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 |
|---|---|---|---|---|---|---|---|---|---|
2019 | T0 | G8 | Geometry | IMO-SL | Let $\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell \in \mathcal{L}$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_{1}, \ell_{2}, \ell_{3}$ passing through $X$, the points $f\left(\ell_{1}\right), f\left(\ell_{2}\r... | For any point $X$, denote by $t(X)$ the line tangent to $g(X)$ at $X$; notice that $f(t(X))=X$, so $f$ is surjective. Step 1: We find a point $P$ for which there are at least two different lines $p_{1}$ and $p_{2}$ such that $f\left(p_{i}\right)=P$. Choose any point $X$. If $X$ does not have this property, take any $Y ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 160 | 3,145 |
2019 | T0 | G8 | Geometry | IMO-SL | Let $\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell \in \mathcal{L}$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_{1}, \ell_{2}, \ell_{3}$ passing through $X$, the points $f\left(\ell_{1}\right), f\left(\ell_{2}\r... | We will get an upper bound on $n$ from the speed at which $v_{2}\left(L_{n}\right)$ grows. From $$ L_{n}=\left(2^{n}-1\right)\left(2^{n}-2\right) \cdots\left(2^{n}-2^{n-1}\right)=2^{1+2+\cdots+(n-1)}\left(2^{n}-1\right)\left(2^{n-1}-1\right) \cdots\left(2^{1}-1\right) $$ we read $$ v_{2}\left(L_{n}\right)=1+2+\cdots+(n... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 160 | 818 |
2019 | T0 | G8 | Geometry | IMO-SL | Let $\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell \in \mathcal{L}$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_{1}, \ell_{2}, \ell_{3}$ passing through $X$, the points $f\left(\ell_{1}\right), f\left(\ell_{2}\r... | Like in the previous solution, the cases $n=1,2,3,4$ are checked manually. We will exclude $n \geqslant 5$ by considering the exponents of 3 and 31 in (1). For odd primes $p$ and distinct integers $a, b$, coprime to $p$, with $p \mid a-b$, the Lifting The Exponent lemma asserts that $$ v_{p}\left(a^{k}-b^{k}\right)=v_{... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 160 | 1,062 |
2019 | T0 | N3 | Number Theory | IMO-SL | We say that a set $S$ of integers is rootiful if, for any positive integer $n$ and any $a_{0}, a_{1}, \ldots, a_{n} \in S$, all integer roots of the polynomial $a_{0}+a_{1} x+\cdots+a_{n} x^{n}$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^{a}-2^{b}$ for positive integers ... | The set $\mathbb{Z}$ of all integers is clearly rootiful. We shall prove that any rootiful set $S$ containing all the numbers of the form $2^{a}-2^{b}$ for $a, b \in \mathbb{Z}_{>0}$ must be all of $\mathbb{Z}$. First, note that $0=2^{1}-2^{1} \in S$ and $2=2^{2}-2^{1} \in S$. Now, $-1 \in S$, since it is a root of $2 ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 117 | 548 |
2019 | T0 | N3 | Number Theory | IMO-SL | We say that a set $S$ of integers is rootiful if, for any positive integer $n$ and any $a_{0}, a_{1}, \ldots, a_{n} \in S$, all integer roots of the polynomial $a_{0}+a_{1} x+\cdots+a_{n} x^{n}$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^{a}-2^{b}$ for positive integers ... | As in the previous solution, we can prove that 0,1 and -1 must all be in any rootiful set $S$ containing all numbers of the form $2^{a}-2^{b}$ for $a, b \in \mathbb{Z}_{>0}$. We show that, in fact, every integer $k$ with $|k|>2$ can be expressed as a root of a polynomial whose coefficients are of the form $2^{a}-2^{b}$... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 117 | 681 |
2019 | T0 | N4 | Number Theory | IMO-SL | Let $\mathbb{Z}_{>0}$ be the set of positive integers. A positive integer constant $C$ is given. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$ satisfying $a+b>C$, $$ a+f(b) \mid a^{2}+b f(a) $$ (Croatia) | First, we show that $b \mid f(b)^{2}$ for all $b$. To do this, we choose a large positive integer $n$ so that $n b-f(b) \geqslant C$. Setting $a=n b-f(b)$ in (*) then shows that $$ n b \mid(n b-f(b))^{2}+b f(n b-f(b)) $$ so that $b \mid f(b)^{2}$ as claimed. Now in particular we have that $p \mid f(p)$ for every prime ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 97 | 509 |
2019 | T0 | N4 | Number Theory | IMO-SL | Let $\mathbb{Z}_{>0}$ be the set of positive integers. A positive integer constant $C$ is given. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$ satisfying $a+b>C$, $$ a+f(b) \mid a^{2}+b f(a) $$ (Croatia) | Fix any two distinct positive integers $a$ and $b$. From (*) it follows that the two integers $$ \left(a^{2}+c f(a)\right) \cdot(b+f(c)) \text { and }\left(b^{2}+c f(b)\right) \cdot(a+f(c)) $$ are both multiples of $(a+f(c)) \cdot(b+f(c))$ for all sufficiently large $c$. Taking an appropriate linear combination to elim... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 97 | 553 |
2019 | T0 | N4 | Number Theory | IMO-SL | Let $\mathbb{Z}_{>0}$ be the set of positive integers. A positive integer constant $C$ is given. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$ satisfying $a+b>C$, $$ a+f(b) \mid a^{2}+b f(a) $$ (Croatia) | Let $\Gamma$ denote the set of all points $(a, f(a))$, so that $\Gamma$ is an infinite subset of the upper-right quadrant of the plane. For a point $A=(a, f(a))$ in $\Gamma$, we define a point $A^{\prime}=\left(-f(a),-f(a)^{2} / a\right)$ in the lower-left quadrant of the plane, and let $\Gamma^{\prime}$ denote the set... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 97 | 856 |
2019 | T0 | N5 | Number Theory | IMO-SL | Let $a$ be a positive integer. We say that a positive integer $b$ is $a$-good if $\binom{a n}{b}-1$ is divisible by $a n+1$ for all positive integers $n$ with $a n \geqslant b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime. (Netherlands) | For $p$ a prime and $n$ a nonzero integer, we write $v_{p}(n)$ for the $p$-adic valuation of $n$ : the largest integer $t$ such that $p^{t} \mid n$. We first show that $b$ is $a$-good if and only if $b$ is even, and $p \mid a$ for all primes $p \leqslant b$. To start with, the condition that $a n+1 \left\lvert\,\binom{... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 112 | 864 |
2019 | T0 | N5 | Number Theory | IMO-SL | Let $a$ be a positive integer. We say that a positive integer $b$ is $a$-good if $\binom{a n}{b}-1$ is divisible by $a n+1$ for all positive integers $n$ with $a n \geqslant b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime. (Netherlands) | We show only half of the claim of the previous solution: we show that if $b$ is $a$-good, then $p \mid a$ for all primes $p \leqslant b$. We do this with Lucas' theorem. Suppose that we have $p \leqslant b$ with $p \nmid a$. Then consider the expansion of $b$ in base $p$; there will be some digit (not the final digit) ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 112 | 608 |
2019 | T0 | N6 | Number Theory | IMO-SL | Let $H=\left\{\lfloor i \sqrt{2}\rfloor: i \in \mathbb{Z}_{>0}\right\}=\{1,2,4,5,7, \ldots\}$, and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A \subset\{1,2, \ldots, n\}$ satisfies $|A| \geqslant C \sqrt{n}$, then there exist $a, b \in A$ such that $a-b \in H$. (Brazil) | First, observe that if $n$ is a positive integer, then $n \in H$ exactly when $$ \left\{\frac{n}{\sqrt{2}}\right\}>1-\frac{1}{\sqrt{2}} . $$ To see why, observe that $n \in H$ if and only if $0<i \sqrt{2}-n<1$ for some $i \in \mathbb{Z}_{>0}$. In other words, $0<i-n / \sqrt{2}<1 / \sqrt{2}$, which is equivalent to (1).... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 132 | 1,117 |
2019 | T0 | N6 | Number Theory | IMO-SL | Let $H=\left\{\lfloor i \sqrt{2}\rfloor: i \in \mathbb{Z}_{>0}\right\}=\{1,2,4,5,7, \ldots\}$, and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A \subset\{1,2, \ldots, n\}$ satisfies $|A| \geqslant C \sqrt{n}$, then there exist $a, b \in A$ such that $a-b \in H$. (Brazil) | Let $\alpha=2+\sqrt{2}$, so $(1 / \alpha)+(1 / \sqrt{2})=1$. Thus, $J=\left\{\lfloor i \alpha\rfloor: i \in \mathbb{Z}_{>0}\right\}$ is the complementary Beatty sequence to $H$ (in other words, $H$ and $J$ are disjoint with $H \cup J=\mathbb{Z}_{>0}$ ). Write $A=\left\{a_{1}<a_{2}<\cdots<a_{k}\right\}$. Suppose that $A... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 132 | 855 |
2019 | T0 | N6 | Number Theory | IMO-SL | Let $H=\left\{\lfloor i \sqrt{2}\rfloor: i \in \mathbb{Z}_{>0}\right\}=\{1,2,4,5,7, \ldots\}$, and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A \subset\{1,2, \ldots, n\}$ satisfies $|A| \geqslant C \sqrt{n}$, then there exist $a, b \in A$ such that $a-b \in H$. (Brazil) | Again, define $J=\mathbb{Z}_{>0} \backslash H$, so all differences between elements of $A$ are in $J$. We start by making the following observation. Suppose we have a set $B \subseteq\{1,2, \ldots, n\}$ such that all of the differences between elements of $B$ are in $H$. Then $|A| \cdot|B| \leqslant 2 n$. To see why, o... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 132 | 1,001 |
2019 | T0 | N6 | Number Theory | IMO-SL | Let $H=\left\{\lfloor i \sqrt{2}\rfloor: i \in \mathbb{Z}_{>0}\right\}=\{1,2,4,5,7, \ldots\}$, and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A \subset\{1,2, \ldots, n\}$ satisfies $|A| \geqslant C \sqrt{n}$, then there exist $a, b \in A$ such that $a-b \in H$. (Brazil) | As in Choose $Y$ to be a solution to the Pell-like equation $X^{2}-2 Y^{2}= \pm 1$; such solutions are given by the recurrence $Y_{1}=1, Y_{2}=2$ and $Y_{m}=2 Y_{m-1}+Y_{m-2}$, and so we can choose $Y$ such that $n /(3 \sqrt{2})<Y \leqslant n / \sqrt{2}$. Furthermore, it is known that for such a $Y$ and for $1 \leqslan... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 132 | 749 |
2019 | T0 | N7 | Number Theory | IMO-SL | Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $c n \log (n)$ pairwise coprime $n^{\text {th }}$ powers. (Canada) | Suppose, for an integer $n$, that we can find another integer $N$ satisfying the following property: $n$ is divisible by $\varphi\left(p^{e}\right)$ for every prime power $p^{e}$ exactly dividing $N$. This property ensures that all $n^{\text {th }}$ powers are congruent to 0 or 1 modulo each such prime power $p^{e}$, a... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 67 | 2,037 |
2019 | T0 | N8 | Number Theory | IMO-SL | Let $a$ and $b$ be two positive integers. Prove that the integer $$ a^{2}+\left\lceil\frac{4 a^{2}}{b}\right\rceil $$ is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) | Suppose that $$ a^{2}+\left\lceil\frac{4 a^{2}}{b}\right\rceil=c^{2} $$ with some positive integer $c>a$, so $$ \begin{aligned} & c^{2}-1<a^{2}+\frac{4 a^{2}}{b} \leqslant c^{2} \\ & 0 \leqslant c^{2} b-a^{2}(b+4)<b \end{aligned} $$ Let $d=c^{2} b-a^{2}(b+4), x=c+a$ and $y=c-a$; then we have $c=\frac{x+y}{2}$ and $a=\f... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 72 | 622 |
2019 | T0 | A1 | Algebra | IMO-SL | Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$ f(2 a)+2 f(b)=f(f(a+b)) $$ (South Africa) Answer: The solutions are $f(n)=0$ and $f(n)=2 n+K$ for any constant $K \in \mathbb{Z}$. Common remarks. Most solutions to this ... | Let $K=f(0)$. First, put $a=0$ in (1); this gives $$ f(f(b))=2 f(b)+K $$ for all $b \in \mathbb{Z}$. Now put $b=0$ in (1); this gives $$ f(2 a)+2 K=f(f(a))=2 f(a)+K $$ where the second equality follows from (2). Consequently, $$ f(2 a)=2 f(a)-K $$ for all $a \in \mathbb{Z}$. Substituting (2) and (3) into (1), we obtain... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 130 | 537 |
2019 | T0 | A2 | Algebra | IMO-SL | Let $u_{1}, u_{2}, \ldots, u_{2019}$ be real numbers satisfying $$ u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1 $$ Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that $$ a b \leqslant-\frac{... | Notice first that $b>0$ and $a<0$. Indeed, since $\sum_{i=1}^{2019} u_{i}^{2}=1$, the variables $u_{i}$ cannot be all zero, and, since $\sum_{i=1}^{2019} u_{i}=0$, the nonzero elements cannot be all positive or all negative. Let $P=\left\{i: u_{i}>0\right\}$ and $N=\left\{i: u_{i} \leqslant 0\right\}$ be the indices of... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 173 | 886 |
2019 | T0 | A2 | Algebra | IMO-SL | Let $u_{1}, u_{2}, \ldots, u_{2019}$ be real numbers satisfying $$ u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1 $$ Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that $$ a b \leqslant-\frac{... | As in the previous solution we conclude that $a<0$ and $b>0$. For every index $i$, the number $u_{i}$ is a convex combination of $a$ and $b$, so $$ u_{i}=x_{i} a+y_{i} b \quad \text { with some weights } 0 \leqslant x_{i}, y_{i} \leqslant 1, \text { with } x_{i}+y_{i}=1 \text {. } $$ Let $X=\sum_{i=1}^{2019} x_{i}$ and... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 173 | 790 |
2019 | T0 | A3 | Algebra | IMO-SL | Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2 . Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of $$ \left|1-\sum_{i \in X} a_{i}\right| $$ is minimised. Prove that there ... | This is similar to Suppose there exists $1 \leqslant j \leqslant n-1$ such that $j \in X$ but $j+1 \in X^{c}$. Then $a_{j+1}-a_{j} \geqslant \Delta$, because otherwise considering $X \cup\{j+1\} \backslash\{j\}$ contradicts $X$ being $\left(a_{i}\right)$-minimising. If $a_{j+1}-a_{j}>\Delta$, put $$ b_{i}= \begin{cases... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 718 | 661 |
2019 | T0 | A3 | Algebra | IMO-SL | Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2 . Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of $$ \left|1-\sum_{i \in X} a_{i}\right| $$ is minimised. Prove that there ... | Without loss of generality, assume $\sum_{i \in X} a_{i} \leqslant 1$, so $\Delta \geqslant 0$. If $\Delta=0$ we can take $b_{i}=a_{i}$, so now assume that $\Delta>0$. Suppose that there is some $k \leqslant n$ such that $|X \cap[k, n]|>\left|X^{c} \cap[k, n]\right|$. If we choose the largest such $k$ then $|X \cap[k, ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 718 | 563 |
2019 | T0 | A3 | Algebra | IMO-SL | Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2 . Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of $$ \left|1-\sum_{i \in X} a_{i}\right| $$ is minimised. Prove that there ... | This uses some similar ideas to Note that, for two subsets $X, Y$ of $[1, n]$, the following are equivalent: - $|X \cap[i, n]| \leqslant|Y \cap[i, n]|$ for all $1 \leqslant i \leqslant n$; - $Y$ is at least as large as $X$, and for all $1 \leqslant j \leqslant|Y|$, the $j^{\text {th }}$ largest element of $Y$ is at lea... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 718 | 1,466 |
2019 | T0 | A6 | Algebra | IMO-SL | A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$ P(x, y, z)=P(x, y, x y-z)=P(x, z x-y, z)=P(y z-x, y, z) . $$ Prove that there exists a polynomial $F(t)$ in one variable such that $$ P(x, y, z)=F\left(x^{2}+y^{2}+z^{2}-x y z\right) . $$ (Russia) Common remarks. The pol... | In the first two steps, we deal with any polynomial $P(x, y, z)$ satisfying $P(x, y, z)=$ $P(x, y, x y-z)$. Call such a polynomial weakly symmetric, and call a polynomial satisfying the full conditions in the problem symmetric. Step 1. We start with the description of weakly symmetric polynomials. We claim that they ar... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 200 | 1,164 |
2019 | T0 | A6 | Algebra | IMO-SL | A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$ P(x, y, z)=P(x, y, x y-z)=P(x, z x-y, z)=P(y z-x, y, z) . $$ Prove that there exists a polynomial $F(t)$ in one variable such that $$ P(x, y, z)=F\left(x^{2}+y^{2}+z^{2}-x y z\right) . $$ (Russia) Common remarks. The pol... | We will rely on the well-known identity $$ \cos ^{2} u+\cos ^{2} v+\cos ^{2} w-2 \cos u \cos v \cos w-1=0 \quad \text { whenever } u+v+w=0 $$ Claim 1. The polynomial $P(x, y, z)$ is constant on the surface $$ \mathfrak{S}=\{(2 \cos u, 2 \cos v, 2 \cos w): u+v+w=0\} $$ Proof. Notice that for $x=2 \cos u, y=2 \cos v, z=2... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 200 | 2,857 |
2019 | T0 | A7 | Algebra | IMO-SL | Let $\mathbb{Z}$ be the set of integers. We consider functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$ f(f(x+y)+y)=f(f(x)+y) $$ for all integers $x$ and $y$. For such a function, we say that an integer $v$ is $f$-rare if the set $$ X_{v}=\{x \in \mathbb{Z}: f(x)=v\} $$ is finite and nonempty. (a) Prove... | a) Let $f$ be the function where $f(0)=0$ and $f(x)$ is the largest power of 2 dividing $2 x$ for $x \neq 0$. The integer 0 is evidently $f$-rare, so it remains to verify the functional equation. Since $f(2 x)=2 f(x)$ for all $x$, it suffices to verify the functional equation when at least one of $x$ and $y$ is odd (th... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 171 | 1,758 |
2019 | T0 | C1 | Combinatorics | IMO-SL | The infinite sequence $a_{0}, a_{1}, a_{2}, \ldots$ of (not necessarily different) integers has the following properties: $0 \leqslant a_{i} \leqslant i$ for all integers $i \geqslant 0$, and $$ \binom{k}{a_{0}}+\binom{k}{a_{1}}+\cdots+\binom{k}{a_{k}}=2^{k} $$ for all integers $k \geqslant 0$. Prove that all integer... | We prove by induction on $k$ that every initial segment of the sequence, $a_{0}, a_{1}, \ldots, a_{k}$, consists of the following elements (counted with multiplicity, and not necessarily in order), for some $\ell \geqslant 0$ with $2 \ell \leqslant k+1$ : $$ 0,1, \ldots, \ell-1, \quad 0,1, \ldots, k-\ell $$ For $k=0$ w... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 179 | 836 |
2019 | T0 | C3 | Combinatorics | IMO-SL | Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are $k$ coins showing heads and $k>0$, then he flips the $k^{\text {th }}$ coin over; otherwise he stops the process. (For example, the process starting with THT wo... | We represent the problem using a directed graph $G_{n}$ whose vertices are the length- $n$ strings of $H$ 's and $T$ 's. The graph features an edge from each string to its successor (except for $T T \cdots T T$, which has no successor). We will also write $\bar{H}=T$ and $\bar{T}=H$. The graph $G_{0}$ consists of a sin... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 220 | 832 |
2019 | T0 | C3 | Combinatorics | IMO-SL | Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are $k$ coins showing heads and $k>0$, then he flips the $k^{\text {th }}$ coin over; otherwise he stops the process. (For example, the process starting with THT wo... | We consider what happens with configurations depending on the coins they start and end with. - If a configuration starts with $H$, the last $n-1$ coins follow the given rules, as if they were all the coins, until they are all $T$, then the first coin is turned over. - If a configuration ends with $T$, the last coin wil... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 220 | 623 |
2019 | T0 | C3 | Combinatorics | IMO-SL | Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are $k$ coins showing heads and $k>0$, then he flips the $k^{\text {th }}$ coin over; otherwise he stops the process. (For example, the process starting with THT wo... | Let $H_{i}$ be the number of heads in positions 1 to $i$ inclusive (so $H_{n}$ is the total number of heads), and let $I_{i}$ be 1 if the $i^{\text {th }}$ coin is a head, 0 otherwise. Consider the function $$ t(i)=I_{i}+2\left(\min \left\{i, H_{n}\right\}-H_{i}\right) $$ We claim that $t(i)$ is the total number of tim... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 220 | 956 |
2019 | T0 | C3 | Combinatorics | IMO-SL | Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are $k$ coins showing heads and $k>0$, then he flips the $k^{\text {th }}$ coin over; otherwise he stops the process. (For example, the process starting with THT wo... | Harry has built a Turing machine to flip the coins for him. The machine is initially positioned at the $k^{\text {th }}$ coin, where there are $k$ heads (and the position before the first coin is considered to be the $0^{\text {th }}$ coin). The machine then moves according to the following rules, stopping when it reac... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 220 | 638 |
2019 | T0 | C3 | Combinatorics | IMO-SL | Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are $k$ coins showing heads and $k>0$, then he flips the $k^{\text {th }}$ coin over; otherwise he stops the process. (For example, the process starting with THT wo... | We explicitly describe what happens with an arbitrary sequence $C$ of $n$ coins. Suppose that $C$ contain $k$ heads at positions $1 \leqslant c_{1}<c_{2}<\cdots<c_{k} \leqslant n$. Let $i$ be the minimal index such that $c_{i} \geqslant k$. Then the first few steps will consist of turning over the $k^{\mathrm{th}},(k+1... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 220 | 754 |
2019 | T0 | C4 | Combinatorics | IMO-SL | On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection ... | First we show by induction that the $n$ walls divide the plane into $\binom{n+1}{2}+1$ regions. The claim is true for $n=0$ as, when there are no walls, the plane forms a single region. When placing the $n^{\text {th }}$ wall, it intersects each of the $n-1$ other walls exactly once and hence splits each of $n$ of the ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 294 | 821 |
2019 | T0 | C4 | Combinatorics | IMO-SL | On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection ... | We give another description of a strategy for Merlin to paint the walls so that Morgana can place no more than $n+1$ knights. Merlin starts by building a labyrinth of $n$ walls of his own design. He places walls in turn with increasing positive gradients, placing each so far to the right that all intersection points of... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 294 | 585 |
2019 | T0 | C5 | Combinatorics | IMO-SL | On a certain social network, there are 2019 users, some pairs of which are friends, where friendship is a symmetric relation. Initially, there are 1010 people with 1009 friends each and 1009 people with 1010 friends each. However, the friendships are rather unstable, so events of the following kind may happen repeatedl... | Note that the given graph is connected, since the total degree of any two vertices is at least 2018 and hence they are either adjacent or have at least one neighbour in common. Hence the given graph satisfies the following condition: Every connected component of $G$ with at least three vertices is not complete and has ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 376 | 1,001 |
2019 | T0 | C5 | Combinatorics | IMO-SL | On a certain social network, there are 2019 users, some pairs of which are friends, where friendship is a symmetric relation. Initially, there are 1010 people with 1009 friends each and 1009 people with 1010 friends each. However, the friendships are rather unstable, so events of the following kind may happen repeatedl... | As in the previous solution, note that a refriending preserves the property that a graph has a vertex of odd degree and (trivially) the property that it is not complete; note also that our initial graph is connected. We describe an algorithm to reduce our initial graph to a graph of maximal degree at most 1, proceeding... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 376 | 611 |
2019 | T0 | C6 | Combinatorics | IMO-SL | Let $n>1$ be an integer. Suppose we are given $2 n$ points in a plane such that no three of them are collinear. The points are to be labelled $A_{1}, A_{2}, \ldots, A_{2 n}$ in some order. We then consider the $2 n$ angles $\angle A_{1} A_{2} A_{3}, \angle A_{2} A_{3} A_{4}, \ldots, \angle A_{2 n-2} A_{2 n-1} A_{2 n}, ... | When tracing a cyclic path through the $A_{i}$ in order, with straight line segments between consecutive points, let $\theta_{i}$ be the exterior angle at $A_{i}$, with a sign convention that it is positive if the path turns left and negative if the path turns right. Then $\sum_{i=1}^{2 n} \theta_{i}=360 k^{\circ}$ for... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 304 | 779 |
2019 | T0 | C6 | Combinatorics | IMO-SL | Let $n>1$ be an integer. Suppose we are given $2 n$ points in a plane such that no three of them are collinear. The points are to be labelled $A_{1}, A_{2}, \ldots, A_{2 n}$ in some order. We then consider the $2 n$ angles $\angle A_{1} A_{2} A_{3}, \angle A_{2} A_{3} A_{4}, \ldots, \angle A_{2 n-2} A_{2 n-1} A_{2 n}, ... | First, let $\ell$ be a line in the plane such that there are $n$ points on one side and the other $n$ points on the other side. For convenience, assume $\ell$ is horizontal (otherwise, we can rotate the plane). Then we can use the terms "above", "below", "left" and "right" in the usual way. We denote the $n$ points abo... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 304 | 2,282 |
2019 | T0 | C8 | Combinatorics | IMO-SL | Alice has a map of Wonderland, a country consisting of $n \geqslant 2$ towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be "one way" only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help he... | We will show Alice needs to ask at most $4 n-7$ questions. Her strategy has the following phases. In what follows, $S$ is the set of towns that Alice, so far, does not know to have more than one outgoing road (so initially $|S|=n$ ). Phase 1. Alice chooses any two towns, say $A$ and $B$. Without loss of generality, sup... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 208 | 3,497 |
2019 | T0 | C9 | Combinatorics | IMO-SL | For any two different real numbers $x$ and $y$, we define $D(x, y)$ to be the unique integer $d$ satisfying $2^{d} \leqslant|x-y|<2^{d+1}$. Given a set of reals $\mathcal{F}$, and an element $x \in \mathcal{F}$, we say that the scales of $x$ in $\mathcal{F}$ are the values of $D(x, y)$ for $y \in \mathcal{F}$ with $x \... | We first construct a set $\mathcal{F}$ with $2^{k}$ members, each member having at most $k$ different scales in $\mathcal{F}$. Take $\mathcal{F}=\left\{0,1,2, \ldots, 2^{k}-1\right\}$. The scale between any two members of $\mathcal{F}$ is in the set $\{0,1, \ldots, k-1\}$. We now show that $2^{k}$ is an upper bound on ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 251 | 2,514 |
2019 | T0 | G2 | Geometry | IMO-SL | Let $A B C$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $B C, C A$, and $A B$, respectively. Denote by $\omega_{B}$ and $\omega_{C}$ the incircles of triangles $B D F$ and $C D E$, and let these circles be tangent to segments $D F$ and $D E$ at $M$ and $N$... | Denote the centres of $\omega_{B}$ and $\omega_{C}$ by $O_{B}$ and $O_{C}$, let their radii be $r_{B}$ and $r_{C}$, and let $B C$ be tangent to the two circles at $T$ and $U$, respectively. From the cyclic quadrilaterals $A F D C$ and $A B D E$ we have $$ \angle M D O_{B}=\frac{1}{2} \angle F D B=\frac{1}{2} \angle B A... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 164 | 666 |
2019 | T0 | G4 | Geometry | IMO-SL | Let $P$ be a point inside triangle $A B C$. Let $A P$ meet $B C$ at $A_{1}$, let $B P$ meet $C A$ at $B_{1}$, and let $C P$ meet $A B$ at $C_{1}$. Let $A_{2}$ be the point such that $A_{1}$ is the midpoint of $P A_{2}$, let $B_{2}$ be the point such that $B_{1}$ is the midpoint of $P B_{2}$, and let $C_{2}$ be the poin... | Since $$ \angle A P B+\angle B P C+\angle C P A=2 \pi=(\pi-\angle A C B)+(\pi-\angle B A C)+(\pi-\angle C B A), $$ at least one of the following inequalities holds: $$ \angle A P B \geqslant \pi-\angle A C B, \quad \angle B P C \geqslant \pi-\angle B A C, \quad \angle C P A \geqslant \pi-\angle C B A . $$ Without loss ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 260 | 874 |
2019 | T0 | G4 | Geometry | IMO-SL | Let $P$ be a point inside triangle $A B C$. Let $A P$ meet $B C$ at $A_{1}$, let $B P$ meet $C A$ at $B_{1}$, and let $C P$ meet $A B$ at $C_{1}$. Let $A_{2}$ be the point such that $A_{1}$ is the midpoint of $P A_{2}$, let $B_{2}$ be the point such that $B_{1}$ is the midpoint of $P B_{2}$, and let $C_{2}$ be the poin... | Choose coordinates such that the circumcentre of $\triangle A B C$ is at the origin and the circumradius is 1 . Then we may think of $A, B$, and $C$ as vectors in $\mathbb{R}^{2}$ such that $$ |A|^{2}=|B|^{2}=|C|^{2}=1 $$ $P$ may be represented as a convex combination $\alpha A+\beta B+\gamma C$ where $\alpha, \beta, \... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 260 | 1,193 |
2019 | T0 | G5 | Geometry | IMO-SL | Let $A B C D E$ be a convex pentagon with $C D=D E$ and $\angle E D C \neq 2 \cdot \angle A D B$. Suppose that a point $P$ is located in the interior of the pentagon such that $A P=A E$ and $B P=B C$. Prove that $P$ lies on the diagonal $C E$ if and only if area $(B C D)+\operatorname{area}(A D E)=$ $\operatorname{are... | Let $P^{\prime}$ be the reflection of $P$ across line $A B$, and let $M$ and $N$ be the midpoints of $P^{\prime} E$ and $P^{\prime} C$ respectively. Convexity ensures that $P^{\prime}$ is distinct from both $E$ and $C$, and hence from both $M$ and $N$. We claim that both the area condition and the collinearity conditio... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 131 | 1,330 |
2019 | T0 | G5 | Geometry | IMO-SL | Let $A B C D E$ be a convex pentagon with $C D=D E$ and $\angle E D C \neq 2 \cdot \angle A D B$. Suppose that a point $P$ is located in the interior of the pentagon such that $A P=A E$ and $B P=B C$. Prove that $P$ lies on the diagonal $C E$ if and only if area $(B C D)+\operatorname{area}(A D E)=$ $\operatorname{are... | Along the perpendicular bisector of $C E$, define the linear function $$ f(X)=\operatorname{area}(B C X)+\operatorname{area}(A X E)-\operatorname{area}(A B X)-\operatorname{area}(A B P), $$ where, from now on, we always use signed areas. Thus, we want to show that $C, P, E$ are collinear if and only if $f(D)=0$. Let $P... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 131 | 1,977 |
2019 | T0 | G6 | Geometry | IMO-SL | Let $I$ be the incentre of acute-angled triangle $A B C$. Let the incircle meet $B C, C A$, and $A B$ at $D, E$, and $F$, respectively. Let line $E F$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle D P A+\angle A Q D=\angle Q I P$. (Slovakia) | Let $N$ and $M$ be the midpoints of the arcs $\widehat{B C}$ of the circumcircle, containing and opposite vertex $A$, respectively. By $\angle F A E=\angle B A C=\angle B N C$, the right-angled kites $A F I E$ and $N B M C$ are similar. Consider the spiral similarity $\varphi$ (dilation in case of $A B=A C$ ) that move... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 107 | 700 |
2019 | T0 | G6 | Geometry | IMO-SL | Let $I$ be the incentre of acute-angled triangle $A B C$. Let the incircle meet $B C, C A$, and $A B$ at $D, E$, and $F$, respectively. Let line $E F$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle D P A+\angle A Q D=\angle Q I P$. (Slovakia) | Define the point $M$ and the same spiral similarity $\varphi$ as in the previous solution. (The point $N$ is not necessary.) It is well-known that the centre of the spiral similarity that maps $F, E$ to $B, C$ is the Miquel point of the lines $F E, B C, B F$ and $C E$; that is, the second intersection of circles $A B C... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 107 | 594 |
2019 | T0 | G6 | Geometry | IMO-SL | Let $I$ be the incentre of acute-angled triangle $A B C$. Let the incircle meet $B C, C A$, and $A B$ at $D, E$, and $F$, respectively. Let line $E F$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle D P A+\angle A Q D=\angle Q I P$. (Slovakia) | Denote the circumcircle of triangle $A B C$ by $\Gamma$, and let rays $P D$ and $Q D$ meet $\Gamma$ again at $V$ and $U$, respectively. We will show that $A U \perp I P$ and $A V \perp I Q$. Then the problem statement will follow as $$ \angle D P A+\angle A Q D=\angle V U A+\angle A V U=180^{\circ}-\angle U A V=\angle ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 107 | 573 |
2019 | T0 | G7 | Geometry | IMO-SL | The incircle $\omega$ of acute-angled scalene triangle $A B C$ has centre $I$ and meets sides $B C$, $C A$, and $A B$ at $D, E$, and $F$, respectively. The line through $D$ perpendicular to $E F$ meets $\omega$ again at $R$. Line $A R$ meets $\omega$ again at $P$. The circumcircles of triangles $P C E$ and $P B F$ mee... | Step 1. The external bisector of $\angle B A C$ is the line through $A$ perpendicular to $I A$. Let $D I$ meet this line at $L$ and let $D I$ meet $\omega$ at $K$. Let $N$ be the midpoint of $E F$, which lies on $I A$ and is the pole of line $A L$ with respect to $\omega$. Since $A N \cdot A I=A E^{2}=A R \cdot A P$, t... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 174 | 1,023 |
2019 | T0 | G7 | Geometry | IMO-SL | The incircle $\omega$ of acute-angled scalene triangle $A B C$ has centre $I$ and meets sides $B C$, $C A$, and $A B$ at $D, E$, and $F$, respectively. The line through $D$ perpendicular to $E F$ meets $\omega$ again at $R$. Line $A R$ meets $\omega$ again at $P$. The circumcircles of triangles $P C E$ and $P B F$ mee... | We start as in Step 1. Let $A R$ meet the circumcircle $\Omega$ of $A B C$ again at $X$. The lines $A R$ and $A K$ are isogonal in the angle $B A C$; it is well known that in this case $X$ is the tangency point of $\Omega$ with the $A$-mixtilinear circle. It is also well known that for this point $X$, the line $X I$ cr... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 174 | 1,268 |
2019 | T0 | G8 | Geometry | IMO-SL | Let $\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell \in \mathcal{L}$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_{1}, \ell_{2}, \ell_{3}$ passing through $X$, the points $f\left(\ell_{1}\right), f\left(\ell_{2}\... | We provide a complete characterisation of the functions satisfying the given condition. Write $\angle\left(\ell_{1}, \ell_{2}\right)$ for the directed angle modulo $180^{\circ}$ between the lines $\ell_{1}$ and $\ell_{2}$. Given a point $P$ and an angle $\alpha \in\left(0,180^{\circ}\right)$, for each line $\ell$, let ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 333 | 2,009 |
2019 | T0 | G8 | Geometry | IMO-SL | Let $\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell \in \mathcal{L}$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_{1}, \ell_{2}, \ell_{3}$ passing through $X$, the points $f\left(\ell_{1}\right), f\left(\ell_{2}\... | Note that for any distinct points $X, Y$, the circles $g(X)$ and $g(Y)$ meet on $X Y$ at the point $f(X Y) \in g(X) \cap g(Y) \cap(X Y)$. We write $s(X, Y)$ for the second intersection point of circles $g(X)$ and $g(Y)$. Lemma 1. Suppose that $X, Y$ and $Z$ are not collinear, and that $f(X Y) \notin\{X, Y\}$ and simila... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 333 | 831 |
2019 | T0 | G8 | Geometry | IMO-SL | Let $\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell \in \mathcal{L}$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_{1}, \ell_{2}, \ell_{3}$ passing through $X$, the points $f\left(\ell_{1}\right), f\left(\ell_{2}\... | Notice that, for any two different points $X$ and $Y$, the point $f(X Y)$ lies on both $g(X)$ and $g(Y)$, so any two such circles meet in at least one point. We refer to two circles as cutting only in the case where they cross, and so meet at exactly two points, thus excluding the cases where they are tangent or are th... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 333 | 1,316 |
2019 | T0 | G8 | Geometry | IMO-SL | Let $\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell \in \mathcal{L}$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_{1}, \ell_{2}, \ell_{3}$ passing through $X$, the points $f\left(\ell_{1}\right), f\left(\ell_{2}\... | For any point $X$, denote by $t(X)$ the line tangent to $g(X)$ at $X$; notice that $f(t(X))=X$, so $f$ is surjective. Step 1: We find a point $P$ for which there are at least two different lines $p_{1}$ and $p_{2}$ such that $f\left(p_{i}\right)=P$. Choose any point $X$. If $X$ does not have this property, take any $Y ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 333 | 3,145 |
2019 | T0 | G8 | Geometry | IMO-SL | Let $\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell \in \mathcal{L}$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_{1}, \ell_{2}, \ell_{3}$ passing through $X$, the points $f\left(\ell_{1}\right), f\left(\ell_{2}\... | We will get an upper bound on $n$ from the speed at which $v_{2}\left(L_{n}\right)$ grows. From $$ L_{n}=\left(2^{n}-1\right)\left(2^{n}-2\right) \cdots\left(2^{n}-2^{n-1}\right)=2^{1+2+\cdots+(n-1)}\left(2^{n}-1\right)\left(2^{n-1}-1\right) \cdots\left(2^{1}-1\right) $$ we read $$ v_{2}\left(L_{n}\right)=1+2+\cdots+(n... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 333 | 818 |
2019 | T0 | G8 | Geometry | IMO-SL | Let $\mathcal{L}$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell \in \mathcal{L}$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_{1}, \ell_{2}, \ell_{3}$ passing through $X$, the points $f\left(\ell_{1}\right), f\left(\ell_{2}\... | Like in the previous solution, the cases $n=1,2,3,4$ are checked manually. We will exclude $n \geqslant 5$ by considering the exponents of 3 and 31 in (1). For odd primes $p$ and distinct integers $a, b$, coprime to $p$, with $p \mid a-b$, the Lifting The Exponent lemma asserts that $$ v_{p}\left(a^{k}-b^{k}\right)=v_{... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 333 | 1,062 |
2019 | T0 | N2 | Number Theory | IMO-SL | Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. (Nigeria) Answer: The solutions are $(1,2,3)$ and its permutations. Common remarks. Note that the equation is symmetric. In all solutions, we will assume without loss of generality that $a \geqslant b \geqslant c$, and prove th... | We will start by proving that $c=1$. Note that $$ 3 a^{3} \geqslant a^{3}+b^{3}+c^{3}>a^{3} . $$ So $3 a^{3} \geqslant(a b c)^{2}>a^{3}$ and hence $3 a \geqslant b^{2} c^{2}>a$. Now $b^{3}+c^{3}=a^{2}\left(b^{2} c^{2}-a\right) \geqslant a^{2}$, and so $$ 18 b^{3} \geqslant 9\left(b^{3}+c^{3}\right) \geqslant 9 a^{2} \g... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 133 | 983 |
2019 | T0 | N2 | Number Theory | IMO-SL | Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. (Nigeria) Answer: The solutions are $(1,2,3)$ and its permutations. Common remarks. Note that the equation is symmetric. In all solutions, we will assume without loss of generality that $a \geqslant b \geqslant c$, and prove th... | Again, we will start by proving that $c=1$. Suppose otherwise that $c \geqslant 2$. We have $a^{3}+b^{3}+c^{3} \leqslant 3 a^{3}$, so $b^{2} c^{2} \leqslant 3 a$. Since $c \geqslant 2$, this tells us that $b \leqslant \sqrt{3 a / 4}$. As the right-hand side of the original equation is a multiple of $a^{2}$, we have $a^... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 133 | 832 |
2019 | T0 | N2 | Number Theory | IMO-SL | Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. (Nigeria) Answer: The solutions are $(1,2,3)$ and its permutations. Common remarks. Note that the equation is symmetric. In all solutions, we will assume without loss of generality that $a \geqslant b \geqslant c$, and prove th... | Again, observe that $b^{3}+c^{3}=a^{2}\left(b^{2} c^{2}-a\right)$, so $b \leqslant a \leqslant b^{2} c^{2}-1$. We consider the function $f(x)=x^{2}\left(b^{2} c^{2}-x\right)$. It can be seen that that on the interval $\left[0, b^{2} c^{2}-1\right]$ the function $f$ is increasing if $x<\frac{2}{3} b^{2} c^{2}$ and decre... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 133 | 836 |
2019 | T0 | N3 | Number Theory | IMO-SL | We say that a set $S$ of integers is rootiful if, for any positive integer $n$ and any $a_{0}, a_{1}, \ldots, a_{n} \in S$, all integer roots of the polynomial $a_{0}+a_{1} x+\cdots+a_{n} x^{n}$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^{a}-2^{b}$ for positive integers ... | The set $\mathbb{Z}$ of all integers is clearly rootiful. We shall prove that any rootiful set $S$ containing all the numbers of the form $2^{a}-2^{b}$ for $a, b \in \mathbb{Z}_{>0}$ must be all of $\mathbb{Z}$. First, note that $0=2^{1}-2^{1} \in S$ and $2=2^{2}-2^{1} \in S$. Now, $-1 \in S$, since it is a root of $2 ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 138 | 548 |
2019 | T0 | N3 | Number Theory | IMO-SL | We say that a set $S$ of integers is rootiful if, for any positive integer $n$ and any $a_{0}, a_{1}, \ldots, a_{n} \in S$, all integer roots of the polynomial $a_{0}+a_{1} x+\cdots+a_{n} x^{n}$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^{a}-2^{b}$ for positive integers ... | As in the previous solution, we can prove that 0,1 and -1 must all be in any rootiful set $S$ containing all numbers of the form $2^{a}-2^{b}$ for $a, b \in \mathbb{Z}_{>0}$. We show that, in fact, every integer $k$ with $|k|>2$ can be expressed as a root of a polynomial whose coefficients are of the form $2^{a}-2^{b}$... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 138 | 681 |
2019 | T0 | N4 | Number Theory | IMO-SL | Let $\mathbb{Z}_{>0}$ be the set of positive integers. A positive integer constant $C$ is given. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$ satisfying $a+b>C$, $$ a+f(b) \mid a^{2}+b f(a) $$ (Croatia) Answer: The functions satisfying (*) are ex... | First, we show that $b \mid f(b)^{2}$ for all $b$. To do this, we choose a large positive integer $n$ so that $n b-f(b) \geqslant C$. Setting $a=n b-f(b)$ in (*) then shows that $$ n b \mid(n b-f(b))^{2}+b f(n b-f(b)) $$ so that $b \mid f(b)^{2}$ as claimed. Now in particular we have that $p \mid f(p)$ for every prime ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 395 | 509 |
2019 | T0 | N4 | Number Theory | IMO-SL | Let $\mathbb{Z}_{>0}$ be the set of positive integers. A positive integer constant $C$ is given. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$ satisfying $a+b>C$, $$ a+f(b) \mid a^{2}+b f(a) $$ (Croatia) Answer: The functions satisfying (*) are ex... | Fix any two distinct positive integers $a$ and $b$. From (*) it follows that the two integers $$ \left(a^{2}+c f(a)\right) \cdot(b+f(c)) \text { and }\left(b^{2}+c f(b)\right) \cdot(a+f(c)) $$ are both multiples of $(a+f(c)) \cdot(b+f(c))$ for all sufficiently large $c$. Taking an appropriate linear combination to elim... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 395 | 553 |
2019 | T0 | N4 | Number Theory | IMO-SL | Let $\mathbb{Z}_{>0}$ be the set of positive integers. A positive integer constant $C$ is given. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$ satisfying $a+b>C$, $$ a+f(b) \mid a^{2}+b f(a) $$ (Croatia) Answer: The functions satisfying (*) are ex... | Let $\Gamma$ denote the set of all points $(a, f(a))$, so that $\Gamma$ is an infinite subset of the upper-right quadrant of the plane. For a point $A=(a, f(a))$ in $\Gamma$, we define a point $A^{\prime}=\left(-f(a),-f(a)^{2} / a\right)$ in the lower-left quadrant of the plane, and let $\Gamma^{\prime}$ denote the set... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 395 | 856 |
2019 | T0 | N6 | Number Theory | IMO-SL | Let $H=\left\{\lfloor i \sqrt{2}\rfloor: i \in \mathbb{Z}_{>0}\right\}=\{1,2,4,5,7, \ldots\}$, and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A \subset\{1,2, \ldots, n\}$ satisfies $|A| \geqslant C \sqrt{n}$, then there exist $a, b \in A$ such that $a-b \in H$. (Brazil) Common... | First, observe that if $n$ is a positive integer, then $n \in H$ exactly when $$ \left\{\frac{n}{\sqrt{2}}\right\}>1-\frac{1}{\sqrt{2}} . $$ To see why, observe that $n \in H$ if and only if $0<i \sqrt{2}-n<1$ for some $i \in \mathbb{Z}_{>0}$. In other words, $0<i-n / \sqrt{2}<1 / \sqrt{2}$, which is equivalent to (1).... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 184 | 1,117 |
2019 | T0 | N6 | Number Theory | IMO-SL | Let $H=\left\{\lfloor i \sqrt{2}\rfloor: i \in \mathbb{Z}_{>0}\right\}=\{1,2,4,5,7, \ldots\}$, and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A \subset\{1,2, \ldots, n\}$ satisfies $|A| \geqslant C \sqrt{n}$, then there exist $a, b \in A$ such that $a-b \in H$. (Brazil) Common... | Let $\alpha=2+\sqrt{2}$, so $(1 / \alpha)+(1 / \sqrt{2})=1$. Thus, $J=\left\{\lfloor i \alpha\rfloor: i \in \mathbb{Z}_{>0}\right\}$ is the complementary Beatty sequence to $H$ (in other words, $H$ and $J$ are disjoint with $H \cup J=\mathbb{Z}_{>0}$ ). Write $A=\left\{a_{1}<a_{2}<\cdots<a_{k}\right\}$. Suppose that $A... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 184 | 855 |
2019 | T0 | N6 | Number Theory | IMO-SL | Let $H=\left\{\lfloor i \sqrt{2}\rfloor: i \in \mathbb{Z}_{>0}\right\}=\{1,2,4,5,7, \ldots\}$, and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A \subset\{1,2, \ldots, n\}$ satisfies $|A| \geqslant C \sqrt{n}$, then there exist $a, b \in A$ such that $a-b \in H$. (Brazil) Common... | Again, define $J=\mathbb{Z}_{>0} \backslash H$, so all differences between elements of $A$ are in $J$. We start by making the following observation. Suppose we have a set $B \subseteq\{1,2, \ldots, n\}$ such that all of the differences between elements of $B$ are in $H$. Then $|A| \cdot|B| \leqslant 2 n$. To see why, o... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 184 | 1,001 |
2019 | T0 | N6 | Number Theory | IMO-SL | Let $H=\left\{\lfloor i \sqrt{2}\rfloor: i \in \mathbb{Z}_{>0}\right\}=\{1,2,4,5,7, \ldots\}$, and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A \subset\{1,2, \ldots, n\}$ satisfies $|A| \geqslant C \sqrt{n}$, then there exist $a, b \in A$ such that $a-b \in H$. (Brazil) Common... | As in Choose $Y$ to be a solution to the Pell-like equation $X^{2}-2 Y^{2}= \pm 1$; such solutions are given by the recurrence $Y_{1}=1, Y_{2}=2$ and $Y_{m}=2 Y_{m-1}+Y_{m-2}$, and so we can choose $Y$ such that $n /(3 \sqrt{2})<Y \leqslant n / \sqrt{2}$. Furthermore, it is known that for such a $Y$ and for $1 \leqslan... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 184 | 749 |
2019 | T0 | N8 | Number Theory | IMO-SL | Let $a$ and $b$ be two positive integers. Prove that the integer $$ a^{2}+\left\lceil\frac{4 a^{2}}{b}\right\rceil $$ is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to z.) (Russia) | Suppose that $$ a^{2}+\left\lceil\frac{4 a^{2}}{b}\right\rceil=c^{2} $$ with some positive integer $c>a$, so $$ \begin{aligned} & c^{2}-1<a^{2}+\frac{4 a^{2}}{b} \leqslant c^{2} \\ & 0 \leqslant c^{2} b-a^{2}(b+4)<b \end{aligned} $$ Let $d=c^{2} b-a^{2}(b+4), x=c+a$ and $y=c-a$; then we have $c=\frac{x+y}{2}$ and $a=\f... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 73 | 622 |
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 $x$, $$ \sq... | 2 (for Version 2). Here we present another proof of the inequality (2) for $x>0$, or, equivalently, for $t=(x-1)^{2} / x \geqslant 0$. Instead of finding the coefficients of the polynomial $f_{N}=f_{N}(t)$ we may find its roots, which is in a sense more straightforward. Note that the recurrence (4) and the initial cond... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 152 | 722 |
2020 | T0 | A1 | Algebra | IMO-SL | Version 1. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, $$ \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 $f(1)=0$, $$ f^{\prime}(x)=\left(\frac{x^{2 N}+1}{2}\right)^{\frac{1}{N}-1} x^{2 N-1}-2 a(x-1)-1 \quad \text { and } \quad ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 152 | 573 |
2020 | T0 | A2 | Algebra | IMO-SL | Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as $$ (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 $$ therefore $x^{3} \in ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 172 | 2,349 |
2020 | T0 | A3 | Algebra | IMO-SL | Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d)=a c+b d$. Find the smallest possible value of $$ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} $$ | By homogeneity we may suppose that $a b c d=1$. Let $a b=C, b c=A$ and $c a=B$. Then $a, b, c$ can be reconstructed from $A, B$ and $C$ as $a=\sqrt{B C / A}, b=\sqrt{A C / B}$ and $c=\sqrt{A B / C}$. Moreover, the condition $(a+c)(b+d)=a c+b d$ can be written in terms of $A, B, C$ as $$ A+\frac{1}{A}+C+\frac{1}{C}=b c+... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 66 | 703 |
2020 | T0 | A4 | Algebra | IMO-SL | Let $a, b, c, d$ be four real numbers such that $a \geqslant b \geqslant c \geqslant d>0$ and $a+b+c+d=1$. Prove that $$ (a+2 b+3 c+4 d) a^{a} b^{b} c^{c} d^{d}<1 $$ (Belgium) | From $b \geqslant d$ we get $$ a+2 b+3 c+4 d \leqslant a+3 b+3 c+3 d=3-2 a $$ If $a<\frac{1}{2}$, then the statement can be proved by $$ (a+2 b+3 c+4 d) a^{a} b^{b} c^{c} d^{d} \leqslant(3-2 a) a^{a} a^{b} a^{c} a^{d}=(3-2 a) a=1-(1-a)(1-2 a)<1 $$ From now on we assume $\frac{1}{2} \leqslant a<1$. By $b, c, d<1-a$ we h... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 88 | 970 |
2020 | T0 | A5 | Algebra | 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
} | 146 | 773 |
2020 | T0 | A6 | Algebra | 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 $n \geqslant 0$. | 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
} | 137 | 1,483 |
2020 | T0 | A7 | Algebra | 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 $$ \sum_{i=1}^{n} \frac{a_{i}}{\sqrt{a_{1}^{2}+\ldots+a_{i}^{2}}} \leqslant 4 \sqrt{k n} $$ | 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 $$ \sum_{\ell \in M_{j}} \frac{a_{\ell}}{\sqrt{a_{1}^{2}+\ldots+a_{\ell}^{2}}} \leqslant \sum_{i=1}... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 96 | 1,007 |
2020 | T0 | A7 | Algebra | IMO-SL | Let $n$ and $k$ be positive integers. Prove that for $a_{1}, \ldots, a_{n} \in\left[1,2^{k}\right]$ one has $$ \sum_{i=1}^{n} \frac{a_{i}}{\sqrt{a_{1}^{2}+\ldots+a_{i}^{2}}} \leqslant 4 \sqrt{k n} $$ | Apply induction on $n$. The base $n \leqslant 16$ is clear: our sum does not exceed $n \leqslant 4 \sqrt{n k}$. For the inductive step from $1, \ldots, n-1$ to $n \geqslant 17$ consider two similar cases. Case 1: $n=2 t$. Let $x_{\ell}=\frac{a_{\ell}}{\sqrt{a_{1}^{2}+\ldots+a_{\ell}^{2}}}$. We have $$ \exp \left(-x_{t+... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 96 | 652 |
2020 | T0 | A8 | Algebra | 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 $y$, $$ f(x+f(x y))+y=f(x) f(y)+1 $$ | 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
} | 76 | 850 |
2020 | T0 | A8 | Algebra | 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 $y$, $$ f(x+f(x y))+y=f(x) f(y)+1 $$ | 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
} | 76 | 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) | 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
} | 98 | 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) | 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
} | 98 | 822 |
2020 | T0 | C2 | Combinatorics | IMO-SL | In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100 -gon, so that - the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and - every quadrilateral ... | Call a quadrilateral skew-colored, if it has three corners of one color and one corner of the other color. We will prove the following Claim. If the vertices of a convex $(4 k+1)$-gon $P$ are colored black and white such that each color is used at least $k$ times, then there exist $k$ pairwise disjoint skew-colored qua... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 121 | 958 |
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
} | 235 | 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
} | 137 | 815 |
2020 | T0 | C5 | Combinatorics | IMO-SL | Let $p$ be an odd prime, and put $N=\frac{1}{4}\left(p^{3}-p\right)-1$. The numbers $1,2, \ldots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N$, denote by $r(n)$ the fraction of integers in $\{1,2, \ldots, n\}$ that are red. Prove that there exists a positive integer ... | Denote by $R(n)$ the number of red numbers in $\{1,2, \ldots, n\}$, i.e., $R(n)=n r(n)$. Similarly, denote by $B(n)$ and $b(n)=B(n) / n$ the number and proportion of blue numbers in $\{1,2, \ldots, n\}$, respectively. Notice that $B(n)+R(n)=n$ and $b(n)+r(n)=1$. Therefore, the statement of the problem does not change a... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 156 | 1,229 |
2020 | T0 | C6 | Combinatorics | IMO-SL | $4 n$ coins of weights $1,2,3, \ldots, 4 n$ are given. Each coin is colored in one of $n$ colors and there are four coins of each color. Show that all these coins can be partitioned into two sets with the same total weight, such that each set contains two coins of each color. (Hungary) | As in Introduce a multi-graph (i.e., a graph with multiple edges allowed) $\Gamma$ whose vertices correspond to coins; thus we have $4 n$ vertices of $n$ colors so that there are four vertices of each color. Connect pairs of vertices $\{1,4 n\},\{2,4 n-1\}, \ldots,\{2 n, 2 n+1\}$ by $2 n$ black edges. Further, for each... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 78 | 607 |
2020 | T0 | C7 | Combinatorics | IMO-SL | Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a saddle pair if the following two conditions are satisfied: (i) For each row $r^{\prime}$, there is $... | We say that a pair $\left(R^{\prime}, C^{\prime}\right)$ of nonempty sets is a subpair of a pair $(R, C)$ if $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$. The subpair is proper if at least one of the inclusions is strict. Let $\left(R_{1}, C_{1}\right)$ and $\left(R_{2}, C_{2}\right)$ be two saddle pairs with ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 276 | 1,985 |
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
} | 201 | 2,946 |
2020 | T0 | G4 | Geometry | IMO-SL | In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that $$ O P_{1}+O P_{2}+\ldots+O ... | We will make use of the following lemma. Lemma. Let $D_{1}, \ldots, D_{6}$ be disjoint disks in the plane with radii $R_{1}, \ldots, R_{6}$. Let $P_{i}$ be a point in $D_{i}$, and let $O$ be an arbitrary point. Then there exist indices $i$ and $j$ such that $O P_{i} \geqslant R_{j}$. Proof. Let $O_{i}$ be the center of... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 177 | 1,019 |
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 N... | 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
} | 177 | 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 N... | 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
} | 177 | 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. | 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
} | 112 | 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. | 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
} | 112 | 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
} | 174 | 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
} | 199 | 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
} | 199 | 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
} | 160 | 2,858 |
2020 | T0 | N1 | Number Theory | IMO-SL | Given a positive integer $k$, show that there exists a prime $p$ such that one can choose distinct integers $a_{1}, a_{2}, \ldots, a_{k+3} \in\{1,2, \ldots, p-1\}$ such that $p$ divides $a_{i} a_{i+1} a_{i+2} a_{i+3}-i$ for all $i=1,2, \ldots, k$. (South Africa) | First we choose distinct positive rational numbers $r_{1}, \ldots, r_{k+3}$ such that $$ r_{i} r_{i+1} r_{i+2} r_{i+3}=i \quad \text { for } 1 \leqslant i \leqslant k $$ Let $r_{1}=x, r_{2}=y, r_{3}=z$ be some distinct primes greater than $k$; the remaining terms satisfy $r_{4}=\frac{1}{r_{1} r_{2} r_{3}}$ and $r_{i+4}... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 109 | 970 |
2020 | T0 | N2 | Number Theory | IMO-SL | For each prime $p$, there is a kingdom of $p$-Landia consisting of $p$ islands numbered $1,2, \ldots, p$. Two distinct islands numbered $n$ and $m$ are connected by a bridge if and only if $p$ divides $\left(n^{2}-m+1\right)\left(m^{2}-n+1\right)$. The bridges may pass over each other, but cannot cross. Prove that for ... | We prove that for each prime $p>3$ dividing a number of the form $x^{2}-x+1$ with integer $x$ there are two unconnected islands in $p$-Landia. For brevity's sake, when a bridge connects the islands numbered $m$ and $n$, we shall speak simply that it connects $m$ and $n$. A bridge connects $m$ and $n$ if $n \equiv m^{2}... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 127 | 661 |
2020 | T0 | N2 | Number Theory | IMO-SL | For each prime $p$, there is a kingdom of $p$-Landia consisting of $p$ islands numbered $1,2, \ldots, p$. Two distinct islands numbered $n$ and $m$ are connected by a bridge if and only if $p$ divides $\left(n^{2}-m+1\right)\left(m^{2}-n+1\right)$. The bridges may pass over each other, but cannot cross. Prove that for ... | One can show, by using only arithmetical methods, that for infinitely many $p$, the kingdom of $p$-Ladia contains two islands connected to no other island, except for each other. Let arrows between islands have the same meaning as in the previous solution. Suppose that positive $a<p$ satisfies the congruence $x^{2}-x+1... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2020SL.jsonl",
"solution_match": null
} | 127 | 702 |
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