Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 Iran Team Selection Test
Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of  one of its elements.
2004 Iran Team Selection Test
Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that:

$$ \left(\begin{array}{c}n\\ \hline p\end{array}\right)=\left(\begin{array}{c}n+k\\ \hline p\end{array}\right)$$
2004 Iran Team Selection Test
Suppose that $ p$ is a prime number. Prove that the equation $ x^2-py^2=-1$ has a solution if and only if $ p\equiv1\pmod 4$.
2004 Iran Team Selection Test
Suppose that $ ABCD$ is a convex quadrilateral. Let $ F = AB\cap CD$, $ E = AD\cap BC$ and $ T = AC\cap BD$. Suppose that $ A,B,T,E$ lie on a circle which intersects with $ EF$ at $ P$. Prove that if $ M$ is midpoint of $ AB$, then $ \angle APM = \angle BPT$.
2004 Iran Team Selection Test
Let $ M,M'$ be two conjugates point in triangle $ ABC$ (in the sense that $ \angle MAB=\angle M'AC,\dots$). Let $ P,Q,R,P',Q',R'$ be foots of perpendiculars from $ M$ and $ M'$ to $ BC,CA,AB$. Let $ E=QR\cap Q'R'$, $ F=RP\cap R'P'$ and $ G=PQ\cap P'Q'$. Prove that the lines $ AG, BF, CE$ are parallel.
2004 Iran Team Selection Test
This problem is generalization of this one.

Suppose $G$ is a graph and $S\subset V(G)$. Suppose we have arbitrarily assign real numbers to each element of $S$. Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.
2002 Iran Team Selection Test
$ABCD$ is a convex quadrilateral. We draw its diagnals to divide the quadrilateral to four triabgles. $P$ is the intersection of diagnals. $I_{1},I_{2},I_{3},I_{4}$ are excenters of $PAD,PAB,PBC,PCD$(excenters corresponding vertex $P$). Prove that $I_{1},I_{2},I_{3},I_{4}$ lie on a circle iff $ABCD$ is a tangential quadrilateral.
2002 Iran Team Selection Test
$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.
2002 Iran Team Selection Test
A "2-line" is the  area between two parallel lines. Length of "2-line" is distance of two parallel lines. We have covered unit circle with some "2-lines". Prove sum of lengths of "2-lines" is at least 2.
2002 Iran Team Selection Test
$O$ is a point in triangle $ABC$. We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$. Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$.
2002 Iran Team Selection Test
A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.
2002 Iran Team Selection Test
Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$, $\sum_{i=1}^{n}x_{i}^{2}=n$, $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$. Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$.
2002 Iran Team Selection Test
$S_{1},S_{2},S_{3}$ are three spheres in $\mathbb R^{3}$ that their centers are not collinear. $k\leq8$ is the number of planes that touch three spheres. $A_{i},B_{i},C_{i}$ is the point that $i$-th plane touch the spheres $S_{1},S_{2},S_{3}$. Let $O_{i}$ be circumcenter of $A_{i}B_{i}C_{i}$. Prove that $O_{i}$ are collinear.
2002 Iran Team Selection Test
We call $A_{1},A_{2},A_{3}$ mangool iff there is a permutation $\pi$ that $A_{\pi(2)}\not\subset A_{\pi(1)},A_{\pi(3)}\not\subset A_{\pi(1)}\cup A_{\pi(2)}$. A good family is a family of finite subsets of $\mathbb N$ like $X,A_{1},A_{2},\dots,A_{n}$. To each goo family we correspond a graph with vertices $\{A_{1},A_{2},\dots,A_{n}\}$. Connect $A_{i},A_{j}$ iff $X,A_{i},A_{j}$ are mangool sets. Find all graphs that we can find a good family corresponding to it.
2002 Iran Team Selection Test
$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have  $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?
2002 Iran Team Selection Test
Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.
2002 Iran Team Selection Test
A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.
2002 Iran Team Selection Test
We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ quadratic if there exists at least a perfect square among the numbers $ a_1$, $ a_1 + a_2$, $ ...$, $ a_1 + a_2 + ... + a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are quadratic.



Remark. $ S_{n}$ denotes the $ n$-th symmetric group, the group of permutations on $ n$ elements.
2002 Iran Team Selection Test
Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the side $BC$ at $A^{\prime}$, and the line $AA^{\prime}$ meets the incircle again at a point $P$. Let the lines $CP$ and $BP$ meet the incircle of triangle $ABC$ again at $N$ and $M$, respectively. Prove that the lines $AA^{\prime}$, $BN$ and $CM$ are concurrent.
2016 Israel Team Selection Test
Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.
2016 Israel Team Selection Test
Rothschild the benefactor has a certain number of coins. A man comes, and Rothschild wants to share his coins with him. If he has an even number of coins, he gives half of them to the man and goes away. If he has an odd number of coins, he donates one coin to charity so he can have an even number of coins, but meanwhile another man comes. So now he has to share his coins with two other people. If it is possible to do so evenly, he does so and goes away. Otherwise, he again donates a few coins to charity (no more than 3). Meanwhile, yet another man comes. This goes on until Rothschild is able to divide his coins evenly or until he runs out of money. Does there exist a natural number $N$ such that if Rothschild has at least $N$ coins in the beginning, he will end with at least one coin?
2016 Israel Team Selection Test
Prove that there exists an ellipsoid touching all edges of an octahedron if and only if the octahedron's diagonals intersect. (Here an octahedron is a polyhedron consisting of eight triangular faces, twelve edges, and six vertices such that four faces meat at each vertex. The diagonals of an octahedron are the lines connecting pairs of vertices not connected by an edge).
2016 Israel Team Selection Test
A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?
2016 Israel Team Selection Test
A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.
2016 Israel Team Selection Test
Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying (for all $x,y \in \mathbb{R}$):  $f(x+y)^2 - f(2x^2) = f(y-x)f(y+x) + 2x\cdot f(y)$.
2016 Israel Team Selection Test
On each square of an $n$x$n$ board sleeps a dragon. Two dragons are called neighbors if their squares have a side in common. Each turn, Minnie wakes up a dragon which has a living neighbor and Max directs it towards one of its living neighbors. The dragon than breathes fire on that neighbor and destroys it, and then goes back to sleep.



Minnie's goal is to minimize the snoring of the dragons and leave as few living dragons as possible. Max is a member of PETD (People for the Ethical Treatment of Dragons), and he wants to save as many dragons as he can.



How many dragons will stay alive at the end if



1. $n=4$?

2. $n=5$?
2016 Israel Team Selection Test
Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers.
2009 Italy TST
Let $n,k$ be positive integers such that $n\ge k$. $n$ lamps are placed on a circle, which are all off. In any step we can change the state of $k$ consecutive lamps. In the following three cases, how many states of lamps are there in all $2^n$ possible states that can be obtained from the initial state by a certain series of operations?

i)$k$ is a prime number greater than $2$;

ii) $k$ is odd;

iii) $k$ is even.
2009 Italy TST
$ABC$ is a triangle in the plane. Find the locus of point $P$ for which $PA,PB,PC$ form a triangle whose area is equal to one third of the area of triangle $ABC$.
2009 Italy TST
Find all pairs of integers $(x,y)$ such that

$$ y^3=8x^6+2x^3y-y^2.$$
2009 Italy TST
Let $n$ be an even positive integer. An $n$-degree monic polynomial $P(x)$ has $n$ real roots (not necessarily distinct). Suppose $y$ is a positive real number such that for any real number $t<y$, we have $P(t)>0$. Prove that

$$P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.$$
2009 Italy TST
Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.
2009 Italy TST
Two persons, A and B, set up an incantation contest in which they spell incantations (i.e. a finite sequence of letters) alternately. They must obey the following rules:

i) Any incantation can appear no more than once;

ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it;

iii)The first person who cannot spell an incantation loses the contest. Answer the following questions:

a) If A says '$STAGEPREIMO$' first, then who will win?

b) Let $M$ be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are $2009$ and containing only four letters $A,B,C,D$, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in $M$ such that A has a winning strategy by starting with it.
2007 Italy TST
Let $ABC$ an acute triangle.



(a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$;



(b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.
2007 Italy TST
In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C  }$  if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$.



(a) Find the minimum of cyclical 3-subset (depending on $n$);

(b) Find the maximum of cyclical 3-subset (depending on $n$).
2007 Italy TST
Find all $f: R \longrightarrow R$ such that



$$f(xy+f(x))=xf(y)+f(x)$$



for every pair of real numbers $x,y$.
2007 Italy TST
We have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?
2007 Italy TST
Let $ABC$ a acute triangle.



(a) Find the locus of all the points $P$ such that, calling $O_{a}, O_{b}, O_{c}$ the circumcenters of $PBC$, $PAC$, $PAB$:

$$\frac{ O_{a}O_{b}}{AB}= \frac{ O_{b}O_{c}}{BC}=\frac{ O_{c}O_{a}}{CA}$$



(b) For all points $P$ of the locus in (a), show that the lines $AO_{a}$, $BO_{b}$ , $CO_{c}$ are cuncurrent (in $X$);



(c) Show that the power of $X$ wrt the circumcircle of $ABC$ is:

$$-\frac{ a^{2}+b^{2}+c^{2}-5R^{2}}4$$

Where $a=BC$ , $b=AC$ and $c=AB$.
2007 Italy TST
Let $p \geq 5$  be a prime.



(a) Show that exists a prime $q \neq p$ such that  $q| (p-1)^{p}+1$



(b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that:

$$\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 $$



[Edited by $\zeta X$, see below]
2006 Italy TST
Let $S$ be a string of $99$ characters, $66$ of which are $A$ and $33$ are $B$. We call $S$ good if, for each $n$ such that $1\le n \le 99$, the sub-string made from the first $n$ characters of $S$ has an odd number of distinct permutations. How many good strings are there? Which strings are good?
2006 Italy TST
Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that

$$HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}$$

if and only if $ABC$ is acute-angled.
2006 Italy TST
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$,

$$f(m - n + f(n)) = f(m) + f(n).$$
2006 Italy TST
The circles $\gamma_1$ and $\gamma_2$ intersect at the points $Q$ and $R$ and internally touch a circle $\gamma$ at $A_1$ and $A_2$ respectively. Let $P$ be an arbitrary point on $\gamma$. Segments $PA_1$ and $PA_2$ meet $\gamma_1$ and $\gamma_2$ again at $B_1$ and $B_2$ respectively.

a) Prove that the tangent to $\gamma_{1}$ at $B_{1}$ and the tangent to $\gamma_{2}$ at $B_{2}$ are parallel.

b) Prove that $B_{1}B_{2}$ is the common tangent to $\gamma_{1}$ and $\gamma_{2}$ iff $P$ lies on $QR$.
2006 Italy TST
Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$.

a) Find all $n$ such that $A_{n}\neq \emptyset$

b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero.

c) Is there $n$ such that $|{A_{n}}| = 130$?
2006 Italy TST
Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.
2005 Italy TST
A stage course is attended by $n \ge  4$ students. The day before the final exam, each group of three students conspire against another student to throw him/her out of the exam. Prove that there is a student against whom there are at least $\sqrt[3]{(n-1)(n- 2)} $conspirators.
2005 Italy TST
$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases  of equality.

$(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.
2005 Italy TST
The function $\psi : \mathbb{N}\rightarrow\mathbb{N}$ is defined by $\psi (n)=\sum_{k=1}^n\gcd (k,n)$.



$(a)$ Prove that $\psi (mn)=\psi (m)\psi (n)$ for every two coprime $m,n \in \mathbb{N}$.

$(b)$ Prove that for each $a\in\mathbb{N}$ the equation $\psi (x)=ax$ has a solution.
2005 Italy TST
Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and

$$f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. $$

$(a)$ Prove that $f$ has a fixed point different from $1$.

$(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.
2005 Italy TST
The circle $\Gamma$ and the line $\ell$ have no common points. Let $AB$ be the diameter of $\Gamma$ perpendicular to $\ell$, with $B$ closer to $\ell$ than $A$. An arbitrary point $C\not= A$, $B$ is chosen on $\Gamma$. The line $AC$ intersects $\ell$ at $D$. The line $DE$ is tangent to $\Gamma$ at $E$, with $B$ and $E$ on the same side of $AC$. Let $BE$ intersect $\ell$ at $F$, and let $AF$ intersect $\Gamma$ at $G\not= A$. Let $H$ be the reflection of $G$ in $AB$. Show that $F,C$, and $H$ are collinear.
2005 Italy TST
Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins.

$(a)$ Determine which player has a winning strategy for $N=2005$.

$(b)$ Determine which player has a winning strategy for $N=2004$.

$(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.
2004 Italy TST
At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained?

$(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$

$(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$

$(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$
2004 Italy TST
Let $\mathcal{P}_0=A_0A_1\ldots A_{n-1}$ be a convex polygon such that $A_iA_{i+1}=2^{[i/2]}$ for $i=0, 1,\ldots ,n-1$ (where $A_n=A_0$). Define the sequence of polygons $\mathcal{P}_k=A_0^kA_1^k\ldots A_{n-1}^k$ as follows: $A_i^1$ is symmetric to $A_i$ with respect to $A_0$, $A_i^2$ is symmetric to $A_i^1$ with respect to $A_1^1$, $A_i^3$ is symmetric to $A_i^2$ with respect to $A_2^2$ and so on. Find the values of $n$ for which infinitely many polygons $\mathcal{P}_k$ coincide with $\mathcal{P}_0$.
2004 Italy TST
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$,

$$(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. $$
2004 Italy TST
Two circles $\gamma_1$ and $\gamma_2$ intersect at $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ so that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ in $F$. Let $t$ be the tangent to $\gamma_2$ at $F$.

$(a)$ Prove that $t$ is parallel to $AC$.

$(b)$ Prove that the lines $r,s,t$ are concurrent.
2004 Italy TST
Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.
2003 Italy TST
Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$
2003 Italy TST
Let $B\not= A$ be a point on the tangent to circle $S_1$ through the point $A$ on the circle. A point $C$ outside the circle is chosen so that segment $AC$ intersects the circle in two distinct points. Let $S_2$ be the circle tangent to $AC$ at $C$ and to $S_1$ at some point $D$, where $D$ and $B$ are on the opposite sides of the line $AC$. Let $O$ be the circumcentre of triangle $BCD$. Show that $O$ lies on the circumcircle of triangle $ABC$.
2003 Italy TST
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy

$$f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. $$
2003 Italy TST
The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. The line through $A$ parallel to $DF$ meets the line through $C$ parallel to $EF$ at $G$.

$(a)$ Prove that the quadrilateral $AICG$ is cyclic.

$(b)$ Prove that the points $B,I,G$ are collinear.
2003 Italy TST
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A tromino is an $L$-shape formed by three connected unit squares.

$(a)$ For which values of $n$ is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard?

$(b)$ When it is possible, find the minimum number of trominoes needed.
2003 Italy TST
Let $p(x)$ be a polynomial with integer coefficients and let $n$ be an integer. Suppose that there is a positive integer $k$ for which $f^{(k)}(n) = n$, where $f^{(k)}(x)$ is the polynomial obtained as the composition of $k$ polynomials $f$. Prove that $p(p(n)) = n$.
2002 Italy TST
Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.
2002 Italy TST
Prove that for each prime number $p$ and positive integer $n$, $p^n$ divides

$$\binom{p^n}{p}-p^{n-1}. $$
2002 Italy TST
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions:

$(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$

$(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.
2002 Italy TST
A scalene triangle $ABC$ is inscribed in a circle $\Gamma$. The bisector of angle $A$ meets $BC$ at $E$. Let $M$ be the midpoint of the arc $BAC$. The line $ME$ intersects $\Gamma$ again at $D$. Show that the circumcentre of triangle $AED$ coincides with the intersection point of the tangent to $\Gamma$ at $D$ and the line $BC$.
2002 Italy TST
On a soccer tournament with $n\ge 3$ teams taking part, several matches are played in such a way that among any three teams, some two play a match.

$(a)$ If $n=7$, find the smallest number of matches that must be played.

$(b)$ Find the smallest number of matches in terms of $n$.
2002 Italy TST
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that

$(\text{i})$ $x$ and $y$ are relatively prime;

$(\text{ii})$ $x$ divides $y^2+m;$

$(\text{iii})$ $y$ divides $x^2+m.$
2001 Italy TST
The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC +MA \cdot CD = MB \cdot MD$, prove that $ \angle BKC = \angle CDB$.
2001 Italy TST
Let $0\le a\le b\le c$ be real numbers. Prove that

$$(a+3b)(b+4c)(c+2a)\ge 60abc $$
2001 Italy TST
Find all pairs $ (p, q)$ of prime numbers such that $ p$ divides $ 5^q + 1$ and $ q$ divides $ 5^p + 1$.
2001 Italy TST
We are given $2001$ balloons and a positive integer $k$. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most $k$ balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of $k$ such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps.
2000 Italy TST
Determine all triples $(x,y,z)$ of positive integers such that

$$\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} $$
2000 Italy TST
Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.
2000 Italy TST
Given positive numbers $a_1$ and $b_1$, consider the sequences defined by

$$a_{n+1}=a_n+\frac{1}{b_n},\quad b_{n+1}=b_n+\frac{1}{a_n}\quad (n \ge 1)$$

Prove that $a_{25}+b_{25} \geq 10\sqrt{2}$.
2000 Italy TST
On a mathematical competition $ n$ problems were given. The final results showed that:

(i) on each problem, exactly three contestants scored $ 7$ points;

(ii) for each pair of problems, exactly one contestant scored $ 7$ points on both problems.

Prove that if $ n \geq  8$, then there is a contestant who got $ 7$ points on each problem. Is this statement necessarily true if $ n = 7$?
1999 Italy TST
Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.