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 | T0 | N3 | Number Theory | IMO-SL | Let $n$ be an integer with $n \geqslant 2$. Does there exist a sequence $\left(a_{1}, \ldots, a_{n}\right)$ of positive integers with not all terms being equal such that the arithmetic mean of every two terms is equal to the geometric mean of some (one or more) terms in this sequence? (Estonia) | Suppose that $a_{1}, \ldots, a_{n}$ satisfy the required properties. Let $d=\operatorname{gcd}\left(a_{1} \ldots, a_{n}\right)$. If $d>1$ then replace the numbers $a_{1}, \ldots, a_{n}$ by $\frac{a_{1}}{d}, \ldots, \frac{a_{n}}{d}$; all arithmetic and all geometric means will be divided by $d$, so we obtain another seq... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
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} | 80 | 769 |
2020 | T0 | N3 | Number Theory | IMO-SL | Let $n$ be an integer with $n \geqslant 2$. Does there exist a sequence $\left(a_{1}, \ldots, a_{n}\right)$ of positive integers with not all terms being equal such that the arithmetic mean of every two terms is equal to the geometric mean of some (one or more) terms in this sequence? (Estonia) | Like in the previous solution, we assume that the numbers $a_{1}, \ldots, a_{n}$ have no common divisor greater than 1 . The arithmetic mean of any two numbers in the sequence is half of an integer; on the other hand, it is a (some integer order) root of an integer. This means each pair's mean is an integer, so all ter... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
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} | 80 | 1,226 |
2020 | T0 | N4 | Number Theory | IMO-SL | For any odd prime $p$ and any integer $n$, let $d_{p}(n) \in\{0,1, \ldots, p-1\}$ denote the remainder when $n$ is divided by $p$. We say that $\left(a_{0}, a_{1}, a_{2}, \ldots\right)$ is a $p$-sequence, if $a_{0}$ is a positive integer coprime to $p$, and $a_{n+1}=a_{n}+d_{p}\left(a_{n}\right)$ for $n \geqslant 0$. (... | Fix some odd prime $p$, and let $T$ be the smallest positive integer such that $p \mid 2^{T}-1$; in other words, $T$ is the multiplicative order of 2 modulo $p$. Consider any $p$-sequence $\left(x_{n}\right)=\left(x_{0}, x_{1}, x_{2}, \ldots\right)$. Obviously, $x_{n+1} \equiv 2 x_{n}(\bmod p)$ and therefore $x_{n} \eq... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 331 | 1,600 |
2020 | T0 | N5 | Number Theory | IMO-SL | Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: (i) $f(n) \neq 0$ for at least one $n$; (ii) $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; (iii) there are infinitely many positive integers $n$ such that $f(k)=... | If a number $n$ is a product of primes, $n=p_{1} p_{2} \cdot \ldots \cdot p_{k}$, then $$ f(n)=f\left(p_{1}\right)+\ldots+f\left(p_{k}\right) $$ in particular, $f(1)=0($ since $f(1)=f(1)+f(1))$. It is also clear that $f(n)=0$ implies $f(p)=0$ for all primes $p$ dividing $n$. Let us call positive integer $n \operatornam... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 107 | 608 |
2020 | T0 | N5 | Number Theory | IMO-SL | Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: (i) $f(n) \neq 0$ for at least one $n$; (ii) $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; (iii) there are infinitely many positive integers $n$ such that $f(k)=... | We use the notion of a good number from the previous solution. As above, we also denote by $\nu_{p}(n)$ the exponent of a prime $p$ in the prime decomposition of $n$. Say that a positive integer $k$ is big if $f(k)>0$. Let $\mathcal{B}$ be the set of big primes, and let $p_{1}<p_{2}<\ldots$ list the elements of $\mathc... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
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} | 107 | 749 |
2020 | T0 | N5 | Number Theory | IMO-SL | Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: (i) $f(n) \neq 0$ for at least one $n$; (ii) $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; (iii) there are infinitely many positive integers $n$ such that $f(k)=... | We call a function $f: \mathbb{N} \rightarrow \mathbb{N}_{0}$ satisfying (ii) additive. We call a pair $(f, n)$, where $f$ is an additive function and $n \in \mathbb{N}$, good, if for all $k<n$ it holds $f(k)=f(n-k)$. For an additive function $f$ and a prime number $p$ the number $\frac{f(p)}{\ln p}$ is denoted by $g(f... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
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} | 107 | 801 |
2020 | T0 | N6 | Number Theory | IMO-SL | For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that $$ \frac{\varphi(d(n))}{d(\varphi(n))} \leqslant C $$ for all $n \geqslant 1$ ? (Cyprus) | Fix $N>1$, let $p_{1}, \ldots, p_{k}$ be all primes between 1 and $N$ and $p_{k+1}, \ldots, p_{k+s}$ be all primes between $N+1$ and $2 N$. Since for $j \leqslant k+s$ all prime divisors of $p_{j}-1$ do not exceed $N$, we have $$ \prod_{j=1}^{k+s}\left(p_{j}-1\right)=\prod_{i=1}^{k} p_{i}^{c_{i}} $$ with some fixed exp... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 103 | 1,380 |
2020 | T0 | N6 | Number Theory | IMO-SL | For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that $$ \frac{\varphi(d(n))}{d(\varphi(n))} \leqslant C $$ for all $n \geqslant 1$ ? (Cyprus) | In this solution we will use the Prime Number Theorem which states that $$ \pi(m)=\frac{m}{\log m} \cdot(1+o(1)) $$ as $m$ tends to infinity. Here and below $\pi(m)$ denotes the number of primes not exceeding $m$, and $\log$ the natural logarithm. Let $m>5$ be a large positive integer and let $n:=p_{1} p_{2} \cdot \ldo... | {
"problem_match": null,
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} | 103 | 1,020 |
2020 | T0 | N7 | Number Theory | IMO-SL | Let $\mathcal{S}$ be a set consisting of $n \geqslant 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_{1}, a_{2}, \ldots, a_{n}$ so that $a_{i}$ does not divide $a_{i-1}+a_{i+1}$ for all $i=2,3, \ldots, n-1$. (Ukr... | We prove the following stronger statement. Claim. Let $\mathcal{S}$ be a good set consisting of $n \geqslant 2$ positive integers. Then the elements of $\mathcal{S}$ may be ordered as $a_{1}, a_{2}, \ldots, a_{n}$ so that $a_{i} \nmid a_{i-1}+a_{i+1}$ and $a_{i} \nmid a_{i-1}-a_{i+1}$, for all $i=2,3, \ldots, n-1$. Pro... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
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} | 122 | 873 |
2020 | T0 | N7 | Number Theory | IMO-SL | Let $\mathcal{S}$ be a set consisting of $n \geqslant 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_{1}, a_{2}, \ldots, a_{n}$ so that $a_{i}$ does not divide $a_{i-1}+a_{i+1}$ for all $i=2,3, \ldots, n-1$. (Ukr... | We again prove a stronger statement. Claim. Let $\mathcal{S}$ be an arbitrary set of $n \geqslant 3$ positive integers. Then its elements can be ordered as $a_{1}, \ldots, a_{n}$ so that, if $a_{i} \mid a_{i-1}+a_{i+1}$, then $a_{i}=\max \mathcal{S}$. The claim easily implies what we need to prove, due to Observation A... | {
"problem_match": null,
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} | 122 | 1,079 |
2020 | T0 | A1 | Algebra | IMO-SL | Version 1. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, $$ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x $$ Version 2. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, $$ \sq... | 2 (for Version 2). Here we present another proof of the inequality (2) for $x>0$, or, equivalently, for $t=(x-1)^{2} / x \geqslant 0$. Instead of finding the coefficients of the polynomial $f_{N}=f_{N}(t)$ we may find its roots, which is in a sense more straightforward. Note that the recurrence (4) and the initial cond... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 815 | 722 |
2020 | T0 | A1 | Algebra | IMO-SL | Version 1. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, $$ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x $$ Version 2. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, $$ \sq... | 3 (for Version 2). Here we solve the problem when $N \geqslant 1$ is an arbitrary real number. For a real number $a$ let $$ f(x)=\left(\frac{x^{2 N}+1}{2}\right)^{\frac{1}{N}}-a(x-1)^{2}-x $$ Then $f(1)=0$, $$ f^{\prime}(x)=\left(\frac{x^{2 N}+1}{2}\right)^{\frac{1}{N}-1} x^{2 N-1}-2 a(x-1)-1 \quad \text { and } \quad ... | {
"problem_match": null,
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} | 815 | 573 |
2020 | T0 | A2 | Lemma. | IMO-SL | Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as $$ (x+y+z) P(x, y, z)+(x y+y z+z x) Q(x, y, z)+x y z R(x, y, z) $$ with $P, Q, R \in \mathcal{A}$. Find t... | We start by showing that $n \leqslant 4$, i.e., any monomial $f=x^{i} y^{j} z^{k}$ with $i+j+k \geqslant 4$ belongs to $\mathcal{B}$. Assume that $i \geqslant j \geqslant k$, the other cases are analogous. Let $x+y+z=p, x y+y z+z x=q$ and $x y z=r$. Then $$ 0=(x-x)(x-y)(x-z)=x^{3}-p x^{2}+q x-r $$ therefore $x^{3} \in ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 179 | 2,349 |
2020 | T0 | A3 | Lemma. | IMO-SL | Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d)=a c+b d$. Find the smallest possible value of $$ S=\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} $$ (Israel) Answer: The smallest possible value is 8. | By homogeneity we may suppose that $a b c d=1$. Let $a b=C, b c=A$ and $c a=B$. Then $a, b, c$ can be reconstructed from $A, B$ and $C$ as $a=\sqrt{B C / A}, b=\sqrt{A C / B}$ and $c=\sqrt{A B / C}$. Moreover, the condition $(a+c)(b+d)=a c+b d$ can be written in terms of $A, B, C$ as $$ A+\frac{1}{A}+C+\frac{1}{C}=b c+... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 81 | 703 |
2020 | T0 | A5 | Lemma. | IMO-SL | A magician intends to perform the following trick. She announces a positive integer $n$, along with $2 n$ real numbers $x_{1}<\ldots<x_{2 n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2 n$ values $P\left(x_{1}\right), \ldots,... | Let $x_{1}<x_{2}<\ldots<x_{2 n}$ be real numbers chosen by the magician. We will construct two distinct polynomials $P(x)$ and $Q(x)$, each of degree $n$, such that the member of audience will write down the same sequence for both polynomials. This will mean that the magician cannot distinguish $P$ from $Q$. Claim. The... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 153 | 773 |
2020 | T0 | A6 | Lemma. | IMO-SL | Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $$ f^{a^{2}+b^{2}}(a+b)=a f(a)+b f(b) \quad \text { for every } a, b \in \mathbb{Z} $$ Here, $f^{n}$ denotes the $n^{\text {th }}$ iteration of $f$, i.e., $f^{0}(x)=x$ and $f^{n+1}(x)=f\left(f^{n}(x)\right)$ for all $n \geqslant 0$. (Slovakia) A... | Refer to the main equation as $E(a, b)$. $E(0, b)$ reads as $f^{b^{2}}(b)=b f(b)$. For $b=-1$ this gives $f(-1)=0$. Now $E(a,-1)$ reads as $$ f^{a^{2}+1}(a-1)=a f(a)=f^{a^{2}}(a) . $$ For $x \in \mathbb{Z}$ define the orbit of $x$ by $\mathcal{O}(x)=\{x, f(x), f(f(x)), \ldots\} \subseteq \mathbb{Z}$. We see that the or... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 186 | 1,483 |
2020 | T0 | A7 | Lemma. | IMO-SL | Let $n$ and $k$ be positive integers. Prove that for $a_{1}, \ldots, a_{n} \in\left[1,2^{k}\right]$ one has $$ \sum_{i=1}^{n} \frac{a_{i}}{\sqrt{a_{1}^{2}+\ldots+a_{i}^{2}}} \leqslant 4 \sqrt{k n} $$ (Iran) | Partition the set of indices $\{1,2, \ldots, n\}$ into disjoint subsets $M_{1}, M_{2}, \ldots, M_{k}$ so that $a_{\ell} \in\left[2^{j-1}, 2^{j}\right]$ for $\ell \in M_{j}$. Then, if $\left|M_{j}\right|=: p_{j}$, we have $$ \sum_{\ell \in M_{j}} \frac{a_{\ell}}{\sqrt{a_{1}^{2}+\ldots+a_{\ell}^{2}}} \leqslant \sum_{i=1}... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 100 | 1,007 |
2020 | T0 | A7 | Lemma. | IMO-SL | Let $n$ and $k$ be positive integers. Prove that for $a_{1}, \ldots, a_{n} \in\left[1,2^{k}\right]$ one has $$ \sum_{i=1}^{n} \frac{a_{i}}{\sqrt{a_{1}^{2}+\ldots+a_{i}^{2}}} \leqslant 4 \sqrt{k n} $$ (Iran) | Apply induction on $n$. The base $n \leqslant 16$ is clear: our sum does not exceed $n \leqslant 4 \sqrt{n k}$. For the inductive step from $1, \ldots, n-1$ to $n \geqslant 17$ consider two similar cases. Case 1: $n=2 t$. Let $x_{\ell}=\frac{a_{\ell}}{\sqrt{a_{1}^{2}+\ldots+a_{\ell}^{2}}}$. We have $$ \exp \left(-x_{t+... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 100 | 652 |
2020 | T0 | A8 | Lemma. | IMO-SL | 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 $y$, $$ f(x+f(x y))+y=f(x) f(y)+1 $$ (Ukraine) Answer: $f(x)=x+1$. | A straightforward check shows that $f(x)=x+1$ satisfies (*). We divide the proof of the converse statement into a sequence of steps. Step 1: $f$ is injective. Put $x=1$ in (*) and rearrange the terms to get $$ y=f(1) f(y)+1-f(1+f(y)) $$ Therefore, if $f\left(y_{1}\right)=f\left(y_{2}\right)$, then $y_{1}=y_{2}$. Step 2... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 91 | 850 |
2020 | T0 | A8 | Lemma. | IMO-SL | 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 $y$, $$ f(x+f(x y))+y=f(x) f(y)+1 $$ (Ukraine) Answer: $f(x)=x+1$. | We provide another proof that $f(x)=x+1$ is the only function satisfying (*). Put $a=f(1)$. Define the function $\phi: \mathbb{R}^{+} \rightarrow \mathbb{R}$ by $$ \phi(x)=f(x)-x-1 $$ Then equation (*) reads as $$ \phi(x+f(x y))=f(x) f(y)-f(x y)-x-y $$ Since the right-hand side of (1) is symmetric under swapping $x$ an... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 91 | 2,356 |
2020 | T0 | C1 | Combinatorics | IMO-SL | Let $n$ be a positive integer. Find the number of permutations $a_{1}, a_{2}, \ldots, a_{n}$ of the sequence $1,2, \ldots, n$ satisfying $$ a_{1} \leqslant 2 a_{2} \leqslant 3 a_{3} \leqslant \ldots \leqslant n a_{n} $$ (United Kingdom) Answer: The number of such permutations is $F_{n+1}$, where $F_{k}$ is the $k^{\... | Denote by $P_{n}$ the number of permutations that satisfy (*). It is easy to see that $P_{1}=1$ and $P_{2}=2$. Lemma 1. Let $n \geqslant 3$. If a permutation $a_{1}, \ldots, a_{n}$ satisfies (*) then either $a_{n}=n$, or $a_{n-1}=n$ and $a_{n}=n-1$. Proof. Let $k$ be the index for which $a_{k}=n$. If $k=n$ then we are ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 164 | 778 |
2020 | T0 | C1 | Combinatorics | IMO-SL | Let $n$ be a positive integer. Find the number of permutations $a_{1}, a_{2}, \ldots, a_{n}$ of the sequence $1,2, \ldots, n$ satisfying $$ a_{1} \leqslant 2 a_{2} \leqslant 3 a_{3} \leqslant \ldots \leqslant n a_{n} $$ (United Kingdom) Answer: The number of such permutations is $F_{n+1}$, where $F_{k}$ is the $k^{\... | We claim that all sought permutations are of the following kind. Split $\{1,2, \ldots, n\}$ into singletons and pairs of adjacent numbers. In each pair, swap the two numbers and keep the singletons unchanged. Such permutations correspond to tilings of a $1 \times n$ chessboard using dominoes and unit squares; it is wel... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 164 | 822 |
2020 | T0 | C3 | Combinatorics | IMO-SL | 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 o... | 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 $k=n^{2}-n$. Let company $A$ connect the pairs of checkpoints of the form $(i, i+1)$, where $n \nmid i$. Then all pairs of checkpoints $(i, j)$ linke... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 247 | 724 |
2020 | T0 | C4 | Combinatorics | IMO-SL | 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... | First we show that if a set $S \subset \mathbb{Z}$ satisfies the conditions then $|S| \geqslant \frac{n}{2}+1$. Let $d=\lceil n / 2\rceil$, 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 $S$. For each $1 \leq... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 151 | 815 |
2020 | T0 | C8 | Combinatorics | IMO-SL | 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... | 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 $2020=\overline{11111100100}_{2}$... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 206 | 2,946 |
2020 | T0 | G5 | Geometry | IMO-SL | Let $A B C D$ be a cyclic quadrilateral with no two sides parallel. Let $K, L, M$, and $N$ be points lying on sides $A B, B C, C D$, and $D A$, respectively, such that $K L M N$ is a rhombus with $K L \| A C$ and $L M \| B D$. Let $\omega_{1}, \omega_{2}, \omega_{3}$, and $\omega_{4}$ be the incircles of triangles $A ... | Let $I_{i}$ be the center of $\omega_{i}$, and let $r_{i}$ be its radius for $i=1,2,3,4$. Denote by $T_{1}$ and $T_{3}$ the points of tangency of $\omega_{1}$ and $\omega_{3}$ with $N K$ and $L M$, respectively. Suppose that the internal common tangents to $\omega_{1}$ and $\omega_{3}$ meet at point $S$, which is the c... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 178 | 2,239 |
2020 | T0 | G5 | Geometry | IMO-SL | Let $A B C D$ be a cyclic quadrilateral with no two sides parallel. Let $K, L, M$, and $N$ be points lying on sides $A B, B C, C D$, and $D A$, respectively, such that $K L M N$ is a rhombus with $K L \| A C$ and $L M \| B D$. Let $\omega_{1}, \omega_{2}, \omega_{3}$, and $\omega_{4}$ be the incircles of triangles $A ... | This solution is based on the following general Lemma. Lemma 2. Let $E$ and $F$ be distinct points, and let $\omega_{i}, i=1,2,3,4$, be circles lying in the same halfplane with respect to $E F$. For distinct indices $i, j \in\{1,2,3,4\}$, denote by $O_{i j}^{+}$ (respectively, $O_{i j}^{-}$) the center of homothety wit... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 178 | 934 |
2020 | T0 | G6 | Geometry | IMO-SL | Let $I$ and $I_{A}$ be the incenter and the $A$-excenter of an acute-angled triangle $A B C$ with $A B<A C$. Let the incircle meet $B C$ at $D$. The line $A D$ meets $B I_{A}$ and $C I_{A}$ at $E$ and $F$, respectively. Prove that the circumcircles of triangles $A I D$ and $I_{A} E F$ are tangent to each other. (Slova... | We use the notation of circles $\Gamma, \omega$, and $\Omega$ as in the previous solution. Let $L$ be the point opposite to $I$ in circle $\Omega$. Then $\angle I A L=\angle I D L=90^{\circ}$, which means that $L$ is the foot of the external bisector of $\angle A$ in triangle $A B C$. Let $L I$ cross $\Gamma$ again at ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 118 | 840 |
2020 | T0 | G6 | Geometry | IMO-SL | Let $I$ and $I_{A}$ be the incenter and the $A$-excenter of an acute-angled triangle $A B C$ with $A B<A C$. Let the incircle meet $B C$ at $D$. The line $A D$ meets $B I_{A}$ and $C I_{A}$ at $E$ and $F$, respectively. Prove that the circumcircles of triangles $A I D$ and $I_{A} E F$ are tangent to each other. (Slova... | We also use the notation of circles $\omega$, and $\Omega$ from the previous solutions. Perform an inversion centered at $D$. The images of the points will be denoted by primes, e.g., $A^{\prime}$ is the image of $A$. For convenience, we use the notation $\angle B I D=\beta, \angle C I D=\gamma$, and $\alpha=180^{\circ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 118 | 1,998 |
2020 | T0 | G7 | Geometry | IMO-SL | Let $P$ be a point on the circumcircle of an acute-angled triangle $A B C$. Let $D$, $E$, and $F$ be the reflections of $P$ in the midlines of triangle $A B C$ parallel to $B C, C A$, and $A B$, respectively. Denote by $\omega_{A}, \omega_{B}$, and $\omega_{C}$ the circumcircles of triangles $A D P, B E P$, and $C F P... | Let $A A_{1}, B B_{1}$, and $C C_{1}$ be the altitudes in triangle $A B C$, and let $m_{A}, m_{B}$, and $m_{C}$ be the midlines parallel to $B C, C A$, and $A B$, respectively. We always denote by $\Varangle(p, q)$ the directed angle from a line $p$ to a line $q$, taken modulo $180^{\circ}$. Step 1: Circles $\omega_{A}... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 175 | 1,496 |
2020 | T0 | G8 | Geometry | IMO-SL | Let $\Gamma$ and $I$ be the circumcircle and the incenter of an acute-angled triangle $A B C$. Two circles $\omega_{B}$ and $\omega_{C}$ passing through $B$ and $C$, respectively, are tangent at $I$. Let $\omega_{B}$ meet the shorter arc $A B$ of $\Gamma$ and segment $A B$ again at $P$ and $M$, respectively. Similarly... | Let $A I, B I$, and $C I$ meet $\Gamma$ again at $D, E$, and $F$, respectively. Let $\ell$ be the common tangent to $\omega_{B}$ and $\omega_{C}$ at $I$. We always denote by $\Varangle(p, q)$ the directed angle from a line $p$ to a line $q$, taken modulo $180^{\circ}$. Step 1: We show that $Y$ lies on $\Gamma$. Recall ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 200 | 1,000 |
2020 | T0 | G8 | Geometry | IMO-SL | Let $\Gamma$ and $I$ be the circumcircle and the incenter of an acute-angled triangle $A B C$. Two circles $\omega_{B}$ and $\omega_{C}$ passing through $B$ and $C$, respectively, are tangent at $I$. Let $\omega_{B}$ meet the shorter arc $A B$ of $\Gamma$ and segment $A B$ again at $P$ and $M$, respectively. Similarly... | Perform an inversion centered at $I$; the images of the points are denoted by primes, e.g., $A^{\prime}$ is the image of $A$. On the inverted figure, $I$ and $\Gamma^{\prime}$ are the orthocenter and the circumcircle of triangle $A^{\prime} B^{\prime} C^{\prime}$, respectively. The points $P^{\prime}$ and $Q^{\prime}$ ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 200 | 1,349 |
2020 | T0 | G9 | Geometry | IMO-SL | Prove that there exists a positive constant $c$ such that the following statement is true: Assume that $n$ is an integer with $n \geqslant 2$, and let $\mathcal{S}$ be a set of $n$ points in the plane such that the distance between any two distinct points in $\mathcal{S}$ is at least 1 . Then there is a line $\ell$ se... | We prove that the desired statement is true with $c=\frac{1}{8}$. Set $\delta=\frac{1}{8} n^{-1 / 3}$. For any line $\ell$ and any point $X$, let $X_{\ell}$ denote the projection of $X$ to $\ell$; a similar notation applies to sets of points. Suppose that, for some line $\ell$, the set $\mathcal{S}_{\ell}$ contains two... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 164 | 2,858 |
2020 | T0 | N3 | Number Theory | IMO-SL | Let $n$ be an integer with $n \geqslant 2$. Does there exist a sequence $\left(a_{1}, \ldots, a_{n}\right)$ of positive integers with not all terms being equal such that the arithmetic mean of every two terms is equal to the geometric mean of some (one or more) terms in this sequence? (Estonia) Answer: No such sequence... | Suppose that $a_{1}, \ldots, a_{n}$ satisfy the required properties. Let $d=\operatorname{gcd}\left(a_{1} \ldots, a_{n}\right)$. If $d>1$ then replace the numbers $a_{1}, \ldots, a_{n}$ by $\frac{a_{1}}{d}, \ldots, \frac{a_{n}}{d}$; all arithmetic and all geometric means will be divided by $d$, so we obtain another seq... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 87 | 769 |
2020 | T0 | N3 | Number Theory | IMO-SL | Let $n$ be an integer with $n \geqslant 2$. Does there exist a sequence $\left(a_{1}, \ldots, a_{n}\right)$ of positive integers with not all terms being equal such that the arithmetic mean of every two terms is equal to the geometric mean of some (one or more) terms in this sequence? (Estonia) Answer: No such sequence... | Like in the previous solution, we assume that the numbers $a_{1}, \ldots, a_{n}$ have no common divisor greater than 1 . The arithmetic mean of any two numbers in the sequence is half of an integer; on the other hand, it is a (some integer order) root of an integer. This means each pair's mean is an integer, so all ter... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 87 | 1,226 |
2020 | T0 | N4 | Number Theory | IMO-SL | For any odd prime $p$ and any integer $n$, let $d_{p}(n) \in\{0,1, \ldots, p-1\}$ denote the remainder when $n$ is divided by $p$. We say that $\left(a_{0}, a_{1}, a_{2}, \ldots\right)$ is a $p$-sequence, if $a_{0}$ is a positive integer coprime to $p$, and $a_{n+1}=a_{n}+d_{p}\left(a_{n}\right)$ for $n \geqslant 0$. (... | Fix some odd prime $p$, and let $T$ be the smallest positive integer such that $p \mid 2^{T}-1$; in other words, $T$ is the multiplicative order of 2 modulo $p$. Consider any $p$-sequence $\left(x_{n}\right)=\left(x_{0}, x_{1}, x_{2}, \ldots\right)$. Obviously, $x_{n+1} \equiv 2 x_{n}(\bmod p)$ and therefore $x_{n} \eq... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 339 | 1,600 |
2020 | T0 | N5 | Number Theory | IMO-SL | Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: (i) $f(n) \neq 0$ for at least one $n$; (ii) $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; (iii) there are infinitely many positive integers $n$ such that $f(k)=... | If a number $n$ is a product of primes, $n=p_{1} p_{2} \cdot \ldots \cdot p_{k}$, then $$ f(n)=f\left(p_{1}\right)+\ldots+f\left(p_{k}\right) $$ in particular, $f(1)=0($ since $f(1)=f(1)+f(1))$. It is also clear that $f(n)=0$ implies $f(p)=0$ for all primes $p$ dividing $n$. Let us call positive integer $n \operatornam... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 172 | 608 |
2020 | T0 | N5 | Number Theory | IMO-SL | Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: (i) $f(n) \neq 0$ for at least one $n$; (ii) $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; (iii) there are infinitely many positive integers $n$ such that $f(k)=... | We use the notion of a good number from the previous solution. As above, we also denote by $\nu_{p}(n)$ the exponent of a prime $p$ in the prime decomposition of $n$. Say that a positive integer $k$ is big if $f(k)>0$. Let $\mathcal{B}$ be the set of big primes, and let $p_{1}<p_{2}<\ldots$ list the elements of $\mathc... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 172 | 749 |
2020 | T0 | N5 | Number Theory | IMO-SL | Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: (i) $f(n) \neq 0$ for at least one $n$; (ii) $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; (iii) there are infinitely many positive integers $n$ such that $f(k)=... | We call a function $f: \mathbb{N} \rightarrow \mathbb{N}_{0}$ satisfying (ii) additive. We call a pair $(f, n)$, where $f$ is an additive function and $n \in \mathbb{N}$, good, if for all $k<n$ it holds $f(k)=f(n-k)$. For an additive function $f$ and a prime number $p$ the number $\frac{f(p)}{\ln p}$ is denoted by $g(f... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 172 | 801 |
2020 | T0 | N6 | Number Theory | IMO-SL | For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that $$ \frac{\varphi(d(n))}{d(\varphi(n))} \leqslant C $$ for all $n \geqslant 1$ ? (Cyprus) Answe... | Fix $N>1$, let $p_{1}, \ldots, p_{k}$ be all primes between 1 and $N$ and $p_{k+1}, \ldots, p_{k+s}$ be all primes between $N+1$ and $2 N$. Since for $j \leqslant k+s$ all prime divisors of $p_{j}-1$ do not exceed $N$, we have $$ \prod_{j=1}^{k+s}\left(p_{j}-1\right)=\prod_{i=1}^{k} p_{i}^{c_{i}} $$ with some fixed exp... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 113 | 1,380 |
2020 | T0 | N6 | Number Theory | IMO-SL | For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that $$ \frac{\varphi(d(n))}{d(\varphi(n))} \leqslant C $$ for all $n \geqslant 1$ ? (Cyprus) Answe... | In this solution we will use the Prime Number Theorem which states that $$ \pi(m)=\frac{m}{\log m} \cdot(1+o(1)) $$ as $m$ tends to infinity. Here and below $\pi(m)$ denotes the number of primes not exceeding $m$, and $\log$ the natural logarithm. Let $m>5$ be a large positive integer and let $n:=p_{1} p_{2} \cdot \ldo... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 113 | 1,020 |
2020 | T0 | N7 | Number Theory | IMO-SL | Let $\mathcal{S}$ be a set consisting of $n \geqslant 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_{1}, a_{2}, \ldots, a_{n}$ so that $a_{i}$ does not divide $a_{i-1}+a_{i+1}$ for all $i=2,3, \ldots, n-1$. (Ukr... | We prove the following stronger statement. Claim. Let $\mathcal{S}$ be a good set consisting of $n \geqslant 2$ positive integers. Then the elements of $\mathcal{S}$ may be ordered as $a_{1}, a_{2}, \ldots, a_{n}$ so that $a_{i} \nmid a_{i-1}+a_{i+1}$ and $a_{i} \nmid a_{i-1}-a_{i+1}$, for all $i=2,3, \ldots, n-1$. Pro... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 252 | 873 |
2020 | T0 | N7 | Number Theory | IMO-SL | Let $\mathcal{S}$ be a set consisting of $n \geqslant 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_{1}, a_{2}, \ldots, a_{n}$ so that $a_{i}$ does not divide $a_{i-1}+a_{i+1}$ for all $i=2,3, \ldots, n-1$. (Ukr... | We again prove a stronger statement. Claim. Let $\mathcal{S}$ be an arbitrary set of $n \geqslant 3$ positive integers. Then its elements can be ordered as $a_{1}, \ldots, a_{n}$ so that, if $a_{i} \mid a_{i-1}+a_{i+1}$, then $a_{i}=\max \mathcal{S}$. The claim easily implies what we need to prove, due to Observation A... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 252 | 1,079 |
2021 | T0 | A2 | Algebra | IMO-SL | For every integer $n \geqslant 1$ consider the $n \times n$ table with entry $\left\lfloor\frac{i j}{n+1}\right\rfloor$ at the intersection of row $i$ and column $j$, for every $i=1, \ldots, n$ and $j=1, \ldots, n$. Determine all integers $n \geqslant 1$ for which the sum of the $n^{2}$ entries in the table is equal to... | To simplify the calculation with indices, extend the table by adding a phantom column of index 0 with zero entries (which will not change the sum of the table). Fix a row $i$ with $1 \leqslant i \leqslant n$, and let $d:=\operatorname{gcd}(i, n+1)$ and $k:=(n+1) / d$. For columns $j=0, \ldots, n$, define the remainder ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 128 | 1,041 |
2021 | T0 | A3 | Algebra | IMO-SL | Given a positive integer $n$, find the smallest value of $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor$ over all permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $(1,2, \ldots, n)$. | Suppose that $2^{k} \leqslant n<2^{k+1}$ with some nonnegative integer $k$. First we show a permutation $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ such that $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor=k+1$; then we will prove that $\... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 109 | 1,508 |
2021 | T0 | A3 | Algebra | IMO-SL | Given a positive integer $n$, find the smallest value of $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor$ over all permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $(1,2, \ldots, n)$. | We present a different proof for the lower bound. Assume again $2^{k} \leqslant n<2^{k+1}$, and let $P=\left\{2^{0}, 2^{1}, \ldots, 2^{k}\right\}$ be the set of powers of 2 among $1,2, \ldots, n$. Call an integer $i \in\{1,2, \ldots, n\}$ and the interval $\left[i, a_{i}\right]$ good if $a_{i} \geqslant i$. Lemma 1. Th... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 109 | 595 |
2021 | T0 | A4 | Algebra | IMO-SL | Show that for all real numbers $x_{1}, \ldots, x_{n}$ the following inequality holds: $$ \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}-x_{j}\right|} \leqslant \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}+x_{j}\right|} $$ | If we add $t$ to all the variables then the left-hand side remains constant and the right-hand side becomes $$ H(t):=\sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}+x_{j}+2 t\right|} $$ Let $T$ be large enough such that both $H(-T)$ and $H(T)$ are larger than the value $L$ of the lefthand side of the inequality we want... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 103 | 569 |
2021 | T0 | A4 | Algebra | IMO-SL | Show that for all real numbers $x_{1}, \ldots, x_{n}$ the following inequality holds: $$ \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}-x_{j}\right|} \leqslant \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}+x_{j}\right|} $$ | For real $p$ consider the integral $$ I(p)=\int_{0}^{\infty} \frac{1-\cos (p x)}{x \sqrt{x}} d x $$ which clearly converges to a strictly positive number. By changing the variable $y=|p| x$ one notices that $I(p)=\sqrt{|p|} I(1)$. Hence, by using the trigonometric formula $\cos (\alpha-\beta)-\cos (\alpha+$ $\beta)=2 \... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 103 | 581 |
2021 | T0 | A5 | Algebra | IMO-SL | Let $n \geqslant 2$ be an integer, and let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots+a_{n}=1$. Prove that $$ \sum_{k=1}^{n} \frac{a_{k}}{1-a_{k}}\left(a_{1}+a_{2}+\cdots+a_{k-1}\right)^{2}<\frac{1}{3} $$ | For all $k \leqslant n$, let $$ s_{k}=a_{1}+a_{2}+\cdots+a_{k} \quad \text { and } \quad b_{k}=\frac{a_{k} s_{k-1}^{2}}{1-a_{k}} $$ with the convention that $s_{0}=0$. Note that $b_{k}$ is exactly a summand in the sum we need to estimate. We shall prove the inequality $$ b_{k}<\frac{s_{k}^{3}-s_{k-1}^{3}}{3} $$ Indeed,... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 120 | 696 |
2021 | T0 | A5 | Algebra | IMO-SL | Let $n \geqslant 2$ be an integer, and let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots+a_{n}=1$. Prove that $$ \sum_{k=1}^{n} \frac{a_{k}}{1-a_{k}}\left(a_{1}+a_{2}+\cdots+a_{k-1}\right)^{2}<\frac{1}{3} $$ | First, let us define $$ S\left(a_{1}, \ldots, a_{n}\right):=\sum_{k=1}^{n} \frac{a_{k}}{1-a_{k}}\left(a_{1}+a_{2}+\cdots+a_{k-1}\right)^{2} $$ For some index $i$, denote $a_{1}+\cdots+a_{i-1}$ by $s$. If we replace $a_{i}$ with two numbers $a_{i} / 2$ and $a_{i} / 2$, i.e. replace the tuple $\left(a_{1}, \ldots, a_{n}\... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 120 | 544 |
2021 | T0 | A5 | Algebra | IMO-SL | Let $n \geqslant 2$ be an integer, and let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots+a_{n}=1$. Prove that $$ \sum_{k=1}^{n} \frac{a_{k}}{1-a_{k}}\left(a_{1}+a_{2}+\cdots+a_{k-1}\right)^{2}<\frac{1}{3} $$ | We sketch a probabilistic version of the first solution. Let $x_{1}, x_{2}, x_{3}$ be drawn uniformly and independently at random from the segment [0,1]. Let $I_{1} \cup I_{2} \cup \cdots \cup I_{n}$ be a partition of $[0,1]$ into segments of length $a_{1}, a_{2}, \ldots, a_{n}$ in this order. Let $J_{k}:=I_{1} \cup \c... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 120 | 630 |
2021 | T0 | A6 | Algebra | IMO-SL | Let $A$ be a finite set of (not necessarily positive) integers, and let $m \geqslant 2$ be an integer. Assume that there exist non-empty subsets $B_{1}, B_{2}, B_{3}, \ldots, B_{m}$ of $A$ whose elements add up to the sums $m^{1}, m^{2}, m^{3}, \ldots, m^{m}$, respectively. Prove that $A$ contains at least $m / 2$ elem... | Let $A=\left\{a_{1}, \ldots, a_{k}\right\}$. Assume that, on the contrary, $k=|A|<m / 2$. Let $$ s_{i}:=\sum_{j: a_{j} \in B_{i}} a_{j} $$ be the sum of elements of $B_{i}$. We are given that $s_{i}=m^{i}$ for $i=1, \ldots, m$. Now consider all $m^{m}$ expressions of the form $$ f\left(c_{1}, \ldots, c_{m}\right):=c_{1... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 112 | 523 |
2021 | T0 | A7 | Algebra | IMO-SL | Let $n \geqslant 1$ be an integer, and let $x_{0}, x_{1}, \ldots, x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_{i} x_{i+1}-x_{i-1}^{2} \geqslant 1$ for all $i=1,2, \ldots, n$. Show that $$ x_{0}+x_{1}+\cdots+x_{n}+x_{n+1}>\left(\frac{2 n}{3}\right)^{3 / 2} $$ | Lemma 1.1. If $a, b, c$ are non-negative numbers such that $a b-c^{2} \geqslant 1$, then $$ (a+2 b)^{2} \geqslant(b+2 c)^{2}+6 $$ Proof. $(a+2 b)^{2}-(b+2 c)^{2}=(a-b)^{2}+2(b-c)^{2}+6\left(a b-c^{2}\right) \geqslant 6$. Lemma 1.2. $\sqrt{1}+\cdots+\sqrt{n}>\frac{2}{3} n^{3 / 2}$. Proof. Bernoulli's inequality $(1+t)^{... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 137 | 564 |
2021 | T0 | A7 | Algebra | IMO-SL | Let $n \geqslant 1$ be an integer, and let $x_{0}, x_{1}, \ldots, x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_{i} x_{i+1}-x_{i-1}^{2} \geqslant 1$ for all $i=1,2, \ldots, n$. Show that $$ x_{0}+x_{1}+\cdots+x_{n}+x_{n+1}>\left(\frac{2 n}{3}\right)^{3 / 2} $$ | Say that an index $i \in\{0,1, \ldots, n+1\}$ is good, if $x_{i} \geqslant \sqrt{\frac{2}{3}} i$, otherwise call the index $i$ bad. Lemma 2.1. There are no two consecutive bad indices. Proof. Assume the contrary and consider two bad indices $j, j+1$ with minimal possible $j$. Since 0 is good, we get $j>0$, thus by mini... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 137 | 1,131 |
2021 | T0 | A8 | Algebra | IMO-SL | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$ (f(a)-f(b))(f(b)-f(c))(f(c)-f(a))=f\left(a b^{2}+b c^{2}+c a^{2}\right)-f\left(a^{2} b+b^{2} c+c^{2} a\right) $$ for all real numbers $a, b, c$. | It is straightforward to check that above functions satisfy the equation. Now let $f(x)$ satisfy the equation, which we denote $E(a, b, c)$. Then clearly $f(x)+C$ also does; therefore, we may suppose without loss of generality that $f(0)=0$.We start with proving Lemma. Either $f(x) \equiv 0$ or $f$ is injective. Proof.... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 101 | 2,450 |
2021 | T0 | C1 | Combinatorics | IMO-SL | Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a, b, c, d \in S$ with $\operatorname{gcd}(a, b) \neq \operatorname{gcd}(c, d)$. Prove that there exist three pairwise distinct $x, y, z \in S$ such that $\operatorname{gcd}(x, y)=\operatorname{gcd}(y, z) \neq \operatorname{g... | There exists $\alpha \in S$ so that $\{\operatorname{gcd}(\alpha, s) \mid s \in S, s \neq \alpha\}$ contains at least two elements. Since $\alpha$ has only finitely many divisors, there is a $d \mid \alpha$ such that the set $B=\{\beta \in$ $S \mid \operatorname{gcd}(\alpha, \beta)=d\}$ is infinite. Pick $\gamma \in S$... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 110 | 690 |
2021 | T0 | C4 | Combinatorics | IMO-SL | The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from $X$ to $Y$ is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is ca... | We write $X \rightarrow Y$ (or $Y \leftarrow X$ ) if the road between $X$ and $Y$ goes from $X$ to $Y$. Notice that, if there is any route moving from $X$ to $Y$ (possibly passing through some cities more than once), then there is a path from $X$ to $Y$ consisting of some roads in the route. Indeed, any cycle in the ro... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 211 | 1,277 |
2021 | T0 | C4 | Combinatorics | IMO-SL | The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from $X$ to $Y$ is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is ca... | We recall some graph-theoretical notions. Let $G$ be a finite graph, and let $V$ be the set of its vertices; fix two distinct vertices $s, t \in V$. An $(s, t)$-cut is a partition of $V$ into two parts $V=S \sqcup T$ such that $s \in S$ and $t \in T$. The cut-edges in the cut $(S, T)$ are the edges going from $S$ to $T... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 211 | 959 |
2021 | T0 | C5 | Combinatorics | IMO-SL | Let $n$ and $k$ be two integers with $n>k \geqslant 1$. There are $2 n+1$ students standing in a circle. Each student $S$ has $2 k$ neighbours - namely, the $k$ students closest to $S$ on the right, and the $k$ students closest to $S$ on the left. Suppose that $n+1$ of the students are girls, and the other $n$ are boy... | We replace the girls by 1's, and the boys by 0 's, getting the numbers $a_{1}, a_{2}, \ldots, a_{2 n+1}$ arranged in a circle. We extend this sequence periodically by letting $a_{2 n+1+k}=a_{k}$ for all $k \in \mathbb{Z}$. We get an infinite periodic sequence $$ \ldots, a_{1}, a_{2}, \ldots, a_{2 n+1}, a_{1}, a_{2}, \l... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 121 | 1,953 |
2021 | T0 | C6 | Combinatorics | IMO-SL | A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent ... | A central idea is that several colourings $C_{1}, C_{2}, \ldots, C_{k}$ can be merged together into a single product colouring $C_{1} \times C_{2} \times \cdots \times C_{k}$ as follows: the colours in the product colouring are ordered tuples $\left(c_{1}, \ldots, c_{n}\right)$ of colours, where $c_{i}$ is a colour use... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 126 | 659 |
2021 | T0 | C8 | Combinatorics | IMO-SL | Determine the largest $N$ for which there exists a table $T$ of integers with $N$ rows and 100 columns that has the following properties: (i) Every row contains the numbers $1,2, \ldots, 100$ in some order. (ii) For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r, c)-T(s, c)| \geqslant 2$. Her... | Non-existence of a larger table. Let us consider some fixed row in the table, and let us replace (for $k=1,2, \ldots, 50$ ) each of two numbers $2 k-1$ and $2 k$ respectively by the symbol $x_{k}$. The resulting pattern is an arrangement of 50 symbols $x_{1}, x_{2}, \ldots, x_{50}$, where every symbol occurs exactly tw... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 131 | 1,604 |
2021 | T0 | C8 | Combinatorics | IMO-SL | Determine the largest $N$ for which there exists a table $T$ of integers with $N$ rows and 100 columns that has the following properties: (i) Every row contains the numbers $1,2, \ldots, 100$ in some order. (ii) For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r, c)-T(s, c)| \geqslant 2$. Her... | We provide a bit different proof why the example from Lemma. Let $\pi_{1}$ and $\pi_{2}$ be two permutations of the set $\{1,2, \ldots, n\}$ such that $\left|\pi_{1}(i)-\pi_{2}(i)\right| \leqslant 1$ for every $i$. Then there exists a set of disjoint pairs $(i, i+1)$ such that $\pi_{2}$ is obtained from $\pi_{1}$ by sw... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 131 | 664 |
2021 | T0 | G2 | Geometry | IMO-SL | Let $A B C D$ be a convex quadrilateral circumscribed around a circle with centre $I$. Let $\omega$ be the circumcircle of the triangle $A C I$. The extensions of $B A$ and $B C$ beyond $A$ and $C$ meet $\omega$ at $X$ and $Z$, respectively. The extensions of $A D$ and $C D$ beyond $D$ meet $\omega$ at $Y$ and $T$, re... | The point $I$ is the intersection of the external bisector of the angle $T C Z$ with the circumcircle $\omega$ of the triangle $T C Z$, so $I$ is the midpoint of the $\operatorname{arc} T C Z$ and $I T=I Z$. Similarly, $I$ is the midpoint of the arc $Y A X$ and $I X=I Y$. Let $O$ be the centre of $\omega$. Then $X$ and... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 138 | 909 |
2021 | T0 | G4 | Geometry | IMO-SL | Let $A B C D$ be a quadrilateral inscribed in a circle $\Omega$. Let the tangent to $\Omega$ at $D$ intersect the rays $B A$ and $B C$ at points $E$ and $F$, respectively. A point $T$ is chosen inside the triangle $A B C$ so that $T E \| C D$ and $T F \| A D$. Let $K \neq D$ be a point on the segment $D F$ such that $T... | Let the segments $T E$ and $T F$ cross $A C$ at $P$ and $Q$, respectively. Since $P E \| C D$ and $E D$ is tangent to the circumcircle of $A B C D$, we have $$ \angle E P A=\angle D C A=\angle E D A, $$ and so the points $A, P, D$, and $E$ lie on some circle $\alpha$. Similarly, the points $C, Q, D$, and $F$ lie on som... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 133 | 760 |
2021 | T0 | G4 | Geometry | IMO-SL | Let $A B C D$ be a quadrilateral inscribed in a circle $\Omega$. Let the tangent to $\Omega$ at $D$ intersect the rays $B A$ and $B C$ at points $E$ and $F$, respectively. A point $T$ is chosen inside the triangle $A B C$ so that $T E \| C D$ and $T F \| A D$. Let $K \neq D$ be a point on the segment $D F$ such that $T... | Consider the spiral similarity $\phi$ centred at $D$ which maps $B$ to $F$. Recall that for any two points $X$ and $Y$, the triangles $D X \phi(X)$ and $D Y \phi(Y)$ are similar. Define $T^{\prime}=\phi(E)$. Then $$ \angle C D F=\angle F B D=\angle \phi(B) B D=\angle \phi(E) E D=\angle T^{\prime} E D, $$ so $C D \| T^{... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 133 | 542 |
2021 | T0 | G5 | Geometry | IMO-SL | Let $A B C D$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcentre of $A B C D$. The internal angle bisectors of $\angle A B C$ and $\angle A D C$ meet $A C$ at $B_{1}$ and $D_{1}$, respectively. Let $O_{B}$ be the centre of the circle which passes through $B$ and is tangen... | Let the diagonals $A C$ and $B D$ cross at $H$. Consider the homothety $h$ centred at $H$ and mapping $B$ to $D$. Since $B D_{1} \| D B_{1}$, we have $h\left(D_{1}\right)=B_{1}$. Let the tangents to $\Omega$ at $B$ and $D$ meet $A C$ at $L_{B}$ and $L_{D}$, respectively. We have $$ \angle L_{B} B B_{1}=\angle L_{B} B C... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 177 | 1,400 |
2021 | T0 | G5 | Geometry | IMO-SL | Let $A B C D$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcentre of $A B C D$. The internal angle bisectors of $\angle A B C$ and $\angle A D C$ meet $A C$ at $B_{1}$ and $D_{1}$, respectively. Let $O_{B}$ be the centre of the circle which passes through $B$ and is tangen... | Let $B D_{1}$ and $T_{B} D_{1}$ meet $\Omega$ again at $X_{B}$ and $Y_{B}$, respectively. Then $$ \angle B D_{1} C=\angle B T_{B} D_{1}=\angle B T_{B} Y_{B}=\angle B X_{B} Y_{B} $$ which shows that $X_{B} Y_{B} \| A C$. Similarly, let $D B_{1}$ and $T_{D} B_{1}$ meet $\Omega$ again at $X_{D}$ and $Y_{D}$, respectively;... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 177 | 735 |
2021 | T0 | G6 | Geometry | IMO-SL | Determine all integers $n \geqslant 3$ satisfying the following property: every convex $n$-gon whose sides all have length 1 contains an equilateral triangle of side length 1. (Every polygon is assumed to contain its boundary.) | First we show that for every even $n \geqslant 4$ there exists a polygon violating the required statement. Consider a regular $k$-gon $A_{0} A_{1}, \ldots A_{k-1}$ with side length 1 . Let $B_{1}, B_{2}, \ldots, B_{n / 2-1}$ be the points symmetric to $A_{1}, A_{2}, \ldots, A_{n / 2-1}$ with respect to the line $A_{0} ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 54 | 2,481 |
2021 | T0 | G7 | Geometry | IMO-SL | A point $D$ is chosen inside an acute-angled triangle $A B C$ with $A B>A C$ so that $\angle B A D=\angle D A C$. A point $E$ is constructed on the segment $A C$ so that $\angle A D E=\angle D C B$. Similarly, a point $F$ is constructed on the segment $A B$ so that $\angle A D F=\angle D B C$. A point $X$ is chosen on ... | Let $T A$ intersect the circle $(A B C)$ again at $M$. Due to the circles ( $B C E F$ ) and $(A M C B)$, and using the above Claim, we get $T M \cdot T A=T F \cdot T E=T B \cdot T C=T D^{2}$; in particular, the points $A, M, E$, and $F$ are concyclic. Under the inversion with centre $T$ and radius $T D$, the point $M$ ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 172 | 1,032 |
2021 | T0 | G7 | Geometry | IMO-SL | A point $D$ is chosen inside an acute-angled triangle $A B C$ with $A B>A C$ so that $\angle B A D=\angle D A C$. A point $E$ is constructed on the segment $A C$ so that $\angle A D E=\angle D C B$. Similarly, a point $F$ is constructed on the segment $A B$ so that $\angle A D F=\angle D B C$. A point $X$ is chosen on ... | We use only the first part of the Common remarks, namely, the facts that the tuples $(C, D, Q, E)$ and $(B, C, E, F)$ are both concyclic. We also introduce the point $T=$ $B C \cap E F$. Let the circle $(C D E)$ meet $B C$ again at $E_{1}$. Since $\angle E_{1} C Q=\angle D C E$, the $\operatorname{arcs} D E$ and $Q E_{... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 172 | 882 |
2021 | T0 | G7 | Geometry | IMO-SL | A point $D$ is chosen inside an acute-angled triangle $A B C$ with $A B>A C$ so that $\angle B A D=\angle D A C$. A point $E$ is constructed on the segment $A C$ so that $\angle A D E=\angle D C B$. Similarly, a point $F$ is constructed on the segment $A B$ so that $\angle A D F=\angle D B C$. A point $X$ is chosen on ... | Notice that $\angle A Q E=\angle Q C B$ and $\angle A Q F=\angle Q B C$; so, if we replace the point $D$ with $Q$ in the problem set up, the points $E, F$, and $T$ remain the same. So, by the Claim, we have $T Q^{2}=T B \cdot T C=T D^{2}$. Thus, there exists a circle $\Gamma$ centred at $T$ and passing through $D$ and ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 172 | 1,775 |
2021 | T0 | G8 | Geometry | IMO-SL | Let $\omega$ be the circumcircle of a triangle $A B C$, and let $\Omega_{A}$ be its excircle which is tangent to the segment $B C$. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_{A}$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_{A}$ at $X$ and $Y$, respectively. The ... | Let $D$ be the point of tangency of $B C$ and $\Omega_{A}$. Let $D^{\prime}$ be the point such that $D D^{\prime}$ is a diameter of $\Omega_{A}$. Let $R^{\prime}$ be (the unique) point such that $A R^{\prime} \perp B C$ and $R^{\prime} D^{\prime} \| B C$. We shall prove that $R^{\prime}$ coincides with $R$. Let $P X$ i... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 152 | 647 |
2021 | T0 | G8 | Geometry | IMO-SL | Let $\omega$ be the circumcircle of a triangle $A B C$, and let $\Omega_{A}$ be its excircle which is tangent to the segment $B C$. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_{A}$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_{A}$ at $X$ and $Y$, respectively. The ... | Let $J$ and $r$ be the center and the radius of $\Omega_{A}$. Denote the diameter of $\omega$ by $d$ and its center by $O$. By Euler's formula, $O J^{2}=(d / 2)^{2}+d r$, so the power of $J$ with respect to $\omega$ equals $d r$. Let $J X$ intersect $\omega$ again at $L$. Then $J L=d$. Let $L K$ be a diameter of $\omeg... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 152 | 1,218 |
2021 | T0 | N2 | Number Theory | IMO-SL | Let $n \geqslant 100$ be an integer. The numbers $n, n+1, \ldots, 2 n$ are written on $n+1$ cards, one number per card. The cards are shuffled and divided into two piles. Prove that one of the piles contains two cards such that the sum of their numbers is a perfect square. | To solve the problem it suffices to find three squares and three cards with numbers $a, b, c$ on them such that pairwise sums $a+b, b+c, a+c$ are equal to the chosen squares. By choosing the three consecutive squares $(2 k-1)^{2},(2 k)^{2},(2 k+1)^{2}$ we arrive at the triple $$ (a, b, c)=\left(2 k^{2}-4 k, \quad 2 k^{... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 81 | 819 |
2021 | T0 | N3 | Number Theory | IMO-SL | Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $\left(d_{1}, d_{2}, \ldots, d_{k}\right)$ such that for every $i=1,2, \ldots, k$, the number $d_{1}+\cdots+d_{i}$ is a perfect square. | For $i=1,2, \ldots, k$ let $d_{1}+\ldots+d_{i}=s_{i}^{2}$, and define $s_{0}=0$ as well. Obviously $0=s_{0}<s_{1}<s_{2}<\ldots<s_{k}$, so $$ s_{i} \geqslant i \quad \text { and } \quad d_{i}=s_{i}^{2}-s_{i-1}^{2}=\left(s_{i}+s_{i-1}\right)\left(s_{i}-s_{i-1}\right) \geqslant s_{i}+s_{i-1} \geqslant 2 i-1 $$ The number ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 82 | 1,158 |
2021 | T0 | N4 | Number Theory | IMO-SL | Alice is given a rational number $r>1$ and a line with two points $B \neq R$, where point $R$ contains a red bead and point $B$ contains a blue bead. Alice plays a solitaire game by performing a sequence of moves. In every move, she chooses a (not necessarily positive) integer $k$, and a bead to move. If that bead is p... | Denote the red and blue beads by $\mathcal{R}$ and $\mathcal{B}$, respectively. Introduce coordinates on the line and identify the points with their coordinates so that $R=0$ and $B=1$. Then, during the game, the coordinate of $\mathcal{R}$ is always smaller than the coordinate of $\mathcal{B}$. Moreover, the distance ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 178 | 1,927 |
2021 | T0 | N5 | Number Theory | IMO-SL | Prove that there are only finitely many quadruples $(a, b, c, n)$ of positive integers such that $$ n!=a^{n-1}+b^{n-1}+c^{n-1} . $$ | For fixed $n$ there are clearly finitely many solutions; we will show that there is no solution with $n>100$. So, assume $n>100$. By the AM-GM inequality, $$ \begin{aligned} n! & =2 n(n-1)(n-2)(n-3) \cdot(3 \cdot 4 \cdots(n-4)) \\ & \leqslant 2(n-1)^{4}\left(\frac{3+\cdots+(n-4)}{n-6}\right)^{n-6}=2(n-1)^{4}\left(\frac... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 52 | 1,483 |
2021 | T0 | N6 | Number Theory | IMO-SL | Determine all integers $n \geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_{1}, a_{2}, \ldots, a_{n}$ so that $n$ divides $1 \cdot a_{1}+2 \cdot a_{2}+\cdots+n \cdot a_{n}$. | If $n=2^{k} a$, where $a \geqslant 3$ is odd and $k$ is a positive integer, we can consider a set containing the number $2^{k}+1$ and $n-1$ numbers congruent to 1 modulo $n$. The sum of these numbers is congruent to $2^{k}$ modulo $n$ and therefore is not divisible by $n$; for any permutation $\left(a_{1}, a_{2}, \ldot... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 91 | 1,866 |
2021 | T0 | N7 | Number Theory | IMO-SL | Let $a_{1}, a_{2}, a_{3}, \ldots$ be an infinite sequence of positive integers such that $a_{n+2 m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m$. Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_{n}=a_{n+d}$ for all $n>N$. | We will make repeated use of the following simple observation: Lemma 1. If a positive integer $d$ divides $a_{n}$ and $a_{n-m}$ for some $m$ and $n>2 m$, it also divides $a_{n-2 m}$. If $d$ divides $a_{n}$ and $a_{n-2 m}$, it also divides $a_{n-m}$. Proof. Both parts are obvious since $a_{n}$ divides $a_{n-2 m}+a_{n-m}... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 99 | 2,572 |
2021 | T0 | N8 | Number Theory | IMO-SL | For a polynomial $P(x)$ with integer coefficients let $P^{1}(x)=P(x)$ and $P^{k+1}(x)=$ $P\left(P^{k}(x)\right)$ for $k \geqslant 1$. Find all positive integers $n$ for which there exists a polynomial $P(x)$ with integer coefficients such that for every integer $m \geqslant 1$, the numbers $P^{m}(1), \ldots, P^{m}(n)$ ... | Denote the set of residues modulo $\ell$ by $\mathbb{Z}_{\ell}$. Observe that $P$ can be regarded as a function $\mathbb{Z}_{\ell} \rightarrow \mathbb{Z}_{\ell}$ for any positive integer $\ell$. Denote the cardinality of the set $P^{m}\left(\mathbb{Z}_{\ell}\right)$ by $f_{m, \ell}$. Note that $f_{m, n}=\left\lceil n /... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 136 | 3,097 |
2021 | T0 | N8 | Number Theory | IMO-SL | For a polynomial $P(x)$ with integer coefficients let $P^{1}(x)=P(x)$ and $P^{k+1}(x)=$ $P\left(P^{k}(x)\right)$ for $k \geqslant 1$. Find all positive integers $n$ for which there exists a polynomial $P(x)$ with integer coefficients such that for every integer $m \geqslant 1$, the numbers $P^{m}(1), \ldots, P^{m}(n)$ ... | Observe that $f$ can be regarded as a function $\mathbb{Z}_{\ell} \rightarrow \mathbb{Z}_{\ell}$ for any positive integer $\ell$. We use notations $f^{m}$ and $f_{m, \ell}$ as in the above solution. Part 1. There exists a function $f: \mathbb{Z}_{p^{k}} \rightarrow \mathbb{Z}_{p^{k}}$ satisfying the desired properties.... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 136 | 1,215 |
2021 | T0 | A2 | Algebra | IMO-SL | For every integer $n \geqslant 1$ consider the $n \times n$ table with entry $\left\lfloor\frac{i j}{n+1}\right\rfloor$ at the intersection of row $i$ and column $j$, for every $i=1, \ldots, n$ and $j=1, \ldots, n$. Determine all integers $n \geqslant 1$ for which the sum of the $n^{2}$ entries in the table is equal to... | To simplify the calculation with indices, extend the table by adding a phantom column of index 0 with zero entries (which will not change the sum of the table). Fix a row $i$ with $1 \leqslant i \leqslant n$, and let $d:=\operatorname{gcd}(i, n+1)$ and $k:=(n+1) / d$. For columns $j=0, \ldots, n$, define the remainder ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 147 | 1,041 |
2021 | T0 | A3 | Algebra | IMO-SL | Given a positive integer $n$, find the smallest value of $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor$ over all permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $(1,2, \ldots, n)$. Answer: The minimum of such sums is $\left... | Suppose that $2^{k} \leqslant n<2^{k+1}$ with some nonnegative integer $k$. First we show a permutation $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ such that $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor=k+1$; then we will prove that $\... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 168 | 1,508 |
2021 | T0 | A3 | Algebra | IMO-SL | Given a positive integer $n$, find the smallest value of $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor$ over all permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $(1,2, \ldots, n)$. Answer: The minimum of such sums is $\left... | We present a different proof for the lower bound. Assume again $2^{k} \leqslant n<2^{k+1}$, and let $P=\left\{2^{0}, 2^{1}, \ldots, 2^{k}\right\}$ be the set of powers of 2 among $1,2, \ldots, n$. Call an integer $i \in\{1,2, \ldots, n\}$ and the interval $\left[i, a_{i}\right]$ good if $a_{i} \geqslant i$. Lemma 1. Th... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 168 | 595 |
2021 | T0 | A8 | Lemma 3. | IMO-SL | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$ (f(a)-f(b))(f(b)-f(c))(f(c)-f(a))=f\left(a b^{2}+b c^{2}+c a^{2}\right)-f\left(a^{2} b+b^{2} c+c^{2} a\right) $$ for all real numbers $a, b, c$. Answer: $f(x)=\alpha x+\beta$ or $f(x)=\alpha x^{3}+\beta$ where $\alpha \in\{-1,0,1\}$ and $\... | It is straightforward to check that above functions satisfy the equation. Now let $f(x)$ satisfy the equation, which we denote $E(a, b, c)$. Then clearly $f(x)+C$ also does; therefore, we may suppose without loss of generality that $f(0)=0$.We start with proving Lemma. Either $f(x) \equiv 0$ or $f$ is injective. Proof.... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 153 | 2,450 |
2021 | T0 | C6 | Step 2: Red and green colouring. | IMO-SL | A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent ... | A central idea is that several colourings $C_{1}, C_{2}, \ldots, C_{k}$ can be merged together into a single product colouring $C_{1} \times C_{2} \times \cdots \times C_{k}$ as follows: the colours in the product colouring are ordered tuples $\left(c_{1}, \ldots, c_{n}\right)$ of colours, where $c_{i}$ is a colour use... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 144 | 659 |
2021 | T0 | C8 | Step 2: Red and green colouring. | IMO-SL | Determine the largest $N$ for which there exists a table $T$ of integers with $N$ rows and 100 columns that has the following properties: (i) Every row contains the numbers $1,2, \ldots, 100$ in some order. (ii) For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r, c)-T(s, c)| \geqslant 2$. Her... | Non-existence of a larger table. Let us consider some fixed row in the table, and let us replace (for $k=1,2, \ldots, 50$ ) each of two numbers $2 k-1$ and $2 k$ respectively by the symbol $x_{k}$. The resulting pattern is an arrangement of 50 symbols $x_{1}, x_{2}, \ldots, x_{50}$, where every symbol occurs exactly tw... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 152 | 1,604 |
2021 | T0 | C8 | Step 2: Red and green colouring. | IMO-SL | Determine the largest $N$ for which there exists a table $T$ of integers with $N$ rows and 100 columns that has the following properties: (i) Every row contains the numbers $1,2, \ldots, 100$ in some order. (ii) For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r, c)-T(s, c)| \geqslant 2$. Her... | We provide a bit different proof why the example from Lemma. Let $\pi_{1}$ and $\pi_{2}$ be two permutations of the set $\{1,2, \ldots, n\}$ such that $\left|\pi_{1}(i)-\pi_{2}(i)\right| \leqslant 1$ for every $i$. Then there exists a set of disjoint pairs $(i, i+1)$ such that $\pi_{2}$ is obtained from $\pi_{1}$ by sw... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 152 | 664 |
2021 | T0 | G2 | Geometry | IMO-SL | Let $A B C D$ be a convex quadrilateral circumscribed around a circle with centre $I$. Let $\omega$ be the circumcircle of the triangle $A C I$. The extensions of $B A$ and $B C$ beyond $A$ and $C$ meet $\omega$ at $X$ and $Z$, respectively. The extensions of $A D$ and $C D$ beyond $D$ meet $\omega$ at $Y$ and $T$, res... | The point $I$ is the intersection of the external bisector of the angle $T C Z$ with the circumcircle $\omega$ of the triangle $T C Z$, so $I$ is the midpoint of the $\operatorname{arc} T C Z$ and $I T=I Z$. Similarly, $I$ is the midpoint of the arc $Y A X$ and $I X=I Y$. Let $O$ be the centre of $\omega$. Then $X$ and... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 137 | 909 |
2021 | T0 | G5 | Geometry | IMO-SL | Let $A B C D$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcentre of $A B C D$. The internal angle bisectors of $\angle A B C$ and $\angle A D C$ meet $A C$ at $B_{1}$ and $D_{1}$, respectively. Let $O_{B}$ be the centre of the circle which passes through $B$ and is tangen... | Let the diagonals $A C$ and $B D$ cross at $H$. Consider the homothety $h$ centred at $H$ and mapping $B$ to $D$. Since $B D_{1} \| D B_{1}$, we have $h\left(D_{1}\right)=B_{1}$. Let the tangents to $\Omega$ at $B$ and $D$ meet $A C$ at $L_{B}$ and $L_{D}$, respectively. We have $$ \angle L_{B} B B_{1}=\angle L_{B} B C... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 664 | 1,400 |
2021 | T0 | G5 | Geometry | IMO-SL | Let $A B C D$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcentre of $A B C D$. The internal angle bisectors of $\angle A B C$ and $\angle A D C$ meet $A C$ at $B_{1}$ and $D_{1}$, respectively. Let $O_{B}$ be the centre of the circle which passes through $B$ and is tangen... | Let $B D_{1}$ and $T_{B} D_{1}$ meet $\Omega$ again at $X_{B}$ and $Y_{B}$, respectively. Then $$ \angle B D_{1} C=\angle B T_{B} D_{1}=\angle B T_{B} Y_{B}=\angle B X_{B} Y_{B} $$ which shows that $X_{B} Y_{B} \| A C$. Similarly, let $D B_{1}$ and $T_{D} B_{1}$ meet $\Omega$ again at $X_{D}$ and $Y_{D}$, respectively;... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 664 | 735 |
2021 | T0 | G6 | Geometry | IMO-SL | Determine all integers $n \geqslant 3$ satisfying the following property: every convex $n$-gon whose sides all have length 1 contains an equilateral triangle of side length 1. (Every polygon is assumed to contain its boundary.) Answer: All odd $n \geqslant 3$. | First we show that for every even $n \geqslant 4$ there exists a polygon violating the required statement. Consider a regular $k$-gon $A_{0} A_{1}, \ldots A_{k-1}$ with side length 1 . Let $B_{1}, B_{2}, \ldots, B_{n / 2-1}$ be the points symmetric to $A_{1}, A_{2}, \ldots, A_{n / 2-1}$ with respect to the line $A_{0} ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 68 | 2,481 |
2021 | T0 | G7 | Geometry | IMO-SL | A point $D$ is chosen inside an acute-angled triangle $A B C$ with $A B>A C$ so that $\angle B A D=\angle D A C$. A point $E$ is constructed on the segment $A C$ so that $\angle A D E=\angle D C B$. Similarly, a point $F$ is constructed on the segment $A B$ so that $\angle A D F=\angle D B C$. A point $X$ is chosen on ... | Let $T A$ intersect the circle $(A B C)$ again at $M$. Due to the circles ( $B C E F$ ) and $(A M C B)$, and using the above Claim, we get $T M \cdot T A=T F \cdot T E=T B \cdot T C=T D^{2}$; in particular, the points $A, M, E$, and $F$ are concyclic. Under the inversion with centre $T$ and radius $T D$, the point $M$ ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 685 | 1,032 |
2021 | T0 | G7 | Geometry | IMO-SL | A point $D$ is chosen inside an acute-angled triangle $A B C$ with $A B>A C$ so that $\angle B A D=\angle D A C$. A point $E$ is constructed on the segment $A C$ so that $\angle A D E=\angle D C B$. Similarly, a point $F$ is constructed on the segment $A B$ so that $\angle A D F=\angle D B C$. A point $X$ is chosen on ... | We use only the first part of the Common remarks, namely, the facts that the tuples $(C, D, Q, E)$ and $(B, C, E, F)$ are both concyclic. We also introduce the point $T=$ $B C \cap E F$. Let the circle $(C D E)$ meet $B C$ again at $E_{1}$. Since $\angle E_{1} C Q=\angle D C E$, the $\operatorname{arcs} D E$ and $Q E_{... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 685 | 882 |
2021 | T0 | G7 | Geometry | IMO-SL | A point $D$ is chosen inside an acute-angled triangle $A B C$ with $A B>A C$ so that $\angle B A D=\angle D A C$. A point $E$ is constructed on the segment $A C$ so that $\angle A D E=\angle D C B$. Similarly, a point $F$ is constructed on the segment $A B$ so that $\angle A D F=\angle D B C$. A point $X$ is chosen on ... | Notice that $\angle A Q E=\angle Q C B$ and $\angle A Q F=\angle Q B C$; so, if we replace the point $D$ with $Q$ in the problem set up, the points $E, F$, and $T$ remain the same. So, by the Claim, we have $T Q^{2}=T B \cdot T C=T D^{2}$. Thus, there exists a circle $\Gamma$ centred at $T$ and passing through $D$ and ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 685 | 1,775 |
2021 | T0 | N3 | Number Theory | IMO-SL | Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $\left(d_{1}, d_{2}, \ldots, d_{k}\right)$ such that for every $i=1,2, \ldots, k$, the number $d_{1}+\cdots+d_{i}$ is a perfect square. Answer: $n=1$ and $n=3$. | For $i=1,2, \ldots, k$ let $d_{1}+\ldots+d_{i}=s_{i}^{2}$, and define $s_{0}=0$ as well. Obviously $0=s_{0}<s_{1}<s_{2}<\ldots<s_{k}$, so $$ s_{i} \geqslant i \quad \text { and } \quad d_{i}=s_{i}^{2}-s_{i-1}^{2}=\left(s_{i}+s_{i-1}\right)\left(s_{i}-s_{i-1}\right) \geqslant s_{i}+s_{i-1} \geqslant 2 i-1 $$ The number ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 96 | 1,158 |
2021 | T0 | N4 | Number Theory | IMO-SL | Alice is given a rational number $r>1$ and a line with two points $B \neq R$, where point $R$ contains a red bead and point $B$ contains a blue bead. Alice plays a solitaire game by performing a sequence of moves. In every move, she chooses a (not necessarily positive) integer $k$, and a bead to move. If that bead is p... | Denote the red and blue beads by $\mathcal{R}$ and $\mathcal{B}$, respectively. Introduce coordinates on the line and identify the points with their coordinates so that $R=0$ and $B=1$. Then, during the game, the coordinate of $\mathcal{R}$ is always smaller than the coordinate of $\mathcal{B}$. Moreover, the distance ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 208 | 1,927 |
2021 | T0 | N6 | Number Theory | IMO-SL | Determine all integers $n \geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_{1}, a_{2}, \ldots, a_{n}$ so that $n$ divides $1 \cdot a_{1}+2 \cdot a_{2}+\cdots+n \cdot a_{n}$. Answer: All odd integers and all powers of 2. | If $n=2^{k} a$, where $a \geqslant 3$ is odd and $k$ is a positive integer, we can consider a set containing the number $2^{k}+1$ and $n-1$ numbers congruent to 1 modulo $n$. The sum of these numbers is congruent to $2^{k}$ modulo $n$ and therefore is not divisible by $n$; for any permutation $\left(a_{1}, a_{2}, \ldot... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 104 | 1,866 |
2021 | T0 | N8 | Number Theory | IMO-SL | For a polynomial $P(x)$ with integer coefficients let $P^{1}(x)=P(x)$ and $P^{k+1}(x)=$ $P\left(P^{k}(x)\right)$ for $k \geqslant 1$. Find all positive integers $n$ for which there exists a polynomial $P(x)$ with integer coefficients such that for every integer $m \geqslant 1$, the numbers $P^{m}(1), \ldots, P^{m}(n)$ ... | Denote the set of residues modulo $\ell$ by $\mathbb{Z}_{\ell}$. Observe that $P$ can be regarded as a function $\mathbb{Z}_{\ell} \rightarrow \mathbb{Z}_{\ell}$ for any positive integer $\ell$. Denote the cardinality of the set $P^{m}\left(\mathbb{Z}_{\ell}\right)$ by $f_{m, \ell}$. Note that $f_{m, n}=\left\lceil n /... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 148 | 3,097 |
2021 | T0 | N8 | Number Theory | IMO-SL | For a polynomial $P(x)$ with integer coefficients let $P^{1}(x)=P(x)$ and $P^{k+1}(x)=$ $P\left(P^{k}(x)\right)$ for $k \geqslant 1$. Find all positive integers $n$ for which there exists a polynomial $P(x)$ with integer coefficients such that for every integer $m \geqslant 1$, the numbers $P^{m}(1), \ldots, P^{m}(n)$ ... | Observe that $f$ can be regarded as a function $\mathbb{Z}_{\ell} \rightarrow \mathbb{Z}_{\ell}$ for any positive integer $\ell$. We use notations $f^{m}$ and $f_{m, \ell}$ as in the above solution. Part 1. There exists a function $f: \mathbb{Z}_{p^{k}} \rightarrow \mathbb{Z}_{p^{k}}$ satisfying the desired properties.... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2021SL.jsonl",
"solution_match": null
} | 148 | 1,215 |
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