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 |
|---|---|---|---|---|---|---|---|---|---|
2018 | T0 | A2 | Algebra | IMO-SL | Find all positive integers $n \geqslant 3$ for which there exist real numbers $a_{1}, a_{2}, \ldots, a_{n}$, $a_{n+1}=a_{1}, a_{n+2}=a_{2}$ such that $$ a_{i} a_{i+1}+1=a_{i+2} $$ for all $i=1,2, \ldots, n$. (Slovakia) | For the sake of convenience, extend the sequence $a_{1}, \ldots, a_{n+2}$ to an infinite periodic sequence with period $n$. ( $n$ is not necessarily the shortest period.) If $n$ is divisible by 3 , then $\left(a_{1}, a_{2}, \ldots\right)=(-1,-1,2,-1,-1,2, \ldots)$ is an obvious solution. We will show that in every peri... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 104 | 933 |
2018 | T0 | A3 | Algebra | IMO-SL | Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x \in F} 1 / x=\sum_{x \in G} 1 / x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x \in F} 1 / x \neq r$ for all fini... | Argue indirectly. Agree, as usual, that the empty sum is 0 to consider rationals in $[0,1)$; adjoining 0 causes no harm, since $\sum_{x \in F} 1 / x=0$ for no nonempty finite subset $F$ of $S$. For every rational $r$ in $[0,1)$, let $F_{r}$ be the unique finite subset of $S$ such that $\sum_{x \in F_{r}} 1 / x=r$. The ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 120 | 861 |
2018 | T0 | A3 | Algebra | IMO-SL | Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x \in F} 1 / x=\sum_{x \in G} 1 / x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x \in F} 1 / x \neq r$ for all fini... | A finite $S$ clearly satisfies (2), so let $S$ be infinite. If $S$ fails both conditions, so does $S \backslash\{1\}$. We may and will therefore assume that $S$ consists of integers greater than 1 . Label the elements of $S$ increasingly $x_{1}<x_{2}<\cdots$, where $x_{1} \geqslant 2$. We first show that $S$ satisfies ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 120 | 738 |
2018 | T0 | A4 | Algebra | IMO-SL | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying $$ a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k} $$ Find the maximal possible value of $a_{2018}-a_{2017}$. (Belgium) | The claimed maximal value is achieved at $$ \begin{gathered} a_{1}=a_{2}=\cdots=a_{2016}=1, \quad a_{2017}=\frac{a_{2016}+\cdots+a_{0}}{2017}=1-\frac{1}{2017}, \\ a_{2018}=\frac{a_{2017}+\cdots+a_{1}}{2017}=1-\frac{1}{2017^{2}} . \end{gathered} $$ Now we need to show that this value is optimal. For brevity, we use the ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 125 | 1,963 |
2018 | T0 | A4 | Algebra | IMO-SL | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying $$ a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k} $$ Find the maximal possible value of $a_{2018}-a_{2017}$. (Belgium) | We present a different proof of the estimate $a_{2018}-a_{2017} \leqslant \frac{2016}{2017^{2}}$. We keep the same notations of $S(n, k), m_{n}$ and $M_{n}$ from the previous solution. Notice that $S(n, n)=S(n, n-1)$, as $a_{0}=0$. Also notice that for $0 \leqslant k \leqslant \ell \leqslant n$ we have $S(n, \ell)=S(n,... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 125 | 2,184 |
2018 | T0 | A5 | Algebra | IMO-SL | Determine all functions $f:(0, \infty) \rightarrow \mathbb{R}$ satisfying $$ \left(x+\frac{1}{x}\right) f(y)=f(x y)+f\left(\frac{y}{x}\right) $$ for all $x, y>0$. (South Korea) | Fix a real number $a>1$, and take a new variable $t$. For the values $f(t), f\left(t^{2}\right)$, $f(a t)$ and $f\left(a^{2} t^{2}\right)$, the relation (1) provides a system of linear equations: $$ \begin{array}{llll} x=y=t: & \left(t+\frac{1}{t}\right) f(t) & =f\left(t^{2}\right)+f(1) \\ x=\frac{t}{a}, y=a t: & \left... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 71 | 953 |
2018 | T0 | A5 | Algebra | IMO-SL | Determine all functions $f:(0, \infty) \rightarrow \mathbb{R}$ satisfying $$ \left(x+\frac{1}{x}\right) f(y)=f(x y)+f\left(\frac{y}{x}\right) $$ for all $x, y>0$. (South Korea) | We start with an observation. If we substitute $x=a \neq 1$ and $y=a^{n}$ in (1), we obtain $$ f\left(a^{n+1}\right)-\left(a+\frac{1}{a}\right) f\left(a^{n}\right)+f\left(a^{n-1}\right)=0 . $$ For the sequence $z_{n}=a^{n}$, this is a homogeneous linear recurrence of the second order, and its characteristic polynomial ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 71 | 1,328 |
2018 | T0 | A6 | Algebra | IMO-SL | Let $m, n \geqslant 2$ be integers. Let $f\left(x_{1}, \ldots, x_{n}\right)$ be a polynomial with real coefficients such that $$ f\left(x_{1}, \ldots, x_{n}\right)=\left\lfloor\frac{x_{1}+\ldots+x_{n}}{m}\right\rfloor \text { for every } x_{1}, \ldots, x_{n} \in\{0,1, \ldots, m-1\} $$ Prove that the total degree of $... | We transform the problem to a single variable question by the following Lemma. Let $a_{1}, \ldots, a_{n}$ be nonnegative integers and let $G(x)$ be a nonzero polynomial with $\operatorname{deg} G \leqslant a_{1}+\ldots+a_{n}$. Suppose that some polynomial $F\left(x_{1}, \ldots, x_{n}\right)$ satisfies $$ F\left(x_{1}, ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 144 | 2,635 |
2018 | T0 | A7 | Algebra | IMO-SL | Find the maximal value of $$ S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}} $$ where $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$. | Since the value $8 / \sqrt[3]{7}$ is reached, it suffices to prove that $S \leqslant 8 / \sqrt[3]{7}$. Assume that $x, y, z, t$ is a permutation of the variables, with $x \leqslant y \leqslant z \leqslant t$. Then, by the rearrangement inequality, $$ S \leqslant\left(\sqrt[3]{\frac{x}{t+7}}+\sqrt[3]{\frac{t}{x+7}}\righ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 93 | 751 |
2018 | T0 | A7 | Algebra | IMO-SL | Find the maximal value of $$ S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}} $$ where $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$. | We present a different proof for the estimate $S \leqslant 8 / \sqrt[3]{7}$. Start by using Hölder's inequality: $$ S^{3}=\left(\sum_{\mathrm{cyc}} \frac{\sqrt[6]{a} \cdot \sqrt[6]{a}}{\sqrt[3]{b+7}}\right)^{3} \leqslant \sum_{\mathrm{cyc}}(\sqrt[6]{a})^{3} \cdot \sum_{\mathrm{cyc}}(\sqrt[6]{a})^{3} \cdot \sum_{\mathrm... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 93 | 595 |
2018 | T0 | C1 | Combinatorics | IMO-SL | Let $n \geqslant 3$ be an integer. Prove that there exists a set $S$ of $2 n$ positive integers satisfying the following property: For every $m=2,3, \ldots, n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$. (Iceland) | We show that one of possible examples is the set $$ S=\left\{1 \cdot 3^{k}, 2 \cdot 3^{k}: k=1,2, \ldots, n-1\right\} \cup\left\{1, \frac{3^{n}+9}{2}-1\right\} $$ It is readily verified that all the numbers listed above are distinct (notice that the last two are not divisible by 3 ). The sum of elements in $S$ is $$ \S... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 85 | 1,050 |
2018 | T0 | C2 | Combinatorics | IMO-SL | Queenie and Horst play a game on a $20 \times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when someb... | We show two strategies, one for Horst to place at least 100 knights, and another strategy for Queenie that prevents Horst from putting more than 100 knights on the board. A strategy for Horst: Put knights only on black squares, until all black squares get occupied. Colour the squares of the board black and white in the... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 119 | 558 |
2018 | T0 | C4 | Combinatorics | IMO-SL | An anti-Pascal pyramid is a finite set of numbers, placed in a triangle-shaped array so that the first row of the array contains one number, the second row contains two numbers, the third row contains three numbers and so on; and, except for the numbers in the bottom row, each number equals the absolute value of the di... | Let $T$ be an anti-Pascal pyramid with $n$ rows, containing every integer from 1 to $1+2+\cdots+n$, and let $a_{1}$ be the topmost number in $T$ (Figure 1). The two numbers below $a_{1}$ are some $a_{2}$ and $b_{2}=a_{1}+a_{2}$, the two numbers below $b_{2}$ are some $a_{3}$ and $b_{3}=a_{1}+a_{2}+a_{3}$, and so on and... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 214 | 870 |
2018 | T0 | C5 | Combinatorics | IMO-SL | Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For eve... | Enumerate the days of the tournament $1,2, \ldots,\left(\begin{array}{c}2 k \\ 2\end{array}\right)$. Let $b_{1} \leqslant b_{2} \leqslant \cdots \leqslant b_{2 k}$ be the days the players arrive to the tournament, arranged in nondecreasing order; similarly, let $e_{1} \geqslant \cdots \geqslant e_{2 k}$ be the days the... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 120 | 1,626 |
2018 | T0 | C5 | Combinatorics | IMO-SL | Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For eve... | Consider any tournament schedule. Label players $P_{1}, P_{2}, \ldots, P_{2 k}$ in order of their arrival, and label them again $Q_{2 k}, Q_{2 k-1}, \ldots, Q_{1}$ in order of their departure, to define a permutation $a_{1}, a_{2}, \ldots, a_{2 k}$ of $1,2, \ldots, 2 k$ by $P_{i}=Q_{a_{i}}$. We first describe an optima... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 120 | 1,212 |
2018 | T0 | C6 | Combinatorics | IMO-SL | Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. (i) If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. (ii) If no such pair exists, we write down two tim... | We may assume $\operatorname{gcd}(a, b)=1$; otherwise we work in the same way with multiples of $d=\operatorname{gcd}(a, b)$. Suppose that after $N$ moves of type (ii) and some moves of type $(i)$ we have to add two new zeros. For each integer $k$, denote by $f(k)$ the number of times that the number $k$ appeared on th... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 121 | 691 |
2018 | T0 | C6 | Combinatorics | IMO-SL | Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. (i) If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. (ii) If no such pair exists, we write down two tim... | We start by showing that the result of the process in the problem does not depend on the way the operations are performed. For that purpose, it is convenient to modify the process a bit. Claim 1. Suppose that the board initially contains a finite number of nonnegative integers, and one starts performing type $(i)$ move... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 121 | 1,099 |
2018 | T0 | C7 | Combinatorics | IMO-SL | Consider 2018 pairwise crossing circles no three of which are concurrent. These circles subdivide the plane into regions bounded by circular edges that meet at vertices. Notice that there are an even number of vertices on each circle. Given the circle, alternately colour the vertices on that circle red and blue. In doi... | Letting $n=2018$, we will show that, if every region has at least one non-yellow vertex, then every circle contains at most $n+\lfloor\sqrt{n-2}\rfloor-2$ yellow points. In the case at hand, the latter equals $2018+44-2=2060$, contradicting the hypothesis. Consider the natural geometric graph $G$ associated with the co... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 141 | 1,317 |
2018 | T0 | C7 | Combinatorics | IMO-SL | Consider 2018 pairwise crossing circles no three of which are concurrent. These circles subdivide the plane into regions bounded by circular edges that meet at vertices. Notice that there are an even number of vertices on each circle. Given the circle, alternately colour the vertices on that circle red and blue. In doi... | The first two lemmata in Call the circles from the two classes white and black, respectively. Call a region yellow if its vertices are all yellow. Let $w$ and $b$ be the numbers of white and black circles, respectively; clearly, $w+b=n$. Assume that $w \geqslant b$, and that there is no yellow region. Clearly, $b \geqs... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 141 | 587 |
2018 | T0 | G2 | Geometry | IMO-SL | Let $A B C$ be a triangle with $A B=A C$, and let $M$ be the midpoint of $B C$. Let $P$ be a point such that $P B<P C$ and $P A$ is parallel to $B C$. Let $X$ and $Y$ be points on the lines $P B$ and $P C$, respectively, so that $B$ lies on the segment $P X, C$ lies on the segment $P Y$, and $\angle P X M=\angle P Y M$... | Since $A B=A C, A M$ is the perpendicular bisector of $B C$, hence $\angle P A M=$ $\angle A M C=90^{\circ}$. Now let $Z$ be the common point of $A M$ and the perpendicular through $Y$ to $P C$ (notice that $Z$ lies on to the ray $A M$ beyond $M$ ). We have $\angle P A Z=\angle P Y Z=90^{\circ}$. Thus the points $P, A,... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 134 | 956 |
2018 | T0 | G3 | Geometry | IMO-SL | A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold: (i) each triangle from $T$ is inscribed in $\omega$; (ii) no two triangles from $T$ have a common interior point. Determine all positive real numbers $t$ such that, for each positive integer $n$, t... | First, we show how to construct a good collection of $n$ triangles, each of perimeter greater than 4 . This will show that all $t \leqslant 4$ satisfy the required conditions. Construct inductively an $(n+2)$-gon $B A_{1} A_{2} \ldots A_{n} C$ inscribed in $\omega$ such that $B C$ is a diameter, and $B A_{1} A_{2}, B A... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 107 | 821 |
2018 | T0 | G4 | Geometry | IMO-SL | A point $T$ is chosen inside a triangle $A B C$. Let $A_{1}, B_{1}$, and $C_{1}$ be the reflections of $T$ in $B C, C A$, and $A B$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_{1} B_{1} C_{1}$. The lines $A_{1} T, B_{1} T$, and $C_{1} T$ meet $\Omega$ again at $A_{2}, B_{2}$, and $C_{2}$, respect... | By $\Varangle(\ell, n)$ we always mean the directed angle of the lines $\ell$ and $n$, taken modulo $180^{\circ}$. Let $C C_{2}$ meet $\Omega$ again at $K$ (as usual, if $C C_{2}$ is tangent to $\Omega$, we set $T=C_{2}$ ). We show that the line $B B_{2}$ contains $K$; similarly, $A A_{2}$ will also pass through $K$. F... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 163 | 1,581 |
2018 | T0 | G5 | Geometry | IMO-SL | Let $A B C$ be a triangle with circumcircle $\omega$ and incentre $I$. A line $\ell$ intersects the lines $A I, B I$, and $C I$ at points $D, E$, and $F$, respectively, distinct from the points $A, B, C$, and $I$. The perpendicular bisectors $x, y$, and $z$ of the segments $A D, B E$, and $C F$, respectively determine ... | Claim 1. The reflections $\ell_{a}, \ell_{b}$ and $\ell_{c}$ of the line $\ell$ in the lines $x, y$, and $z$, respectively, are concurrent at a point $T$ which belongs to $\omega$. Proof. Notice that $\Varangle\left(\ell_{b}, \ell_{c}\right)=\Varangle\left(\ell_{b}, \ell\right)+\Varangle\left(\ell, \ell_{c}\right)=2 \V... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 126 | 1,665 |
2018 | T0 | G5 | Geometry | IMO-SL | Let $A B C$ be a triangle with circumcircle $\omega$ and incentre $I$. A line $\ell$ intersects the lines $A I, B I$, and $C I$ at points $D, E$, and $F$, respectively, distinct from the points $A, B, C$, and $I$. The perpendicular bisectors $x, y$, and $z$ of the segments $A D, B E$, and $C F$, respectively determine ... | As mentioned in the preamble, it is sufficient to prove that the centre $T$ of the homothety taking $X Y Z$ to $X_{0} Y_{0} Z_{0}$ belongs to $\omega$. Thus, it suffices to prove that $\Varangle\left(T X_{0}, T Y_{0}\right)=$ $\Varangle\left(Z_{0} X_{0}, Z_{0} Y_{0}\right)$, or, equivalently, $\Varangle\left(X X_{0}, Y... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 126 | 657 |
2018 | T0 | G5 | Geometry | IMO-SL | Let $A B C$ be a triangle with circumcircle $\omega$ and incentre $I$. A line $\ell$ intersects the lines $A I, B I$, and $C I$ at points $D, E$, and $F$, respectively, distinct from the points $A, B, C$, and $I$. The perpendicular bisectors $x, y$, and $z$ of the segments $A D, B E$, and $C F$, respectively determine ... | Let $I_{a}, I_{b}$, and $I_{c}$ be the excentres of triangle $A B C$ corresponding to $A, B$, and $C$, respectively. Also, let $u, v$, and $w$ be the lines through $D, E$, and $F$ which are perpendicular to $A I, B I$, and $C I$, respectively, and let $U V W$ be the triangle determined by these lines, where $u=V W, v=U... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 126 | 663 |
2018 | T0 | G6 | Geometry | IMO-SL | A convex quadrilateral $A B C D$ satisfies $A B \cdot C D=B C \cdot D A$. A point $X$ is chosen inside the quadrilateral so that $\angle X A B=\angle X C D$ and $\angle X B C=\angle X D A$. Prove that $\angle A X B+$ $\angle C X D=180^{\circ}$. (Poland) | Let $B^{\prime}$ be the reflection of $B$ in the internal angle bisector of $\angle A X C$, so that $\angle A X B^{\prime}=\angle C X B$ and $\angle C X B^{\prime}=\angle A X B$. If $X, D$, and $B^{\prime}$ are collinear, then we are done. Now assume the contrary. On the ray $X B^{\prime}$ take a point $E$ such that $X... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 89 | 686 |
2018 | T0 | G6 | Geometry | IMO-SL | A convex quadrilateral $A B C D$ satisfies $A B \cdot C D=B C \cdot D A$. A point $X$ is chosen inside the quadrilateral so that $\angle X A B=\angle X C D$ and $\angle X B C=\angle X D A$. Prove that $\angle A X B+$ $\angle C X D=180^{\circ}$. (Poland) | The solution consists of two parts. In Part 1 we show that it suffices to prove that $$ \frac{X B}{X D}=\frac{A B}{C D} $$ and $$ \frac{X A}{X C}=\frac{D A}{B C} $$ In Part 2 we establish these equalities. Part 1. Using the sine law and applying (1) we obtain $$ \frac{\sin \angle A X B}{\sin \angle X A B}=\frac{A B}{X ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 89 | 1,402 |
2018 | T0 | G7 | Geometry | IMO-SL | Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $A B C$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A, B, C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $A O P, B O P$, and $C O P$ by $O_{A}, O_{B}$, and $O_{C}$, respectively. The ... | As usual, we denote the directed angle between the lines $a$ and $b$ by $\Varangle(a, b)$. We frequently use the fact that $a_{1} \perp a_{2}$ and $b_{1} \perp b_{2}$ yield $\Varangle\left(a_{1}, b_{1}\right)=\Varangle\left(a_{2}, b_{2}\right)$. Let the lines $\ell_{B}$ and $\ell_{C}$ meet at $L_{A}$; define the points... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 197 | 1,667 |
2018 | T0 | N2 | Number Theory | IMO-SL | Let $n>1$ be a positive integer. Each cell of an $n \times n$ table contains an integer. Suppose that the following conditions are satisfied: (i) Each number in the table is congruent to 1 modulo $n$; (ii) The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^{2}$. ... | Let $A_{i, j}$ be the entry in the $i^{\text {th }}$ row and the $j^{\text {th }}$ column; let $P$ be the product of all $n^{2}$ entries. For convenience, denote $a_{i, j}=A_{i, j}-1$ and $r_{i}=R_{i}-1$. We show that $$ \sum_{i=1}^{n} R_{i} \equiv(n-1)+P \quad\left(\bmod n^{4}\right) $$ Due to symmetry of the problem ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 188 | 613 |
2018 | T0 | N2 | Number Theory | IMO-SL | Let $n>1$ be a positive integer. Each cell of an $n \times n$ table contains an integer. Suppose that the following conditions are satisfied: (i) Each number in the table is congruent to 1 modulo $n$; (ii) The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^{2}$. ... | We present a more straightforward (though lengthier) way to establish (1). We also use the notation of $a_{i, j}$. By condition (i), all the $a_{i, j}$ are divisible by $n$. Therefore, we have $$ \begin{aligned} P=\prod_{i=1}^{n} \prod_{j=1}^{n}\left(1+a_{i, j}\right) \equiv 1+\sum_{(i, j)} a_{i, j} & +\sum_{\left(i_{1... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 188 | 1,334 |
2018 | T0 | N3 | Number Theory | IMO-SL | Define the sequence $a_{0}, a_{1}, a_{2}, \ldots$ by $a_{n}=2^{n}+2^{\lfloor n / 2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way. (Serbia) | Call a nonnegative integer representable if it equals the sum of several (possibly 0 or 1) distinct terms of the sequence. We say that two nonnegative integers $b$ and $c$ are equivalent (written as $b \sim c$ ) if they are either both representable or both non-representable. One can easily compute $$ S_{n-1}:=a_{0}+\c... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 96 | 1,372 |
2018 | T0 | N3 | Number Theory | IMO-SL | Define the sequence $a_{0}, a_{1}, a_{2}, \ldots$ by $a_{n}=2^{n}+2^{\lfloor n / 2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way. (Serbia) | We keep the notion of representability and the notation $S_{n}$ from the previous solution. We say that an index $n$ is good if $a_{n}$ writes as a sum of smaller terms from the sequence $a_{0}, a_{1}, \ldots$. Otherwise we say it is bad. We must prove that there are infinitely many good indices, as well as infinitely ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 96 | 2,160 |
2018 | T0 | N4 | Number Theory | IMO-SL | Let $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ be a sequence of positive integers such that $$ \frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}+\frac{a_{n}}{a_{1}} $$ is an integer for all $n \geqslant k$, where $k$ is some positive integer. Prove that there exists a positive integer $m$ such that $a_{... | The argument hinges on the following two facts: Let $a, b, c$ be positive integers such that $N=b / c+(c-b) / a$ is an integer. (1) If $\operatorname{gcd}(a, c)=1$, then $c$ divides $b$; and (2) If $\operatorname{gcd}(a, b, c)=1$, then $\operatorname{gcd}(a, b)=1$. To prove (1), write $a b=c(a N+b-c)$. Since $\operator... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 151 | 742 |
2018 | T0 | N4 | Number Theory | IMO-SL | Let $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ be a sequence of positive integers such that $$ \frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}+\frac{a_{n}}{a_{1}} $$ is an integer for all $n \geqslant k$, where $k$ is some positive integer. Prove that there exists a positive integer $m$ such that $a_{... | We use the same notation $s_{n}$. This time, we explore the exponents of primes in the prime factorizations of the $a_{n}$ for $n \geqslant k$. To start, for every $n \geqslant k$, we know that the number $$ s_{n+1}-s_{n}=\frac{a_{n}}{a_{n+1}}+\frac{a_{n+1}}{a_{1}}-\frac{a_{n}}{a_{1}} $$ is integer. Multiplying it by $... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 151 | 1,735 |
2018 | T0 | N5 | Number Theory | IMO-SL | Four positive integers $x, y, z$, and $t$ satisfy the relations $$ x y-z t=x+y=z+t $$ Is it possible that both $x y$ and $z t$ are perfect squares? (Russia) | Arguing indirectly, assume that $x y=a^{2}$ and $z t=c^{2}$ with $a, c>0$. Suppose that the number $x+y=z+t$ is odd. Then $x$ and $y$ have opposite parity, as well as $z$ and $t$. This means that both $x y$ and $z t$ are even, as well as $x y-z t=x+y$; a contradiction. Thus, $x+y$ is even, so the number $s=\frac{x+y}{2... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 51 | 662 |
2018 | T0 | N5 | Number Theory | IMO-SL | Four positive integers $x, y, z$, and $t$ satisfy the relations $$ x y-z t=x+y=z+t $$ Is it possible that both $x y$ and $z t$ are perfect squares? (Russia) | We start with a complete description of all 4-tuples $(x, y, z, t)$ of positive integers satisfying (*). As in the solution above, we notice that the numbers $$ s=\frac{x+y}{2}=\frac{z+t}{2}, \quad p=\frac{x-y}{2}, \quad \text { and } \quad q=\frac{z-t}{2} $$ are integers (we may, and will, assume that $p, q \geqslant ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 51 | 842 |
2018 | T0 | N6 | Number Theory | IMO-SL | Let $f:\{1,2,3, \ldots\} \rightarrow\{2,3, \ldots\}$ be a function such that $f(m+n) \mid f(m)+f(n)$ for all pairs $m, n$ of positive integers. Prove that there exists a positive integer $c>1$ which divides all values of $f$. (Mexico) | For every positive integer $m$, define $S_{m}=\{n: m \mid f(n)\}$. Lemma. If the set $S_{m}$ is infinite, then $S_{m}=\{d, 2 d, 3 d, \ldots\}=d \cdot \mathbb{Z}_{>0}$ for some positive integer $d$. Proof. Let $d=\min S_{m}$; the definition of $S_{m}$ yields $m \mid f(d)$. Whenever $n \in S_{m}$ and $n>d$, we have $m|f(... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 84 | 1,466 |
2018 | T0 | N6 | Number Theory | IMO-SL | Let $f:\{1,2,3, \ldots\} \rightarrow\{2,3, \ldots\}$ be a function such that $f(m+n) \mid f(m)+f(n)$ for all pairs $m, n$ of positive integers. Prove that there exists a positive integer $c>1$ which divides all values of $f$. (Mexico) | Let $d_{n}=\operatorname{gcd}(f(n), f(1))$. From $d_{n+1} \mid f(1)$ and $d_{n+1}|f(n+1)| f(n)+f(1)$, we can see that $d_{n+1} \mid f(n)$; then $d_{n+1} \mid \operatorname{gcd}(f(n), f(1))=d_{n}$. So the sequence $d_{1}, d_{2}, \ldots$ is nonincreasing in the sense that every element is a divisor of the previous elemen... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 84 | 1,315 |
2018 | T0 | N7 | Number Theory | IMO-SL | Let $n \geqslant 2018$ be an integer, and let $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ be pairwise distinct positive integers not exceeding $5 n$. Suppose that the sequence $$ \frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \ldots, \frac{a_{n}}{b_{n}} $$ forms an arithmetic progression. Prove that the ter... | Suppose that (1) is an arithmetic progression with nonzero difference. Let the difference be $\Delta=\frac{c}{d}$, where $d>0$ and $c, d$ are coprime. We will show that too many denominators $b_{i}$ should be divisible by $d$. To this end, for any $1 \leqslant i \leqslant n$ and any prime divisor $p$ of $d$, say that t... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 128 | 1,250 |
2018 | T0 | A1 | Algebra | IMO-SL | Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}_{>0} \rightarrow \mathbb{Q}_{>0}$ satisfying $$ f\left(x^{2} f(y)^{2}\right)=f(x)^{2} f(y) $$ for all $x, y \in \mathbb{Q}_{>0}$. (Switzerland) Answer: $f(x)=1$ for all $x \in \mathbb{Q}_{>0}$. | Take any $a, b \in \mathbb{Q}_{>0}$. By substituting $x=f(a), y=b$ and $x=f(b), y=a$ into $(*)$ we get $$ f(f(a))^{2} f(b)=f\left(f(a)^{2} f(b)^{2}\right)=f(f(b))^{2} f(a) $$ which yields $$ \frac{f(f(a))^{2}}{f(a)}=\frac{f(f(b))^{2}}{f(b)} \quad \text { for all } a, b \in \mathbb{Q}_{>0} $$ In other words, this shows ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 122 | 812 |
2018 | T0 | A2 | Algebra | IMO-SL | Find all positive integers $n \geqslant 3$ for which there exist real numbers $a_{1}, a_{2}, \ldots, a_{n}$, $a_{n+1}=a_{1}, a_{n+2}=a_{2}$ such that $$ a_{i} a_{i+1}+1=a_{i+2} $$ for all $i=1,2, \ldots, n$. (Slovakia) Answer: $n$ can be any multiple of 3 . | For the sake of convenience, extend the sequence $a_{1}, \ldots, a_{n+2}$ to an infinite periodic sequence with period $n$. ( $n$ is not necessarily the shortest period.) If $n$ is divisible by 3 , then $\left(a_{1}, a_{2}, \ldots\right)=(-1,-1,2,-1,-1,2, \ldots)$ is an obvious solution. We will show that in every peri... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 118 | 933 |
2018 | T0 | A3 | Algebra | IMO-SL | Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x \in F} 1 / x=\sum_{x \in G} 1 / x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x \in F} 1 / x \neq r$ for all fini... | Argue indirectly. Agree, as usual, that the empty sum is 0 to consider rationals in $[0,1)$; adjoining 0 causes no harm, since $\sum_{x \in F} 1 / x=0$ for no nonempty finite subset $F$ of $S$. For every rational $r$ in $[0,1)$, let $F_{r}$ be the unique finite subset of $S$ such that $\sum_{x \in F_{r}} 1 / x=r$. The ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 121 | 861 |
2018 | T0 | A3 | Algebra | IMO-SL | Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x \in F} 1 / x=\sum_{x \in G} 1 / x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x \in F} 1 / x \neq r$ for all fini... | A finite $S$ clearly satisfies (2), so let $S$ be infinite. If $S$ fails both conditions, so does $S \backslash\{1\}$. We may and will therefore assume that $S$ consists of integers greater than 1 . Label the elements of $S$ increasingly $x_{1}<x_{2}<\cdots$, where $x_{1} \geqslant 2$. We first show that $S$ satisfies ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 121 | 738 |
2018 | T0 | A4 | Algebra | IMO-SL | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying $$ a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k} $$ Find the maximal possible value of $a_{2018}-a_{2017}$. (Belgium) Answer: The maximal value is ... | The claimed maximal value is achieved at $$ \begin{gathered} a_{1}=a_{2}=\cdots=a_{2016}=1, \quad a_{2017}=\frac{a_{2016}+\cdots+a_{0}}{2017}=1-\frac{1}{2017}, \\ a_{2018}=\frac{a_{2017}+\cdots+a_{1}}{2017}=1-\frac{1}{2017^{2}} . \end{gathered} $$ Now we need to show that this value is optimal. For brevity, we use the ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 148 | 1,963 |
2018 | T0 | A4 | Algebra | IMO-SL | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying $$ a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k} $$ Find the maximal possible value of $a_{2018}-a_{2017}$. (Belgium) Answer: The maximal value is ... | We present a different proof of the estimate $a_{2018}-a_{2017} \leqslant \frac{2016}{2017^{2}}$. We keep the same notations of $S(n, k), m_{n}$ and $M_{n}$ from the previous solution. Notice that $S(n, n)=S(n, n-1)$, as $a_{0}=0$. Also notice that for $0 \leqslant k \leqslant \ell \leqslant n$ we have $S(n, \ell)=S(n,... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 148 | 2,184 |
2018 | T0 | A5 | Algebra | IMO-SL | Determine all functions $f:(0, \infty) \rightarrow \mathbb{R}$ satisfying $$ \left(x+\frac{1}{x}\right) f(y)=f(x y)+f\left(\frac{y}{x}\right) $$ for all $x, y>0$. (South Korea) Answer: $f(x)=C_{1} x+\frac{C_{2}}{x}$ with arbitrary constants $C_{1}$ and $C_{2}$. | Fix a real number $a>1$, and take a new variable $t$. For the values $f(t), f\left(t^{2}\right)$, $f(a t)$ and $f\left(a^{2} t^{2}\right)$, the relation (1) provides a system of linear equations: $$ \begin{array}{llll} x=y=t: & \left(t+\frac{1}{t}\right) f(t) & =f\left(t^{2}\right)+f(1) \\ x=\frac{t}{a}, y=a t: & \left... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 108 | 953 |
2018 | T0 | A5 | Algebra | IMO-SL | Determine all functions $f:(0, \infty) \rightarrow \mathbb{R}$ satisfying $$ \left(x+\frac{1}{x}\right) f(y)=f(x y)+f\left(\frac{y}{x}\right) $$ for all $x, y>0$. (South Korea) Answer: $f(x)=C_{1} x+\frac{C_{2}}{x}$ with arbitrary constants $C_{1}$ and $C_{2}$. | We start with an observation. If we substitute $x=a \neq 1$ and $y=a^{n}$ in (1), we obtain $$ f\left(a^{n+1}\right)-\left(a+\frac{1}{a}\right) f\left(a^{n}\right)+f\left(a^{n-1}\right)=0 . $$ For the sequence $z_{n}=a^{n}$, this is a homogeneous linear recurrence of the second order, and its characteristic polynomial ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 108 | 1,328 |
2018 | T0 | A7 | Algebra | IMO-SL | Find the maximal value of $$ S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}} $$ where $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$. (Taiwan) Answer: $\frac{8}{\sqrt[3]{7}}$, reached when $(a, b, c, d)$ is a cyclic permutation of $(1,49,1,49)... | Since the value $8 / \sqrt[3]{7}$ is reached, it suffices to prove that $S \leqslant 8 / \sqrt[3]{7}$. Assume that $x, y, z, t$ is a permutation of the variables, with $x \leqslant y \leqslant z \leqslant t$. Then, by the rearrangement inequality, $$ S \leqslant\left(\sqrt[3]{\frac{x}{t+7}}+\sqrt[3]{\frac{t}{x+7}}\righ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 142 | 751 |
2018 | T0 | A7 | Algebra | IMO-SL | Find the maximal value of $$ S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}} $$ where $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$. (Taiwan) Answer: $\frac{8}{\sqrt[3]{7}}$, reached when $(a, b, c, d)$ is a cyclic permutation of $(1,49,1,49)... | We present a different proof for the estimate $S \leqslant 8 / \sqrt[3]{7}$. Start by using Hölder's inequality: $$ S^{3}=\left(\sum_{\mathrm{cyc}} \frac{\sqrt[6]{a} \cdot \sqrt[6]{a}}{\sqrt[3]{b+7}}\right)^{3} \leqslant \sum_{\mathrm{cyc}}(\sqrt[6]{a})^{3} \cdot \sum_{\mathrm{cyc}}(\sqrt[6]{a})^{3} \cdot \sum_{\mathrm... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 142 | 595 |
2018 | T0 | C2 | Combinatorics | IMO-SL | Queenie and Horst play a game on a $20 \times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when someb... | We show two strategies, one for Horst to place at least 100 knights, and another strategy for Queenie that prevents Horst from putting more than 100 knights on the board. A strategy for Horst: Put knights only on black squares, until all black squares get occupied. Colour the squares of the board black and white in the... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 163 | 558 |
2018 | T0 | C4 | Combinatorics | IMO-SL | An anti-Pascal pyramid is a finite set of numbers, placed in a triangle-shaped array so that the first row of the array contains one number, the second row contains two numbers, the third row contains three numbers and so on; and, except for the numbers in the bottom row, each number equals the absolute value of the di... | Let $T$ be an anti-Pascal pyramid with $n$ rows, containing every integer from 1 to $1+2+\cdots+n$, and let $a_{1}$ be the topmost number in $T$ (Figure 1). The two numbers below $a_{1}$ are some $a_{2}$ and $b_{2}=a_{1}+a_{2}$, the two numbers below $b_{2}$ are some $a_{3}$ and $b_{3}=a_{1}+a_{2}+a_{3}$, and so on and... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 220 | 870 |
2018 | T0 | C5 | Combinatorics | IMO-SL | Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For eve... | Enumerate the days of the tournament $1,2, \ldots,\left(\begin{array}{c}2 k \\ 2\end{array}\right)$. Let $b_{1} \leqslant b_{2} \leqslant \cdots \leqslant b_{2 k}$ be the days the players arrive to the tournament, arranged in nondecreasing order; similarly, let $e_{1} \geqslant \cdots \geqslant e_{2 k}$ be the days the... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 148 | 1,626 |
2018 | T0 | C5 | Combinatorics | IMO-SL | Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For eve... | Consider any tournament schedule. Label players $P_{1}, P_{2}, \ldots, P_{2 k}$ in order of their arrival, and label them again $Q_{2 k}, Q_{2 k-1}, \ldots, Q_{1}$ in order of their departure, to define a permutation $a_{1}, a_{2}, \ldots, a_{2 k}$ of $1,2, \ldots, 2 k$ by $P_{i}=Q_{a_{i}}$. We first describe an optima... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 148 | 1,212 |
2018 | T0 | G3 | Geometry | IMO-SL | A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold: (i) each triangle from $T$ is inscribed in $\omega$; (ii) no two triangles from $T$ have a common interior point. Determine all positive real numbers $t$ such that, for each positive integer $n$, t... | First, we show how to construct a good collection of $n$ triangles, each of perimeter greater than 4 . This will show that all $t \leqslant 4$ satisfy the required conditions. Construct inductively an $(n+2)$-gon $B A_{1} A_{2} \ldots A_{n} C$ inscribed in $\omega$ such that $B C$ is a diameter, and $B A_{1} A_{2}, B A... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 120 | 821 |
2018 | T0 | G5 | Geometry | IMO-SL | Let $A B C$ be a triangle with circumcircle $\omega$ and incentre $I$. A line $\ell$ intersects the lines $A I, B I$, and $C I$ at points $D, E$, and $F$, respectively, distinct from the points $A, B, C$, and $I$. The perpendicular bisectors $x, y$, and $z$ of the segments $A D, B E$, and $C F$, respectively determine ... | Claim 1. The reflections $\ell_{a}, \ell_{b}$ and $\ell_{c}$ of the line $\ell$ in the lines $x, y$, and $z$, respectively, are concurrent at a point $T$ which belongs to $\omega$. Proof. Notice that $\Varangle\left(\ell_{b}, \ell_{c}\right)=\Varangle\left(\ell_{b}, \ell\right)+\Varangle\left(\ell, \ell_{c}\right)=2 \V... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 472 | 1,665 |
2018 | T0 | G5 | Geometry | IMO-SL | Let $A B C$ be a triangle with circumcircle $\omega$ and incentre $I$. A line $\ell$ intersects the lines $A I, B I$, and $C I$ at points $D, E$, and $F$, respectively, distinct from the points $A, B, C$, and $I$. The perpendicular bisectors $x, y$, and $z$ of the segments $A D, B E$, and $C F$, respectively determine ... | As mentioned in the preamble, it is sufficient to prove that the centre $T$ of the homothety taking $X Y Z$ to $X_{0} Y_{0} Z_{0}$ belongs to $\omega$. Thus, it suffices to prove that $\Varangle\left(T X_{0}, T Y_{0}\right)=$ $\Varangle\left(Z_{0} X_{0}, Z_{0} Y_{0}\right)$, or, equivalently, $\Varangle\left(X X_{0}, Y... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 472 | 657 |
2018 | T0 | G5 | Geometry | IMO-SL | Let $A B C$ be a triangle with circumcircle $\omega$ and incentre $I$. A line $\ell$ intersects the lines $A I, B I$, and $C I$ at points $D, E$, and $F$, respectively, distinct from the points $A, B, C$, and $I$. The perpendicular bisectors $x, y$, and $z$ of the segments $A D, B E$, and $C F$, respectively determine ... | Let $I_{a}, I_{b}$, and $I_{c}$ be the excentres of triangle $A B C$ corresponding to $A, B$, and $C$, respectively. Also, let $u, v$, and $w$ be the lines through $D, E$, and $F$ which are perpendicular to $A I, B I$, and $C I$, respectively, and let $U V W$ be the triangle determined by these lines, where $u=V W, v=U... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 472 | 663 |
2018 | T0 | N1 | Number Theory | IMO-SL | Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the numbers of divisors of $s n$ and of $s k$ are equal. (Ukraine) Answer: All pairs $(n, k)$ such that $n \nmid k$ and $k \nmid n$. | As usual, the number of divisors of a positive integer $n$ is denoted by $d(n)$. If $n=\prod_{i} p_{i}^{\alpha_{i}}$ is the prime factorisation of $n$, then $d(n)=\prod_{i}\left(\alpha_{i}+1\right)$. We start by showing that one cannot find any suitable number $s$ if $k \mid n$ or $n \mid k$ (and $k \neq n$ ). Suppose ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 76 | 1,104 |
2018 | T0 | N5 | Number Theory | IMO-SL | Four positive integers $x, y, z$, and $t$ satisfy the relations $$ x y-z t=x+y=z+t . $$ Is it possible that both $x y$ and $z t$ are perfect squares? (Russia) Answer: No. | Arguing indirectly, assume that $x y=a^{2}$ and $z t=c^{2}$ with $a, c>0$. Suppose that the number $x+y=z+t$ is odd. Then $x$ and $y$ have opposite parity, as well as $z$ and $t$. This means that both $x y$ and $z t$ are even, as well as $x y-z t=x+y$; a contradiction. Thus, $x+y$ is even, so the number $s=\frac{x+y}{2... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 57 | 662 |
2018 | T0 | N5 | Number Theory | IMO-SL | Four positive integers $x, y, z$, and $t$ satisfy the relations $$ x y-z t=x+y=z+t . $$ Is it possible that both $x y$ and $z t$ are perfect squares? (Russia) Answer: No. | We start with a complete description of all 4-tuples $(x, y, z, t)$ of positive integers satisfying (*). As in the solution above, we notice that the numbers $$ s=\frac{x+y}{2}=\frac{z+t}{2}, \quad p=\frac{x-y}{2}, \quad \text { and } \quad q=\frac{z-t}{2} $$ are integers (we may, and will, assume that $p, q \geqslant ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 57 | 842 |
2018 | T0 | N7 | Number Theory | IMO-SL | Let $n \geqslant 2018$ be an integer, and let $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ be pairwise distinct positive integers not exceeding $5 n$. Suppose that the sequence $$ \frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \ldots, \frac{a_{n}}{b_{n}} $$ forms an arithmetic progression. Prove that the ter... | Suppose that (1) is an arithmetic progression with nonzero difference. Let the difference be $\Delta=\frac{c}{d}$, where $d>0$ and $c, d$ are coprime. We will show that too many denominators $b_{i}$ should be divisible by $d$. To this end, for any $1 \leqslant i \leqslant n$ and any prime divisor $p$ of $d$, say that t... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2018SL.jsonl",
"solution_match": null
} | 133 | 1,250 |
2019 | T0 | A1 | Algebra | IMO-SL | Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$ f(2 a)+2 f(b)=f(f(a+b)) $$ (South Africa) | Let $K=f(0)$. First, put $a=0$ in (1); this gives $$ 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 $$ for all $a \in \mathbb{Z}$. Substituting (2) and (3) into (1), we obtain... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 68 | 537 |
2019 | T0 | A2 | Algebra | IMO-SL | Let $u_{1}, u_{2}, \ldots, u_{2019}$ be real numbers satisfying $$ 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-\fra... | 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
} | 174 | 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 $$ a b \leqslant-\fra... | As in the previous solution we conclude that $a<0$ and $b>0$. For every index $i$, the number $u_{i}$ is a convex combination of $a$ and $b$, so $$ 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 {. } $$ Let $X=\sum_{i=1}^{2019} x_{i}$ and... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 174 | 790 |
2019 | T0 | A3 | Algebra | IMO-SL | Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2 . Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of $$ \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
} | 173 | 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
} | 173 | 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
} | 173 | 1,466 |
2019 | T0 | A4 | Algebra | IMO-SL | Let $n \geqslant 2$ be a positive integer and $a_{1}, a_{2}, \ldots, a_{n}$ be real numbers such that $$ a_{1}+a_{2}+\cdots+a_{n}=0 $$ Define the set $A$ by $$ A=\left\{(i, j)\left|1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\} .\right. $$ Prove that, if $A$ is not empty, then $$ \sum_{(i... | Define sets $B$ and $C$ by $$ \begin{aligned} & B=\left\{(i, j)\left|1 \leqslant i, j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\},\right. \\ & C=\left\{(i, j)\left|1 \leqslant i, j \leqslant n,\left|a_{i}-a_{j}\right|<1\right\} .\right. \end{aligned} $$ We have $$ \begin{aligned} \sum_{(i, j) \in A} a_{i} ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 164 | 699 |
2019 | T0 | A4 | Algebra | IMO-SL | Let $n \geqslant 2$ be a positive integer and $a_{1}, a_{2}, \ldots, a_{n}$ be real numbers such that $$ a_{1}+a_{2}+\cdots+a_{n}=0 $$ Define the set $A$ by $$ A=\left\{(i, j)\left|1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\} .\right. $$ Prove that, if $A$ is not empty, then $$ \sum_{(i... | Consider $P, Q, R, S$ as in $$ p=\sum_{i \in P} a_{i}, \quad q=\sum_{i \in Q} a_{i}, \quad r=\sum_{i \in R} a_{i}, \quad s=\sum_{i \in S} a_{i}, $$ and let $$ t_{+}=\sum_{(i, j) \in A, a_{i} a_{j} \geqslant 0} a_{i} a_{j}, \quad t_{-}=\sum_{(i, j) \in A, a_{i} a_{j} \leqslant 0} a_{i} a_{j} . $$ We know that $p+q+r+s=0... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 164 | 572 |
2019 | T0 | A6 | Algebra | IMO-SL | A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$ 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) . $$ | In the first two steps, we deal with any polynomial $P(x, y, z)$ satisfying $P(x, y, z)=$ $P(x, y, x y-z)$. Call such a polynomial weakly symmetric, and call a polynomial satisfying the full conditions in the problem symmetric. Step 1. We start with the description of weakly symmetric polynomials. We claim that they ar... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 108 | 1,164 |
2019 | T0 | A6 | Algebra | IMO-SL | A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$ 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) . $$ | We will rely on the well-known identity $$ \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\} $$ Proof. Notice that for $x=2 \cos u, y=2 \cos v, z=2... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 108 | 2,857 |
2019 | T0 | A7 | Algebra | IMO-SL | Let $\mathbb{Z}$ be the set of integers. We consider functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$ 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\} $$ is finite and nonempty. (a) Prove... | a) Let $f$ be the function where $f(0)=0$ and $f(x)$ is the largest power of 2 dividing $2 x$ for $x \neq 0$. The integer 0 is evidently $f$-rare, so it remains to verify the functional equation. Since $f(2 x)=2 f(x)$ for all $x$, it suffices to verify the functional equation when at least one of $x$ and $y$ is odd (th... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 167 | 1,758 |
2019 | T0 | C1 | Combinatorics | IMO-SL | The infinite sequence $a_{0}, a_{1}, a_{2}, \ldots$ of (not necessarily different) integers has the following properties: $0 \leqslant a_{i} \leqslant i$ for all integers $i \geqslant 0$, and $$ \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
} | 178 | 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
} | 181 | 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
} | 181 | 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
} | 181 | 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
} | 181 | 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
} | 181 | 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 o... | 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
} | 267 | 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 o... | 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
} | 267 | 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
} | 184 | 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
} | 184 | 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
} | 234 | 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
} | 234 | 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 her... | 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
} | 207 | 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
} | 192 | 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
} | 163 | 666 |
2019 | T0 | G3 | Geometry | IMO-SL | In triangle $A B C$, let $A_{1}$ and $B_{1}$ be two points on sides $B C$ and $A C$, and let $P$ and $Q$ be two points on segments $A A_{1}$ and $B B_{1}$, respectively, so that line $P Q$ is parallel to $A B$. On ray $P B_{1}$, beyond $B_{1}$, let $P_{1}$ be a point so that $\angle P P_{1} C=\angle B A C$. Similarly, ... | First consider the case when lines $P P_{1}$ and $Q Q_{1}$ intersect each other at some point $R$. Let line $P Q$ meet the sides $A C$ and $B C$ at $E$ and $F$, respectively. Then $$ \angle P P_{1} C=\angle B A C=\angle P E C $$ so points $C, E, P, P_{1}$ lie on a circle; denote that circle by $\omega_{P}$. It follows ... | {
"problem_match": null,
"resource_path": "IMO_SL/segmented/en-IMO2019SL.jsonl",
"solution_match": null
} | 188 | 917 |
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
} | 186 | 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
} | 186 | 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 $\operatorname{area}(B C D)+\operatorname{area}(A D E)=$ $\ope... | 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
} | 134 | 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 $\operatorname{area}(B C D)+\operatorname{area}(A D E)=$ $\ope... | 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
} | 134 | 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
} | 106 | 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
} | 106 | 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
} | 106 | 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 $P B F$ meet... | 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
} | 142 | 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 $P B F$ meet... | 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
} | 142 | 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}\r... | 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
} | 160 | 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}\r... | 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
} | 160 | 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}\r... | 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
} | 160 | 1,316 |
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