Navvye/Olympiad-Problems-CSV huggingface dataset:
1992 Yugoslav Team Selection Test
Periodic sequences $(a_n),(b_n),(c_n)$ and $(d_n)$ satisfy the following conditions:

$$a_{n+1}=a_n+b_n,\enspace\enspace b_{n+1}=b_n+c_n,$$$$c_{n+1}=c_n+d_n,\enspace\enspace d_{n+1}=d_n+a_n,$$for $n=1,2,\ldots$. Prove that $a_2=b_2=c_2=d_2=0$.
1992 Yugoslav Team Selection Test
Does it exist a permutation of the numbers $1,2,\ldots,1992$ such that the arithmetic mean of arbitrary two of the numbers is not equal to any of the numbers which is placed between these two numbers in the permutation?
1982 Yugoslav Team Selection Test
Let $p>2$ be a prime number. For $k=1,2,\ldots,p-1$ we denote by $a_k$ the remainder when $k^p$ is divided by $p^2$. Prove that

$$a_1+a_2+\ldots+a_{p-1}=\frac{p^3-p^2}2.$$
1982 Yugoslav Team Selection Test
Find all polynomials $P_n(x)$ of the form

$$P_n(x)=n!x^n+a_{n-1}x^{n-1}+\ldots+a_1x+(-1)^nn(n+1),$$with integer coefficients, such that its roots $x_1,x_2,\ldots,x_n$ satisfy $k\le x_k\le k+1$ for $k=1,2,\ldots,n$.
1982 Yugoslav Team Selection Test
Let there be given real numbers $x_i>1~(i=1,2,\ldots,2n)$. Prove that the interval $[0,2]$ contains at most $\binom{2n}n$ sums of the form $\alpha_1x_1+\ldots+\alpha_{2n}x_{2n}$, where $\alpha_i\in\{-1,1\}$ for all $i$.
1981 Yugoslav Team Selection Test
Let $n\ge3$ be a natural number. For a set $S$ of $n$ real numbers, $A(S)$ denotes the set of all strictly increasing arithmetic sequences of three terms in $S$. At most, how many elements can the set $A(S)$ have?
1981 Yugoslav Team Selection Test
Suppose that there is a point $S$ inside a quadrilateral $ABCD$ such that segments $SA,SB,SC,SD$ divide the quadrilateral into four triangles of equal areas. Prove that one of the diagonals of the quadrilateral bisects the other one.
1981 Yugoslav Team Selection Test
Let $a,b$ be nonnegative integers. Prove that $5a>7b$ if and only if there exist nonnegative integers $x,y,z,t$ such that

\begin{align*}

x+2y+3z+7t&=a,\\
y+2z+5t&=b.

\end{align*}
1980 Yugoslav Team Selection Test
Circles $k$ and $l$ intersect at points $P$ and $Q$. Let $A$ be an arbitrary point on $k$ distinct from $P$ and $Q$. Lines $AP$ and $AQ$ meet $l$ again at $B$ and $C$. Prove that the altitude from $A$ in triangle $ABC$ passes through a point that does not depend on $A$.
1980 Yugoslav Team Selection Test
Let $a,b,c,m$ be integers, where $m>1$. Prove that if

$$a^n+bn+c\equiv0\pmod m$$for each natural number $n$, then $b^2\equiv0\pmod m$. Must $b\equiv0\pmod m$ also hold?
1980 Yugoslav Team Selection Test
A sequence $(x_n)$ satisfies $x_{n+1}=\frac{x_n^2+a}{x_{n-1}}$ for all $n\in\mathbb N$. Prove that if $x_0,x_1$, and $\frac{x_0^2+x_1^2+a}{x_0x_1}$ are integers, then all the terms of sequence $(x_n)$ are integers.
1979 Yugoslav Team Selection Test
Find all integers $n$ with $1<n<1979$ having the following property: If $m$ is an integer coprime with $n$ and $1<m<n$, then $m$ is a prime number.
1979 Yugoslav Team Selection Test
There are two circles of perimeter $1979$. Let $1979$ points be marked on the first circle, and several arcs with the total length of $1$ on the second. Show that it is possible to place the second circle onto the first in such a way that none of the marked points is covered by a marked arc.
1978 Yugoslav Team Selection Test
Find all integers $x,y,z$ such that $x^2(x^2+y)=y^{z+1}$.
1978 Yugoslav Team Selection Test
Let $k_0$ be a unit semi-circle with diameter $AB$. Assume that $k_1$ is a circle of radius

$r_1=\frac12$ that is tangent to both $k_0$ and $AB$. The circle $k_{n+1}$ of radius $r_{n+1}$ touches

$k_n,k_0$, and $AB$. Prove that:



(a) For each $n\in\{2,3,\ldots\}$ it holds that $\frac1{r_{n+1}}+\frac1{r_{n-1}}=\frac6{r_n}-4$.

(b) $\frac1{r_n}$ is either a square of an even integer, or twice a square of an odd integer.
1978 Yugoslav Team Selection Test
Let $F$ be the collection of subsets of a set with $n$ elements such that no element of $F$ is a subset of another of its elements. Prove that

$$|F|\le\binom n{\lfloor n/2\rfloor}.$$
1977 Yugoslav Team Selection Test
Determine the set of all real numbers $\alpha$ with the following property: For each positive $c$ there exists a rational number $\frac mn~(m\in\mathbb Z,n\in\mathbb N)$ different than $\alpha$ such that

$$\left|\alpha-\frac mn\right|<\frac cn.$$
1977 Yugoslav Team Selection Test
Determine all $6$-tuples $(p,q,r,x,y,z)$ where $p,q,r$ are prime, and $x,y,z$ natural numbers such that $p^{2x}=q^yr^z+1$.
1977 Yugoslav Team Selection Test
Assume that the equality $2BC=AB+AC$ holds in $\triangle ABC$. Prove that:



(a) The vertex $A$, the midpoints $M$ and $N$ of $AB$ and $AC$ respectively, the incenter $I$, and the circumcenter $O$ belong to a circle $k$.

(b) The line $GI$, where $G$ is the centroid of $\triangle ABC$ is a tangent to $k$.
1976 Yugoslav Team Selection Test
Prove that for a given convex polygon of area $A$ and perimeter $P$ there exists a circle of radius $\frac AP$ that is contained in the interior of the polygon.
1976 Yugoslav Team Selection Test
Assume that $2n+1$ positive integers satisfy the following: If we remove any of these integers, the remaining $2n$ integers can be partitioned in two groups of $n$ numbers in each, such that the sum of the numbers in one group is equal to the sum of the numbers in the other. Prove that all of these numbers must be equal.
1976 Yugoslav Team Selection Test
Find the minimum and maximum values of the function

$$f(x,y,z,t)=\frac{ax^2+by^2}{ax+by}+\frac{az^2+bt^2}{az+bt},~(a>0,b>0),$$given that $x+z=y+t=1$, and $x,y,z,t\ge0$.
1974 Yugoslav Team Selection Test
Assume that $a$ is a given irrational number.



(a) Prove that for each positive real number $\epsilon$ there exists at least one integer $q\ge0$ such that $aq-\lfloor aq\rfloor<\epsilon$.

(b) Prove that for given $\epsilon>0$ there exist infinitely many rational numbers $\frac pq$ such that $q>0$ and $\left|a-\frac pq\right|<\frac\epsilon q$.
1974 Yugoslav Team Selection Test
Given two directly congruent triangles $ABC$ and $A'B'C'$ in a plane, assume that the circles with centers $C$ and $C'$ and radii $CA$ and $C'A'$ intersect. Denote by $\mathcal M$ the transformation that maps $\triangle ABC$ to $\triangle A'B'C'$. Prove that $\mathcal M$ can be expressed as a composition of at most three rotations in the following way: The first rotation has the center in one of $A,B,C$ and maps $\triangle ABC$ to $\triangle A_1B_1C_1$; The second rotation has the center in one of $A_1,B_1,C_1$, and maps $\triangle A_1B_1C_1$ to $\triangle A_2B_2C_2$; The third rotation has the center in one of $A_2,B_2,C_2$ and maps $\triangle A_2B_2C_2$ to $\triangle A'B'C'$.
1974 Yugoslav Team Selection Test
Let $S$ be a set of $n$ points $P_1,P_2,\ldots,P_n$ in a plane such that no three of the

points are collinear. Let $\alpha$ be the smallest of the angles $\angle P_iP_jP_k$ ($i\ne j\ne k\ne i,i,j,k\in\{1,2,\ldots,n\}$). Find $\max_S\alpha$ and determine those sets $S$ for which this maximal value is attained.
1973 Yugoslav Team Selection Test
All sides of a rectangle are odd positive integers. Prove that there does not exist a point inside the rectangle whose distance to each of the vertices is an integer.
1973 Yugoslav Team Selection Test
A circle $k$ is drawn using a given disc (e.g. a coin). A point $A$ is chosen on $k$. Using just the given disc, determine the point $B$ on $k$ so that $AB$ is a diameter of $k$. (You are allowed to choose an arbitrary point in one of the drawn circles, and using the given disc it is possible to construct either of the two circles that passes through the points at a distance that is smaller than the radius of the circle.)
1973 Yugoslav Team Selection Test
Several points are denoted on a white piece of paper. The distance between each two of the points is greater than $24$. A drop of ink was sprinkled over the paper covering an area smaller than $\pi$. Prove that there exists a vector $\overrightarrow v$ with $\overrightarrow v<1$, such that after translating all of the points by $v$ none of them is covered in ink.
1972 Yugoslav Team Selection Test
Given non-zero real numbers $u,v,w,x,y,z$, how many different possibilities are there for the signs of these numbers if

$$(u+ix)(v+iy)(w+iz)=i?$$
1972 Yugoslav Team Selection Test
If a convex set of points in the line has at least two diameters, say $AB$ and $CD$, prove that $AB$ and $CD$ have a common point.
1972 Yugoslav Team Selection Test
Assume that the numbers from the table

$$\begin{matrix}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{matrix}$$satisfy the inequality:

$$\sum_{j=1}^n|a_{j1}x_1+a_{j2}x_2+\ldots+a_{jn}x_n|\le M,$$for each choice $x_j=\pm1$. Prove that

$$|a_{11}+a_{22}+\ldots+a_{nn}|\le M.$$
1972 Yugoslav Team Selection Test
Determine the largest integer $k(n)$ with the following properties: There exist $k(n)$ different subsets of a given set with $n$ elements such that each two of them have a non-empty intersection.
1970 Yugoslav Team Selection Test
Positive integers $a$ and $b$ have $n$ digits each in their decimal representation. Assume that $m$ is a positive integer such that $\frac n2<m<n$ and assume that each of the leftmost $m$ digits of $a$ is equal to the corresponding digit of $b$. Prove that

$$a^{\frac1n}-b^{\frac1n}<\frac1n.$$
1970 Yugoslav Team Selection Test
Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.
1970 Yugoslav Team Selection Test
If all edges of a non-planar quadrilateral tangent the faces of a sphere, prove that all of the points of tangency belong to a plane.
1969 Yugoslav Team Selection Test
Given real numbers $a_i,b_i~(i=1,2,\ldots,n)$ such that

\begin{align*}

&a_1\ge a_2\ge\ldots\ge a_n>0,\\
&b_1\ge a_1,\\
&b_1b_2\ge a_1a_2,\\
&\vdots\\
&b_1b_2\cdots b_n\ge a_1a_2\cdots a_n,

\end{align*}prove that $b_1+b_2+\ldots+b_n\ge a_1+a_2+\ldots+a_n$.
1969 Yugoslav Team Selection Test
Let $f(x)$ and $g(x)$ be degree $n$ polynomials, and $x_0,x_1,\ldots,x_n$ be real numbers such that

$$f(x_0)=g(x_0),f'(x_1)=g'(x_1),f''(x_2)=g''(x_2),\ldots,f^{(n)}(x_n)=g^{(n)}(x_n).$$Prove that $f(x)=g(x)$ for all $x$.
1969 Yugoslav Team Selection Test
Points $A$ and $B$ move with a constant speed along lines $a$ and $b$. Two corresponding positions of these points $A_1,B_1$, and $A_2,B_2$ are known. Find the position of $A$ and $B$ for which the length of $AB$ is minimal.
1969 Yugoslav Team Selection Test
Let $a$ and $b$ be two natural numbers such that $a<b$. Prove that in each set of $b$ consecutive positive integers there are two numbers whose product is divisible by $ab$.
1969 Yugoslav Team Selection Test
Prove that the product of the sines of two opposite dihedrals in a tetrahedron is proportional to the product of the lengths of the edges of these dihedrals.
1969 Yugoslav Team Selection Test
Let $E$ be the set of $n^2+1$ closed intervals on the real axis. Prove that there exists a subset of $n+1$ intervals that are monotonically increasing with respect to inclusion, or a subset of $n+1$ intervals none of which contains any other interval from the subset.
1968 Yugoslav Team Selection Test
Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.
1968 Yugoslav Team Selection Test
Let $n>3$ be a positive integer. Prove that $n$ is prime if and only if there exists a positive integer $\alpha$ such that $n!=n(n-1)(\alpha n+1)$.
1968 Yugoslav Team Selection Test
Each side of a triangle $ABC$ is divided into three equal parts, and the middle segment in each of the sides is painted green. In the exterior of $\triangle ABC$ three equilateral triangles are constructed, in such a way that the three green segments are sides of these triangles. Denote by $A',B',C'$ the vertices of these new equilateral triangles that don’t belong to the edges of $\triangle ABC$, respectively. Let $A'',B'',C''$ be the points symmetric to $A',B',C'$ with respect to $BC,CA,AB$.



(a) Prove that $\triangle A'B'C'$ and $\triangle A''B''C''$ are equilateral.

(b) Prove that $ABC,A'B'C'$, and $A''B''C''$ have a common centroid.
1968 Yugoslav Team Selection Test
If a polynomial of degree n has integer values when evaluated in each of $k,k+1,\ldots,k+n$, where $k$ is an integer, prove that the polynomial has integer values when evaluated at each integer $x$.
1968 Yugoslav Team Selection Test
Let $n$ be an integer greater than $1$. Let $x\in\mathbb R$.



(a) Evaluate $S(x,n)=\sum(x+p)(x+q)$, where the summation is over all pairs $(p,q)$ of different numbers from $\{1,2,\ldots,n\}$.

(b) Do there exist integers $x,n$ for which $S(x,n)=0$?
1968 Yugoslav Team Selection Test
Prove that the incenter coincides with the circumcenter of a tetrahedron if and only if each pair of opposite edges are of equal length.
2009 Singapore Team Selection Test
Determine the smallest positive integer $\ N $ such that there exists 6 distinct integers  $\ a_1, a_2, a_3, a_4, a_5, a_6 > 0 $ satisfying:

(i) $\ N = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 $

(ii) $\ N - a_i$  is a perfect square for $\ i = 1,2,3,4,5,6 $.
2009 Singapore Team Selection Test
Let $S=\{a+np : n=0,1,2,3,... \}$ where $a$ is a positive integer and $p$ is a prime. Suppose there exist positive integers $x$ and $y$ st $x^{41}$ and $y^{49}$ are in $S$. Determine if there exists a positive integer $z$ st $z^{2009}$ is in $S$.
2009 Singapore Team Selection Test
Let $H$ be the orthocentre of  $\triangle ABC$ and let $P$ be a point on the circumcircle of $\triangle ABC$, distinct from $A,B,C$. Let $E$ and $F$ be the feet of altitudes from $H$ onto $AC$ and $AB$ respectively. Let $PAQB$ and $PARC$ be parallelograms. Suppose $QA$ meets $RH$ at $X$ and $RA$ meets $QH$ at $Y$. Prove that $XE$ is parallel to $YF$.
2008 Singapore Team Selection Test
In triangle $ABC$, $D$ is a point on $AB$ and $E$ is a point on $AC$ such that $BE$ and $CD$ are bisectors of $\angle B$ and $\angle C$ respectively. Let $Q,M$ and $N$ be the feet of perpendiculars from the midpoint $P$ of $DE$ onto $BC,AB$ and $AC$, respectively. Prove that $PQ=PM+PN$.
2008 Singapore Team Selection Test
Let $ x_1, x_2,\ldots , x_n$ be positive real numbers such that $ x_1x_2\cdots x_n = 1$. Prove that

$$\sum_{i = 1}^n \frac {1}{n - 1 + x_i}\le 1.$$
2008 Singapore Team Selection Test
Find all odd primes $ p$, if any, such that $ p$ divides $ \sum_{n=1}^{103}n^{p-1}$
2008 Singapore Team Selection Test
Let $(O)$ be a circle, and let $ABP$ be a line segment such that $A,B$ lie on $(O)$ and $P$ is a point outside $(O)$. Let $C$ be a point on $(O)$ such that $PC$ is tangent to $(O)$ and let $D$ be the point on $(O)$ such that $CD$ is a diameter of $(O)$ and intersects $AB$ inside $(O)$. Suppose that the lines $DB$ and $OP$ intersect at $E$. Prove that $AC$ is perpendicular to $CE$.
2008 Singapore Team Selection Test
Find all functions $ f : \mathbb R \rightarrow \mathbb R$ such that $ (x + y)(f(x) - f(y)) = (x -y)f(x + y)$ for all $ x, y\in \mathbb R$
2008 Singapore Team Selection Test
Fifty teams participate in a round robin competition over 50 days. Moreover, all the teams (at least two) that show up in any day must play against each other. Prove that on every pair of consecutive days, there is a team that has to play on those two days.
2007 Singapore Team Selection Test
Find all pairs of nonnegative integers $ (x, y)$ satisfying $ (14y)^x + y^{x+y} = 2007$.
2007 Singapore Team Selection Test
Let $ABCD$ be a convex quadrilateral inscribed in a circle with $M$ and $N$ the midpoints of the diagonals $AC$ and $BD$ respectively. Suppose that $AC$ bisects $\angle BMD$. Prove that $BD$ bisects $\angle ANC$.
2007 Singapore Team Selection Test
Let $ a_1, a_2,\ldots  ,a_8$ be $8$ distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every $4$ of the $8$ points form a quadrilateral which is called a quad. If two chords, each joining a pair of the $8$ points, intersect, the point of intersection is called a bullet. Suppose some of the bullets are coloured red. For each pair $(i j)$, with $ 1 \le i < j \le 8$, let $r(i,j)$ be the number of quads, each containing $ a_i, a_j$ as vertices, whose diagonals intersect at a red bullet. Determine the smallest positive integer $n$ such that it is possible to colour $n$ of the bullets red so that $r(i,j)$ is a constant for all pairs $(i,j)$.
2007 Singapore Team Selection Test
Two circles $ (O_1)$ and $ (O_2)$ touch externally at the point $C$ and internally at the points $A$ and $B$ respectively with another circle $(O)$. Suppose that the common tangent of $ (O_1)$ and $ (O_2)$ at $C$ meets $(O)$ at $P$ such that $PA=PB$. Prove that $PO$ is perpendicular to $AB$.
2007 Singapore Team Selection Test
Prove the inequality

$$\sum_{i<j}  \frac{a_ia_j}{a_i + a_j} \le \frac{n}{2(a_1 + a_2 +\cdots + a_n)}\sum_{i<j} a_ia_j$$

for all positive real numbers $ a_1, a_2,\ldots , a_n$.
2007 Singapore Team Selection Test
Let $A,B,C$ be $3$ points on the plane with integral coordinates. Prove that there exists a point $P$ with integral coordinates distinct from $A,B$ and $C$ such that the interiors of the segments $PA,PB$ and $PC$ do not contain points with integral coordinates.
2006 Singapore Team Selection Test
Let $ANC$, $CLB$ and $BKA$ be triangles erected on the outside of the triangle $ABC$ such that $\angle NAC = \angle KBA = \angle LCB$ and $\angle NCA = \angle KAB = \angle LBC$. Let $D$, $E$, $G$ and $H$ be the midpoints of $AB$, $LK$, $CA$ and $NA$ respectively. Prove that $DEGH$ is a parallelogram.
2006 Singapore Team Selection Test
Let n be an integer greater than 1 and let $x_1, x_2, . . . , x_n$ be real numbers such that

$|x_1| + |x_2| + ... + |x_n| = 1$ and $x_1 + x_2 + ... + x_n = 0$

Prove that

$\left| \frac{x_1}{1}+\frac{x_2}{2}+\cdots+\frac{x_n}{n} \right| \leq \frac{1}{2} \left(1-\frac{1}{n}\right)$
2006 Singapore Team Selection Test
A pile of n pebbles is placed in a vertical column. This configuration is

modified according to the following rules. A pebble can be moved if it is at

the top of a column which contains at least two more pebbles than the column

immediately to its right. (If there are no pebbles to the right, think of this as

a column with 0 pebbles.) At each stage, choose a pebble from among those

that can be moved (if there are any) and place it at the top of the column

to its right. If no pebbles can be moved, the configuration is called a final

configuration. For each n, show that, no matter what choices are made at each

stage, the final configuration obtained is unique. Describe that configuration

in terms of n.
2006 Singapore Team Selection Test
In the plane containing a triangle $ABC$, points $A'$, $B'$ and $C'$ distinct from the vertices of $ABC$ lie on the lines $BC$, $AC$ and $AB$ respectively such that $AA'$, $BB'$ and $CC'$ are concurrent at $G$ and $AG/GA' = BG/GB' = CG/GC'$.



Prove that $G$ is the centroid of $ABC$.
2006 Singapore Team Selection Test
Let S be a set of sequences of length 15 formed by using the letters a and b

such that every pair of sequences in S differ in at least 3 places. What is the

maximum number of sequences in S?
2006 Singapore Team Selection Test
Let $n$ be a positive integer such that the sum of all its positive divisors (inclusive of $n$) equals to $2n + 1$. Prove that $n$ is an odd perfect square.



related:



 (Putnam 1976)
2004 Singapore Team Selection Test
Let $D$ be a point in the interior of $\bigtriangleup ABC$ such that $AB = ab$, $AC = ac$, $AD = ad$, $BC = bc$, $BD = bd$ and $CD = cd$.  Prove that $\angle ABD + \angle ACD = \frac{\pi}{3}$.
2004 Singapore Team Selection Test
Let $0 < a, b, c < 1$ with $ab + bc + ca = 1$.  Prove that

$$\frac{a}{1-a^2} + \frac{b}{1-b^2} + \frac{c}{1-c^2} \geq \frac {3 \sqrt{3}}{2}.$$



Determine when equality holds.
2004 Singapore Team Selection Test
Let $p \geq 5$ be a prime number.  Prove that there exist at least 2 distinct primes $q_1, q_2$ satisfying $1 < q_i < p - 1$ and $q_i^{p-1} \not\equiv 1 \mbox{ (mod }p^2)$, for $i = 1, 2$.
2004 Singapore Team Selection Test
Let $x_0, x_1, x_2, \ldots$ be the sequence defined by

$x_i= 2^i$ if $0 \leq i \leq 2003$

$x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$



Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.
2004 Singapore Team Selection Test
Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying

$$ f\left(\frac {x + y}{x - y}\right) = \frac {f\left(x\right) + f\left(y\right)}{f\left(x\right) - f\left(y\right)}

$$

for all $ x \neq y$.
2018 Slovenia Team Selection Test
Let $n$ be a positive integer. On the table, we have $n^2$ ornaments in $n$ different colours, not necessarily $n$ of each colour. Prove that we can hang the ornaments on $n$ Christmas trees in such a way that there are exactly $n$ ornaments on each tree and the ornaments on every tree are of at most $2$ different colours.
2018 Slovenia Team Selection Test
Ana and Bojan are playing a game: Ana chooses positive integers $a$ and $b$ and each one gets $2016$ pieces of paper, visible to both - Ana gets the pieces with the numbers $a+1$, $a+2$, $\ldots$, $a+2016$ and Bojan gets the pieces with the numbers $b+1$, $b+2$, $\ldots$, $b+2016$ on them. Afterwards, one of them writes the number $a+b$ on the board. In every move, Ana chooses one of her pieces of paper and hands it to Bojan who chooses one of his own, writes their sum on the board and removes them both from the game. When they run out of pieces, they multiply the numbers on the board together. If the result has the same remainder than $a+b$ when divided by $2017$, Bojan wins, otherwise, Ana wins. Who has the winning strategy?
2018 Slovenia Team Selection Test
Let $a$, $b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that the following inequality holds:

$$\frac{a+b+c}{3}\geq\frac{a}{a^2b+2}+\frac{b}{b^2c+2}+\frac{c}{c^2a+2}.$$
2018 Slovenia Team Selection Test
Let $\mathcal{K}$ be a circle centered in $A$. Let $p$ be a line tangent to $\mathcal{K}$ in $B$ and let a line parallel to $p$ intersect $\mathcal{K}$ in $C$ and $D$. Let the line $AD$ intersect $p$ in $E$ and let $F$ be the intersection of the lines $CE$ and $AB$. Prove that the line through $D$, parallel to the tangent through $A$ to the circumcircle of $AFD$ intersects the line $CF$ on $\mathcal{K}$.
2018 Slovenia Team Selection Test
Let $n$ be a positive integer. We are given a regular $4n$-gon in the plane. We divide its vertices in $2n$ pairs and connect the two vertices in each pair by a line segment. What is the maximal possible amount of distinct intersections of the segments?
2019 Switzerland Team Selection Test
Let $ABC$ be a triangle and $D, E, F$ be the foots of altitudes drawn from $A,B,C$ respectively. Let $H$ be the orthocenter of $ABC$. Lines $EF$ and $AD$ intersect at $G$. Let $K$ the point on circumcircle of $ABC$ such that $AK$ is a diameter of this circle.  $AK$ cuts $BC$ in $M$. Prove that $GM$ and $HK$ are parallel.
2019 Switzerland Team Selection Test
Find the largest prime $p$ such that there exist positive integers $a,b$ satisfying $$p=\frac{b}{2}\sqrt{\frac{a-b}{a+b}}.$$
2019 Switzerland Team Selection Test
Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties:

$(a)$ $P(x)>x$ for all positive integers $x$.

$(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.