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 |
|---|---|---|---|---|---|---|---|---|---|
2025 | T1 | 1 | null | Dutch_TST | Beschouw de rij \(y_0, y_1, \ldots\) met \(y_0 = -\frac{1}{4}\) en \(y_1 = 0\) en die verder voldoet aan
\[y_{n+1} + y_{n-1} = 4y_n + 1\]
voor alle \(n \ge 1\). Bewijs dat voor alle \(n \ge 0\) de uitdrukking \(2y_{2n} + \frac{3}{2}\)
a) een positief geheel getal is en
b) het kwadraat van een geheel getal is. | Voor een alternatieve aanpak van onderdeel (b) rekenen we uit dat
\[ x_{n+1}^2 - x_{n-1}^2 = (x_{n+1} + x_{n-1})(x_{n+1} - x_{n-1}) = 4x_n(x_{n+1} - x_{n-1}). \]
Dit betekent dat \(x_{n+1}^2 - 4x_{n+1}x_n + x_n^2 = x_n^2 - 4x_nx_{n-1} + x_{n-1}^2\). Met inductie naar \(n\) betekent dit dat \(x_{n+1}^2 - 4x_{n+1}x_n ... | {
"problem_match": "\nOpgave 1.",
"resource_path": "Dutch_TST/segmented/nl-2025-C2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing II."
} | 149 | 737 |
2025 | T1 | 2 | null | Dutch_TST | Beschouw een rechthoekig bord van \(m \times n\) vakjes met \(m, n \ge 1\). De hoekpunten van de vakjes vormen een \((m+1) \times (n+1)\)-grid. We noemen een driehoek met hoekpunten punten van het grid *laag* als die minstens één zijde heeft die parallel is met een zijde van het bord zo dat de hoogte van de driehoek op... | Als \(m, n \ge 2\) en minstens één van de twee is even, dan is het antwoord 0. Anders (minstens één van de twee is 1, of ze zijn beide oneven) is het antwoord 2.
We tekenen eerst een voorbeeld voor \(n = 1\) en \(m \ge 1\) met twee bijzondere driehoeken, een voorbeeld voor \(n = 2\) en \(m \ge 3\) met nul bijzondere ... | {
"problem_match": "\nOpgave 2.",
"resource_path": "Dutch_TST/segmented/nl-2025-C2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing."
} | 200 | 969 |
2025 | T1 | 3 | null | Dutch_TST | Zij \(\triangle ABC\) een scherphoekige driehoek zo dat \(|AB| + |BC| = 4|AC|\) en \(|AB| < |BC|\). Zij \(D\) het snijpunt van de bissectrice van \(\angle ABC\) met de zijde \(AC\). Punten \(P\) en \(Q\) liggen op het lijnstuk \(BD\) zo dat \(|BP| = 2|DQ|\). Zij \(\ell\) de lijn door \(P\) parallel aan \(AC\). De lijn ... | Laat \(R\), \(S\) en \(T\) respectievelijk de punten zijn waar de mier de eerste keer op \(AC\) is, op \(\ell\) is en de tweede keer op \(AC\) is. Laat \(\ell'\) en \(S'\) de spiegelingen van \(\ell\) en \(S\) in \(AC\) zijn, en zij \(Y'\) de spiegeling van \(Y\) in \(\ell'\). Dan geldt wegens de driehoekongelijkheid v... | {
"problem_match": "\nOpgave 3.",
"resource_path": "Dutch_TST/segmented/nl-2025-C2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing I."
} | 351 | 908 |
2025 | T1 | 2 | null | Dutch_TST | Zij \(\triangle ABC\) een scherphoekige driehoek met \(|AB| > |AC|\), zij \(\omega\) de omgeschreven cirkel van \(\triangle ABC\) met middelpunt \(O\). De hoogtelijn vanuit \(A\) snijdt \(BC\) in \(D\) en snijdt \(\omega\) een tweede keer in \(P\). Definieer \(H\) als het hoogtepunt van \(\triangle ABC\) en zij \(K\) h... | Zij \(O\) het middelpunt van \(\omega\). Wegens \(|BD| = |KC|\) vallen de middelloodlijnen van \(BC\) en \(KD\) samen en in het bijzonder geldt dus dat \(|OK| = |OD|\). Zij \(T'\) de spiegeling van \(K\) in \(O\). Wegens Thales geldt dan dat \(\angle KDT' = 90^\circ\), dus \(T'\) ligt op \(AD\).
Het is een standaardp... | {
"problem_match": "\nOpgave 2.",
"resource_path": "Dutch_TST/segmented/nl-2025-D2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing I."
} | 292 | 630 |
2025 | T1 | 3 | null | Dutch_TST | Johan en Quintijn spelen het volgende spel, waarbij ze om en om aan de beurt zijn en Johan begint. Aan het begin staan op een bord de getallen \(1, 2, \dots, 2024\) geschreven. In elke beurt veegt de speler die aan de beurt is twee getallen \(a\) en \(b\) die op het bord staan uit, en schrijft het (mogelijk negatieve) ... | We gaan bewijzen dat Quintijn een winnende strategie heeft. Merk allereerst op dat alleen het aantal getallen in elke restklasse modulo 3 van belang is. Als \(a \equiv i \mod 3\), \(b \equiv j \mod 3\) en \(a - b \equiv k \mod 3\), dan noteren we de zet met \(a\) en \(b\) als \((i, j) \to k\). Zij \(x\) het aantal geta... | {
"problem_match": "\nOpgave 3.",
"resource_path": "Dutch_TST/segmented/nl-2025-D2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing I."
} | 177 | 898 |
2025 | T1 | 3 | null | Dutch_TST | Johan en Quintijn spelen het volgende spel, waarbij ze om en om aan de beurt zijn en Johan begint. Aan het begin staan op een bord de getallen \(1, 2, \dots, 2024\) geschreven. In elke beurt veegt de speler die aan de beurt is twee getallen \(a\) en \(b\) die op het bord staan uit, en schrijft het (mogelijk negatieve) ... | Voor \(i = 0, 1, 2\) noteren we het aantal getallen op het bord dat congruent is aan \(i\) modulo 3 met \(x_i\). Merk op dat Johan de laatst overgebleven getallen die niet deelbaar zijn door 3 alleen weg kan spelen als \(x_1 = 2\) en \(x_2 = 0\), of \(x_1 = 0\) en \(x_2 = 2\). We beweren nu dat, als \(x_1\) en \(x_2\) ... | {
"problem_match": "\nOpgave 3.",
"resource_path": "Dutch_TST/segmented/nl-2025-D2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing II."
} | 177 | 602 |
2025 | T1 | 4 | null | Dutch_TST | Vind alle functies \(f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}\) zo dat voor alle positieve gehele getallen \(m, n\) geldt dat
\[(f(m))^2 + 2mf(n) + f(n^2)\]
het kwadraat van een geheel getal is. | Vul in \(m = n = 1\), dan moet \(f(1)^2 + 3f(1)\) een kwadraat zijn. Aangezien
\[(f(1) + 1)^2 \le f(1)^2 + 3f(1) < (f(1) + 2)^2,\]
moet er links gelijkheid gelden, en dus is \(f(1) = 1\). Laat nu \(p = 2k+1\) een oneven priemgetal en vul in \(m = k = (p-1)/2\) en \(n = 1\). Dan zien we dat \(f(k)^2 + p\) een kwadraa... | {
"problem_match": "\nOpgave 4.",
"resource_path": "Dutch_TST/segmented/nl-2025-D2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing."
} | 81 | 674 |
2025 | T1 | 2 | null | Dutch_TST | We noemen een geheel getal \(n \ge 3\) polypythagorees als er \(n\) verschillende positieve getallen zijn die je een cirkel achter elkaar kan zetten zo dat de som van de kwadraten van elk paar opvolgende getallen een kwadraat is. Zo is 3 een polypythagorees getal omdat je bijvoorbeeld met 44, 117 en 240 een drietal heb... | We bewijzen met inductie dat alle gehele getallen groter of gelijk aan 2 polypythagorees zijn, waarbij we de definitie uitbreiden naar \(n = 2\) op de logische manier. Als inductiebasis nemen we (3, 4) voor \(n = 2\) en (44, 117, 240) uit het voorbeeld voor \(n = 3\).
Stel nu dat \(n\) polypythagorees is met als getu... | {
"problem_match": "\nOpgave 2.",
"resource_path": "Dutch_TST/segmented/nl-2025-E2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing."
} | 183 | 700 |
2025 | T1 | 3 | null | Dutch_TST | Bepaal alle drietallen \((x, y, p)\) van positieve gehele getallen zo dat \(p\) een
priemgetal is, \(x^2 = p - 1\) en \(y^2 = 2p^2 - 1\). | Het enige drietal dat voldoet is \((2, 7, 5)\).
We rekenen eerst uit dat
\[ (y + x)(y - x) = y^2 - x^2 = (2p^2 - 1) - (p - 1) = 2p^2 - p = p(2p - 1). \quad (1) \]
Dat betekent in het bijzonder dat \(p \mid x + y\) of \(p \mid x - y\).
Stel dat \(p \mid y + x\). Dan geldt dat \(y = kp - x\) voor een zekere \(k \i... | {
"problem_match": "\nOpgave 3.",
"resource_path": "Dutch_TST/segmented/nl-2025-E2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing."
} | 63 | 633 |
2025 | T1 | 4 | null | Dutch_TST | We zeggen dat een rij \(a_1, \dots, a_n\) van reële getallen afnemend stijgend is als voor alle \(1 < i < n\) geldt dat \(0 < a_{i+1} - a_i < a_i - a_{i-1}\). Vind voor elk positief geheel getal \(m\) het kleinste positieve gehele getal \(k\) waarvoor er een afnemend stijgende rij bestaat van lengte \(k\) zo dat 1 op z... | We bewijzen eerst dat \(k \ge 2m\). We definiëren \(b_i = a_{i+1} - a_i\). Dan is \(b_1, b_2, \dots, b_{k-1}\) een dalende rij positieve reële getallen. En elk van de manieren om 1 te schrijven is in deze schrijfwijze een som van opvolgende elementen in deze rij \(b_j + b_{j+1} + \dots + b_{j+t-1} = a_{j+t} - a_j = 1\)... | {
"problem_match": "\nOpgave 4.",
"resource_path": "Dutch_TST/segmented/nl-2025-E2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing I."
} | 155 | 962 |
2025 | T1 | 4 | null | Dutch_TST | We zeggen dat een rij \(a_1, \dots, a_n\) van reële getallen afnemend stijgend is als voor alle \(1 < i < n\) geldt dat \(0 < a_{i+1} - a_i < a_i - a_{i-1}\). Vind voor elk positief geheel getal \(m\) het kleinste positieve gehele getal \(k\) waarvoor er een afnemend stijgende rij bestaat van lengte \(k\) zo dat 1 op z... | We presenteren een alternatief voorbeeld. We kiezen eerst \(b_1 = 1\). Nu nemen we een \(0 < \epsilon_1 < \frac{1}{6}\) en definiëren we \(b_2 = \frac{1}{2} + \epsilon_1\) en \(b_3 = \frac{1}{2} - \epsilon_1\). Dan geldt er dat \(0 < \epsilon_1 < \frac{b_1 - b_2}{2}\).
We definièren de rest van de rij recursief. Stel ... | {
"problem_match": "\nOpgave 4.",
"resource_path": "Dutch_TST/segmented/nl-2025-E2025_uitwerkingen.jsonl",
"solution_match": "\nOplossing II."
} | 155 | 541 |
2012 | T2 | 1 | null | EGMO | Let $A B C$ be a triangle with circumcentre $O$. The points $D, E$ and $F$ lie in the interiors of the sides $B C, C A$ and $A B$ respectively, such that $D E$ is perpendicular to $C O$ and $D F$ is perpendicular to $B O$. (By interior we mean, for example, that the point $D$ lies on the line $B C$ and $D$ is between $... | Denote by $\ell_{A}, \ell_{B}$ and $\ell_{C}$ the tangents at $A, B$ and $C$ to the circumcircle of $\triangle A B C$. Let $A^{\prime}$ be the point of intersection of $\ell_{B}$ and $\ell_{C}$ and define $B^{\prime}$ and $C^{\prime}$ analogously. As in the first solution, we find that $D E \| \ell_{C}$ and $D F \| \el... | {
"problem_match": "\nProblem 1.",
"resource_path": "EGMO/segmented/en-2012-solutions-day1.jsonl",
"solution_match": "\nSolution 3 (submitter)."
} | 227 | 649 |
2012 | T2 | 2 | null | EGMO | Let $n$ be a positive integer. Find the greatest possible integer $m$, in terms of $n$, with the following property: a table with $m$ rows and $n$ columns can be filled with real numbers in such a manner that for any two different rows $\left[a_{1}, a_{2}, \ldots, a_{n}\right]$ and $\left[b_{1}, b_{2}, \ldots, b_{n}\ri... | The largest possible $m$ is equal to $2^{n}$.
In order to see that the value $2^{n}$ can be indeed achieved, consider all binary vectors of length $n$ as rows of the table. We now proceed with proving that this is the maximum value.
Let $\left[a_{k}^{i}\right]$ be a feasible table, where $i=1, \ldots, m$ and $k=1, \ld... | {
"problem_match": "\nProblem 2.",
"resource_path": "EGMO/segmented/en-2012-solutions-day1.jsonl",
"solution_match": "\nSolution 1 (submitter)."
} | 166 | 543 |
2012 | T2 | 4 | null | EGMO | A set $A$ of integers is called sum-full if $A \subseteq A+A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b, c \in A$. A set $A$ of integers is said to be zero-sum-free if 0 is the only integer that cannot be expressed as the sum of the elements of a finite nonempty sub... | The set $A=\left\{F_{2 n}: n=1,2, \ldots\right\} \cup\left\{-F_{2 n+1}: n=1,2, \ldots\right\}$, where $F_{k}$ is the $k^{\text {th }}$ Fibonacci number $\left(F_{1}=1, F_{2}=1, F_{k+2}=F_{k+1}+F_{k}\right.$ for $k \geq 1$ ) qualifies for an example. We then have $F_{2 n}=F_{2 n+2}+\left(-F_{2 n+1}\right)$ and $-F_{2 n+... | {
"problem_match": "\nProblem 4.",
"resource_path": "EGMO/segmented/en-2012-solutions-day1.jsonl",
"solution_match": "\nSolution (submitter, adapted)."
} | 148 | 592 |
2012 | T2 | 5 | null | EGMO | The numbers $p$ and $q$ are prime and satisfy
$$
\frac{p}{p+1}+\frac{q+1}{q}=\frac{2 n}{n+2}
$$
for some positive integer $n$. Find all possible values of $q-p$.
Origin. Luxembourg (Pierre Haas). | Rearranging the equation, $2 q n(p+1)=(n+2)(2 p q+p+q+1)$. The left hand side is even, so either $n+2$ or $p+q+1$ is even, so either $p=2$ or $q=2$ since $p$ and $q$ are prime, or $n$ is even.
If $p=2,6 q n=(n+2)(5 q+3)$, so $(q-3)(n-10)=36$. Considering the divisors of 36 for which $q$ is prime, we find the possible ... | {
"problem_match": "\nProblem 5.",
"resource_path": "EGMO/segmented/en-2012-solutions-day2.jsonl",
"solution_match": "\nSolution 1 (submitter)."
} | 72 | 623 |
2012 | T2 | 7 | null | EGMO | Let $A B C$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $B C$ from $A$. Let $L$ be the reflection of $K$ in the line $A B$, and let $M$ be the reflection of $K$ in the line $B C$. Let $E$ be the second point of intersection of $\Gamma$ ... | Since the quadrilateral $B M E L$ is cyclic, we have $\angle B E M=\angle B L M$. By construction, $|B K|=|B L|=|B M|$, and so (using directed angles)
$$
\begin{aligned}
\angle B L M & =90^{\circ}-\frac{1}{2} \angle M B L=90^{\circ}-\left(180^{\circ}-\frac{1}{2} \angle L B K-\frac{1}{2} \angle K B M\right) \\
& =\left... | {
"problem_match": "\nProblem 7.",
"resource_path": "EGMO/segmented/en-2012-solutions-day2.jsonl",
"solution_match": "\nSolution 1 (submitter)."
} | 158 | 521 |
2012 | T2 | 8 | null | EGMO | A word is a finite sequence of letters from some alphabet. A word is repetitive if it is a concatenation of at least two identical subwords (for example, $a b a b a b$ and $a b c a b c$ are repetitive, but $a b a b a$ and $a a b b$ are not). Prove that if a word has the property that swapping any two adjacent letters m... | In this and the subsequent solutions we refer to a word with all letters identical as constant.
Let us consider a nonconstant word $W$, of length $|W|=w$, and reach a contradiction. Since the word $W$ must contain two distinct adjacent letters, be it $W=A a b B$ with $a \neq b$, we may assume $B=c C$ to be non-empty, ... | {
"problem_match": "\nProblem 8.",
"resource_path": "EGMO/segmented/en-2012-solutions-day2.jsonl",
"solution_match": "\nSolution 1 (submitter)."
} | 122 | 950 |
2012 | T2 | 8 | null | EGMO | A word is a finite sequence of letters from some alphabet. A word is repetitive if it is a concatenation of at least two identical subwords (for example, $a b a b a b$ and $a b c a b c$ are repetitive, but $a b a b a$ and $a a b b$ are not). Prove that if a word has the property that swapping any two adjacent letters m... | We will take over from the solution above, just before invoking the Wilf-Fine Theorem, by replacing it with a weaker lemma, also built upon a seminal result of combinatorics on words.
Lemma. Let $p, q$ be positive integers, and let $N$ be a word of length $n$, which is both $p$-periodic and $q$ periodic. If $n \geq p+... | {
"problem_match": "\nProblem 8.",
"resource_path": "EGMO/segmented/en-2012-solutions-day2.jsonl",
"solution_match": "\nSolution 2 (submitter)."
} | 122 | 1,324 |
2012 | T2 | 8 | null | EGMO | A word is a finite sequence of letters from some alphabet. A word is repetitive if it is a concatenation of at least two identical subwords (for example, $a b a b a b$ and $a b c a b c$ are repetitive, but $a b a b a$ and $a a b b$ are not). Prove that if a word has the property that swapping any two adjacent letters m... | We define the distance between two words of the same length to be the number of positions in which those two words have different letters. Any two words related by a transposition have distance 0 or 2 ; any two words related by a sequence of two transpositions have distance $0,2,3$ or 4 .
Say the period of a repetitiv... | {
"problem_match": "\nProblem 8.",
"resource_path": "EGMO/segmented/en-2012-solutions-day2.jsonl",
"solution_match": "\nSolution 3 (PSC)."
} | 122 | 1,819 |
2013 | T2 | 2 | null | EGMO | Determine all integers $m$ for which the $m \times m$ square can be dissected into five rectangles, the side lengths of which are the integers $1,2,3, \ldots, 10$ in some order. | The solution naturally divides into three different parts: we first obtain some bounds on $m$. We then describe the structure of possible dissections, and finally, we deal with the few remaining cases.
In the first part of the solution, we get rid of the cases with $m \leqslant 10$ or $m \geqslant 14$. Let $\ell_{1}, ... | {
"problem_match": "\nProblem 2. (Proposed by Matti Lehtinen, Finland)\n",
"resource_path": "EGMO/segmented/en-2013-solutions.jsonl",
"solution_match": "\nSolution:"
} | 52 | 1,937 |
2013 | T2 | 3 | null | EGMO | Let $\boldsymbol{n}$ be a positive integer.
(a) Prove that there exists a set $S$ of $6 n$ pairwise different positive integers, such that the least common multiple of any two elements of $S$ is no larger than $32 n^{2}$.
(b) Prove that every set $T$ of $6 n$ pairwise different positive integers contains two elements ... | (a) Let the set $A$ consist of the $4 n$ integers $1,2, \ldots, 4 n$ and let the set $B$ consist of the $2 n$ even integers $4 n+2,4 n+4, \ldots, 8 n$. We claim that the $6 n$-element set $S=A \cup B$ has the desired property.
Indeed, the least common multiple of two (even) elements of $B$ is no larger than $8 n \cdot... | {
"problem_match": "\nProblem 3. (Proposed by Dan Schwarz, Romania)\n",
"resource_path": "EGMO/segmented/en-2013-solutions.jsonl",
"solution_match": "\nSolution:"
} | 101 | 1,060 |
2013 | T2 | 5 | null | EGMO | Let $\Omega$ be the circumcircle of the triangle $A B C$. The circle $\omega$ is tangent to the sides $A C$ and $B C$, and it is internally tangent to $\Omega$ at the point $P$. A line parallel to $A B$ and intersecting the interior of triangle $A B C$ is tangent to $\omega$ at $Q$.
Prove that $\angle A C P=\angle Q C... | Assume that $\omega$ is tangent to $A C$ and $B C$ at $E$ and $F$, respectively and let $P E, P F, P Q$ meet $\Omega$ at $K, L, M$, respectively. Let $I$ and $O$ denote the respective centres of $\omega$ and $\Omega$, and consider the homethety $\mathscr{H}$ that maps $\omega$ onto $\Omega$. Now $K$ is the image of $E$... | {
"problem_match": "\nProblem 5. (Proposed by Waldemar Pompe, Poland)\n",
"resource_path": "EGMO/segmented/en-2013-solutions.jsonl",
"solution_match": "\nSolution 1: "
} | 95 | 1,112 |
2013 | T2 | 5 | null | EGMO | Let $\Omega$ be the circumcircle of the triangle $A B C$. The circle $\omega$ is tangent to the sides $A C$ and $B C$, and it is internally tangent to $\Omega$ at the point $P$. A line parallel to $A B$ and intersecting the interior of triangle $A B C$ is tangent to $\omega$ at $Q$.
Prove that $\angle A C P=\angle Q C... | Let $I$ and $O$ denote the respective centres of $\omega$ and $\Omega$. Observe that $C I$ is the angle bisector of angle $\angle C$, because $\omega$ is tangent to $A C$ and $B C$. Consider the homethety $\mathscr{H}$ that maps $\omega$ onto $\Omega$. Let $M$ be the image of $Q$ under $\mathscr{H}$. By construction, $... | {
"problem_match": "\nProblem 5. (Proposed by Waldemar Pompe, Poland)\n",
"resource_path": "EGMO/segmented/en-2013-solutions.jsonl",
"solution_match": "\nSolution 2: "
} | 95 | 851 |
2013 | T2 | 5 | null | EGMO | Let $\Omega$ be the circumcircle of the triangle $A B C$. The circle $\omega$ is tangent to the sides $A C$ and $B C$, and it is internally tangent to $\Omega$ at the point $P$. A line parallel to $A B$ and intersecting the interior of triangle $A B C$ is tangent to $\omega$ at $Q$.
Prove that $\angle A C P=\angle Q C... | Let the tangent to $\omega$ at $Q$ meet $A C$ and $B C$ at $X$ and $Y$, respectively. Then $A C / X C=B C / Y C$, and thus there is a radius $r$ such that $r^{2}=A C \cdot Y C=B C \cdot X C$. Let $\Gamma$ denote the circle with centre $C$ and radius $r$, and consider the inversion $\mathscr{I}$ in the circle $\Gamma$. ... | {
"problem_match": "\nProblem 5. (Proposed by Waldemar Pompe, Poland)\n",
"resource_path": "EGMO/segmented/en-2013-solutions.jsonl",
"solution_match": "\nSolution 4: "
} | 95 | 603 |
2013 | T2 | 5 | null | EGMO | Let $\Omega$ be the circumcircle of the triangle $A B C$. The circle $\omega$ is tangent to the sides $A C$ and $B C$, and it is internally tangent to $\Omega$ at the point $P$. A line parallel to $A B$ and intersecting the interior of triangle $A B C$ is tangent to $\omega$ at $Q$.
Prove that $\angle A C P=\angle Q C... | Assume that $\omega$ is tangent to $A C$ and $B C$ at $E$ and $F$, respectively. Assume that $C P$ meets $\omega$ at $D$. Let $I$ and $O$ denote the respective centres of $\omega$ and $\Omega$. To set up a solution in the complex plane, we take the circle $\omega$ as the unit circle centered at the origin of the comple... | {
"problem_match": "\nProblem 5. (Proposed by Waldemar Pompe, Poland)\n",
"resource_path": "EGMO/segmented/en-2013-solutions.jsonl",
"solution_match": "\nSolution 6: "
} | 95 | 1,720 |
2013 | T2 | 6 | null | EGMO | Snow White and the Seven Dwarves are living in their house in the forest. On each of 16 consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecutive) ... | We define $V$ as the set of all 128 vectors of length 7 with entries in $\{0,1\}$. Every such vector encodes the work schedule of a single day: if the $i$-th entry is 0 then the $i$-th dwarf works in the mine, and if this entry is 1 then the $i$-th dwarf collects berries. The 16 working days correspond to 16 vectors $d... | {
"problem_match": "\nProblem 6. (Proposed by Emil Kolev, Bulgaria)\n",
"resource_path": "EGMO/segmented/en-2013-solutions.jsonl",
"solution_match": "\nSolution 1: "
} | 120 | 815 |
2013 | T2 | 6 | null | EGMO | Snow White and the Seven Dwarves are living in their house in the forest. On each of 16 consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecutive) ... | If a dwarf $X$ performs the same type of work on three days $D_{1}, D_{2}, D_{3}$, then we say that this triple of days is monotonous for $X$. We claim that the following configuration cannot occur: There are three dwarves $X_{1}, X_{2}, X_{3}$ and three days $D_{1}, D_{2}$, $D_{3}$, such that the triple $\left(D_{1}, ... | {
"problem_match": "\nProblem 6. (Proposed by Emil Kolev, Bulgaria)\n",
"resource_path": "EGMO/segmented/en-2013-solutions.jsonl",
"solution_match": "\nSolution 2: "
} | 120 | 1,606 |
2014 | T2 | 1 | null | EGMO | Determine all real constants $t$ such that whenever $a, b, c$ are the lengths of the sides of a triangle, then so are $a^{2}+b c t, b^{2}+c a t, c^{2}+a b t$.
Proposed by S. Khan, UNK
The answer is the interval $[2 / 3,2]$. | If $t<2 / 3$, take a triangle with sides $c=b=1$ and $a=2-\epsilon$. Then $b^{2}+c a t+c^{2}+$ $a b t-a^{2}-b c t=3 t-2+\epsilon(4-2 t-\epsilon) \leq 0$ for small positive $\epsilon$; for instance, for any $0<\epsilon<(2-3 t) /(4-2 t)$.
On the other hand, if $t>2$, then take a triangle with sides $b=c=1$ and $a=\epsil... | {
"problem_match": "\n1.",
"resource_path": "EGMO/segmented/en-2014-solutions-day1.jsonl",
"solution_match": "# Solution 1."
} | 85 | 513 |
2014 | T2 | 2 | null | EGMO | Let $D$ and $E$ be two points on the sides $A B$ and $A C$, respectively, of a triangle $A B C$, such that $D B=B C=C E$, and let $F$ be the point of intersection of the lines $C D$ and $B E$. Prove that the incenter $I$ of the triangle $A B C$, the orthocenter $H$ of the triangle $D E F$ and the midpoint $M$ of the $\... | Let the points $K, L, U, V$ be as in Solution 1. Le $P$ be the point of intersection of $D U$ and $E I$, and let $Q$ be the point of intersection of $E V$ and $D I$.
Since $D B=B C=C E$, the points $C I$ and $B I$ are perpendicular to $B E$ and $C D$, respectively. Hence the lines $B I$ and $E V$ are parallel and $\an... | {
"problem_match": "\n2.",
"resource_path": "EGMO/segmented/en-2014-solutions-day1.jsonl",
"solution_match": "# Solution 2."
} | 143 | 669 |
2014 | T2 | 2 | null | EGMO | Let $D$ and $E$ be two points on the sides $A B$ and $A C$, respectively, of a triangle $A B C$, such that $D B=B C=C E$, and let $F$ be the point of intersection of the lines $C D$ and $B E$. Prove that the incenter $I$ of the triangle $A B C$, the orthocenter $H$ of the triangle $D E F$ and the midpoint $M$ of the $\... | Suppose that we have a coordinate system and $\left(b_{x}, b_{y}\right),\left(c_{x}, c_{y}\right),\left(d_{x}, d_{y}\right),\left(e_{x}, e_{y}\right)$ are the coordinates of the points $B, C, D, E$, respectively. From $\overrightarrow{B I} \cdot \overrightarrow{C D}=0, \overrightarrow{C I} \cdot \overrightarrow{B E}=$ ... | {
"problem_match": "\n2.",
"resource_path": "EGMO/segmented/en-2014-solutions-day1.jsonl",
"solution_match": "# Solution 3."
} | 143 | 801 |
2014 | T2 | 3 | null | EGMO | We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n)=k$ and $d(n)$ does not divide $d\left(a^{2}+b^{2}\right)$ for an... | We will show that any number of the form $n=2^{p-1} m$ where $m$ is a positive integer that has exactly $k-1$ prime factors all of which are greater than 3 and $p$ is a prime number such that $(5 / 4)^{(p-1) / 2}>m$ satisfies the given condition.
Suppose that $a$ and $b$ are positive integers such that $a+b=n$ and $d(... | {
"problem_match": "\n3.",
"resource_path": "EGMO/segmented/en-2014-solutions-day1.jsonl",
"solution_match": "# Solution."
} | 111 | 614 |
2014 | T2 | 4 | null | EGMO | Determine all integers $n \geq 2$ for which there exist integers $x_{1}, x_{2}, \ldots, x_{n-1}$ satisfying the condition that if $0<i<n, 0<j<n, i \neq j$ and $n$ divides $2 i+j$, then $x_{i}<x_{j}$.
Proposed by Merlijn Staps, NLD
The answer is that $n=2^{k}$ with $k \geq 1$ or $n=3 \cdot 2^{k}$ where $k \geq 0$. | Let $E=\{n / 3, n / 2,2 n / 3\} \cap\{1,2, \ldots, n-1\}, D=\{1,2, \ldots, n-1\} \backslash E$, and let $f: D \rightarrow\{1,2, \ldots, n-1\}$ be the function sending $i$ in $D$ to the unique $f(i)$ in $\{1,2, \ldots, n-1\}$ such that $f(i) \equiv-2 i(\bmod n)$.
Then the condition of the problem is that $x_{i}<x_{f(i)... | {
"problem_match": "\n4.",
"resource_path": "EGMO/segmented/en-2014-solutions-day2.jsonl",
"solution_match": "# Solution 2."
} | 131 | 589 |
2014 | T2 | 4 | null | EGMO | Determine all integers $n \geq 2$ for which there exist integers $x_{1}, x_{2}, \ldots, x_{n-1}$ satisfying the condition that if $0<i<n, 0<j<n, i \neq j$ and $n$ divides $2 i+j$, then $x_{i}<x_{j}$.
Proposed by Merlijn Staps, NLD
The answer is that $n=2^{k}$ with $k \geq 1$ or $n=3 \cdot 2^{k}$ where $k \geq 0$. | Suppose that $x_{1}, x_{2}, \ldots, x_{k-1}$ satisfy the condition of the problem for $n=k$. Let $y_{2 i}=x_{i}$ for $1 \leq i \leq k-1$ and choose $y_{2 i-1}$ for $1 \leq i \leq k$ to be less than $\min \left\{x_{1}, x_{2}, \ldots, x_{k-1}\right\}$. Now suppose that for $n=2 k$ we have $0<i<n, 0<j<n$, $i \neq j, n$ di... | {
"problem_match": "\n4.",
"resource_path": "EGMO/segmented/en-2014-solutions-day2.jsonl",
"solution_match": "# Solution 3."
} | 131 | 689 |
2014 | T2 | 5 | null | EGMO | Let $n$ be a positive integer. We have $n$ boxes where each box contains a nonnegative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called solvable if it is... | Number the boxes from 1 through $n$ and denote a configuration by $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ where $x_{i}$ is the number of pebbles in the $i$ th box. Let
$$
D(x)=\sum_{i=1}^{n}\left\lfloor\frac{x_{i}-1}{2}\right\rfloor
$$
for a configuration $x$. We can rewrite this in the form
$$
D(x)=\frac{1}{2}... | {
"problem_match": "\n5.",
"resource_path": "EGMO/segmented/en-2014-solutions-day2.jsonl",
"solution_match": "\nSolution 1. "
} | 173 | 730 |
2014 | T2 | 5 | null | EGMO | Let $n$ be a positive integer. We have $n$ boxes where each box contains a nonnegative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called solvable if it is... | Let $x$ be a configuration and $\tilde{x}$ be another configuration obtained from $x$ by removing two pebbles from a box and depositing them in another box.
Claim 1: $\tilde{x}$ is solvable if and only if $x$ is solvable.
Let us call two configurations equivalent if they have the same total number of pebbles and pariti... | {
"problem_match": "\n5.",
"resource_path": "EGMO/segmented/en-2014-solutions-day2.jsonl",
"solution_match": "\nSolution 2. "
} | 173 | 679 |
2014 | T2 | 6 | null | EGMO | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition
$$
f\left(y^{2}+2 x f(y)+f(x)^{2}\right)=(y+f(x))(x+f(y))
$$
for all real numbers $x$ and $y$.
Proposed by Daniël Kroes, NLD
The answer is the functions $f(x)=x, f(x)=-x$ and $f(x)=\frac{1}{2}-x$. | It can be easily checked that the functions $f(x)=x, f(x)=-x$ and $f(x)=\frac{1}{2}-x$ satisfy the given condition. We will show that these are the only functions doing so. Let $y=-f(x)$ in the original equation to obtain
$$
f\left(2 f(x)^{2}+2 x f(-f(x))\right)=0
$$
for all $x$. In particular, 0 is a value of $f$. S... | {
"problem_match": "\n6.",
"resource_path": "EGMO/segmented/en-2014-solutions-day2.jsonl",
"solution_match": "# Solution."
} | 112 | 2,091 |
2015 | T2 | 2 | null | EGMO | A domino is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^{2}$ dominoes can be placed without overlapping on a $2 n \times 2 n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column.
(Turkey) | The answer is $\binom{2 n}{n}^{2}$.
Divide the schessboard into $2 \times 2$ squares. There are exactly $n^{2}$ such squares on the chessboard. Each of these squares can have at most two unit squares covered by the dominos. As the dominos cover exactly $2 n^{2}$ squares, each of them must have exactly two unit squares ... | {
"problem_match": "\nProblem 2.",
"resource_path": "EGMO/segmented/en-2015-solutions.jsonl",
"solution_match": "\nSolution:"
} | 85 | 816 |
2015 | T2 | 3 | null | EGMO | Let $n, m$ be integers greater than 1 , and let $a_{1}, a_{2}, \ldots, a_{m}$ be positive integers not greater than $n^{m}$. Prove that there exist positive integers $b_{1}, b_{2}, \ldots, b_{m}$ not greater than $n$, such that
$$
\operatorname{gcd}\left(a_{1}+b_{1}, a_{2}+b_{2}, \ldots, a_{m}+b_{m}\right)<n
$$
where... | Suppose without loss of generality that $a_{1}$ is the smallest of the $a_{i}$. If $a_{1} \geq n^{m}-1$, then the problem is simple: either all the $a_{i}$ are equal, or $a_{1}=n^{m}-1$ and $a_{j}=n^{m}$ for some $j$. In the first case, we can take (say) $b_{1}=1, b_{2}=2$, and the rest of the $b_{i}$ can be arbitrary,... | {
"problem_match": "\nProblem 3.",
"resource_path": "EGMO/segmented/en-2015-solutions.jsonl",
"solution_match": "\nSolution 1: "
} | 173 | 719 |
2015 | T2 | 4 | null | EGMO | Determine whether there exists an infinite sequence $a_{1}, a_{2}, a_{3}, \ldots$ of positive integers which satisfies the equality
$$
a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}}
$$
for every positive integer n.
(Japan) | 3: We will show that there is no sequence $\left(a_{n}\right)$ of positive integers which consists of $N>5$ members and satisfies
$$
a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}}
$$
for all $n=1, \ldots, N-2$. Moreover, we will describe all such sequences with five members.
Since every $a_{i}$ is a positive integer it follows... | {
"problem_match": "\nProblem 4.",
"resource_path": "EGMO/segmented/en-2015-solutions.jsonl",
"solution_match": "\nSolutions"
} | 72 | 703 |
2015 | T2 | 5 | null | EGMO | Let $m, n$ be positive integers with $m>1$. Anastasia partitions the integers $1,2, \ldots, 2 m$ into $m$ pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to $n$.
(Netherlands) | 1A: Define the following ordered partitions:
$$
\begin{aligned}
& P_{1}=(\{1,2\},\{3,4\}, \ldots,\{2 m-1,2 m\}), \\
& P_{2}=(\{1, m+1\},\{2, m+2\}, \ldots,\{m, 2 m\}), \\
& P_{3}=(\{1,2 m\},\{2, m+1\},\{3, m+2\}, \ldots,\{m, 2 m-1\})
\end{aligned}
$$
For each $P_{j}$ we will compute the possible values for the expres... | {
"problem_match": "\nProblem 5.",
"resource_path": "EGMO/segmented/en-2015-solutions.jsonl",
"solution_match": "\nSolution "
} | 81 | 1,013 |
2015 | T2 | 6 | null | EGMO | Let $H$ be the orthocenter and $G$ be the centroid of acute-angled triangle $\triangle A B C$ with $A B \neq A C$. The line $A G$ intersects the circumcircle of $\triangle A B C$ at $A$ and $P$. Let $P^{\prime}$ be the reflection of $P$ in the line $B C$. Prove that $\angle C A B=60^{\circ}$ if and only if $H G=G P^{\p... | Let $\omega$ be the circumcircle of $\triangle A B C$. Reflecting $\omega$ in line $B C$, we obtain circle $\omega^{\prime}$ which, obviously, contains points $H$ and $P^{\prime}$. Let $M$ be the midpoint of $B C$. As triangle $\triangle A B C$ is acute-angled, then $H$ and $O$ lie inside this triangle.
Let us assume ... | {
"problem_match": "\nProblem 6.",
"resource_path": "EGMO/segmented/en-2015-solutions.jsonl",
"solution_match": "\nSolution 1: "
} | 189 | 1,803 |
2015 | T2 | 6 | null | EGMO | Let $H$ be the orthocenter and $G$ be the centroid of acute-angled triangle $\triangle A B C$ with $A B \neq A C$. The line $A G$ intersects the circumcircle of $\triangle A B C$ at $A$ and $P$. Let $P^{\prime}$ be the reflection of $P$ in the line $B C$. Prove that $\angle C A B=60^{\circ}$ if and only if $H G=G P^{\p... | Let $O^{\prime}$ and $G^{\prime}$ denote the reflection of $O$ and $G$, respectively, with respect to the line $B C$. We then need to show $\angle C A B=60^{\circ}$ iff $G^{\prime} H^{\prime}=G^{\prime} P$. Note that $\triangle H^{\prime} O P$ is isosceles and hence
$G^{\prime} H^{\prime}=G^{\prime} P$ is equivalent to... | {
"problem_match": "\nProblem 6.",
"resource_path": "EGMO/segmented/en-2015-solutions.jsonl",
"solution_match": "\nSolution 2: "
} | 189 | 658 |
2015 | T2 | 6 | null | EGMO | Let $H$ be the orthocenter and $G$ be the centroid of acute-angled triangle $\triangle A B C$ with $A B \neq A C$. The line $A G$ intersects the circumcircle of $\triangle A B C$ at $A$ and $P$. Let $P^{\prime}$ be the reflection of $P$ in the line $B C$. Prove that $\angle C A B=60^{\circ}$ if and only if $H G=G P^{\p... | Let $H^{\prime}$ and $G^{\prime}$ denote the reflection of points $H$ and $G$ with respect to the line $B C$. It is known that $H^{\prime}$ belongs to the circumcircle of $\triangle A B C$. The equality $H G=G P^{\prime}$ is equivalent to $H^{\prime} G^{\prime}=G^{\prime} P$. As in the Solution 2, it is equivalent to t... | {
"problem_match": "\nProblem 6.",
"resource_path": "EGMO/segmented/en-2015-solutions.jsonl",
"solution_match": "\nSolution 3: "
} | 189 | 929 |
2016 | T2 | 3 | null | EGMO | Let $m$ be a positive integer. Consider a $4 m \times 4 m$ array of square unit cells. Two different cells are related to each other if they are in either the same row or in the same column. No cell is related to itself. Some cells are coloured blue, such that every cell is related to at least two blue cells. Determine... | The required minimum is 6 m and is achieved by a diagonal string of $m 4 \times 4$ blocks of the form below (bullets mark centres of blue cells):
In particular, this configuration shows that the required minimum does not exceed 6 m .
We now show that any configuration of blue cells satisfying the condition in the sta... | {
"problem_match": "\nProblem 3.",
"resource_path": "EGMO/segmented/en-2016-solutions.jsonl",
"solution_match": "\nSolution 1 (Israel)."
} | 82 | 558 |
2016 | T2 | 3 | null | EGMO | Let $m$ be a positive integer. Consider a $4 m \times 4 m$ array of square unit cells. Two different cells are related to each other if they are in either the same row or in the same column. No cell is related to itself. Some cells are coloured blue, such that every cell is related to at least two blue cells. Determine... | To prove that a minimal configuration of blue cells satisfying the condition in the statement has cardinality at least 6 m , consider a bipartite graph whose vertex parts are the rows and the columns of the array, respectively, a row and a column being joined by an edge if and only if the two cross at a blue cell. Clea... | {
"problem_match": "\nProblem 3.",
"resource_path": "EGMO/segmented/en-2016-solutions.jsonl",
"solution_match": "\nSolution 2. "
} | 82 | 673 |
2016 | T2 | 5 | null | EGMO | Let $k$ and $n$ be integers such that $k \geq 2$ and $k \leq n \leq 2 k-1$. Place rectangular tiles, each of size $1 \times k$ or $k \times 1$, on an $n \times n$ chessboard so that each tile covers exactly $k$ cells, and no two tiles overlap. Do this until
no further tile can be placed in this way. For each such $k$ a... | The required minimum is $n$ if $n=k$, and it is $\min (n, 2 n-2 k+2)$ if $k<n<2 k$.
The case $n=k$ being clear, assume henceforth $k<n<2 k$. Begin by describing maximal arrangements on the board $[0, n] \times[0, n]$, having the above mentioned cardinalities.
If $k<n<2 k-1$, then $\min (n, 2 n-2 k+2)=2 n-2 k+2$. To o... | {
"problem_match": "\nProblem 5.",
"resource_path": "EGMO/segmented/en-2016-solutions.jsonl",
"solution_match": "\nSolution."
} | 121 | 1,316 |
2016 | T2 | 6 | null | EGMO | Let $S$ be the set of all positive integers $n$ such that $n^{4}$ has a divisor in the range $n^{2}+1, n^{2}+2, \ldots, n^{2}+2 n$. Prove that there are infinitely many elements of $S$ of each of the forms $7 m, 7 m+1,7 m+2,7 m+5,7 m+6$ and no elements of $S$ of the form $7 m+3$ or $7 m+4$, where $m$ is an integer. | The conclusion is a consequence of the lemma below which actually provides a recursive description of $S$. The proof of the lemma is at the end of the solution.
Lemma. The fourth power of a positive integer $n$ has a divisor in the range $n^{2}+1, n^{2}+2, \ldots, n^{2}+2 n$ if and only if at least one of the numbers ... | {
"problem_match": "\nProblem 6.",
"resource_path": "EGMO/segmented/en-2016-solutions.jsonl",
"solution_match": "\nSolution."
} | 127 | 1,276 |
2017 | T2 | 2 | null | EGMO | Find the smallest positive integer $k$ for which there exist a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ with the following two properties:
(i) For all positive integers $m, n$ of the same colour, $f(m+n)=f(m)+f(n)$.
(ii) There ... | The answer is $k=3$.
First we show that there is such a function and coloring for $k=3$. Consider $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ given by $f(n)=n$ for all $n \equiv 1$ or 2 modulo 3 , and $f(n)=2 n$ for $n \equiv 0$ modulo 3 . Moreover, give a positive integer $n$ the $i$-th color if $n \equiv i$ (3).... | {
"problem_match": "# Problem 2",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "# Solution 1:"
} | 185 | 1,195 |
2017 | T2 | 2 | null | EGMO | Find the smallest positive integer $k$ for which there exist a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ with the following two properties:
(i) For all positive integers $m, n$ of the same colour, $f(m+n)=f(m)+f(n)$.
(ii) There ... | We prove that $k \leq 3$ just as in first solution.
Next we show that there is no such function and coloring for $k=2$.
Consider any coloring of $\mathbb{Z}_{>0}$ with 2 colors and any function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ satisfying conditions (i) and (ii). We first notice with $m=n$ that $f(2 n)=2... | {
"problem_match": "# Problem 2",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "# Solution 2:"
} | 185 | 831 |
2017 | T2 | 2 | null | EGMO | Find the smallest positive integer $k$ for which there exist a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ with the following two properties:
(i) For all positive integers $m, n$ of the same colour, $f(m+n)=f(m)+f(n)$.
(ii) There ... | As before we prove that $k \leq 3$ and for any such function and colouring we have $f(2 n)=2 f(n)$.
Now we show that there is no such function and coloring for $k=2$.
Consider any coloring of $\mathbb{Z}_{>0}$ with 2 colors and any function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ satisfying conditions (i) and ... | {
"problem_match": "# Problem 2",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "# Solution 3:"
} | 185 | 1,013 |
2017 | T2 | 3 | null | EGMO | There are 2017 lines in a plane such that no 3 of them go through the same point. Turbo the snail can slide along the lines in the following fashion: she initially moves on one of the lines and continues moving on a given line until she reaches an intersection of 2 lines. At the intersection, she follows her journey on... | Let us color in red all intersection points of the given lines and let us choose one of two possible directions on each segment (draw an arrow on each segment). Consider a red point $R$ where two given lines $a$ and $b$ meet, and the four segments $a_{1}, a_{2}, b_{1}, b_{2}$ with endpoint $R$ (so that $a_{i} \subset a... | {
"problem_match": "# Problem 3",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "\n## 4. Solution"
} | 129 | 626 |
2017 | T2 | 4 | null | EGMO | Let $n \geq 1$ be an integer and let $t_{1}<t_{2}<\ldots<t_{n}$ be positive integers. In a group of $t_{n}+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following conditions to hold at the same time:
i) The number of games played by each person... | Let $\mathcal{T}=\left\{t_{1}, \ldots, t_{n}\right\}$. The proof proceeds by induction on $n=|\mathcal{T}|$. If $n=1$ and $\mathcal{T}=\{t\}$, we choose a group of $t+1$ people such that everyone plays with
everyone else. If $n=2$ and $\mathcal{T}=\left\{t_{1}, t_{2}\right\}$ with $t_{1}<t_{2}$, divide the $t_{2}+1$ pe... | {
"problem_match": "# Problem 4",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "\n## 2. Solution"
} | 241 | 561 |
2017 | T2 | 4 | null | EGMO | Let $n \geq 1$ be an integer and let $t_{1}<t_{2}<\ldots<t_{n}$ be positive integers. In a group of $t_{n}+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following conditions to hold at the same time:
i) The number of games played by each person... | The proof proceeds by induction on $\left|t_{n}\right|$. If $t_{n}=1$ we have $n=1$ and we can consider two persons that play against each other. Then every player has played 1 game and the conditions of the problem are satisfied. If $t_{n}>1$ we distinguish the two cases $t_{1}>1$ and $t_{1}=1$. If $t_{1}>1$ there exi... | {
"problem_match": "# Problem 4",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "\n## 3. Solution"
} | 241 | 546 |
2017 | T2 | 4 | null | EGMO | Let $n \geq 1$ be an integer and let $t_{1}<t_{2}<\ldots<t_{n}$ be positive integers. In a group of $t_{n}+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following conditions to hold at the same time:
i) The number of games played by each person... | We generalize the construction for $\mathcal{T}=\{1, \ldots, n\}$
## Construction
Take sets of people $A_{1}, \ldots, A_{n}$. Let all people of $A_{i}$ play chess with all people in $A_{j}$ with $j \geq n-i+1$
Now the number of games played by anyone in $A_{i}$ is
$\left(\sum_{j \geq n-i+1}\left|A_{j}\right|\right)$... | {
"problem_match": "# Problem 4",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "\n## 4. Solution"
} | 241 | 1,085 |
2017 | T2 | 5 | null | EGMO | Let $n \geq 2$ be an integer. An $n$-tuple $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of positive integers is expensive if there exists a positive integer $k$ such that
$$
\left(a_{1}+a_{2}\right)\left(a_{2}+a_{3}\right) \cdots \cdots\left(a_{n-1}+a_{n}\right)\left(a_{n}+a_{1}\right)=2^{2 k-1} .
$$
a) Find all positi... | 1
a) Notice that for odd integers $n>2$, the tuple $(1,1, \ldots, 1)$ is expensive. We will prove that there are no expensive $n$-tuples for even $n$.
Lemma 0.1. If an expensive $n$-tuple exists for some $n \geq 4$, then also an expensive $n-2$-tuple.
Proof. In what follows all indices are considered modulo $n$. Let ... | {
"problem_match": "# Problem 5",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "# Solution "
} | 217 | 817 |
2017 | T2 | 5 | null | EGMO | Let $n \geq 2$ be an integer. An $n$-tuple $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of positive integers is expensive if there exists a positive integer $k$ such that
$$
\left(a_{1}+a_{2}\right)\left(a_{2}+a_{3}\right) \cdots \cdots\left(a_{n-1}+a_{n}\right)\left(a_{n}+a_{1}\right)=2^{2 k-1} .
$$
a) Find all positi... | 2
a) For odd $n$ the tuple $(1,1, \ldots, 1)$ is a solution.
Now consider $n$ even. Since the product $\prod\left(a_{i}+a_{i+1}\right)$ is a power of two, every factor needs to be a power of two. We are going to prove that for all tuples $\left(a_{1}, \ldots, a_{n}\right)$ such that $a_{i}+a_{i+1}$ is always a power ... | {
"problem_match": "# Problem 5",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "# Solution "
} | 217 | 1,586 |
2017 | T2 | 6 | null | EGMO | Let $A B C$ be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid $G$ and the circumcentre $O$ of $A B C$ in its sides $B C, C A, A B$ are denoted by $G_{1}, G_{2}, G_{3}$, and $O_{1}, O_{2}, O_{3}$, respectively. Show that the circumcircles of the triangles $G_{1} G_{2... | Let $H$ denote the orthocenter of $A B C$, and let $e$ denote its Euler line. Let $e_{1}, e_{2}, e_{3}$ denote the respective reflections of $e$ in $B C, C A, A B$. The proof naturally divides into two parts: we first show that pairwise intersections of the circles in question correspond to pairwise intersections of $e... | {
"problem_match": "# Problem 6",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "# Solution 1 (Euler lines)"
} | 211 | 1,383 |
2017 | T2 | 6 | null | EGMO | Let $A B C$ be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid $G$ and the circumcentre $O$ of $A B C$ in its sides $B C, C A, A B$ are denoted by $G_{1}, G_{2}, G_{3}$, and $O_{1}, O_{2}, O_{3}$, respectively. Show that the circumcircles of the triangles $G_{1} G_{2... | 2
The proof consists of two parts. First, we show that if $P$ is any point inside the triangle $A B C$ and $P_{1}, P_{2}, P_{3}$ are its reflections in the sides $B C, C A, A B$, then the circumcircles of the triangles $P_{1} P_{2} C, P_{1} P_{3} B, P_{2} P_{3} A$ intersect in a point $T_{P}$ on the circumcircle of th... | {
"problem_match": "# Problem 6",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "# Solution "
} | 211 | 833 |
2017 | T2 | 6 | null | EGMO | Let $A B C$ be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid $G$ and the circumcentre $O$ of $A B C$ in its sides $B C, C A, A B$ are denoted by $G_{1}, G_{2}, G_{3}$, and $O_{1}, O_{2}, O_{3}$, respectively. Show that the circumcircles of the triangles $G_{1} G_{2... | For every point $P$, let $p$ denote the corresponding complex number. Set $O$ to be the origin, so $o=0$, and without loss of generality we can assume that $a, b$ and $c$ lie on the unit circle. Then the centroid can be expressed as $g=\frac{a+b+c}{3}$.
The segments $O o_{1}$ and $b c$ have a common midpoint, so $o_{1}... | {
"problem_match": "# Problem 6",
"resource_path": "EGMO/segmented/en-2017-solutions.jsonl",
"solution_match": "# Solution 3 (complex numbers)"
} | 211 | 1,460 |
2018 | T2 | 1 | null | EGMO | Let $A B C$ be a triangle with $C A=C B$ and $\angle A C B=120^{\circ}$, and let $M$ be the midpoint of $A B$. Let $P$ be a variable point on the circumcircle of $A B C$, and let $Q$ be the point on the segment $C P$ such that $Q P=2 Q C$. It is given that the line through $P$ and perpendicular to $A B$ intersects the ... | Let $O$ be the circumcenter of $A B C$. From the assumption that $\angle A C B=120^{\circ}$ it follows that $M$ is the midpoint of $C O$.
Let $\omega$ denote the circle with center in $C$ and radius $C O$. This circle in the image of the circumcircle of $A B C$ through the translation that sends $O$ to $C$. We claim t... | {
"problem_match": "\nProblem 1",
"resource_path": "EGMO/segmented/en-2018-solutions.jsonl",
"solution_match": "\nSolution "
} | 153 | 890 |
2018 | T2 | 2 | null | EGMO | Consider the set
$$
A=\left\{1+\frac{1}{k}: k=1,2,3, \ldots\right\}
$$
(a) Prove that every integer $x \geq 2$ can be written as the product of one or more elements of $A$, which are not necessarily different.
(b) For every integer $x \geq 2$, let $f(x)$ denote the minimum integer such that $x$ can be written as the ... | Every integer $x \geq 2$ can be written as the telescopic product of $x-1$ elements of $A$ as
$$
x=\left(1+\frac{1}{x-1}\right) \cdot\left(1+\frac{1}{x-2}\right) \cdot \ldots \cdot\left(1+\frac{1}{2}\right) \cdot\left(1+\frac{1}{1}\right)
$$
which is enough to establish part (a). We now consider part (b). Notice that... | {
"problem_match": "\nProblem 2",
"resource_path": "EGMO/segmented/en-2018-solutions.jsonl",
"solution_match": "\nSolution "
} | 236 | 1,775 |
2018 | T2 | 3 | null | EGMO | The $n$ contestants of an EGMO are named $C_{1}, \ldots, C_{n}$. After the competition they queue in front of the restaurant according to the following rules.
- The Jury chooses the initial order of the contestants in the queue.
- Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$.
- If contestant $C... | The maximal number of euros is $2^{n}-n-1$.
To begin with, we show that it is possible for the Jury to collect this number of euros. We argue by induction. Let us assume that the Jury can collect $M_{n}$ euros in a configuration with $n$ contestants. Then we show that the Jury can collect at least $2 M_{n}+n$ moves in ... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2018-solutions.jsonl",
"solution_match": "\nSolution "
} | 203 | 1,382 |
2018 | T2 | 4 | null | EGMO | A domino is a $1 \times 2$ or $2 \times 1$ tile.
Let $n \geq 3$ be an integer. Dominoes are placed on an $n \times n$ board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap.
The value of a row or column is the number of dominoes that cover at least one cell of this row ... | The minimal number of dominoes required in a balanced configuration is $2 n / 3$ if $n$ is a multiple of 3 , and $2 n$ otherwise.
In order to show that this number is necessary, we count in two different ways the number of elements of the set $S$ of all pairs $(\ell, d)$, where $\ell$ is a row or a column of the board... | {
"problem_match": "\nProblem 4",
"resource_path": "EGMO/segmented/en-2018-solutions.jsonl",
"solution_match": "\nSolution "
} | 172 | 525 |
2018 | T2 | 6 | null | EGMO | (a) Prove that for every real number $t$ such that $0<t<\frac{1}{2}$ there exists a positive integer $n$ with the following property: for every set $S$ of $n$ positive integers there exist two different elements $x$ and $y$ of $S$, and a non-negative integer $m$ (i.e. $m \geq 0$ ), such that
$$
|x-m y| \leq t y
$$
(b... | Part (a) Let $n$ be any positive integer such that
$$
(1+t)^{n-1} \geq \frac{1}{t}
$$
(this inequality is actually true for every large enough $n$ due to Bernoulli's inequality).
Let $S$ be any set of $n$ distinct positive integers, which we denote by
$$
s_{1}<s_{2}<\ldots<s_{n}
$$
We distinguish two cases.
- If $... | {
"problem_match": "# Problem 6",
"resource_path": "EGMO/segmented/en-2018-solutions.jsonl",
"solution_match": "# Solution"
} | 198 | 2,263 |
2019 | T2 | 1 | null | EGMO | (Netherlands). Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b
$$ | First suppose that $a=0$. Then we have $b c=1$ and $c=b^{2} c=b$. So $b=c$, which implies $b^{2}=1$ and hence $b= \pm 1$. This leads to the solutions $(a, b, c)=(0,1,1)$ and $(a, b, c)=(0,-1,-1)$. Similarly, $b=0$ gives the solutions $(a, b, c)=(1,0,1)$ and $(a, b, c)=(-1,0,-1)$, while $c=0$ gives $(a, b, c)=(1,1,0)$ a... | {
"problem_match": "\nProblem 1",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution 1. "
} | 54 | 864 |
2019 | T2 | 1 | null | EGMO | (Netherlands). Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b
$$ | by Achilleas Sinefakopoulos, Greece. We have
$$
c\left(1-b^{2}\right)=a(1-a b)=a(b c+c a)=c\left(a b+a^{2}\right)
$$
and so
$$
c\left(a^{2}+a b+b^{2}-1\right)=0
$$
Similarly, we have
$$
b\left(a^{2}+a c+c^{2}-1\right)=0 \quad \text { and } \quad a\left(b^{2}+b c+c^{2}-1\right)=0
$$
If $c=0$, then we get $a b=1$ a... | {
"problem_match": "\nProblem 1",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 54 | 558 |
2019 | T2 | 1 | null | EGMO | (Netherlands). Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b
$$ | by Eirini Miliori (HEL2). It is $a b+b c+c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b
$$
We have
$$
\begin{aligned}
a^{2} b+c=b^{2} c+a & \Longleftrightarrow a^{2} b-a=b^{2} c-c \\
& \Longleftrightarrow a(a b-1)=c\left(b^{2}-1\right) \\
& \Longleftrightarrow a(-b c-a c)=c\left(b^{2}-1\right) \\
& \Longleftrightarrow-... | {
"problem_match": "\nProblem 1",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 54 | 1,383 |
2019 | T2 | 1 | null | EGMO | (Netherlands). Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and
$$
a^{2} b+c=b^{2} c+a=c^{2} a+b
$$ | by ISR5. First, homogenize the condition $a^{2} b+c=b^{2} c+a=c^{2} a+b$ by replacing $c$ by $c(a b+b c+c a)$ (etc.), yielding
$$
a^{2} b+c=a^{2} b+a b c+b c^{2}+c^{2} a=a b c+\sum_{c y c} a^{2} b+\left(c^{2} b-b^{2} c\right)=a b c+\sum_{c y c} a^{2} b+b c(c-b) .
$$
Thus, after substracting the cyclicly symmetric par... | {
"problem_match": "\nProblem 1",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 54 | 615 |
2019 | T2 | 2 | null | EGMO | (Luxembourg). Let $n$ be a positive integer. Dominoes are placed on a $2 n \times 2 n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way.
(A domino is a tile of size $2 \times 1$ or ... | Let $M$ denote the maximal number of dominoes that can be placed on the chessboard. We claim that $M=n(n+1) / 2$. The proof naturally splits into two parts: we first prove that $n(n+1) / 2$ dominoes can be placed on the board, and then show that $M \leq n(n+1) / 2$ to complete the proof.
We construct placings of the d... | {
"problem_match": "\nProblem 2",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution 1. "
} | 143 | 967 |
2019 | T2 | 3 | null | EGMO | (Poland). Let $A B C$ be a triangle such that $\angle C A B>\angle A B C$, and let $I$ be its incentre. Let $D$ be the point on segment $B C$ such that $\angle C A D=\angle A B C$. Let $\omega$ be the circle tangent to $A C$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the... | Let $S$ be the intersection point of $B C$ and the angle bisector of $\angle B A D$, and let $T$ be the intersection point of $B C$ and the angle bisector of $\angle B X C$. We will prove that both quadruples $A, I, B, S$ and $A, I, B, T$ are concyclic, which yields $S=T$.
Firstly denote by $M$ the middle of $\operato... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution 1. "
} | 139 | 617 |
2019 | T2 | 3 | null | EGMO | (Poland). Let $A B C$ be a triangle such that $\angle C A B>\angle A B C$, and let $I$ be its incentre. Let $D$ be the point on segment $B C$ such that $\angle C A D=\angle A B C$. Let $\omega$ be the circle tangent to $A C$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the... | Let $\angle B A C=\alpha, \angle A B C=\beta, \angle B C A=\gamma \angle A C X=\phi$. Denote by $W_{1}$ and $W_{2}$ the intersections of segment $B C$ with the angle bisectors of $\angle B X C$ and $\angle B A D$ respectively. Then $B W_{1} / W_{1} C=B X / X C$ and $B W_{2} / W_{2} D=B A / A D$. We shall show that $B W... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution 2. "
} | 139 | 1,501 |
2019 | T2 | 3 | null | EGMO | (Poland). Let $A B C$ be a triangle such that $\angle C A B>\angle A B C$, and let $I$ be its incentre. Let $D$ be the point on segment $B C$ such that $\angle C A D=\angle A B C$. Let $\omega$ be the circle tangent to $A C$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the... | by Achilleas Sinefakopoulos, Greece. We first note that
$$
\angle B A D=\angle B A C-\angle D A C=\angle A-\angle B .
$$
Let $C X$ and $A D$ meet at $K$. Then $\angle C X A=\angle A B C=\angle K A C$. Also, we have $\angle I X A=$ $\angle A / 2$, since $\omega$ is tangent to $A C$ at $A$. Therefore,
$$
\angle D A I=... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 139 | 794 |
2019 | T2 | 3 | null | EGMO | (Poland). Let $A B C$ be a triangle such that $\angle C A B>\angle A B C$, and let $I$ be its incentre. Let $D$ be the point on segment $B C$ such that $\angle C A D=\angle A B C$. Let $\omega$ be the circle tangent to $A C$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the... | based on that by Eirini Miliori (HEL2), edited by A. Sinefakopoulos, Greece. It is $\angle A B D=\angle D A C$, and so $\overline{A C}$ is tangent to the circumcircle of $\triangle B A D$ at $A$. Hence $C A^{2}=C D \cdot C B$.
Triangle $\triangle A B C$ is similar to triangle $\triangle C A D$, because $\angle C$ is a... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 139 | 718 |
2019 | T2 | 3 | null | EGMO | (Poland). Let $A B C$ be a triangle such that $\angle C A B>\angle A B C$, and let $I$ be its incentre. Let $D$ be the point on segment $B C$ such that $\angle C A D=\angle A B C$. Let $\omega$ be the circle tangent to $A C$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the... | based on the work of Artemis-Chrysanthi Savva (HEL4), completed by A. Sinefakopoulos, Greece. Let $G$ be the point of intersection of $\overline{A D}$ and $\overline{C X}$. Since the quadrilateral $A X B C$ is cyclic, it is $\angle A X C=\angle A B C$.
Let the line $\overline{A D}$ meet $\omega$ at $K$. Then it is $\a... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 139 | 712 |
2019 | T2 | 3 | null | EGMO | (Poland). Let $A B C$ be a triangle such that $\angle C A B>\angle A B C$, and let $I$ be its incentre. Let $D$ be the point on segment $B C$ such that $\angle C A D=\angle A B C$. Let $\omega$ be the circle tangent to $A C$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the... | by IRL1 and IRL 5. Let $\omega$ denote the circle through $A$ and $I$ tangent to $A C$. Let $Y$ be the second point of intersection of the circle $\omega$ with the line $A D$. Let $L$
be the intersection of $B C$ with the angle bisector of $\angle B A D$. We will prove $\angle L X C=$ $1 / 2 \angle B A C=1 / 2 \angle B... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 139 | 509 |
2019 | T2 | 3 | null | EGMO | (Poland). Let $A B C$ be a triangle such that $\angle C A B>\angle A B C$, and let $I$ be its incentre. Let $D$ be the point on segment $B C$ such that $\angle C A D=\angle A B C$. Let $\omega$ be the circle tangent to $A C$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the... | by IRL 5. Let $M$ be the midpoint of the $\operatorname{arc} B C$. Let $\omega$ denote the circle through $A$ and $I$ tangent to $A C$. Let $N$ be the second point of intersection of $\omega$ with $A B$ and $L$ the intersection of $B C$ with the angle bisector of $\angle B A D$. We know $\frac{D L}{L B}=\frac{A D}{A B}... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 139 | 656 |
2019 | T2 | 3 | null | EGMO | (Poland). Let $A B C$ be a triangle such that $\angle C A B>\angle A B C$, and let $I$ be its incentre. Let $D$ be the point on segment $B C$ such that $\angle C A D=\angle A B C$. Let $\omega$ be the circle tangent to $A C$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the... | by ISR5 (with help from IRL5). Let $M, N$ be the midpoints of arcs $B C, B A$ of the circumcircle $A B C$, respectively. Let $Y$ be the second intersection of $A D$ and circle $A B C$. Let $E$ be the incenter of triangle $A B Y$ and note that $E$ lies on the angle bisectors of the triangle, which are the lines $Y N$ (i... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 139 | 1,056 |
2019 | T2 | 3 | null | EGMO | (Poland). Let $A B C$ be a triangle such that $\angle C A B>\angle A B C$, and let $I$ be its incentre. Let $D$ be the point on segment $B C$ such that $\angle C A D=\angle A B C$. Let $\omega$ be the circle tangent to $A C$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the... | inspired by ISR2. Let $W$ be the midpoint of arc $B C$, let $D^{\prime}$ be the second intersection point of $A D$ and the circle $A B C$. Let $P$ be the intersection of the angle bisector $X W$ of $\angle C X B$ with $B C$; we wish to prove that $A P$ is the angle bisector of $D A B$. Denote $\alpha=\frac{\angle C A B... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 139 | 607 |
2019 | T2 | 3 | null | EGMO | (Poland). Let $A B C$ be a triangle such that $\angle C A B>\angle A B C$, and let $I$ be its incentre. Let $D$ be the point on segment $B C$ such that $\angle C A D=\angle A B C$. Let $\omega$ be the circle tangent to $A C$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the... | by inversion, by JPN Observer A, Satoshi Hayakawa. Let $E$ be the intersection of the bisector of $\angle B A D$ and $B C$, and $N$ be the middle point of arc $B C$ of the circumcircle of $A B C$. Then it suffices to show that $E$ is on line $X N$.
We consider the inversion at $A$. Let $P^{*}$ be the image of a point ... | {
"problem_match": "\nProblem 3",
"resource_path": "EGMO/segmented/en-2019-solutions-day1.jsonl",
"solution_match": "\nSolution "
} | 139 | 538 |
2019 | T2 | 4 | null | EGMO | (Poland). Let $A B C$ be a triangle with incentre $I$. The circle through $B$ tangent to $A I$ at $I$ meets side $A B$ again at $P$. The circle through $C$ tangent to $A I$ at $I$ meets side $A C$ again at $Q$. Prove that $P Q$ is tangent to the incircle of $A B C$. | by Achilleas Sinefakopoulos, Greece. From the power of a point theorem, we have
$$
A P \cdot A B=A I^{2}=A Q \cdot A C
$$
Hence $P B C Q$ is cyclic, and so, $\angle A P Q=\angle B C A$. Let $K$ be the circumcenter of $\triangle B I P$ and let $L$ be the circumcenter of $\triangle Q I C$. Then $\overline{K L}$ is perp... | {
"problem_match": "\nProblem 4",
"resource_path": "EGMO/segmented/en-2019-solutions-day2.jsonl",
"solution_match": "\nSolution "
} | 91 | 528 |
2019 | T2 | 4 | null | EGMO | (Poland). Let $A B C$ be a triangle with incentre $I$. The circle through $B$ tangent to $A I$ at $I$ meets side $A B$ again at $P$. The circle through $C$ tangent to $A I$ at $I$ meets side $A C$ again at $Q$. Prove that $P Q$ is tangent to the incircle of $A B C$. | by Eirini Miliori (HEL2). Let $D$ be the point of intersection of $\overline{A I}$ and $\overline{B C}$ and let $R$ be the point of intersection of $\overline{A I}$ and $\overline{P Q}$. We have $\angle R I P=\angle P B I=\frac{\angle B}{2}$, $\angle R I Q=\angle I C Q=\frac{\angle C}{2}, \angle I Q C=\angle D I C=x$ a... | {
"problem_match": "\nProblem 4",
"resource_path": "EGMO/segmented/en-2019-solutions-day2.jsonl",
"solution_match": "\nSolution "
} | 91 | 1,187 |
2019 | T2 | 5 | null | EGMO | (Netherlands).
Let $n \geq 2$ be an integer, and let $a_{1}, a_{2}, \ldots, a_{n}$ be positive integers. Show that there exist positive integers $b_{1}, b_{2}, \ldots, b_{n}$ satisfying the following three conditions:
1). $a_{i} \leq b_{i}$ for $i=1,2, \ldots, n$;
2). the remainders of $b_{1}, b_{2}, \ldots, b_{n}$ o... | Note that the problem is invariant under each of the following operations:
- adding a multiple of $n$ to some $a_{i}$ (and the corresponding $b_{i}$ );
- adding the same integer to all $a_{i}$ (and all $b_{i}$ );
- permuting the index set $1,2, \ldots, n$.
We may therefore remove the restriction that our $a_{i}$ and ... | {
"problem_match": "# Problem 5",
"resource_path": "EGMO/segmented/en-2019-solutions-day2.jsonl",
"solution_match": "\nSolution 2. "
} | 225 | 974 |
2019 | T2 | 5 | null | EGMO | (Netherlands).
Let $n \geq 2$ be an integer, and let $a_{1}, a_{2}, \ldots, a_{n}$ be positive integers. Show that there exist positive integers $b_{1}, b_{2}, \ldots, b_{n}$ satisfying the following three conditions:
1). $a_{i} \leq b_{i}$ for $i=1,2, \ldots, n$;
2). the remainders of $b_{1}, b_{2}, \ldots, b_{n}$ o... | We will prove the required statement for all sequences of non-negative integers $a_{i}$ by induction on $n$.
Case $n=1$ is obvious, just set $b_{1}=a_{1}$.
Now suppose that the statement is true for some $n \geq 1$; we shall prove it for $n+1$.
First note that, by subtracting a multiple of $n+1$ to each $a_{i}$ and pos... | {
"problem_match": "# Problem 5",
"resource_path": "EGMO/segmented/en-2019-solutions-day2.jsonl",
"solution_match": "\nSolution 4. "
} | 225 | 1,141 |
2019 | T2 | 5 | null | EGMO | (Netherlands).
Let $n \geq 2$ be an integer, and let $a_{1}, a_{2}, \ldots, a_{n}$ be positive integers. Show that there exist positive integers $b_{1}, b_{2}, \ldots, b_{n}$ satisfying the following three conditions:
1). $a_{i} \leq b_{i}$ for $i=1,2, \ldots, n$;
2). the remainders of $b_{1}, b_{2}, \ldots, b_{n}$ o... | We can assume that all $a_{i} \in\{0,1, \ldots, n-1\}$, as we can deduct $n$ from both $a_{i}$ and $b_{i}$ for arbitrary $i$ without violating any of the three conditions from the problem statement. We shall also assume that $a_{1} \leq \ldots \leq a_{n}$.
Now let us provide an algorithm for constructing $b_{1}, \ldots... | {
"problem_match": "# Problem 5",
"resource_path": "EGMO/segmented/en-2019-solutions-day2.jsonl",
"solution_match": "\nSolution 5. "
} | 225 | 3,927 |
2019 | T2 | 6 | null | EGMO | (United Kingdom).
On a circle, Alina draws 2019 chords, the endpoints of which are all different. A point is considered marked if it is either
(i) one of the 4038 endpoints of a chord; or
(ii) an intersection point of at least two chords.
Alina labels each marked point. Of the 4038 points meeting criterion (i), Alina... | First we prove the following:
Lemma: if we color all of the points white or black, then the number of white-black edges, which we denote $E_{W B}$, is equal modulo 2 to the number of white (or black) points on the circumference, which we denote $C_{W}$, resp. $C_{B}$.
Observe that changing the colour of any interior p... | {
"problem_match": "# Problem 6",
"resource_path": "EGMO/segmented/en-2019-solutions-day2.jsonl",
"solution_match": "\nSolution 1. "
} | 247 | 521 |
2021 | T2 | 2 | null | EGMO | Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that the equation
$$
f(x f(x)+y)=f(y)+x^{2}
$$
holds for all rational numbers $x$ and $y$.
Here, $\mathbb{Q}$ denotes the set of rational numbers.
(Slovakia, Patrik Bak)
Answer: $f(x)=x$ and $f(x)=-x$. | Denote the equation from the statement by (1). Let $x f(x)=A$ and $x^{2}=B$. The equation (1) is of the form
$$
f(A+y)=f(y)+B
$$
Also, if we put $y \rightarrow-A+y$, we have $f(A-A+y)=f(-A+y)+B$. Therefore
$$
f(-A+y)=f(y)-B
$$
We can easily show that for any integer $n$ we even have
$$
f(n A+y)=f(y)+n B
$$
Indeed... | {
"problem_match": "\nProblem 2.",
"resource_path": "EGMO/segmented/en-2021-solutions.jsonl",
"solution_match": "\nSolution."
} | 95 | 760 |
2021 | T2 | 3 | null | EGMO | Let $A B C$ be a triangle with an obtuse angle at $A$. Let $E$ and $F$ be the intersections of the external bisector of angle $A$ with the altitudes of $A B C$ through $B$ and $C$ respectively. Let $M$ and $N$ be the points on the segments $E C$ and $F B$ respectively such that $\angle E M A=\angle B C A$ and $\angle A... | The first solution is based on the main Lemma. We present this Lemma with two different proofs.
Lemma: Let $A B C$ be an acute triangle with $A B=B C$. Let $P$ be any point on $A C$. Line passing through $P$ perpendicular to $A B$, intersects ray $B C$ in point $T$. If the line $A T$ intersects the circumscribed circle... | {
"problem_match": "\nProblem 3.",
"resource_path": "EGMO/segmented/en-2021-solutions.jsonl",
"solution_match": "# Solution 1."
} | 127 | 788 |
2021 | T2 | 5 | null | EGMO | A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane, such that
(i) no three points in $P$ lie on a line and
(ii) no two points in $P$ lie on a line through the origin.
A triangle with vertices in $P$ is fat, if $O$ is strictly inside the triangle. Find the maximum number of ... | We will count minimal number of triangles that are not fat. Let $F$ set of fat triangles, and S set of triangles that are not fat. If triangle $X Y Z \in S$, we call $X$ and $Z$ good vertices if $O Y$ is located between $O X$ and $O Z$. For $A \in P$ let $S_{A} \subseteq S$ be set of triangles in $S$ for which $A$ is o... | {
"problem_match": "\nProblems 5.",
"resource_path": "EGMO/segmented/en-2021-solutions.jsonl",
"solution_match": "# Solution"
} | 125 | 715 |
2021 | T2 | 6 | null | EGMO | Does there exist a nonnegative integer $a$ for which the equation
$$
\left\lfloor\frac{m}{1}\right\rfloor+\left\lfloor\frac{m}{2}\right\rfloor+\left\lfloor\frac{m}{3}\right\rfloor+\cdots+\left\lfloor\frac{m}{m}\right\rfloor=n^{2}+a
$$
has more than one million different solutions $(m, n)$ where $m$ and $n$ are positi... | Denote the equation from the statement by (1). The left hand side of (1) depends only on $m$, and will throughout be denoted by $L(m)$. Fix an integer $q>10^{7}$ and note that for $m=q^{3}$
$$
L\left(q^{3}\right)=\sum_{k=1}^{q^{3}}\left[\frac{q^{3}}{k}\right] \leq \sum_{k=1}^{q^{3}} \frac{q^{3}}{k} \leq q^{3} \cdot \s... | {
"problem_match": "\nProblems 6.",
"resource_path": "EGMO/segmented/en-2021-solutions.jsonl",
"solution_match": "# Solution."
} | 214 | 685 |
2022 | T2 | 1 | null | EGMO | Let $A B C$ be an acute-angled triangle in which $B C<A B$ and $B C<C A$. Let point $P$ lie on segment $A B$ and point $Q$ lie on segment $A C$ such that $P \neq B, Q \neq C$ and $B Q=B C=C P$. Let $T$ be the circumcentre of triangle $A P Q, H$ the orthocentre of triangle $A B C$, and $S$ the point of intersection of t... | We show that $T$ and $H$ are both on the angle bisector $\ell$ of $\angle B S C$.
We first prove that $H \in \ell$. The altitude $C H$ in triangle $A B C$ is also the altitude in isosceles triangle $P B C$ with $C P=C B$. Therefore, $C H$ is also the angle bisector of $\angle P C B$ and hence also of $\angle S C B$. An... | {
"problem_match": "# Problem 1.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "# Solution 1."
} | 144 | 739 |
2022 | T2 | 2 | null | EGMO | Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(a b)=f(a) f(b)$, and
(2) at least two of the numbers $f(a), f(b)$ and $f(a+b)$ are equal.
Proposed... | First, all such functions $f$ satisfy the conditions, as $v_{p}(a) \neq v_{p}(b)$ implies $v_{p}(a+b)=$ $\min \left(v_{p}(a), v_{p}(b)\right)$. Plugging $a, b=1$ into (1) gives $f(1)=1$. Also, a simple induction gives that $f\left(\prod_{i=1}^{k} p_{i}^{a_{i}}\right)=\prod_{i=1}^{k} f\left(p_{i}\right)^{a_{i}}$. Let $S... | {
"problem_match": "# Problem 2.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "\nSolution 1. "
} | 172 | 556 |
2022 | T2 | 2 | null | EGMO | Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(a b)=f(a) f(b)$, and
(2) at least two of the numbers $f(a), f(b)$ and $f(a+b)$ are equal.
Proposed... | Suppose there exists a positive integer $n$ such that $f(n) \neq 1$ and $f(n+1) \neq 1$. We know that $f(1)=1$, so by $(2)$ two of $f(n) \neq 1, f(n+1) \neq 1$ and $f(1)=1$ are equal, therefore $f(n)=f(n+1)$.
$f\left(n^{2}\right)=f(n)^{2} \neq 1$ and $f\left(n^{2}-1\right)=f(n+1) f(n-1)=f(n) f(n-1) \neq 1$, but by (2) ... | {
"problem_match": "# Problem 2.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "# Solution 2 (for the main part)."
} | 172 | 527 |
2022 | T2 | 2 | null | EGMO | Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(a b)=f(a) f(b)$, and
(2) at least two of the numbers $f(a), f(b)$ and $f(a+b)$ are equal.
Proposed... | (for the second part).
Claim 3.1. If $f(m)>1$ for some $m \in \mathbb{N}$, then there are less than $m$ different prime numbers $p_{i}$ with $f\left(p_{i}\right)>1$.
Proof. On the one hand, if $f(m)=c>1$, then for any $n \in \mathbb{N}$, there is some $k \in\{n+1, n+2 \ldots, n+m\}$ such that $f(k) \in\{1, c\}$. To sh... | {
"problem_match": "# Problem 2.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "# Solution 3 (Joseph Myers) "
} | 172 | 677 |
2022 | T2 | 2 | null | EGMO | Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(a b)=f(a) f(b)$, and
(2) at least two of the numbers $f(a), f(b)$ and $f(a+b)$ are equal.
Proposed... | 4B (for the main part). Let $p$ the minimal (prime) number with $f(p)=c>1$.
Lemma 4B.1. For any $k \in \mathbb{N}, f\left(1+p+p^{2}+\ldots+p^{k}\right)=1$.
Proof of the Lemma, by induction. $k=0$ is trivial.
$k=1$ is also easy, as $p+1$ cannot be a prime, except in the case of $p=2$, which is also easy. (If $f(2)=c>1$,... | {
"problem_match": "# Problem 2.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "\nSolution "
} | 172 | 575 |
2022 | T2 | 3 | null | EGMO | An infinite sequence of positive integers $a_{1}, a_{2}, \ldots$ is called good if
(1) $a_{1}$ is a perfect square, and
(2) for any integer $n \geq 2, a_{n}$ is the smallest positive integer such that
$$
n a_{1}+(n-1) a_{2}+\ldots+2 a_{n-1}+a_{n}
$$
is a perfect square.
Prove that for any good sequence $a_{1}, a_{2},... | Define the following auxiliary sequences:
$$
\begin{array}{ll}
b_{1}=a_{1}, & b_{n}=a_{1}+a_{2}+\cdots+a_{n} \\
c_{1}=b_{1}, & c_{n}=b_{1}+b_{2}+\cdots+b_{n}
\end{array}
$$
Observe that
$$
\begin{gathered}
c_{n}-c_{n-1}=b_{n} \\
b_{n}-b_{n-1}=\left(c_{n}-c_{n-1}\right)-\left(c_{n-1}-c_{n-2}\right)=a_{n}
\end{gathere... | {
"problem_match": "# Problem 3.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "\nSolution."
} | 167 | 737 |
2022 | T2 | 3 | null | EGMO | An infinite sequence of positive integers $a_{1}, a_{2}, \ldots$ is called good if
(1) $a_{1}$ is a perfect square, and
(2) for any integer $n \geq 2, a_{n}$ is the smallest positive integer such that
$$
n a_{1}+(n-1) a_{2}+\ldots+2 a_{n-1}+a_{n}
$$
is a perfect square.
Prove that for any good sequence $a_{1}, a_{2},... | We write:
$$
s_{n}^{2}=S_{n}=a_{1}+\left(a_{1}+a_{2}\right)+\ldots+\left(a_{1}+\ldots+a_{n}\right)
$$
So, setting $b_{n}:=a_{1}+\ldots+a_{n}$, we have $S_{n}=b_{1}+b_{2}+\ldots+b_{n}$ and, in particular $S_{n+1}=S_{n}+b_{n+1}$. Now, we study the quantity $S_{n} b_{n}=+b_{1}+b_{2}+\ldots+b_{n}+b_{n}$ in two different ... | {
"problem_match": "# Problem 3.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "\nSolution."
} | 167 | 933 |
2022 | T2 | 4 | null | EGMO | Given a positive integer $n \geq 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_{0}, a_{1}, \ldots, a_{N}$ such that
(1) $a_{0}+a_{1}=-\frac{1}{n}$, and
(2) $\left(a_{k}+a_{k-1}\right)\left(a_{k}+a_{k+1}\right)=a_{k-1}-a_{k+1}$ for $1 \leq k \leq N-1$.
Proposed by: Romania | $\left(a_{k}+a_{k-1}\right)\left(a_{k}+a_{k+1}\right)=a_{k-1}-a_{k+1}$ is equivalent to $\left(a_{k}+a_{k-1}+1\right)\left(a_{k}+a_{k+1}-1\right)=-1$. Let $b_{k}=a_{k}+a_{k+1}$. Thus we need $b_{0}, b_{1}, \ldots$ the following way: $b_{0}=-\frac{1}{n}$ and $\left(b_{k-1}+1\right)\left(b_{k}-1\right)=-1$. There is a pr... | {
"problem_match": "# Problem 4.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "\nSolution 1. "
} | 139 | 574 |
2022 | T2 | 4 | null | EGMO | Given a positive integer $n \geq 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_{0}, a_{1}, \ldots, a_{N}$ such that
(1) $a_{0}+a_{1}=-\frac{1}{n}$, and
(2) $\left(a_{k}+a_{k-1}\right)\left(a_{k}+a_{k+1}\right)=a_{k-1}-a_{k+1}$ for $1 \leq k \leq N-1$.
Proposed by: Romania | The required maximum is $N=n$.
To rule out the case $N \geq n+1$, it is clearly sufficient to rule out the case $N=n+1$.
Assume for contradiction that $a_{0}, a_{1}, \ldots, a_{n+1}$ are real numbers satisfying both conditions in the statement. It is sufficient to show that $a_{k}+a_{k+1}=0$ for some $k \leq n$, becaus... | {
"problem_match": "# Problem 4.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "\nSolution 2. "
} | 139 | 543 |
2022 | T2 | 5 | null | EGMO | For all positive integers $n, k$, let $f(n, 2 k)$ be the number of ways an $n \times 2 k$ board can be fully covered by $n k$ dominoes of size $2 \times 1$. (For example, $f(2,2)=2$ and $f(3,2)=3$.)
Find all positive integers $n$ such that for every positive integer $k$, the number $f(n, 2 k)$ is odd.
Proposed by: U.S.... | Color the board as a chessboard. Consider the bipartite graph whose vertices are the squares and the neighbors are connected by an edge. Notice that a domino tiling described in the problem corresponds to a perfect matching in this bipartite graph, so we are interested in the number of perfect matchings.
And that is t... | {
"problem_match": "# Problem 5.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "# Solution 2."
} | 115 | 529 |
2022 | T2 | 6 | null | EGMO | Let $A B C D$ be a cyclic quadrilateral with circumcentre $O$. Let the internal angle bisectors at $A$ and $B$ meet at $X$, the internal angle bisectors at $B$ and $C$ meet at $Y$, the internal angle bisectors at $C$ and $D$ meet at $Z$, and the internal angle bisectors at $D$ and $A$ meet at $W$. Further, let $A C$ an... | Let $\Omega$ be the circumcircle of the quadrilateral $A B C D$ and let $r$ be its radius.
First, notice that the points $X, Y, Z, W$ are concyclic. Indeed, using oriented (modulo $180^{\circ}$ ) angles,
$$
\angle(X W, X Y)+\angle(Z Y, Z W)=\angle(X A, X B)+\angle(Z C, Z D)=-\frac{\angle A+\angle B}{2}-\frac{\angle C+... | {
"problem_match": "# Problem 6.",
"resource_path": "EGMO/segmented/en-2022-solutions.jsonl",
"solution_match": "\nSolution 1. "
} | 183 | 1,006 |
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