Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Danube Mathematical Competition
Consider the real numbers $a_1,a_2,...,a_{2n}$ whose sum is equal to $0$.

Prove that among pairs $(a_i,a_j) , i<j$ where $ i,j \in \{1,2,...,2n\} $

there are at least $2n-1$ pairs with the property that  $a_i+a_j\ge 0$.
2014 Danube Mathematical Competition
Two circles $\gamma_1$ and $\gamma_2$ cross one another at two points; let $A$ be one of these points. The tangent to $\gamma_1$ at $A$ meets again $\gamma_2$ at $B$, the tangent to $\gamma_2$ at $A$ meets again $\gamma_1$ at $C$, and the line $BC$ meets again $\gamma_1$ and $\gamma_2$ at $D_1$ and $D_2$, respectively. Let $E_1$ and $E_2$ be interior points of the segments $AD_1$ and $AD_2$, respectively, such that $AE_1 = AE_2$. The lines $BE_1$ and $AC$ meet at $M$, the lines $CE_2$ and $AB$ meet at $N$, and the lines $MN$ and $BC$ meet at $P$. Show that the line $PA$ is tangent to the circle $ABC$.
2014 Danube Mathematical Competition
Let $S$ be a set of positive integers such that  $\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor $  for all $x, y \in S$.

Show that the products $xy$, where $x, y \in S$, are pairwise distinct.
2014 Danube Mathematical Competition
Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.
2014 Danube Mathematical Competition
Let $n$ be a positive integer and let $\triangle$ be the closed triangular domain with vertices at the lattice points $(0, 0), (n, 0)$ and $(0, n)$. Determine the maximal cardinality a set $S$ of lattice points in $\triangle$ may have, if the line through every pair of distinct points in $S$ is parallel to no side of $\triangle$.
2013 Danube Mathematical Competition
Determine the natural numbers $n\ge 2$ for which  exist $x_1,x_2,...,x_n \in R^*$,

such that $x_1+x_2+...+x_n=\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}=0$
2013 Danube Mathematical Competition
Consider $64$ distinct natural numbers, at most equal to $2012$.

Show that it is possible to choose four of them, denoted as $a,b,c,d$  such that $ a+b-c-d$ to be a multiple of $2013$
2013 Danube Mathematical Competition
Determine the natural numbers $m,n$ such as $85^m-n^4=4$
2013 Danube Mathematical Competition
Let $ABCD$ be a rectangle   with $AB \ne BC$ and  the center the point $O$. Perpendicular from $O$ on $BD$ intersects lines $AB$ and $BC$ in points $E$ and $F$ respectively. Points $M$ and $N$ are midpoints of segments $[CD]$ and  $[AD]$ respectively. Prove that $FM \perp EN$ .
2013 Danube Mathematical Competition
Given six points on a circle, $A, a, B, b, C, c$, show that the Pascal lines of the hexagrams $AaBbCc, AbBcCa, AcBaCb$ are concurrent.
2013 Danube Mathematical Competition
Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$.  for instance, $a \equiv b \equiv  -1 (mod 3), c  \equiv  1 (mod 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.
2013 Danube Mathematical Competition
Show that, for every integer $r \ge  2$, there exists an $r$-chromatic simple graph (no loops, nor multiple edges) which has no cycle of less than $6$ edges
2013 Danube Mathematical Competition
Show that there exists a proper non-empty subset $S$ of the set of real numbers such that, for every real number $x$, the set $\{nx + S : n \in  N\}$ is finite, where $nx + S =\{nx + s : s \in  S\}$
2012 Danube Mathematical Competition
a) Exist $a, b, c, \in N$, such that the numbers $ab+1,bc+1$  and $ca+1$ are simultaneously even perfect squares ?

b) Show that there is an infinity of natural numbers (distinct two by two) $a, b, c$ and $d$, so that the numbers $ab+1,bc+1, cd+1$ and $da+1$ are simultaneously perfect squares.
2012 Danube Mathematical Competition
Consider the natural number prime $p, p> 5$. From the decimal number $\frac1p$,  randomly remove $2012$ numbers, after the comma. Show that the remaining number can be represented as  $\frac{a}{b}$  , where $a$ and $b$ are coprime numbers , and $b$ is multiple of $p$.
2012 Danube Mathematical Competition
Let $ABC$ be a triangle with $\angle BAC = 90^o$. Angle bisector of the $\angle CBA$  intersects the segment $(AB)$ at point $E$. If there exists $D \in (CE)$ so  that $\angle DAC = \angle BDE =x^o$ , calculate $x$.
2012 Danube Mathematical Competition
Let $A$ be a subset with seven elements of the set $\{1,2,3, ...,26\}$.

Show that there are two distinct elements of $A$, having the same sum of their elements.
2012 Danube Mathematical Competition
Given a positive integer $n$, determine the maximum number of lattice points in the plane a square of side length $n +\frac{1}{2n+1}$ may cover.
2012 Danube Mathematical Competition
Let ABC be an acute triangle and let $A_1$, $B_1$, $C_1$ be points on the sides $BC, CA$ and $AB$, respectively. Show that the triangles $ABC$ and $A_1B_1C_1$ are similar ($\angle A = \angle A_1, \angle B = \angle B_1,\angle C = \angle C_1$) if and only if the orthocentre of the triangle $A_1B_1C_1$ and the circumcentre of the triangle $ABC$ coincide.
2012 Danube Mathematical Competition
Let $p$ and $q, p < q,$ be two primes such that $1 + p + p^2+...+p^m$ is a power of $q$ for some positive integer $m$, and $1 + q + q^2+...+q^n$ is a power of $p$ for some positive integer $n$. Show that $p = 2$ and $q = 2^t-1$ where $t$ is prime.
2012 Danube Mathematical Competition
Given a positive integer $n$, show that the set $\{1,2,...,n\}$ can be partitioned into $m$ sets, each with the same sum, if and only if m is a divisor of $\frac{n(n + 1)}{2}$ which does not exceed $\frac{n + 1}{2}$.
2011 Danube Mathematical Competition
Let $ABCM$ be a quadrilateral and $D$ be an interior point such that $ABCD$ is a parallelogram. It is known that $\angle  AMB =\angle CMD$. Prove that $\angle  MAD =\angle  MCD$.
2011 Danube Mathematical Competition
Let S be a set of positive integers such that: min { lcm (x, y) : x, y ∈ S, $x \neq y$ } $\ge$ 2 + max S.

Prove that $\displaystyle\sum\limits_{x \in S} \frac{1}{x} \le \frac{3}{2} $.
2011 Danube Mathematical Competition
Determine all positive integer numbers $n$ satisfying the following condition:

the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.
2011 Danube Mathematical Competition
Given a positive integer number $n$, determine the maximum number of edges a triangle-free Hamiltonian simple graph on $n$ vertices may have.
2010 Danube Mathematical Olympiad
Determine all integer numbers $n\ge 3$ such that the regular $n$-gon can be decomposed into isosceles triangles by non-intersecting diagonals.
2010 Danube Mathematical Olympiad
Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.
2010 Danube Mathematical Olympiad
All sides and diagonals of a convex $n$-gon, $n\ge 3$, are coloured one of two colours. Show that there exist $\left[\frac{n+1}{3}\right]$ pairwise disjoint monochromatic segments.



(Two segments are disjoint if they do not share an endpoint or an interior point).
2010 Danube Mathematical Olympiad
Let $p$ be a prime number of the form $4k+3$. Prove that there are no integers $w,x,y,z$ whose product is not divisible by $p$, such that:



$$

w^{2p}+x^{2p}+y^{2p}=z^{2p}.

$$
2010 Danube Mathematical Olympiad
Let $n\ge3$ be a positive integer. Find the real numbers $x_1\ge0,\ldots,x_n\ge 0$, with $x_1+x_2+\ldots +x_n=n$, for which the expression $$(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n$$ takes a minimal value.
2009 Danube Mathematical Competition
Let be $\triangle ABC$ .Let $A'$, $B'$, $C'$ be the foot of perpendiculars from $A$, $B$ and $C$ respectively.

The points $E$ and $F$ are on the sides $CB'$ and $BC'$ respectively, such that $B'E\cdot C'F = BF\cdot CE$.

Show that $AEA'F$ is cyclic.
2009 Danube Mathematical Competition
Prove that all the positive integer numbers , except for the powers of $2$, can be written as the sum of (at least two) consecutive natural numbers .
2009 Danube Mathematical Competition
Let $n$ be a natural number.

Determine the number minimal of equilateral triangles of side $1$ to cover the surface of an equilateral triangle of $n +\frac{1}{2n}$.
2009 Danube Mathematical Competition
Let be $ a,b,c $ positive integers.Prove that $ |a-b\sqrt{c}|<\frac{1}{2b} $ is true if and only if $ |a^{2}-b^{2}c|<\sqrt{c} $.
2009 Danube Mathematical Competition
Let $\sigma, \tau$ be  two permutations of the quantity $\{1, 2,. . . , n\}$.

Prove that there is a function $f: \{1, 2,. . . , n\} \to \{-1, 1\}$ such that  for any $1 \le  i \le j \le  n$,

we have $\left|\sum_{k=i}^{j} f(\sigma (k)) \right| \le 2$ and  $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$
2008 Danube Mathematical Competition
$x,y,z,t \in \mathbb R_+^*$:



$$   (xy)^{1/2}+(yz)^{1/2}+(zt)^{1/2}+(tx)^{1/2}+(xz)^{1/2}+(yt)^{1/2} \ge 3(xyz+xyt+xzt+yzt)^{\frac{1}{3}} $$
2008 Danube Mathematical Competition
In a triangle $ABC$  let  $A_1$ be the midpoint of side $BC$. Draw circles with centers $A, A1$ and radii $AA_1, BC$   respectively  and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$. Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.
2008 Danube Mathematical Competition
On a semicircle centred at $O$ and with radius $1$ choose the respective points $A_1,A_2,...,A_{2n}$ , for $n \in N^*$.

The lenght of the projection of the vector $\overrightarrow {u}=\overrightarrow{OA_1} +\overrightarrow{OA_2}+...+\overrightarrow{OA_{2n}}$  on the diameter is an odd integer.

Show that the projection of that vector on the diameter  is at least $1$.
2008 Danube Mathematical Competition
Let $ n\geq 2$ be a positive integer. Find the maximum number of segments with lenghts greater than $ 1,$ determined by $ n$ points which lie on a closed disc with radius $ 1.$
2007 Danube Mathematical Competition
Let $ n\ge2$ be a positive integer and denote by $ S_n$ the set of all permutations of the set $ \{1,2,\ldots,n\}$. For $ \sigma\in S_n$ define $ l(\sigma)$ to be $ \displaystyle\min_{1\le i\le n-1}\left|\sigma(i+1)-\sigma(i)\right|$. Determine $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$.
2007 Danube Mathematical Competition
Let $ ABCD$ be an inscribed quadrilateral and let $ E$ be the midpoint of the diagonal $ BD$. Let $ \Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be the circumcircles of triangles $ AEB$, $ BEC$, $ CED$ and $ DEA$ respectively. Prove that if $ \Gamma_4$ is tangent to the line $ CD$, then $ \Gamma_1,\Gamma_2,\Gamma_3$ are tangent to the lines $ BC,AB,AD$ respectively.
2007 Danube Mathematical Competition
For each positive integer $ n$, define $ f(n)$ as the exponent of the $ 2$ in the decomposition in prime factors of the number $ n!$. Prove that the equation $ n-f(n)=a$ has infinitely many solutions for any positive integer $ a$.
2007 Danube Mathematical Competition
Let $ a,n$ be positive integers such that $ a\ge(n-1)!$. Prove that there exist $ n$ distinct prime numbers $ p_1,\ldots,p_n$ so that $ p_i|a+i$, for all $ i=\overline{1,\ldots,n}$.
2005 Danube Mathematical Olympiad
Prove that the equation $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x$ and $y$ with $x\leq 2005$.
2005 Danube Mathematical Olympiad
Prove that the sum:



$$ S_n=\binom{n}{1}+\binom{n}{3}\cdot 2005+\binom{n}{5}\cdot 2005^2+...=\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\binom{n}{2k+1}\cdot 2005^k $$



is divisible by $2^{n-1}$ for any positive integer $n$.
2005 Danube Mathematical Olympiad
Let $\mathcal{C}$ be a circle with center $O$, and let $A$ be a point outside the circle. Let the two tangents from the point $A$ to the circle $\mathcal{C}$ meet this circle at the points $S$ and $T$, respectively. Given a point $M$ on the circle $\mathcal{C}$ which is different from the points $S$ and $T$, let the line $MA$ meet the perpendicular from the point $S$ to the line $MO$ at $P$.



Prove that the reflection of the point $S$ in the point $P$ lies on the line $MT$.
2005 Danube Mathematical Olympiad
Let $k$ and $n$ be positive integers. Consider an array of $2\left(2^n-1\right)$ rows by $k$ columns. A $2$-coloring of the elements of the array is said to be acceptable if any two columns agree on less than $2^n-1$ entries on the same row.



Given $n$, determine the maximum value of $k$ for an acceptable $2$-coloring to exist.
2021 EGMO
The number 2021 is fantabulous. For any positive integer $m$, if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?
2021 EGMO
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that the equation

$$f(xf(x)+y) = f(y) + x^2$$holds for all rational numbers $x$ and $y$.



Here, $\mathbb{Q}$ denotes the set of rational numbers.
2021 EGMO
Let $ABC$ 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 $ABC$ through $B$ and $C$ respectively. Let $M$ and $N$ be the points on the segments $EC$ and $FB$ respectively such that $\angle EMA = \angle BCA$ and $\angle ANF = \angle ABC$. Prove that the points $E, F, N, M$ lie on a circle.
2021 EGMO
Let $ABC$ be a triangle with incenter $I$ and let $D$ be an arbitrary point on the side $BC$. Let the line through $D$ perpendicular to $BI$ intersect $CI$ at $E$. Let the line through $D$ perpendicular to $CI$ intersect $BI$ at $F$. Prove that the reflection of $A$ across the line $EF$ lies on the line $BC$.
2021 EGMO
A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane such that



no three points in $P$ lie on a line and

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 fat triangles.
2021 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 positive integers?



The expression $\lfloor x\rfloor$ denotes the integer part (or floor) of the real number $x$. Thus $\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42,$ and $\lfloor 0 \rfloor = 0$.
2020 EGMO
The positive integers $a_0, a_1, a_2, \ldots, a_{3030}$ satisfy $$2a_{n + 2} = a_{n + 1} + 4a_n \text{ for } n = 0, 1, 2, \ldots, 3028.$$

Prove that at least one of the numbers $a_0, a_1, a_2, \ldots, a_{3030}$ is divisible by $2^{2020}$.
2020 EGMO
Find all lists $(x_1, x_2, \ldots, x_{2020})$ of non-negative real numbers such that the following three conditions are all satisfied:



$x_1 \le x_2 \le \ldots \le x_{2020}$;

$x_{2020} \le x_1  + 1$;

there is a permutation $(y_1, y_2, \ldots, y_{2020})$ of $(x_1, x_2, \ldots, x_{2020})$ such that $$\sum_{i = 1}^{2020} ((x_i + 1)(y_i + 1))^2 = 8 \sum_{i = 1}^{2020} x_i^3.$$





A permutation of a list is a list of the same length, with the same entries, but the entries are allowed to be in any order.  For example, $(2, 1, 2)$ is a permutation of $(1, 2, 2)$, and they are both permutations of $(2, 2, 1)$.  Note that any list is a permutation of itself.
2020 EGMO
Let $ABCDEF$ be a convex hexagon such that $\angle A = \angle C = \angle E$ and $\angle B = \angle D = \angle F$ and the (interior) angle bisectors of $\angle A, ~\angle C,$ and $\angle E$ are concurrent.



Prove that the (interior) angle bisectors of $\angle B, ~\angle D, $ and $\angle F$ must also be concurrent.



Note that $\angle A = \angle FAB$.  The other interior angles of the hexagon are similarly described.
2020 EGMO
A permutation of the integers $1, 2, \ldots, m$ is called fresh if there exists no positive integer $k < m$ such that the first $k$ numbers in the permutation are $1, 2, \ldots, k$ in some order.  Let $f_m$ be the number of fresh permutations of the integers $1, 2, \ldots, m$.



Prove that $f_n \ge n \cdot f_{n - 1}$ for all $n \ge 3$.



For example, if $m = 4$, then the permutation $(3, 1, 4, 2)$ is fresh, whereas the permutation $(2, 3, 1, 4)$ is not.
2020 EGMO
Consider the triangle $ABC$ with $\angle BCA > 90^{\circ}$.  The circumcircle $\Gamma$ of $ABC$ has radius $R$.  There is a point $P$ in the interior of the line segment $AB$ such that $PB = PC$ and the length of $PA$ is $R$.  The perpendicular bisector of $PB$ intersects $\Gamma$ at the points $D$ and $E$.



Prove $P$ is the incentre of triangle $CDE$.
2020 EGMO
Let $m > 1$ be an integer.  A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$, $a_3 = 4$, and for all $n \ge 4$, $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$

Determine all integers $m$ such that every term of the sequence is a square.
2019 EGMO
Find all triples $(a, b, c)$ of real numbers such that $ab + bc + ca = 1$ and



$$a^2b + c = b^2c + a = c^2a + b.$$
2019 EGMO
Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ 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 $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
2019 EGMO
Let $ABC$ be a triangle such that $\angle CAB > \angle ABC$, and let $I$ be its incentre. Let $D$ be the point on segment $BC$ such that $\angle CAD = \angle ABC$. Let $\omega$ be the circle tangent to $AC$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the circumcircle of $ABC$. Prove that the angle bisectors of $\angle DAB$ and $\angle CXB$ intersect at a point on line $BC$.
2019 EGMO
Let $ABC$ be a triangle with incentre $I$. The circle through $B$ tangent to $AI$ at $I$ meets side $AB$ again at $P$. The circle through $C$ tangent to $AI$ at $I$ meets side $AC$ again at $Q$. Prove that $PQ$ is tangent to the incircle of $ABC.$
2019 EGMO
Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions:



$\text{(A)} \ a_i\le b_i$ for $i=1, 2, \cdots , n;$



$\text{(B)} \ $ the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and



$\text{(C)} \ $ $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$



(Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)
2019 EGMO
On a circle, Alina draws $2019$ chords, the endpoints of which are all different. A point is considered marked if it is either



$\text{(i)}$ one of the $4038$ endpoints of a chord; or



$\text{(ii)}$ an intersection point of at least two chords.



Alina labels each marked point. Of the $4038$ points meeting criterion $\text{(i)}$, Alina labels $2019$ points with a $0$ and the other $2019$ points with a $1$. She labels each point meeting criterion $\text{(ii)}$ with an arbitrary integer (not necessarily positive).

Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with $k$ marked points has $k-1$ such segments.) She labels each such segment in yellow with the sum of the labels of its two endpoints and in blue with the absolute value of their difference.

Alina finds that the $N + 1$ yellow labels take each value $0, 1, . . . , N$ exactly once. Show that at least one blue label is a multiple of $3$.

(A chord is a line segment joining two different points on a circle.)
2018 EGMO
Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$, and let $M$ be the midpoint of $AB$. Let $P$ be a variable point of the circumcircle of $ABC$, and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$. It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$.

Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$.
2018 EGMO
Consider the set

$$A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.$$



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.



For every integer $x \geq 2$ let $f(x)$ denote the minimum integer such that $x$ can be written as the

product of $f(x)$ elements of $A$, which are not necessarily different.

Prove that there exist infinitely many pairs $(x,y)$ of integers with $x\geq 2$, $y \geq 2$, and $$f(xy)<f(x)+f(y).$$(Pairs $(x_1,y_1)$ and $(x_2,y_2)$ are different if $x_1 \neq x_2$ or $y_1 \neq y_2$).
2018 EGMO
The $n$ contestant of EGMO are named $C_1, C_2, \cdots C_n$. 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 $C_i$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions.

If contestant $C_i$ has fewer than $i$ other contestants in front of her, the restaurant opens and process ends.









Prove that the process cannot continue indefinitely, regardless of the Jury’s choices.

Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.
2018 EGMO
A domino is a $ 1 \times 2 $ or $ 2 \times 1 $ tile.

Let $n \ge 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 or column. The configuration is called balanced if there exists some $k \ge 1 $ such that each row and each column has a value of $k$. Prove that a balanced configuration exists for every $n \ge 3 $, and find the minimum number of dominoes needed in such a configuration.
2018 EGMO
Let $\Gamma $ be the circumcircle of triangle $ABC$. A circle $\Omega$ is tangent to the line segment $AB$ and is tangent to $\Gamma$ at a point lying on the same side of the line $AB$ as $C$. The angle bisector of $\angle BCA$ intersects $\Omega$ at two different points $P$ and $Q$.

Prove that $\angle ABP = \angle QBC$.
2018 EGMO




Prove that for every real number $t$ such that $0 < t < \tfrac{1}{2}$ 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 \ge 0 $), such that $$ |x-my|\leq ty.$$

Determine whether for every real number $t$ such that $0 < t < \tfrac{1}{2} $ there exists an infinite set $S$ of positive integers such that $$|x-my| > ty$$for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m > 0$).
2017 EGMO
Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.