateral $A B C D$ in its interior points $R$ and $S$ respectively, and intersects ray $D C$ beyond point $C$ at $Q$, and ray $B A$ beyond point $A$ at $P$. Circumcircles of the triangles $Q C R$ and $Q D S$ intersect at $N \neq Q$, while circumcircles of the ... | We start with the following lemma.
Lemma 1. Points $M, N, P, Q$ are concyclic.
Point $M$ is the Miquel point of lines $A P=A B, P S=\ell, A S=A D$, and $B R=B C$, and point $N$ is the Miquel point of lines $C Q=C D, R C=B C, Q R=\ell$, and $D S=A D$. Both points $M$ and $N$ are on the circumcircle of the triangle deter... | {
"problem_match": "# Problem 5",
"resource_path": "APMO/segmented/en-apmo2024_sol.jsonl",
"solution_match": "# Solution 1"
} | 187 | 739 |
2024 | T1 | 5 | null | APMO | Line $\ell$ intersects sides $B C$ and $A D$ of cyclic quadrilateral $A B C D$ in its interior points $R$ and $S$ respectively, and intersects ray $D C$ beyond point $C$ at $Q$, and ray $B A$ beyond point $A$ at $P$. Circumcircles of the triangles $Q C R$ and $Q D S$ intersect at $N \neq Q$, while circumcircles of the ... | Barycentric coordinates are a viable way to solve the problem, but even the solution we have found had some clever computations. Here is an outline of this solution.
Lemma 2. Denote by $\operatorname{pow}_{\omega} X$ the power of point $X$ with respect to circle $\omega$. Let $\Gamma_{1}$ and $\Gamma_{2}$ be circles w... | {
"problem_match": "# Problem 5",
"resource_path": "APMO/segmented/en-apmo2024_sol.jsonl",
"solution_match": "# Solution 2"
} | 187 | 1,134 |
2025 | T1 | 1 | null | APMO | Let \(A B C\) be an acute triangle inscribed in a circle \(\Gamma\) . Let \(A_{1}\) be the orthogonal projection of \(A\) onto \(B C\) so that \(A A_{1}\) is an altitude. Let \(B_{1}\) and \(C_{1}\) be the orthogonal projections of \(A_{1}\) onto \(A B\) and \(A C\) , respectively. Point \(P\) is such that quadrilatera... | First notice that, since angles \(\angle A A_{1}B_{1}\) and \(\angle A A_{1}C_{1}\) are both right, the points \(B_{1}\) and \(C_{1}\) lie on the circle with \(A A_{1}\) as a diameter. Therefore, \(A C_{1} = A A_{1}\sin \angle A A_{1}C_{1} = A A_{1}\sin (90^{\circ}-\) \(\angle A_{1}A C) = A A_{1}\sin \angle C\) , simil... | {
"problem_match": "# Problem 1 ",
"resource_path": "APMO/segmented/en-apmo2025_sol.jsonl",
"solution_match": "# Solution \n\n"
} | 156 | 679 |
2025 | T1 | 2 | null | APMO | Let \(\alpha\) and \(\beta\) be positive real numbers. Emerald makes a trip in the coordinate plane, starting off from the origin \((0,0)\) . Each minute she moves one unit up or one unit to the right, restricting herself to the region \(|x - y|< 2025\) , in the coordinate plane. By the time she visits a point \((x,y)\... | Let \((x_{n},y_{n})\) be the point that Emerald visits after \(n\) minutes. Then \((x_{n + 1},y_{n + 1})\in \{(x_{n}+\) \(1,y_{n}),(x_{n},y_{n} + 1)\}\) . Either way, \(x_{n + 1} + y_{n + 1} = x_{n} + y_{n} + 1\) , and since \(x_{0} + y_{0} = 0 + 0 = 0\) \(x_{n} + y_{n} = n\)
The \(n\) - th number would be then
\[z... | {
"problem_match": "# Problem 2 ",
"resource_path": "APMO/segmented/en-apmo2025_sol.jsonl",
"solution_match": "# Solution \n\n"
} | 172 | 831 |
2025 | T1 | 4 | null | APMO | Let \(n \geq 3\) be an integer. There are \(n\) cells on a circle, and each cell is assigned either 0 or 1. There is a rooster on one of these cells, and it repeats the following operations:
If the rooster is on a cell assigned 0, it changes the assigned number to 1 and moves to the next cell counterclockwise. If th... | Reformulate the problem as a \(n\) - string of numbers in \(\{0,1\}\) and a position at which the action described in the problem is performed, and add 1 or 2 modulo \(n\) to the position according to the action. Say that a lap is complete for each time the position resets to 0 or 1. We will prove that the statement cl... | {
"problem_match": "# Problem 4 ",
"resource_path": "APMO/segmented/en-apmo2025_sol.jsonl",
"solution_match": "# Solution 1"
} | 193 | 1,990 |
2025 | T1 | 4 | null | APMO | Let \(n \geq 3\) be an integer. There are \(n\) cells on a circle, and each cell is assigned either 0 or 1. There is a rooster on one of these cells, and it repeats the following operations:
If the rooster is on a cell assigned 0, it changes the assigned number to 1 and moves to the next cell counterclockwise. If th... | Define positions, laps, stoppings, and bypassing as in Solution 1. This other pair of lemmata also solves the problem.
Lemma 3. There is a position and a lap in which the rooster stops twice and bypasses once (in some order) in the next three laps.
Proof. There is a position \(j\) the rooster stops for infinitely m... | {
"problem_match": "# Problem 4 ",
"resource_path": "APMO/segmented/en-apmo2025_sol.jsonl",
"solution_match": "# Solution 2"
} | 193 | 514 |
2025 | T1 | 4 | null | APMO | Let \(n \geq 3\) be an integer. There are \(n\) cells on a circle, and each cell is assigned either 0 or 1. There is a rooster on one of these cells, and it repeats the following operations:
If the rooster is on a cell assigned 0, it changes the assigned number to 1 and moves to the next cell counterclockwise. If th... | Let us reformulate the problem in terms of Graphs: we have a directed graph \(G\) with \(V = \{v_{1}, v_{2}, \ldots , v_{n}\}\) representing positions and \(E = \{v_{i} \to v_{i + 1}, v_{i} \to v_{i + 2} \mid 1 \le i \le n\}\) representing moves. Indices are taken mod \(n\) . Note that each vertex has in- degree and ou... | {
"problem_match": "# Problem 4 ",
"resource_path": "APMO/segmented/en-apmo2025_sol.jsonl",
"solution_match": "# Solution 3"
} | 193 | 1,854 |
2025 | T1 | 5 | null | APMO | Consider an infinite sequence \(a_{1},a_{2},\ldots\) of positive integers such that
\[100!(a_{m} + a_{m + 1} + \cdot \cdot \cdot +a_{n})\quad \mathrm{is~a~multiple~of}\quad a_{n - m + 1}a_{n + m}\]
for all positive integers \(m,n\) such that \(m\leq n\)
Prove that the sequence is either bounded or linear.
O... | Let \(c = 100!\) . Suppose that \(n\geq m + 2\) . Then \(a_{m + n} = a_{(m + 1) + (n - 1)}\) divides both \(c(a_{m} + a_{m + 1}+\) \(\cdot \cdot \cdot +a_{n - 1} + a_{n})\) and \(c(a_{m + 1} + \cdot \cdot \cdot +a_{n - 1})\) , so it also divides the difference \(c(a_{m} + a_{n})\) . Notice that if \(n = m + 1\) then \(... | {
"problem_match": "# Problem 5 ",
"resource_path": "APMO/segmented/en-apmo2025_sol.jsonl",
"solution_match": "# Solution \n\n"
} | 192 | 2,860 |
2008 | T1 | 3 | null | Balkan_MO | Let $n$ be a positive integer. The rectangle $A B C D$ with side lengths $A B=90 n+1$ and $B C=90 n+5$ is partitioned into unit squares with sides parallel to the sides of $A B C D$. Let $S$ be the set of all points which are vertices of these unit squares. Prove that the number of lines which pass through at least two... | Denote $90 n+1=m$. We investigate the number of the lines modulo 4 consecutively reducing different types of lines.
The vertical and horizontal lines are
$(m+5)+(m+1)=2(m+3)$ which is divisible to 4.
Moreover, every line which makes an acute angle to the axe $O x$ (i.e. that line has a positive angular coefficient) cor... | {
"problem_match": "# Problem 3",
"resource_path": "Balkan_MO/segmented/en-2008-BMO-type1.jsonl",
"solution_match": "# Solution."
} | 102 | 535 |
2008 | T1 | 4 | null | Balkan_MO | Let $c$ be a positive integer. The sequence $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ is defined by $a_{1}=c$, and $a_{n+1}=a_{n}^{2}+a_{n}+c^{3}$, for every positive integer $n$. Find all values of $c$ for which there exist some integers $k \geq 1$ and $m \geq 2$, such that $a_{k}^{2}+c^{3}$ is the $m^{\text {th }}$ power... | First, notice:
$$
a_{n+1}^{2}+c^{3}=\left(a_{n}^{2}+a_{n}+c^{3}\right)^{2}+c^{3}=\left(a_{n}^{2}+c^{3}\right)\left(a_{n}^{2}+2 a_{n}+1+c^{3}\right)
$$
We first prove that $a_{n}^{2}+c^{3}$ and $a_{n}^{2}+2 a_{n}+1+c^{3}$ are coprime.
We prove by induction that $4 c^{3}+1$ is coprime with $2 a_{n}+1$, for every $n \ge... | {
"problem_match": "# Problem 4",
"resource_path": "Balkan_MO/segmented/en-2008-BMO-type1.jsonl",
"solution_match": "# Solution."
} | 137 | 736 |
2009 | T1 | 1 | null | Balkan_MO | We start by observing that $z$ must be even, so $z^{2}=3^{x}-5^{y} \equiv(-1)^{x}-1(\bmod 4)$ is divisible by 4 , which implies that $x$ is even, say $x=2 t$. Then our equation can be rewritten as $\left(3^{t}-z\right)\left(3^{t}+z\right)=5^{y}$, which means that both $3^{t}-z=5^{k}$ and $3^{t}+z=5^{y-k}$ for some nonn... | We start by observing that $f$ is injective. From the known identity
$$
\left(a^{2}+2 b^{2}\right)\left(c^{2}+2 d^{2}\right)=(a c \pm 2 b d)^{2}+2(a d \mp b c)^{2}
$$
we obtain $f(a c+2 b d)^{2}+2 f(a d-b c)^{2}=f(a c-2 b d)^{2}+2 f(a d+b c)^{2}$, assuming that the arguments are positive integers. Specially, for $b=c... | {
"problem_match": "\n1.",
"resource_path": "Balkan_MO/segmented/en-2009-BMO-type2.jsonl",
"solution_match": "\n4."
} | 321 | 762 |
2009 | T1 | 2 | null | Balkan_MO | In a triangle $A B C$, points $M$ and $N$ on the sides $A B$ and $A C$ respectively are such that $M N \| B C$. Let $B N$ and $C M$ intersect at point $P$. The circumcircles of triangles $B M P$ and $C N P$ intersect at two distinct points $P$ and $Q$. Prove that $\angle B A Q=\angle C A P$.
(Moldova) | Since the quadrilaterals $B M P Q$ and $C N P Q$ are cyclic, we have $\angle B Q N=\angle B Q P+$ $\angle P Q N=\angle A M C+\angle M C A=180^{\circ}-$ $\angle C A B$, so $A B Q N$ is cyclic as well. Hence $\frac{\sin \angle B A Q}{\sin \angle N A Q}=\frac{B Q}{N Q}$. Moreover, triangles $M B Q$ and $C N Q$ are similar... | {
"problem_match": "\n2.",
"resource_path": "Balkan_MO/segmented/en-2009-BMO-type2.jsonl",
"solution_match": "\n2."
} | 103 | 524 |
2009 | T1 | 3 | null | Balkan_MO | A $9 \times 12$ rectangle is divided into unit squares. The centers of all the unit squares, except the four corner squares and the eight squares adjacent (by side) to them, are colored red. Is it possible to numerate the red centers by $C_{1}, C_{2}, \ldots, C_{96}$ so that the following two conditions are fulfilled:
... | Place the given rectangle into the coordinate plane so that the center of the square at the intersection of $i$-th column and $j$-th row has the coordinates $(i, j)$. Suppose that a desired numeration of the red points exists; it corresponds to a path, i.e. a closed poligonal line consisting of 96 segments of length $\... | {
"problem_match": "\n3.",
"resource_path": "Balkan_MO/segmented/en-2009-BMO-type2.jsonl",
"solution_match": "\n3."
} | 179 | 518 |
2009 | T1 | 4 | null | Balkan_MO | Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying
$$
f\left(f(m)^{2}+2 f(n)^{2}\right)=m^{2}+2 n^{2} \quad \text { for all } m, n \in \mathbb{N} . \quad \text { (Bulgaria) }
$$
Time allowed: 270 minutes.
Each problem is worth 10 points.
## SOLUTIONS | We start by observing that $f$ is injective. From the known identity
$$
\left(a^{2}+2 b^{2}\right)\left(c^{2}+2 d^{2}\right)=(a c \pm 2 b d)^{2}+2(a d \mp b c)^{2}
$$
we obtain $f(a c+2 b d)^{2}+2 f(a d-b c)^{2}=f(a c-2 b d)^{2}+2 f(a d+b c)^{2}$, assuming that the arguments are positive integers. Specially, for $b=c... | {
"problem_match": "\n4.",
"resource_path": "Balkan_MO/segmented/en-2009-BMO-type2.jsonl",
"solution_match": "\n4."
} | 108 | 762 |
2010 | T1 | 1 | null | Balkan_MO | The left-hand side is equal to
$$
\frac{a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3}-a^{3} b^{2} c-b^{3} c^{2} a-c^{3} a^{2} b}{(a+b)(b+c)(c+a)}
$$
so it is enough to show that $a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3} \geq a^{3} b^{2} c+b^{3} c^{2} a+c^{3} a^{2} b$. The AM-GM inequality gives us $a^{3} b^{3}+a^{3} b^{3}+a^{3} c^{... | There are $n+1-\varphi(n)$ nonnegative integers not coprime with $n$, and whenever $r$ is among them, so is $n-r$. This gives us the formula $f(n)=\frac{1}{2} n(n+1-\varphi(n))$. Suppose that $f(n)=f(n+p)$. We observe first that $n$ and $n+p$ divide $2 f(n)<n(n+p)$, so $n$ and $n+p$ are not coprime, which implies that ... | {
"problem_match": "\n1.",
"resource_path": "Balkan_MO/segmented/en-2010-BMO-type2.jsonl",
"solution_match": "\n4."
} | 254 | 864 |
2010 | T1 | 4 | null | Balkan_MO | For every integer $n \geq 2$, denote by $f(n)$ the sum of positive integers not exceeding $n$ that are not coprime to $n$. Prove that $f(n+p) \neq f(n)$ for any such $n$ and any prime number $p$.
(Turkey)
Time allowed: 270 minutes.
Each problem is worth 10 points.
## SOLUTIONS | There are $n+1-\varphi(n)$ nonnegative integers not coprime with $n$, and whenever $r$ is among them, so is $n-r$. This gives us the formula $f(n)=\frac{1}{2} n(n+1-\varphi(n))$. Suppose that $f(n)=f(n+p)$. We observe first that $n$ and $n+p$ divide $2 f(n)<n(n+p)$, so $n$ and $n+p$ are not coprime, which implies that ... | {
"problem_match": "\n4.",
"resource_path": "Balkan_MO/segmented/en-2010-BMO-type2.jsonl",
"solution_match": "\n4."
} | 90 | 864 |
2011 | T1 | 2 | null | Balkan_MO | Given real numbers $x, y, z$ such that $x+y+z=0$, show that
$$
\frac{x(x+2)}{2 x^{2}+1}+\frac{y(y+2)}{2 y^{2}+1}+\frac{z(z+2)}{2 z^{2}+1} \geq 0 .
$$
When does equality hold? | The inequality is clear if $x y z=0$, in which case equality holds if and only if $x=y=z=0$.
Henceforth assume $x y z \neq 0$ and rewrite the inequality as
$$
\frac{(2 x+1)^{2}}{2 x^{2}+1}+\frac{(2 y+1)^{2}}{2 y^{2}+1}+\frac{(2 z+1)^{2}}{2 z^{2}+1} \geq 3 .
$$
Notice that (exactly) one of the products $x y, y z, z x... | {
"problem_match": "# PROBLEM 2",
"resource_path": "Balkan_MO/segmented/en-2011-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 87 | 588 |
2011 | T1 | 3 | null | Balkan_MO | Let $S$ be a finite set of positive integers which has the following property: if $x$ is a member of $S$, then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is good if whenever $x, y \in T$ and $x<y$, the ratio $y / x$ is a power of a prime number. A non-empty subset $T$ of $S$ is bad if whenever $... | Notice first that a bad subset of $S$ contains at most one element from a good one, to deduce that a partition of $S$ into bad subsets has at least as many members as a maximal good subset.
Notice further that the elements of a good subset of $S$ must be among the terms of a geometric sequence whose ratio is a prime: ... | {
"problem_match": "# PROBLEM 3",
"resource_path": "Balkan_MO/segmented/en-2011-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 182 | 568 |
2011 | T1 | 4 | null | Balkan_MO | Let $A B C D E F$ be a convex hexagon of area 1 , whose opposite sides are parallel. The lines $A B, C D$ and $E F$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $B C, D E$ and $F A$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these tw... | Unless otherwise stated, throughout the proof indices take on values from 0 to 5 and are reduced modulo 6 . Label the vertices of the hexagon in circular order, $A_{0}, A_{1}, \cdots, A_{5}$, and let the lines of support of the alternate sides $A_{i} A_{i+1}$ and $A_{i+2} A_{i+3}$ meet at $B_{i}$. To show that the area... | {
"problem_match": "# PROBLEM 4",
"resource_path": "Balkan_MO/segmented/en-2011-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 98 | 1,105 |
2012 | T1 | 3 | null | Balkan_MO | Let $n$ be a positive integer. Let $P_{n}=\left\{2^{n}, 2^{n-1} \cdot 3,2^{n-2} \cdot 3^{2}, \ldots, 3^{n}\right\}$. For each subset $X$ of $P_{n}$, we write $S_{X}$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with ... | Note that $3^{m+1}-2^{m+1}=(3-2)\left(3^{m}+3^{m-1} \cdot 2+\cdots+3 \cdot 2^{m-1}+2^{m}\right)=S_{P_{m}}$. Dividing every element of $P_{m}$ by $2^{m}$ gives us the following equivalent problem:
Let $m$ be a positive integer, $a=3 / 2$, and $Q_{m}=\left\{1, a, a^{2}, \ldots, a^{m}\right\}$. Show that for any real num... | {
"problem_match": "\n3. ",
"resource_path": "Balkan_MO/segmented/en-2012-BMO-type3.jsonl",
"solution_match": "\nSolution 2."
} | 179 | 642 |
2012 | T1 | 4 | null | Balkan_MO | \quad$ Let $\mathbb{Z}^{+}$be the set of positive integers. Find all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$such that the following conditions both hold:
(i) $f(n!)=f(n)$ ! for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers. | There are three such functions: the constant functions 1, 2 and the identity function $\mathrm{id}_{\mathbf{Z}^{+}}$. These functions clearly satisfy the conditions in the hypothesis. Let us prove that there are only ones.
Consider such a function $f$ and suppose that it has a fixed point $a \geq 3$, that is $f(a)=a$.... | {
"problem_match": "\n4. ",
"resource_path": "Balkan_MO/segmented/en-2012-BMO-type3.jsonl",
"solution_match": "\nSolution 1."
} | 100 | 684 |
2012 | T1 | 4 | null | Balkan_MO | \quad$ Let $\mathbb{Z}^{+}$be the set of positive integers. Find all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$such that the following conditions both hold:
(i) $f(n!)=f(n)$ ! for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers. | Note first that if $f\left(n_{0}\right)=n_{0}$, then $m-n_{0} \mid f(m)-m$ for all $m \in \mathbf{Z}^{+}$. If $f\left(n_{0}\right)=n_{0}$ for infinitely many $n_{0} \in \mathbf{Z}^{+}$, then $f(m)-m$ has infinitely many divisors, hence $f(m)=m$ for all $m \in \mathbf{Z}^{+}$. On the other hand, if $f\left(n_{0}\right)=... | {
"problem_match": "\n4. ",
"resource_path": "Balkan_MO/segmented/en-2012-BMO-type3.jsonl",
"solution_match": "\nSolution 2."
} | 100 | 559 |
2014 | T1 | 1 | null | Balkan_MO | Let $x, y$ and $z$ be positive real numbers such that $x y+y z+z x=3 x y z$. Prove that
$$
x^{2} y+y^{2} z+z^{2} x \geq 2(x+y+z)-3
$$
and determine when equality holds. | The given condition can be rearranged to $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=3$. Using this, we obtain:
$$
\begin{aligned}
x^{2} y+y^{2} z+z^{2} x-2(x+y+z)+3 & =x^{2} y-2 x+\frac{1}{y}+y^{2} z-2 y+\frac{1}{z}+z^{2} x-2 x+\frac{1}{x}= \\
& =y\left(x-\frac{1}{y}\right)^{2}+z\left(y-\frac{1}{z}\right)^{2}+x\left(z-\frac... | {
"problem_match": "\nProblem 1.",
"resource_path": "Balkan_MO/segmented/en-2014-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 69 | 620 |
2014 | T1 | 2 | null | Balkan_MO | A special number is a positive integer $n$ for which there exist positive integers $a, b, c$ and $d$ with
$$
n=\frac{a^{3}+2 b^{3}}{c^{3}+2 d^{3}}
$$
Prove that:
(a) there are infinitely many special numbers;
(b) 2014 is not a special number. | (a) Every perfect cube $k^{3}$ of a positive integer is special because we can write
$$
k^{3}=k^{3} \frac{a^{3}+2 b^{3}}{a^{3}+2 b^{3}}=\frac{(k a)^{3}+2(k b)^{3}}{a^{3}+2 b^{3}}
$$
for some positive integers $a, b$.
(b) Observe that $2014=2.19 .53$. If 2014 is special, then we have,
$$
x^{3}+2 y^{3}=2014\left(u^{3}... | {
"problem_match": "\nProblem 2.",
"resource_path": "Balkan_MO/segmented/en-2014-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 84 | 620 |
2015 | T1 | 1 | null | Balkan_MO | Let $a, b$ and $c$ be positive real numbers. Prove that
$$
a^{3} b^{6}+b^{3} c^{6}+c^{3} a^{6}+3 a^{3} b^{3} c^{3} \geq a b c\left(a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3}\right)+a^{2} b^{2} c^{2}\left(a^{3}+b^{3}+c^{3}\right)
$$ | After dividing both sides of the given inequality by $a^{3} b^{3} c^{3}$ it becomes
$$
\left(\frac{b}{c}\right)^{3}+\left(\frac{c}{a}\right)^{3}+\left(\frac{a}{b}\right)^{3}+3 \geq\left(\frac{a}{c} \cdot \frac{b}{c}+\frac{b}{a} \cdot \frac{c}{a}+\frac{c}{b} \cdot \frac{a}{b}\right)+\left(\frac{a}{b} \cdot \frac{a}{c}+... | {
"problem_match": "\nProblem 1.",
"resource_path": "Balkan_MO/segmented/en-2015-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 127 | 667 |
2015 | T1 | 2 | null | Balkan_MO | Let $A B C$ be a scalene triangle with incentre $I$ and circumcircle ( $\omega$ ). The lines $A I, B I, C I$ intersect $(\omega)$ for the second time at the points $D, E, F$, respectively. The lines through $I$ parallel to the sides $B C, A C, A B$ intersect the lines $E F, D F, D E$ at the points $K, L, M$, respective... | First we will prove that $K A$ is tangent to $(\omega)$.
Indeed, it is a well-known fact that $F A=F B=F I$ and $E A=E C=E I$, so $F E$ is the perpendicular bisector of $A I$. It follows that $K A=K I$ and
$$
\angle K A F=\angle K I F=\angle F C B=\angle F E B=\angle F E A,
$$
so $K A$ is tangent to $(\omega)$. Simil... | {
"problem_match": "\nProblem 2.",
"resource_path": "Balkan_MO/segmented/en-2015-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 119 | 550 |
2015 | T1 | 3 | null | Balkan_MO | A jury of 3366 film critics are judging the Oscars. Each critic makes a single vote for his favourite actor, and a single vote for his favourite actress. It turns out that for every integer $n \in\{1,2, \ldots, 100\}$ there is an actor or actress who has been voted for exactly $n$ times. Show that there are two critics... | Let us assume that every critic votes for a different pair of actor and actress. We'll arrive at a contradiction proving the required result. Indeed:
Call the vote of each critic, i.e his choice for the pair of an actor and an actress, as a double-vote, and call as a single-vote each one of the two choices he makes, i... | {
"problem_match": "\nProblem 3.",
"resource_path": "Balkan_MO/segmented/en-2015-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 97 | 1,047 |
2016 | T1 | 2 | null | Balkan_MO | Let $A B C D$ be a cyclic quadrilateral with $A B<C D$. The diagonals intersect at the point $F$ and lines $A D$ and $B C$ intersect at the point $E$. Let $K$ and $L$ be the orthogonal projections of $F$ onto lines $A D$ and $B C$ respectively, and let $M, S$ and $T$ be the midpoints of $E F, C F$ and $D F$ respectivel... | Let $N$ be the midpoint of $C D$. We will prove that the circumcircles of the triangles $M K T$ and $M L S$ pass through $N$. (1)
First will prove that the circumcircle of $M L S$ passes through $N$.
Let $Q$ be the midpoint of $E C$. Note that the circumcircle of $M L S$ is the Euler circle (2) of the triangle $E F C$,... | {
"problem_match": "# Problem 2.",
"resource_path": "Balkan_MO/segmented/en-2016-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 139 | 513 |
2016 | T1 | 3 | null | Balkan_MO | Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer.
Note: A monic polynomial has leading coefficient equal to 1. | If $f$ is a constant polynomial then it's obvious that the condition cannot hold for
$$
p \geq 5 \text { since } f(p)=1
$$
From the divisibility relation $p \mid 2(f(p))$ ! +1 we conclude that:
$$
f(p)<p, \text { for all primes } p>N \quad(*)
$$
In fact, if for some prime number $p$ we have $f(p) \geq p$, then $p \... | {
"problem_match": "# Problem 3.",
"resource_path": "Balkan_MO/segmented/en-2016-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 71 | 546 |
2017 | T1 | 2 | null | Balkan_MO | Let $A B C$ be an acute triangle with with $A B<A C$ and let $\Gamma$ be its circumcircle. Let the tangents to $\Gamma$ at $B$ and $C$ be $t_{B}$ and $t_{C}$ respectively and let their point of intersection be $L$. The line through $B$ parallel to $A C$ intersects $t_{C}$ at $D$. The line through $C$ parallel to $A B$ ... | How we attack this problem depends on how much triangle geometry we can effortlessly recall - a good knowledge of some standard results helps a great deal.
We might instantly note that $A L$ is a symmedian of $A B C$, and so divides the line $B C$ in the ratio $c^{2}: b^{2}$. Now the plan is to show that $S T$ also di... | {
"problem_match": "\n2. ",
"resource_path": "Balkan_MO/segmented/en-2017-BMO-type3.jsonl",
"solution_match": "\nSolution."
} | 200 | 594 |
2017 | T1 | 3 | null | Balkan_MO | Let $\mathbb{N}$ be the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that:
$$
n+f(m) \text { divides } f(n)+n f(m)
$$ | The striking thing about this problem is that the relation concerns divisibility rather than equality. How can we exploit this? We are given that $n+f(m) \mid f(n)+n f(m)$ but we can certainly add or subtract multiples of the left hand side from the right hand side and preserve the divisibility. This leads to a key ide... | {
"problem_match": "\n3. ",
"resource_path": "Balkan_MO/segmented/en-2017-BMO-type3.jsonl",
"solution_match": "\nSolution."
} | 56 | 1,387 |
2017 | T1 | 4 | null | Balkan_MO | There are $n>2$ students sitting at a round table. Initially each student has exactly one candy. At each step, each student chooses one of the following operations:
(a) Pass one candy to the student on their left or the student on their right.
(b) Divide all their candies into two, possibly empty, sets and pass one set... | One possible initial reaction to this problem is that there is rather too much movement of caramels ${ }^{2}$ at each step to keep track of easily. This leads to the question: 'How little can I do in, say, two steps?' If every student passes all their caramels left on one step using (b), and all their caramels right on... | {
"problem_match": "\n4. ",
"resource_path": "Balkan_MO/segmented/en-2017-BMO-type3.jsonl",
"solution_match": "\nSolution."
} | 142 | 1,206 |
2018 | T1 | 2 | null | Balkan_MO | Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n=1,2, \ldots)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^{n}$ metres. After a whole number of minutes, they are at the same point in the p... | Answer: $q=1$.
Let $x_{A}^{(n)}$ (resp. $x_{B}^{(n)}$ ) be the $x$-coordinates of the first (resp. second) ant's position after $n$ minutes. Then $x_{A}^{(n)}-x_{A}^{(n-1)} \in\left\{q^{n},-q^{n}, 0\right\}$, and so $x_{A}^{(n)}, x_{B}^{(n)}$ are given by polynomials in $q$ with coefficients in $\{-1,0,1\}$. So if the ... | {
"problem_match": "\n2. ",
"resource_path": "Balkan_MO/segmented/en-2018-BMO-type3.jsonl",
"solution_match": "\nSolution 1."
} | 115 | 821 |
2018 | T1 | 3 | null | Balkan_MO | Alice and Bob play the following game: They start with two non-empty piles of coins. Taking turns, with Alice playing first, each player chooses a pile with an even number of coins and moves half of the coins of this pile to the other pile. The game ends if a player cannot move, in which case the other player wins.
Det... | By $v_{2}(n)$ we denote the largest nonnegative integer $r$ such that $2^{r} \mid n$.
A position $(a, b)$ (i.e. two piles of sizes $a$ and $b$ ) is said to be $k$-happy if $v_{2}(a)=v_{2}(b)=k$ for some integer $k \geqslant 0$, and $k$-unhappy if $\min \left\{v_{2}(a), v_{2}(b)\right\}=k<\max \left\{v_{2}(a), v_{2}(b)\... | {
"problem_match": "\n3. ",
"resource_path": "Balkan_MO/segmented/en-2018-BMO-type3.jsonl",
"solution_match": "\nSolution."
} | 105 | 805 |
2018 | T1 | 4 | null | Balkan_MO | Find all primes $p$ and $q$ such that $3 p^{q-1}+1$ divides $11^{p}+17^{p}$.
Time allowed: 4 hours and 30 minutes.
Each problem is worth 10 points. | Answer: $(p, q)=(3,3)$.
For $p=2$ it is directly checked that there are no solutions. Assume that $p>2$.
Observe that $N=11^{p}+17^{p} \equiv 4(\bmod 8)$, so $8 \nmid 3 p^{q-1}+1>4$. Consider an odd prime divisor $r$ of $3 p^{q-1}+1$. Obviously, $r \notin\{3,11,17\}$. There exists $b$ such that $17 b \equiv 1$ $(\bmod ... | {
"problem_match": "\n4. ",
"resource_path": "Balkan_MO/segmented/en-2018-BMO-type3.jsonl",
"solution_match": "\nSolution."
} | 58 | 845 |
2019 | T1 | 1 | null | Balkan_MO | Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f: \mathbb{P} \rightarrow \mathbb{P}$ such that
$$
f(p)^{f(q)}+q^{p}=f(q)^{f(p)}+p^{q}
$$
holds for all $p, q \in \mathbb{P}$.
Proposed by Albania | Obviously, the identical function $f(p)=p$ for all $p \in \mathbb{P}$ is a solution. We will show that this is the only one.
First we will show that $f(2)=2$. Taking $q=2$ and $p$ any odd prime number, we have
$$
f(p)^{f(2)}+2^{p}=f(2)^{f(p)}+p^{2}
$$
Assume that $f(2) \neq 2$. It follows that $f(2)$ is odd and so $... | {
"problem_match": "# Problem 1.",
"resource_path": "Balkan_MO/segmented/en-2019-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 86 | 564 |
2019 | T1 | 3 | null | Balkan_MO | Let $A B C$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $B C$ such that $\angle C A X=\angle Y A B$. Suppose that:
1) $K$ and $S$ are the feet of perpendiculars from $B$ to the lines $A X$ and $A Y$ respectively;
2) $T$ and $L$ are the feet of perpendiculars from $C$ to... | Denote $\phi=\widehat{X A B}=\widehat{Y A C}, \alpha=\widehat{C A X}=\widehat{B A Y}$. Then, because the quadrilaterals ABSK and ACTL are cyclic, we have
$$
\widehat{B S K}+\widehat{B A K}=180^{\circ}=\widehat{B S K}+\phi=\widehat{L A C}+\widehat{L T C}=\widehat{L T C}+\phi
$$
so, due to the 90-degree angles formed, ... | {
"problem_match": "# Problem 3.",
"resource_path": "Balkan_MO/segmented/en-2019-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 140 | 595 |
2019 | T1 | 4 | null | Balkan_MO | A grid consists of all points of the form $(m, n)$ where $m$ and $n$ are integers with $|m| \leqslant 2019$, $|n| \leqslant 2019$ and $|m|+|n|<4038$. We call the points $(m, n)$ of the grid with either $|m|=2019$ or $|n|=2019$ the boundary points. The four lines $x= \pm 2019$ and $y= \pm 2019$ are called boundary lines... | Anna does not have a winning strategy. We will provide a winning strategy for Bob. It is enough to describe his strategy for the deletions on the line $y=2019$.
Bob starts by deleting $(0,2019)$ and $(-1,2019)$. Once Anna completes her turn, he deletes the next two available points on the left if Anna decreased her $x... | {
"problem_match": "# Problem 4.",
"resource_path": "Balkan_MO/segmented/en-2019-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 282 | 1,130 |
2020 | T1 | 3 | null | Balkan_MO | Let $k$ be a positive integer. Determine the least integer $n \geqslant k+1$ for which the game below can be played indefinitely:
Consider $n$ boxes, labelled $b_{1}, b_{2}, \ldots, b_{n}$. For each index $i$, box $b_{i}$ contains initially exactly $i$ coins. At each step, the following three substeps are performed in... | The required minimum is $n=2^{k}+k-1$.
In this case the game can be played indefinitely by choosing the last $k+1$ boxes, $b_{2^{k}-1}, b_{2^{k}}, \ldots, b_{2^{k}+k-1}$, at each step: At step $r$, if box $b_{2^{k}+i-1}$ has exactly $m_{i}$ coins, then $\left\lceil m_{i} / 2\right\rceil$ coins are removed from that box... | {
"problem_match": "## 2020 BMO, Problem 3",
"resource_path": "Balkan_MO/segmented/en-2020-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 178 | 1,566 |
2020 | T1 | 4 | null | Balkan_MO | Let $a_{1}=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_{n}$ that has more positive divisors than $a_{n}$. Prove that $2 a_{n+1}=3 a_{n}$ only for finitely many indices $n$. | Begin with a mere remark on the terms of the sequence under consideration.
Lemma 1. Each $a_{n}$ is minimal amongst all positive integers having the same number of positive divisors as $a_{n}$.
Proof. Suppose, if possible, that for some $n$, some positive integer $b<a_{n}$ has as many positive divisors as $a_{n}$. The... | {
"problem_match": "## 2020 BMO, Problem 4",
"resource_path": "Balkan_MO/segmented/en-2020-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 76 | 1,461 |
2021 | T1 | 1 | null | Balkan_MO | Let $A B C$ be a triangle with $A B<A C$. Let $\omega$ be a circle passing through $B, C$ and assume that $A$ is inside $\omega$. Suppose $X, Y$ lie on $\omega$ such that $\angle B X A=\angle A Y C$. Suppose also that $X$ and $C$ lie on opposite sides of the line $A B$ and that $Y$ and $B$ lie on opposite sides of the ... | . Let $B^{\prime}$ and $C^{\prime}$ be the points of intersection of the lines $A B$ and $A C$ with $\omega$ respectively and let $\omega_{1}$ be the circumcircle of the triangle $A B^{\prime} C^{\prime}$. Let $\varepsilon$ be the tangent to $\omega_{1}$ at the point $A$. Because $A B<A C$ the lines $B^{\prime} C^{\pri... | {
"problem_match": "## BMO 2021 - Problem 1",
"resource_path": "Balkan_MO/segmented/en-2021-BMO-type1.jsonl",
"solution_match": "\nSolution 2"
} | 133 | 565 |
2021 | T1 | 2 | null | Balkan_MO | Find all functions $f:(0,+\infty) \rightarrow(0,+\infty)$ such that
$$
f(x+f(x)+f(y))=2 f(x)+y
$$
holds for all $x, y \in(0,+\infty)$. | . We will show that $f(x)=x$ for every $x \in \mathbb{R}^{+}$. It is easy to check that this function satisfies the equation.
We write $P(x, y)$ for the assertion that $f(x+f(x)+f(y))=2 f(x)+y$.
We first show that $f$ is injective. So assume $f(a)=f(b)$. Now $P(1, a)$ and $P(1, b)$ show that
$$
2 f(1)+a=f(1+f(1)+f(a)... | {
"problem_match": "## BMO 2021 - Problem 2",
"resource_path": "Balkan_MO/segmented/en-2021-BMO-type1.jsonl",
"solution_match": "\nSolution 1"
} | 60 | 565 |
2021 | T1 | 2 | null | Balkan_MO | Find all functions $f:(0,+\infty) \rightarrow(0,+\infty)$ such that
$$
f(x+f(x)+f(y))=2 f(x)+y
$$
holds for all $x, y \in(0,+\infty)$. | . As in Solution 1, $f$ is injective. Furthermore, letting $m=2 f(1)$ we have that the image of $f$ contains $(m, \infty)$. Indeed, if $t>m$, say $t=m+y$ for some $y>0$, then $P(1, y)$ shows that $f(1+f(1)+f(y))=t$.
Let $a, b \in \mathbb{R}$. We will show that $f(a)-a=f(b)-b$. Define $c=2 f(a)-2 f(b)$ and $d=a+f(a)-b-... | {
"problem_match": "## BMO 2021 - Problem 2",
"resource_path": "Balkan_MO/segmented/en-2021-BMO-type1.jsonl",
"solution_match": "\nSolution 2"
} | 60 | 658 |
2021 | T1 | 3 | null | Balkan_MO | Let $a, b$ and $c$ be positive integers satisfying the equation
$$
(a, b)+[a, b]=2021^{c} .
$$
If $|a-b|$ is a prime number, prove that the number $(a+b)^{2}+4$ is composite.
Here, $(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $[a, b]$ denotes the least common multiple of $a$ and $b$. | We write $p=|a-b|$ and assume for contradiction that $q=(a+b)^{2}+4$ is a prime number.
Since $(a, b) \mid[a, b]$, we have that $(a, b) \mid 2021^{c}$. As $(a, b)$ also divides $p=|a-b|$, it follows that $(a, b) \in\{1,43,47\}$. We will consider all 3 cases separately:
(1) If $(a, b)=1$, then $1+a b=2021^{c}$, and the... | {
"problem_match": "## BMO 2021 - Problem 3",
"resource_path": "Balkan_MO/segmented/en-2021-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 106 | 844 |
2021 | T1 | 4 | null | Balkan_MO | Angel has a warehouse, which initially contains 100 piles of 100 pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:
(a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.
However, every evening, a demon sneaks into the warehouse... | . We will show that he can do so by the morning of day 199 but not earlier.If we have $n$ piles with at least two pieces of rubbish and $m$ piles with exactly one piece of rubbish, then we define the value of the pile to be
$$
V= \begin{cases}n & m=0 \\ n+\frac{1}{2} & m=1 \\ n+1 & m \geqslant 2\end{cases}
$$
We also... | {
"problem_match": "## BMO 2021 - Problem 4",
"resource_path": "Balkan_MO/segmented/en-2021-BMO-type1.jsonl",
"solution_match": "\nSolution 1"
} | 127 | 1,022 |
2022 | T1 | 2 | null | Balkan_MO | Problem. Let $a, b$ and $n$ be positive integers with $a>b$ such that all of the following hold:
(i) $a^{2021}$ divides $n$,
(ii) $b^{2021}$ divides $n$,
(iii) 2022 divides $a-b$.
Prove that there is a subset $T$ of the set of positive divisors of the number $n$ such that the sum of the elements of $T$ is divisible by... | If $1011 \mid a$, then $1011^{2021} \mid n$ and we can take $T=\left\{1011,1011^{2}\right\}$. So we can assume that $3 \nmid a$ or $337 \nmid a$.
We continue with the following claim:
Claim. If $k$ is a positive integer, then $a^{k} b^{2021-k} \mid n$.
Proof of the Claim. We have that $n^{2021}=n^{k} \cdot n^{2021-k}$... | {
"problem_match": "# Problem 2",
"resource_path": "Balkan_MO/segmented/en-2022-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 127 | 521 |
2022 | T1 | 3 | null | Balkan_MO | Problem. Find all functions $f:(0, \infty) \rightarrow(0, \infty)$ such that
$$
f\left(y(f(x))^{3}+x\right)=x^{3} f(y)+f(x)
$$
for all $x, y>0$. | . Setting $y=\frac{t}{f(x)^{3}}$ we get
$$
f(x+t)=x^{3} f\left(\frac{t}{f(x)^{3}}\right)+f(x)
$$
for every $x, t>0$.
From (1) it is immediate that $f$ is increasing.
Claim. $f(1)=1$
Proof of Claim. Let $c=f(1)$. If $c<1$, taking $x=1$ and $y=\frac{1}{1-c^{3}}$ we have $y-y c^{3}=1$, so $y f(1)^{3}+1=y$ and $f\left(y ... | {
"problem_match": "# Problem 3",
"resource_path": "Balkan_MO/segmented/en-2022-BMO-type1.jsonl",
"solution_match": "\nSolution 1"
} | 64 | 701 |
2022 | T1 | 4 | null | Balkan_MO | Problem. Consider an $n \times n$ grid consisting of $n^{2}$ unit cells, where $n \geqslant 3$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Th... | . Consider an $n \times m$ grid with $n, m \geqslant 3$ being odd. We say that a column is of 'Type A' if, when partitioned into its monochromatic pieces, the first and last piece have the same colour with each one containing at least two cells. Otherwise we say that that it is of 'Type B'.
It is enough to show that t... | {
"problem_match": "# Problem 4",
"resource_path": "Balkan_MO/segmented/en-2022-BMO-type1.jsonl",
"solution_match": "\nSolution 2"
} | 195 | 1,812 |
2023 | T1 | 1 | null | Balkan_MO | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$,
$$
x f(x+f(y))=(y-x) f(f(x))
$$ | Answer: For any real $c, f(x)=c-x$ for all $x \in \mathbb{R}$ and $f(x)=0$ for all $x \in \mathbb{R}$.
Let $P(x, y)$ denote the proposition that $x$ and $y$ satisfy the given equation. $P(0,1)$ gives us $f(f(0))=0$.
From $P(x, x)$ we get that $x f(x+f(x))=0$ for all $x \in \mathbb{R}$, which together with $f(f(0))=$ ... | {
"problem_match": "\nProblem 1.",
"resource_path": "Balkan_MO/segmented/en-2023-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 54 | 711 |
2023 | T1 | 2 | null | Balkan_MO | In triangle $A B C$, the incircle touches sides $B C, C A, A B$ at $D, E, F$ respectively. Assume there exists a point $X$ on the line $E F$ such that
$$
\angle X B C=\angle X C B=45^{\circ} .
$$
Let $M$ be the midpoint of the arc $B C$ on the circumcircle of $A B C$ not containing $A$. Prove that the line $M D$ pass... | .
Let $I$ be the incenter of $\triangle A B C$ and let $K$ be the foot of the perpendicular from $D$ to $E F$. We begin by proving that $B K X C$ is cyclic, which can be done in two ways:
First Way. Note that $\angle K F D=90^{\circ}-\frac{\angle C}{2}$ and $\angle K E D=90^{\circ}-\frac{\angle B}{2}$, so by using $K... | {
"problem_match": "\nProblem 2.",
"resource_path": "Balkan_MO/segmented/en-2023-BMO-type1.jsonl",
"solution_match": "# Solution 2"
} | 119 | 993 |
2023 | T1 | 3 | null | Balkan_MO | For each positive integer $n$, denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$ ). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$, then $P(n)$ is also a positive integ... | Answer: All polynomials of the form $f(x)=x^{m}$ for some $m \in \mathbb{Z}^{+}$and $f(x)=c$ for some $c \in \mathbb{Z}^{+}$with $\omega(c) \leq 2023^{2023}+1$.
First of all we prove the following (well-known) Lemma. Lemma: Let $f(x)$ be a nonconstant polynomial with integer coefficients. Then, the number of primes $p... | {
"problem_match": "\nProblem 3.",
"resource_path": "Balkan_MO/segmented/en-2023-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 111 | 1,145 |
2023 | T1 | 4 | null | Balkan_MO | Find the greatest integer $k \leq 2023$ for which the following holds: whenever Alice colours exactly $k$ numbers of the set $\{1,2, \ldots, 2023\}$ in red, Bob can colour some of the remaining uncoloured numbers in blue, such that the sum of the red numbers is the same as the sum of the blue numbers. | Answer: 592.
For $k \geq 593$, Alice can color the greatest 593 numbers $1431,1432, \ldots, 2023$ and any other $(k-593)$ numbers so that their sum $s$ would satisfy
$$
s \geq \frac{2023 \cdot 2024}{2}-\frac{1430 \cdot 1431}{2}>\frac{1}{2} \cdot\left(\frac{2023 \cdot 2024}{2}\right),
$$
thus anyhow Bob chooses his nu... | {
"problem_match": "\nProblem 4.",
"resource_path": "Balkan_MO/segmented/en-2023-BMO-type1.jsonl",
"solution_match": "\nSolution."
} | 85 | 1,341 |
2024 | T1 | 4 | null | Balkan_MO | Let $\mathbb{R}^{+}=(0, \infty)$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0)=0$ which satisfy the equality
$$
f(f(x)+P(y))=f(x-y)+2 y
$$
for all real numbers $x>y>0$. | . Assume that $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$and the polynomial $P$ with non-negative coefficients and $P(0)=0$ satisfy the conditions of the problem. For positive reals with $x>y$, we shall write $Q(x, y)$ for the relation:
$$
f(f(x)+P(y))=f(x-y)+2 y
$$
1. Step 1. $f(x) \geq x$. Assume that this is no... | {
"problem_match": "\nProblem 4.",
"resource_path": "Balkan_MO/segmented/en-2024-BMO-type1.jsonl",
"solution_match": "\nSolution 1"
} | 107 | 1,306 |
2024 | T1 | 4 | null | Balkan_MO | Let $\mathbb{R}^{+}=(0, \infty)$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0)=0$ which satisfy the equality
$$
f(f(x)+P(y))=f(x-y)+2 y
$$
for all real numbers $x>y>0$. | . Assume that the function $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$and the polynomial with non-negative coefficients $P(y)=y P_{1}(y)$ satisfy the given equation. Fix $x=x_{0}>0$ and note that:
$$
f\left(f\left(x_{0}+y\right)+P(y)\right)=f\left(x_{0}+y-y\right)+2 y=f\left(x_{0}\right)+2 y
$$
Assume that $g=0$. ... | {
"problem_match": "\nProblem 4.",
"resource_path": "Balkan_MO/segmented/en-2024-BMO-type1.jsonl",
"solution_match": "\nSolution 2"
} | 107 | 1,938 |
2025 | T1 | 1 | null | Balkan_MO | An integer \(n > 1\) is called good if there exists a permutation \(a_{1},a_{2},a_{3},\ldots ,a_{n}\) of the numbers \(1,2,3,\ldots ,n\) , such that:
- \(a_{i}\) and \(a_{i + 1}\) have different parities for every \(1 \leqslant i \leqslant n - 1\) ;
- the sum \(a_{1} + a_{2} + \dots + a_{k}\) is a quadratic resid... | We will split the problem into two parts - the first one proving there are infinitely many numbers that are not good, and the second part proving there are infinitely many good numbers.
## Infinitely many numbers are not good
## Proof #1
We will show that all numbers \(n = 4^{m}\) with \(m \in \mathbb{Z}^{+}\) ar... | {
"problem_match": "\nProblem 1.",
"resource_path": "Balkan_MO/segmented/en-2025-BMO-type1.jsonl",
"solution_match": "# Solution "
} | 216 | 3,404 |
2025 | T1 | 2 | null | Balkan_MO | Let \(\triangle ABC\) be an acute- angled triangle with orthocentre \(H\) and let \(D\) be an arbitrary interior point on side \(BC\) . Suppose \(E\) and \(F\) are points on the segments \(AB\) and \(AC\) respectively such that the quadrilaterals \(ABDF\) and \(ACDE\) are cyclic, and let \(BF\) and \(CE\) intersect at ... | Let \(H_{A}\) and \(H_{C}\) be the feet of the altitudes from \(A\) and \(C\) , respectively. Also, let \(B A\) and \(B H\) meet \(C L\) at \(Z\) and \(T\) respectively.
We prove that \(B C P H\) is cyclic as in Solution 1. \(C H_{A}H C_{A}\) and \(C D E A\) are cyclic, so \(\angle B H_{A}H_{C} = \angle B D E = \angl... | {
"problem_match": "\nProblem 2.",
"resource_path": "Balkan_MO/segmented/en-2025-BMO-type1.jsonl",
"solution_match": "# Solution 3"
} | 157 | 516 |
2025 | T1 | 3 | null | Balkan_MO | Find all functions \(f: \mathbb{R} \to \mathbb{R}\) such that, for all real numbers \(x\) and \(y\),
\[f(x + y f(x)) + y = x y + f(x + y).\] | .
Let \(P(x,y)\) denote the given relation. If there is an \(a\in \mathbb{R}\) such that \(f(a) = 0\) , then \(P(a,y)\) gives that \(y = a y + f(a + y)\) , and so \(f\) must be linear. Then we can easily check and get that the only linear solutions are \(f(x) = x\) and \(f(x) = 2 - x\) ( \(x\in \mathbb{R}\) ).
Now ... | {
"problem_match": "\nProblem 3.",
"resource_path": "Balkan_MO/segmented/en-2025-BMO-type1.jsonl",
"solution_match": "# Solution 1"
} | 57 | 621 |
2025 | T1 | 3 | null | Balkan_MO | Find all functions \(f: \mathbb{R} \to \mathbb{R}\) such that, for all real numbers \(x\) and \(y\),
\[f(x + y f(x)) + y = x y + f(x + y).\] | .
Let \(P(x,y)\) denote the given relation. Similarly to the first solution, if a root exists ( \(f(a) = 0\) for any \(a\) ), we get that the function is linear and that the two solutions are \(f(x) = x\) and \(f(x) = 2 - x\) . Assertion \(P(x,c - x)\) gives us the following relation:
\[f(x + (c - x)f(x)) = (c - x)... | {
"problem_match": "\nProblem 3.",
"resource_path": "Balkan_MO/segmented/en-2025-BMO-type1.jsonl",
"solution_match": "# Solution 2"
} | 57 | 1,056 |
2025 | T1 | 3 | null | Balkan_MO | Find all functions \(f: \mathbb{R} \to \mathbb{R}\) such that, for all real numbers \(x\) and \(y\),
\[f(x + y f(x)) + y = x y + f(x + y).\] | . (by Stefan Šebez)
Let \(P(x,y)\) denote the given relation. Putting \(P(0,x + y)\) gives us:
\[f((x + y)f(0)) + x + y = 0 + f(x + y)\]
Subtracting this identity from the relation \(P(x,y)\) yields:
\[P(x + yf(x)) - P((x + y)f(0)) = xy + x\]
Suppose that \(f(x)\neq f(0)\) for some \(x\in \mathbf{R}\) (thus i... | {
"problem_match": "\nProblem 3.",
"resource_path": "Balkan_MO/segmented/en-2025-BMO-type1.jsonl",
"solution_match": "# Solution 4"
} | 57 | 686 |
2025 | T1 | 4 | null | Balkan_MO | There are \(n\) cities in a country, where \(n \geqslant 100\) is an integer. Some pairs of cities are connected by direct (two- way) flights. For two cities \(A\) and \(B\) we define:
- a path between \(A\) and \(B\) as a sequence of distinct cities \(A = C_{0}, C_{1}, \ldots , C_{k}, C_{k + 1} = B\) , \(k \geqslan... | Use the obvious graph interpretation. We show that any such graph is one of the following: the full graph \(K_{n}\) , the circular graph \(C_{n}\) , and for \(n\) even, the bipartite graph \(K_{\frac{n}{2}, \frac{n}{2}}\) . First, we show that these graphs satisfy the condition.
- For \(K_{n}\) , we can choose any lo... | {
"problem_match": "\nProblem 4.",
"resource_path": "Balkan_MO/segmented/en-2025-BMO-type1.jsonl",
"solution_match": "# Solution "
} | 318 | 1,827 |
2016 | T1 | A4 | Algebra | Balkan_Shortlist | The positive real numbers $a, b, c$ satisfy the equality $a+b+c=1$. For every natural number $n$ find the minimal possible value of the expression
$$
E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}
$$ | We transform the first term of the expression $E$ in the following way:
$$
\begin{aligned}
\frac{a^{-n}+b}{1-a}=\frac{1+a^{n} b}{a^{n}(b+c)}=\frac{a^{n+1}+a^{n} b+1-a^{n+1}}{a^{n}(b+c)} & =\frac{a^{n}(a+b)+(1-a)\left(1+a+a^{2}+\ldots+a^{n}\right)}{a^{n}(b+c)} \\
\frac{a^{n}(a+b)}{a^{n}(b+c)}+\frac{(b+c)\left(1+a+a^{2}... | {
"problem_match": "\n## A4.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 80 | 841 |
2016 | T1 | A5 | Algebra | Balkan_Shortlist | Let $a, b, c$ and $d$ be real numbers such that $a+b+c+d=2$ and $a b+b c+c d+d a+a c+b d=0$.
Find the minimum value and the maximum value of the product $a b c d$. | Let's find the minimum first.
$$
a^{2}+b^{2}+c^{2}+d^{2}=(a+b+c+d)^{2}-2(a b+b c+c d+d a+a c+b d)=4
$$
By AM-GM, $4=a^{2}+b^{2}+c^{2}+d^{2} \geq 4 \sqrt{|a b c d|} \Rightarrow 1 \geq|a b c d| \Rightarrow a b c d \geq-1$.
Note that if $a=b=c=1$ and $d=-1$, then $a b c d=-1$.
We'll find the maximum. We search for $a b ... | {
"problem_match": "\n## A5.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 60 | 838 |
2016 | T1 | A6 | Algebra | Balkan_Shortlist | Prove that there is no function from positive real numbers to itself, $f:(0,+\infty) \rightarrow(0,+\infty)$ such that:
$$
f(f(x)+y)=f(x)+3 x+y f(y) \quad \text {,for every } \quad x, y \in(0,+\infty)
$$ | First we prove that $f(x) \geq x$ for all $x>0$.
Indeed, if there is an $a>0$ with $f(a)<a$ then from the initial for $x=a$ and $y=a-f(a)>0$ we get that $3 a+(a-f(a)) f(a-f(a))=0$. This is absurd since $3 a+(a-f(a)) f(a-f(a))>0$.
So we have that
$$
f(x) \geq x \quad \text {,for all } \quad x>0
$$
Then using (1) we ha... | {
"problem_match": "\n## A6.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 76 | 756 |
2016 | T1 | A7 | Algebra | Balkan_Shortlist | Find all integers $n \geq 2$ for which there exist the real numbers $a_{k}, 1 \leq k \leq n$, which are satisfying the following conditions:
$$
\sum_{k=1}^{n} a_{k}=0, \sum_{k=1}^{n} a_{k}^{2}=1 \text { and } \sqrt{n} \cdot\left(\sum_{k=1}^{n} a_{k}^{3}\right)=2(b \sqrt{n}-1), \text { where } b=\max _{1 \leq k \leq n}... | We have: $\left(a_{k}+\frac{1}{\sqrt{n}}\right)^{2}\left(a_{k}-b\right) \leq 0 \Rightarrow\left(a_{k}^{2}+\frac{2}{\sqrt{n}} \cdot a_{k}+\frac{1}{n}\right)\left(a_{k}-b\right) \leq 0 \Rightarrow$
$$
a_{k}^{3} \leq\left(b-\frac{2}{\sqrt{n}}\right) \cdot a_{k}^{2}+\left(\frac{2 b}{\sqrt{n}}-\frac{1}{n}\right) \cdot a_{k... | {
"problem_match": "\nA7.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 148 | 1,259 |
2016 | T1 | A8 | Algebra | Balkan_Shortlist | Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ for which $f(g(n))-g(f(n))$ is independent on $n$ for any $g: \mathbb{Z} \rightarrow \mathbb{Z}$. | First observe that if $f(n)=n$, then $f(g(n))-g(f(n))=0$. Therefore the identity function satisfies the problem condition.
If there is $n_{0}$ with $f\left(n_{0}\right) \neq n_{0}$, consider the characteristic function $g$ that is defined as $g\left(f\left(n_{0}\right)\right)=1$ and $g(n)=0$ for $n \neq f\left(n_{0}\r... | {
"problem_match": "\n## A8.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 58 | 658 |
2016 | T1 | C1 | Combinatorics | Balkan_Shortlist | Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_{1}, b_{2}, \ldots, b_{K}$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_{1}, b_{2}, \ldots, b_{K}$. | Let a positive integer $d$ be given. We shall prove that, for each large enough $n$, the number $(n!)^{d-1}$ is a $d$-digit palindrome in all number bases $\frac{n!}{i}-1$ for $1 \leqslant i \leqslant n$. In particular, we shall prove that the digit expansion of $(n!)^{d-1}$ in the base $\frac{n!}{i}-1$ is
$$
\left\la... | {
"problem_match": "\n## C1.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 87 | 510 |
2016 | T1 | C2 | Combinatorics | Balkan_Shortlist | There are 2016 costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)
Find the maximal $k$ such that the following holds:
There are $k$ customers such that either all ... | We show that the maximal $k$ is 45 .
First we show that no larger $k$ can be achieved: We break the day at 45 disjoint time intervals and assume that at each time interval there were exactly 45 costumers who stayed in the shop only during that time interval (except in the last interval in which there were only 36 custo... | {
"problem_match": "\n## C2.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 99 | 537 |
2016 | T1 | C3 | Combinatorics | Balkan_Shortlist | The plane is divided into unit squares by means of two sets of parallel lines. The unit squares are coloured in 1201 colours so that no rectangle of perimeter 100 contains two squares of the same colour. Show that no rectangle of size $1 \times 1201$ contains two squares of the same colour. | Let the centers of the unit squares be the integer points in the plane, and denote each unit square by the coordinates of its center.
Consider the set $D$ of all unit squares $(x, y)$ such that $|x|+|y| \leq 24$. Any translate of $D$ is called a diamond.
Since any two unit squares that belong to the same diamond also... | {
"problem_match": "\n## C3.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 70 | 558 |
2016 | T1 | G1 | Geometry | Balkan_Shortlist | The point $M$ lies on the side $A B$ of the circumscribed quadrilateral $A B C D$. The points $I_{1}, I_{2}$, and $I_{3}$ are the incenters of $\triangle M B C, \triangle M C D$, and $\triangle M D A$. Show that the points $M, I_{1}, I_{2}$, and $I_{3}$ lie on a circle.
![](https://cdn.mathpix.com/cropped/2024_12_07_3d... | Lemma. Let $I$ be the incenter of $\triangle A B C$ and let the points $P$ and $Q$ lie on the lines $A B$ and $A C$. Then the points $A, I, P$, and $Q$ lie on a circle if and only if
$$
\overline{B P}+\overline{C Q}=B C
$$
where $\overline{B P}$ equals $|B P|$ if $P$ lies in the ray $B A \rightarrow$ and $-|B P|$ if ... | {
"problem_match": "\n## G1.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 170 | 1,153 |
2016 | T1 | G3 | Geometry | Balkan_Shortlist | Given that $A B C$ is a triangle where $A B<A C$. On the half-lines $B A$ and $C A$ we take points $F$ and $E$ respectively such that $B F=C E=B C$. Let $M, N$ and $H$ be the mid-points of the segments $B F, C E$ and $B C$ respectively and $K$ and $O$ be the circumcircles of the triangles $A B C$ and $M N H$ respective... | The circumcenter of the triangle $\triangle M N H$ coincides with the incentre of the triangle $\triangle A B C$ because the triangles $\triangle B M H$ and $\triangle N H C$ are isosceles and therefore the perpendiculars of the $M H, H N$ are also the bisectors of the angles $\angle A B C, \angle A C B$, respectively.... | {
"problem_match": "\nG3.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 248 | 719 |
2016 | T1 | N2 | Number Theory | Balkan_Shortlist | Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$ $(d(n)$ is the number of divisors of the number $n$ including 1 and $n$ ). | From $d(n)\left|n, \frac{n}{d(n)}\right| n$ one obtains $\frac{n}{d(n)} \leq d(n)$.
Let $n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{s}^{\alpha_{s}}$ where $p_{i}, 1 \leq i \leq s$ are prime numbers. The number $n$ is odd from where we get $p_{i}>2,1 \leq i \leq s$. The multiplicativity of the function $d(n)$ imp... | {
"problem_match": "\nN2.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 53 | 638 |
2016 | T1 | N5 | Number Theory | Balkan_Shortlist | A positive integer $n$ is downhill if its decimal representation $\overline{a_{k} a_{k-1} \ldots a_{0}}$ satisfies $a_{k} \geq a_{k-1} \geq \ldots \geq a_{0}$. A real-coefficient polynomial $P$ is integer-valued if $P(n)$ is an integer for all integer $n$, and downhill-integervalued if $P(n)$ is an integer for all down... | No, it is not.
A downhill number can always be written as $a-b_{1}-b_{2}-\ldots-b_{9}$, where $a$ is of the form $99 \ldots 99$ and each $b_{i}$ either equals 0 or is of the form $\overline{11 \ldots 11}$.
Let $n$ be a positive integer. The numbers of the form $99 \ldots 99$ yield at most $n$ different remainders upon... | {
"problem_match": "\nN5.",
"resource_path": "Balkan_Shortlist/segmented/en-2016_bmo_shortlist.jsonl",
"solution_match": "\nSolution."
} | 128 | 651 |
2017 | T1 | A1 | Algebra | Balkan_Shortlist | Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that
$$
\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+a^{5}+b^{2}} \leq 1 .
$$ | First we remark that
$$
a^{5}+b^{5} \geq a b\left(a^{3}+b^{3}\right)
$$
Indeed
$$
\begin{aligned}
a^{5}+b^{5} \geq a b\left(a^{3}+b^{3}\right) & \Leftrightarrow a^{5}-a^{4} b-a b^{4}+b^{5} \geq 0 \\
& \Leftrightarrow a^{4}(a-b)-b^{4}(a-b) \geq 0 \\
(a-b)\left(a^{4}-b^{4}\right) \geq 0 & \Leftrightarrow(a-b)^{2}\left... | {
"problem_match": "\n## A1",
"resource_path": "Balkan_Shortlist/segmented/en-2017_bmo_shortlist.jsonl",
"solution_match": "\n## Solution"
} | 91 | 541 |
2017 | T1 | A3 | Algebra | Balkan_Shortlist | Find all the functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that:
$$
n+f(m) \mid f(n)+n f(m)
$$
for any $m, n \in \mathbb{N}$ | We will consider 2 cases, whether the range of the functions is infinite or finite or in other words the function take infinite or finite values.
Case 1. The Function has an infinite range. Let's fix a random natural number $n$ and let $m$ be any natural number. Then using (1) we have
$$
n+f(m)\left|f(n)+n f(m)=f(n)-n... | {
"problem_match": "\n## A3",
"resource_path": "Balkan_Shortlist/segmented/en-2017_bmo_shortlist.jsonl",
"solution_match": "\n## Solution"
} | 55 | 827 |
2017 | T1 | A4 | Algebra | Balkan_Shortlist | Let $M=\left\{(a, b, c) \in \mathbb{R}^{3}: 0<a, b, c<\frac{1}{2}\right.$ with $\left.a+b+c=1\right\}$ and $f: M \rightarrow \mathbb{R}$ given as
$$
f(a, b, c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{a b c}
$$
Find the best (real) bounds $\alpha$ and $\beta$ such that
$$
f(M)=\{f(a, b, c):(a, b, c... | Let $\forall(a, b, c) \in M, \alpha \leq f(a, b, c) \leq \beta$ and supose that there are no better bounds, i.e. $\alpha$ is the largest possible and $\beta$ is the smallest possible. Now,
$$
\begin{aligned}
\alpha \leq f(a, b, c) \leq \beta & \Leftrightarrow \alpha a b c \leq 4(a b+b c+c a)-1 \leq \beta a b c \\
& \L... | {
"problem_match": "\nA4",
"resource_path": "Balkan_Shortlist/segmented/en-2017_bmo_shortlist.jsonl",
"solution_match": "\n## Solution"
} | 176 | 821 |
2017 | T1 | A5 | Algebra | Balkan_Shortlist | Consider integers $m \geq 2$ and $n \geq 1$. Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0), P(1), \ldots, P(n)$ are all perfect powers of $m$. | Let $a_{0}, a_{1}, \ldots, a_{n}$ be integers to be chosen later, and consider the polynomial $P(x)=\frac{1}{n!} Q(x)$ where
$$
Q(x)=\sum_{k=0}^{n}(-1)^{n-k}\binom{n}{k} a_{k} \prod_{\substack{0 \leq i \leq n \\ i \neq k}}(x-i) .
$$
Observe that for $l \in\{0,1, \ldots, n\}$ we have
$$
\begin{aligned}
P(l) & =\frac{... | {
"problem_match": "\n## A5",
"resource_path": "Balkan_Shortlist/segmented/en-2017_bmo_shortlist.jsonl",
"solution_match": "\n## Solution"
} | 65 | 1,195 |
2017 | T1 | A6 | Algebra | Balkan_Shortlist | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying
$$
f\left(x+y f\left(x^{2}\right)\right)=f(x)+x f(x y)
$$
for all real numbers $x$ and $y$. | Let $P(x, y)$ be the assertion $f\left(x+y f\left(x^{2}\right)\right)=f(x)+x f(x y) . P(1,0)$ yields $f(0)=0$. If there exists $x_{0} \neq 0$ satisfying $f\left(x_{0}^{2}\right)=0$, then considering $P\left(x_{0}, y\right)$, we get $f\left(x_{0} y\right)=0$ for all $y \in \mathbb{R}$. In this case, since $x_{0} \neq 0$... | {
"problem_match": "\nA6",
"resource_path": "Balkan_Shortlist/segmented/en-2017_bmo_shortlist.jsonl",
"solution_match": "\n## Solution"
} | 61 | 1,728 |
2017 | T1 | NT2 | Number Theory | Balkan_Shortlist | Find all functions $f: \mathbf{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that the number $x f(x)+f^{2}(y)+2 x f(y)$ is a perfect square for all positive integers $x, y$. | Let $p$ be a prime number. Then for $x=y=p$ the given condition gives us that the number $f^{2}(p)+3 p f(p)$ is a perfect square. Then, $f^{2}(p)+3 p f(p)=k^{2}$ for some positive integer $k$. Completing the square gives us that $(2 f(p)+3 p)^{2}-9 p^{2}=4 k^{2}$, or
$$
(2 f(p)+3 p-2 k)(2 f(p)+3 p+2 k)=9 p^{2} .
$$
S... | {
"problem_match": "\n## NT2",
"resource_path": "Balkan_Shortlist/segmented/en-2017_bmo_shortlist.jsonl",
"solution_match": "\n## Solution"
} | 59 | 663 |
2017 | T1 | G1 | Geometry | Balkan_Shortlist | Let $A B C$ be an acute triangle. Variable points $E$ and $F$ are on sides $A C$ and $A B$ respectively such that $B C^{2}=B A \cdot B F+C E \cdot C A$. As $E$ and $F$ vary prove that the circumcircle of $A E F$ passes through a fixed point other than $A$. | 1
Let $H$ be the ortocenter of $A B C$ and $K, L, M$ be the feet of perpendiculars respectively from $A, B, C$ to their opposite sides of $A B C$. Also let $D$ be the intersection point of lines $B E$ and $C F$. From power of point we have
$$
B A \cdot B M=B C \cdot B K
$$
and
$$
C A \cdot C L=C B \cdot C K
$$
Add... | {
"problem_match": "\n## G1",
"resource_path": "Balkan_Shortlist/segmented/en-2017_bmo_shortlist.jsonl",
"solution_match": "\n## Solution"
} | 84 | 737 |
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