Navvye/Olympiad-Problems-CSV huggingface dataset:
2018 European Mathematical Cup
Let $a, b, c$ be non-zero real numbers such that $a^2+b+c=\frac{1}{a}, b^2+c+a=\frac{1}{b}, c^2+a+b=\frac{1}{c}.$ Prove that at least two of $a, b, c$ are equal.
2018 European Mathematical Cup
Find all pairs $ (x; y) $ of positive integers such that

$$xy | x^2 + 2y -1.$$
2018 European Mathematical Cup
Let $ABC$ be an acute triangle with $ |AB | <  |AC |$and orthocenter $H$. The circle with center A and radius$ |AC |$ intersects the circumcircle of $\triangle ABC$ at point $D$ and the circle with center $A$ and radius$ |AB |$ intersects the segment $\overline{AD}$ at point $K. $ The line through $K$ parallel to $CD $ intersects $BC$ at the point $ L.$  If $M$ is the midpoint of $\overline{BC}$  and N is the foot of the perpendicular from $H$ to $AL, $ prove that the line $ MN $ bisects the segment $\overline{AH}$.
2018 European Mathematical Cup
Let $n$ be a positive integer. Ana and Banana are playing the following game:

First, Ana arranges $2n$ cups in a row on a table, each facing upside-down. She then places a ball under a cup

and makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana,

where each command consists of swapping two adjacent cups in the row.

Her goal is to achieve that the ball has fallen into the hole during the game. Assuming Banana has no information

about the position of the hole and the position of the ball at any point, what is the smallest number of commands

she has to give in order to achieve her goal?
2018 European Mathematical Cup
A partition of a positive integer is even if all its elements are even numbers. Similarly, a partition

is odd if all its elements are odd. Determine all positive integers $n$ such that the number of even partitions of

$n$ is equal to the number of odd partitions of $n$.

Remark: A partition of a positive integer $n$ is a non-decreasing sequence of positive integers whose sum of

elements equals $n$. For example, $(2; 3; 4), (1; 2; 2; 2; 2)$ and $(9) $ are partitions of $9.$
2018 European Mathematical Cup
Let ABC be a triangle with$|AB|< |AC|. $ Let $k$ be the circumcircle of $\triangle ABC$ and let $O$ be the center of $k$. Point $M$ is the midpoint of the arc $BC $ of $k$ not containing $A$. Let $D $ be the second intersection of the perpendicular line from $M$ to $AB$ with $ k$ and $E$ be the second intersection of the perpendicular line from $M$ to $AC $ with $k$. Points $X $and $Y $ are the intersections of $CD$ and $BE$ with $OM$ respectively. Denote by $k_b$ and $k_c$  circumcircles of triangles $BDX$ and $CEY$ respectively. Let $G$ and $H$ be the second intersections of $k_b$ and $k_c $ with $AB$ and $AC$ respectively. Denote by ka the circumcircle of triangle $AGH.$

Prove that $O$ is the circumcenter of $\triangle O_aO_bO_c, $where $O_a, O_b, O_c $ are the centers of $k_a, k_b, k_c$  respectively.
2018 European Mathematical Cup
For which real numbers $k > 1$  does there exist a bounded set of positive real numbers $S$  with at

least $3$ elements such that

$$k(a - b)\in S$$for all $a,b\in  S $ with $a > b?$

Remark: A set of positive real numbers $S$  is bounded if there exists a positive real number $M$ such that

$x < M$  for all $x \in  S.$
2018 European Mathematical Cup
Let $x; y; m; n$ be integers greater than $1$ such that
2017 European Mathematical Cup
Solve in integers the equation :

$x^2y+y^2=x^3$
2017 European Mathematical Cup
A regular hexagon in the plane is called sweet if its area is equal to $1$. Is it possible to place $2000000$ sweet hexagons in the plane such that the union of their interiors is a convex polygon of area at least $1900000$?



Remark: A subset $S$ of the plane is called convex if for every pair of points in $S$, every point on the straight line segment that joins the pair of points also belongs to $S$. The hexagons may overlap.
2017 European Mathematical Cup
Let $ABC$ be an acute triangle. Denote by $H$ and $M$ the orthocenter of $ABC$ and the midpoint

of side $BC,$ respectively. Let $Y$ be a point on $AC$ such that $YH$ is perpendicular to $MH$ and let $Q$ be a point

on $BH$ such that $QA$ is perpendicular to $AM.$ Let $J$ be the second point of intersection of $MQ$ and the circle

with diameter $MY.$ Prove that $HJ$ is perpendicular to $AM.$





(Steve Dinh)
2017 European Mathematical Cup
The real numbers $x,y,z$ satisfy $x^2+y^2+z^2=3.$ Prove that the inequality

$x^3-(y^2+yz+z^2)x+yz(y+z)\le 3\sqrt{3}.$

and find all triples $(x,y,z)$ for which equality holds.
2017 European Mathematical Cup
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$holds for all positive integers $x, y$.



Proposed by Adrian Beker.
2017 European Mathematical Cup
A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times.



a) Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum number of players required?

b) For the number of players found in a), what is the minimum number of substitutions required, so that each player plays the same number of minutes?



Remark: Substitutions can only take place after a positive integer number of minutes, and players who have come off earlier can return to the game as many times as needed. There is no limit to the number of substitutions allowed.



Proposed by Athanasios Kontogeorgis and Demetres Christofides.
2017 European Mathematical Cup
Let $ABC$ be a scalene triangle and let its incircle touch sides $BC$, $CA$ and $AB$ at points $D$, $E$ and

$F$ respectively. Let line $AD$ intersect this incircle at point $X$. Point $M$ is chosen on the line $FX$ so that the

quadrilateral $AFEM$ is cyclic. Let lines $AM$ and $DE$ intersect at point $L$ and let $Q$ be the midpoint of segment

$AE$. Point $T$ is given on the line $LQ$ such that the quadrilateral $ALDT$ is cyclic. Let $S$ be a point such that

the quadrilateral $TFSA$ is a parallelogram, and let $N$ be the second point of intersection of the circumcircle of

triangle $ASX$ and the line $TS$. Prove that the circumcircles of triangles $TAN$ and $LSA$ are tangent to each

other.
2017 European Mathematical Cup
Find all polynomials $P$ with integer coefficients such that $P (0)\ne  0$ and $$P^n(m)\cdot P^m(n)$$is a square of an integer for all nonnegative integers $n, m$.



Remark: For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$.



Proposed by Adrian Beker.
2016 European Mathematical Cup
Is there a sequence $a_{1}, . . . , a_{2016}$ of positive integers, such that every sum

$$a_{r} + a_{r+1} + . . . + a_{s-1} + a_{s}$$(with $1 \le r \le s \le 2016$) is a composite number, but:

a) $GCD(a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$;

b) $GCD(a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$ and $GCD(a_{i}, a_{i+2}) = 1$ for all $i = 1, 2, . . . , 2014$?

$GCD(x, y)$ denotes the greatest common divisor of $x$, $y$.



Proposed by Matija Bucić
2016 European Mathematical Cup
For two positive integers $a$ and $b$, Ivica and Marica play the following game: Given two piles of $a$

and $b$ cookies, on each turn a player takes $2n$ cookies from one of the piles, of which he eats $n$ and puts $n$ of

them on the other pile. Number $n$ is arbitrary in every move. Players take turns alternatively, with Ivica going

first. The player who cannot make a move, loses. Assuming both players play perfectly, determine all pairs of

numbers $(a, b)$ for which Marica has a winning strategy.



Proposed by Petar Orlić
2016 European Mathematical Cup
Determine all functions $f:\mathbb R\to\mathbb R$ such that equality

$$f(x + y + yf(x)) = f(x) + f(y) + xf(y)$$holds for all real numbers $x$, $y$.



Proposed by Athanasios Kontogeorgis
2016 European Mathematical Cup
Let $C_{1}$, $C_{2}$ be circles intersecting in $X$, $Y$ . Let $A$, $D$ be points on $C_{1}$ and $B$, $C$ on $C_2$ such that $A$, $X$, $C$ are collinear and $D$, $X$, $B$ are collinear. The tangent to circle $C_{1}$ at $D$ intersects $BC$ and the tangent to $C_{2}$ at $B$ in $P$, $R$ respectively. The tangent to $C_2$ at $C$ intersects $AD$ and tangent to $C_1$ at $A$, in $Q$, $S$ respectively. Let $W$ be the intersection of $AD$ with the tangent to $C_{2}$ at $B$ and $Z$ the intersection of $BC$ with the tangent to $C_1$ at $A$. Prove that the circumcircles of triangles $YWZ$, $RSY$ and $PQY$ have two points in common, or are tangent in the same point.



Proposed by Misiakos Panagiotis
2016 European Mathematical Cup
A grasshopper is jumping along the number line. Initially it is situated at zero. In $k$-th step, the

length of his jump is $k$.

a) If the jump length is even, then it jumps to the left, otherwise it jumps to the right (for example, firstly

it jumps one step to the right, then two steps to the left, then three steps to the right, then four steps to

the left...). Will it visit on every integer at least once?

b) If the jump length is divisible by three, then it jumps to the left, otherwise it jumps to the right (for

example, firstly it jumps one step to the right, then two steps to the right, then three steps to the left,

then four steps to the right...). Will it visit every integer at least once?



Proposed by Matko Ljulj
2016 European Mathematical Cup
Two circles $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Let $P$, $Q$ be points on circles $C_{1}$, $C_{2}$ respectively, such that $|AP| = |AQ|$. The segment $PQ$ intersects circles $C_{1}$ and $C_{2}$ in points $M$, $N$ respectively. Let $C$ be the center of the arc $BP$ of $C_{1}$ which does not contain point $A$ and let $D$ be the center of arc $BQ$ of $C_{2}$ which does not contain point $A$ Let $E$ be the intersection of $CM$ and $DN$. Prove that $AE$ is perpendicular to $CD$.



Proposed by Steve Dinh
2016 European Mathematical Cup
Prove that for all positive integers $n$ there exist $n$ distinct, positive rational numbers with sum of

their squares equal to $n$.



Proposed by Daniyar Aubekerov
2016 European Mathematical Cup
We will call a pair of positive integers $(n, k)$ with $k > 1$ a $lovely$ $couple$ if there exists a table $nxn$

consisting of ones and zeros with following properties:

• In every row there are exactly $k$ ones.

• For each two rows there is exactly one column such that on both intersections of that column with the

mentioned rows, number one is written.

Solve the following subproblems:

a) Let $d \neq 1$ be a divisor of $n$. Determine all remainders that $d$ can give when divided by $6$.

b) Prove that there exist infinitely many lovely couples.



Proposed by Miroslav Marinov, Daniel Atanasov
2015 European Mathematical Cup
We are given an $n \times n$ board. Rows are labeled with numbers $1$ to $n$ downwards and columns are labeled with numbers $1$ to $n$ from left to right. On each field we write the number $x^2 + y^2$ where $(x, y)$ are its coordinates. We are given a figure and can initially place it on any field. In every step we can move the figure from one field to another if the other field has not already been visited and if at least one of the following

conditions is satisfied:



the numbers in those $2$ fields give the same remainders when divided by $n$,

those fields are point reflected with respect to the center of the board.



Can all the fields be visited in case:



$n = 4$,

$n = 5$?





Josip Pupić
2015 European Mathematical Cup
Let $m, n, p$ be fixed positive real numbers which satisfy $mnp = 8$. Depending on these constants, find the minimum of $$x^2+y^2+z^2+ mxy + nxz + pyz,$$where $x, y, z$ are arbitrary positive real numbers satisfying $xyz = 8$. When is the equality attained?

Solve the problem for:



$m = n = p = 2,$

arbitrary (but fixed) positive real numbers $m, n, p.$





Stijn Cambie
2015 European Mathematical Cup
Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ almost perfect if $f(n) = n$. Find all almost perfect numbers.



Paulius Ašvydis
2015 European Mathematical Cup
Let $ABC$ be an acute angled triangle. Let $B' , A'$ be points on the perpendicular bisectors of $AC, BC$ respectively such that $B'A \perp AB$ and $A'B \perp AB$. Let $P$ be a point on the segment $AB$ and $O$ the circumcenter of the triangle $ABC$. Let $D, E$ be points on $BC, AC$ respectively such that $DP \perp BO$ and $EP \perp AO$. Let $O'$ be the circumcenter of the triangle $CDE$. Prove that $B', A'$ and $O'$ are collinear.



Steve Dinh
2015 European Mathematical Cup
$A = \{a, b, c\}$ is a set containing three positive integers. Prove that we can find a set $B \subset A$, $B = \{x, y\}$ such that for all odd positive integers $m, n$ we have $$10\mid x^my^n-x^ny^m.$$

Tomi Dimovski
2015 European Mathematical Cup
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\frac{a+b+c+3}{4}\geqslant \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.$$

Dimitar Trenevski
2015 European Mathematical Cup
Circles $k_1$ and $k_2$ intersect in points $A$ and $B$, such that $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ in points $K$ and $O$ and $k_2$ in points $L$ and $M$, such that the point $L$ is between $K$ and $O$. The point $P$ is orthogonal projection of the point $L$ to the line $AB$. Prove that the line $KP$ is parallel to the $M-$median of the triangle $ABM$.



Matko Ljulj
2015 European Mathematical Cup
A group of mathematicians is attending a conference. We say that a mathematician is $k-$content if he is in a room with at least $k$ people he admires or if he is admired by at least $k$ other people in the room. It is known that when all participants are in a same room then they are all at least $3k + 1$-content. Prove that you can assign everyone into one of $2$ rooms in a way that everyone is at least $k$-content in his room and neither room is empty. Admiration is not necessarily mutual and no one admires himself.



Matija Bucić
2014 European Mathematical Cup
Which of the following claims are true, and which of them are false? If a fact is true you should prove it, if it isn't, find a counterexample.

a) Let $a,b,c$ be real numbers such that $ a^{2013} + b^{2013} + c^{2013} = 0 $. Then $ a^{2014} + b^{2014} + c^{2014} = 0 $.

b) Let $a,b,c$ be real numbers such that $ a^{2014} + b^{2014} + c^{2014} = 0 $. Then $ a^{2015} + b^{2015} + c^{2015} = 0 $.

c) Let $a,b,c$ be real numbers such that $ a^{2013} + b^{2013} + c^{2013} = 0 $ and $ a^{2015} + b^{2015} + c^{2015} = 0 $. Then $ a^{2014} + b^{2014} + c^{2014} = 0 $.





Proposed by Matko Ljulj
2014 European Mathematical Cup
In each vertex of a regular $n$-gon $A_1A_2...A_n$ there is a unique pawn. In each step it is allowed:

1. to move all pawns one step in the clockwise direction or

2. to swap the pawns at vertices $A_1$ and $A_2$.

Prove that by a finite series of such steps it is possible to swap the pawns at vertices:

a) $A_i$ and $A_{i+1}$ for any $ 1 \leq i < n$ while leaving all other pawns in their initial place

b) $A_i$ and $A_j$ for any $ 1 \leq i < j \leq n$ leaving all other pawns in their initial place.



Proposed by Matija Bucic
2014 European Mathematical Cup
Let ABC be a triangle. The external and internal angle bisectors of ∠CAB intersect side BC at D and E, respectively. Let F be a point on the segment BC. The circumcircle of triangle ADF intersects AB and AC at I and J, respectively. Let N be the mid-point of IJ and H the foot of E on DN. Prove that E is the incenter of triangle AHF, or the center of the excircle.



Proposed by Steve Dinh
2014 European Mathematical Cup
Find all functions $f$ from positive integers to themselves such that:

1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$

2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.
2014 European Mathematical Cup
Prove that there exist infinitely many positive integers which cannot be written in form $a^{d(a)}+b^{d(b)}$ for some positive integers $a$ and $b$

For positive integer $d(a)$ denotes number of positive divisors of $a$



Proposed by Borna Vukorepa
2014 European Mathematical Cup
Jeck and Lisa are playing a game on table dimensions $m \times n$ , where $m,n>2$. Lisa starts so that she puts knight figurine on arbitrary square of table.After that Jeck and Lisa put new figurine on table by the following rules:

1. Jeck puts queen figurine on any empty square of a table which is two squares vertically and one square horizontally distant, or one square vertically and two squares horizontally distant from last knight figurine which Lisa put on the table

2. Lisa puts knight figurine on any empty square of a table which is in the same row, column or diagonal as last queen figurine Jeck put on the table.

Player which cannot put his figurine loses. For which pairs of $(m,n)$ Lisa has winning strategy?



Proposed by Stijn Cambie
2014 European Mathematical Cup
Let $ABCD$ be a cyclic quadrilateral in which internal angle bisectors $\angle ABC$ and $\angle ADC$ intersect on diagonal $AC$. Let $M$ be the midpoint of $AC$. Line parallel to $BC$ which passes through $D$ cuts $BM$ at $E$ and circle $ABCD$ in $F$ ($F \neq D$ ). Prove that $BCEF$ is parallelogram



Proposed by Steve Dinh
2014 European Mathematical Cup
Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that for all $x,y\in{{\mathbb{R}}}$ holds

$f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y))$



Proposed by Matija Bucić
2013 European Mathematical Cup
For $m\in \mathbb{N}$ define $m?$ be the product of first $m$ primes. Determine if there exists positive integers $m,n$ with the following property :

$$ m?=n(n+1)(n+2)(n+3) $$



Proposed by Matko Ljulj
2013 European Mathematical Cup
Let $P$ be a point inside a triangle $ABC$. A line through $P$ parallel to $AB$ meets $BC$ and $CA$ at points $L$ and $F$, respectively. A line through $P$ parallel to $BC$ meets $CA$ and $BA$ at points $M$ and $D$ respectively, and a line through $P$ parallel to $CA$ meets $AB$ and $BC$ at points $N$ and $E$ respectively. Prove

\begin{align*}

[PDBL] \cdot [PECM] \cdot [PFAN]=8\cdot [PFM] \cdot [PEL] \cdot [PDN] \\ \end{align*}



Proposed by Steve Dinh
2013 European Mathematical Cup
We are given a combination lock consisting of $6$ rotating discs. Each disc consists of digits $0, 1, 2,\ldots , 9$ in that order (after digit $9$ comes $0$). Lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction and the lock opens instantly if the current combination is correct. Discs are initially put in the position $000000$, and we know that this combination is not correct.



a) What is the least number of moves necessary to ensure that we have found the correct combination?

b) What is the least number of moves necessary to ensure that we have found the correct combination, if we

know that none of the combinations

$000000, 111111, 222222, \ldots , 999999$ is correct?



Proposed by Ognjen Stipetić and Grgur Valentić
2013 European Mathematical Cup
Let $a,b,c$ be positive reals satisfying :

$$ \frac{a}{1+b+c}+\frac{b}{1+c+a}+\frac{c}{1+a+b}\ge \frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ca}{1+c+a} $$

Then prove that :

$$ \frac{a^2+b^2+c^2}{ab+bc+ca}+a+b+c+2\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}) $$



Proposed by Dimitar Trenevski
2013 European Mathematical Cup
In each field of a  table there is a real number.  We call such $n \times n$ table silly if each entry equals the product of all the numbers in the neighbouring fields.



a) Find all $2 \times 2$ silly tables.

b) Find all $3 \times 3$ silly tables.
2013 European Mathematical Cup
Palindrome is a sequence of digits which doesn't change if we reverse the order of its digits. Prove that a sequence $(x_n)^{\infty}_{n=0}$ defined as



$x_n=2013+317n$



contains infinitely many numbers with their decimal expansions being palindromes.
2013 European Mathematical Cup
We call a sequence of $n$ digits one or zero a code. Subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. (Code $(1101)$ contains $10$ palindromes, of which $6$ are nice.)



a) What is the least number of palindromes in a code?



b) What is the least number of nice palindromes in a code?
2013 European Mathematical Cup
Given a triangle $ABC$ let $D$, $E$, $F$ be orthogonal projections from $A$, $B$, $C$ to the opposite sides respectively. Let $X$, $Y$, $Z$ denote midpoints of $AD$, $BE$, $CF$ respectively. Prove that perpendiculars from $D$ to $YZ$, from $E$ to $XZ$ and from $F$ to $XY$ are concurrent.
2012 European Mathematical Cup
Let $ABC$ be a triangle and $Q$ a point on the internal angle bisector of $\angle BAC $. Circle $\omega_1$ is circumscribed to triangle $BAQ$ and intersects the segment $AC$ in point $P \neq C$. Circle $\omega_2$ is circumscribed to the triangle $CQP$. Radius of the cirlce $\omega_1$ is larger than the radius of $\omega_2$. Circle centered at $Q$ with radius $QA$ intersects the circle $\omega_1$ in points $A$ and $A_1$. Circle centered at $Q$ with radius $QC$ intersects $\omega_1$ in points $C_1$ and $C_2$. Prove $\angle A_1BC_1 = \angle C_2PA $.



Proposed by Matija Bucić.
2012 European Mathematical Cup
Let $S$ be the set of positive integers. For any $a$ and $b$ in the set we have $GCD(a, b)>1$. For any $a$, $b$ and $c$ in the set we have $GCD(a, b, c)=1$. Is it possible that $S$ has $2012$ elements?



Proposed by Ognjen Stipetić.
2012 European Mathematical Cup
Are there positive real numbers $x$, $y$ and $z$ such that



$		x^4 + y^4 + z^4 = 13\text{,} $



$		x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,}  $



$		x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?}  $



Proposed by Matko Ljulj.
2012 European Mathematical Cup
Let $k$ be a positive integer. At the European Chess Cup every pair of players played a game in which somebody won (there were no draws). For any $k$ players there was a player against whom they all lost, and the number of players was the least possible for such $k$. Is it possible that at the Closing Ceremony all the participants were seated at the round table in such a way that every participant was seated next to both a person he won against and a person he lost against.





Proposed by Matija Bucić.
2012 European Mathematical Cup
Find all positive integers $a$, $b$, $n$ and prime numbers $p$ that satisfy



$$	a^{2013} + b^{2013} = p^n\text{.}$$



Proposed by Matija Bucić.
2012 European Mathematical Cup
Let $ABC$ be an acute triangle with orthocenter $H$. Segments $AH$ and $CH$ intersect segments $BC$ and $AB$ in points $A_1$ and $C_1$ respectively. The segments $BH$ and $A_1C_1$ meet at point $D$. Let $P$ be the midpoint of the segment $BH$. Let $D'$ be the reflection of the point $D$ in $AC$. Prove that quadrilateral $APCD'$ is cyclic.



Proposed by Matko Ljulj.
2012 European Mathematical Cup
Prove that the following inequality holds for all positive real numbers $a$, $b$, $c$, $d$, $e$ and $f$



$$\sqrt[3]{\frac{abc}{a+b+d}}+\sqrt[3]{\frac{def}{c+e+f}} < \sqrt[3]{(a+b+d)(c+e+f)} \text{.}$$



Proposed by Dimitar Trenevski.
2012 European Mathematical Cup
Olja writes down $n$ positive integers $a_1, a_2, \ldots, a_n$ smaller than $p_n$ where $p_n$ denotes the $n$-th prime number. Oleg can choose two (not necessarily different) numbers $x$ and $y$ and replace one of them with their product $xy$. If there are two equal numbers Oleg wins. Can Oleg guarantee a win?





Proposed by Matko Ljulj.