Navvye/Olympiad-Problems-CSV huggingface dataset:
2013 Saudi Arabia IMO TST
For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$).  Find, when $n$ is any positive integer, the maximum possible value of $p$.
2013 Saudi Arabia IMO TST
Determine if there exists an infinite sequence of positive integers $a_1,a_2, a_3, ...$ such that

(i) each positive integer occurs exactly once in the sequence, and

(ii) each positive integer occurs exactly once in the sequence $ |a_1 - a_2|, |a_2 - a_3|, ..., |a+k - a_{k+1}|, ...$
2013 Saudi Arabia IMO TST
Adel draws an $m \times  n$ grid of dots on the coordinate plane, at the points of integer coordinates $(a,b)$ where $1 \le  a \le  m$ and $1 \le  b \le n$. He proceeds to draw a closed path along $k$ of these dots, $(a_1, b_1)$,$(a_2,b_2)$,...,$(a_k,b_k)$, such that $(a_i,b_i)$ and $(a_{i+1}, b_{i+1})$ (where $(a_{k+1}, b_{k+1}) = (a_1, b_1)$) are $1$ unit apart for each $1 \le  i \le  k$. Adel makes sure his path does not cross itself, that is, the $k$ dots are distinct. Find, with proof, the maximum possible value of $k$ in terms of $m$ and $n$.
2013 Saudi Arabia IMO TST
Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that

(a) $a_1 + a_2 + .. + a_n \ge n^2$  and

(b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$
2013 Saudi Arabia IMO TST
Let $ABC$ be an acute triangle, $M$ be the midpoint of $BC$ and $P$ be a point on line segment $AM$. Lines $BP$ and $CP$ meet the circumcircle of $ABC$ again at $X$ and $Y$ , respectively, and sides $AC$ at $D$ and $AB$ at $E$, respectively. Prove that the circumcircles of $AXD$ and $AYE$ have a common point $T \ne A$ on line $AM$.
2013 Saudi Arabia IMO TST
Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.
2019 Serbia Team Selection Test
a) Given $2019$ different integers wich have no odd prime divisor less than $37$, prove there exists two of these numbers such that their sum has no odd prime divisor less than $37$.



b)Does the result hold if we change $37$ to $38$ ?
2019 Serbia Team Selection Test
Given triangle $\triangle ABC $ with $AC\neq BC $,and let $D $ be a point inside triangle such that $\measuredangle ADB=90^{\circ} + \frac {1}{2}\measuredangle ACB $.Tangents from $C $ to the circumcircles of $\triangle ABC $ and $\triangle ADC $ intersect $AB $ and $AD $ at $P $ and $Q $ , respectively.Prove that $PQ $ bisects the angle $\measuredangle BPC $.
2019 Serbia Team Selection Test
It is given $n$ a natural number and a circle with circumference $n$. On the circle, in clockwise direction, numbers $0,1,2,\dots n-1$ are written, in this order and in the same distance to each other. Every number is colored red or blue, and there exists a non-zero number of numbers of each color. It is known that there exists a set $S\subsetneq \{0,1,2,\dots n-1\}, |S|\geq 2$, for wich it holds: if $(x,y), x<y$ is a circle sector whose endpoints are of  distinct colors, whose distance $y-x$ is in $S$, then $y$ is in $S$.



Prove that there is a divisor $d$ of $n$ different from $1$ and $n$ for wich holds: if $(x,y),x<y$ are different points of distinct colors, such that their distance is divisible by $d$, then both $x,y$ are divisible by $d$.
2019 Serbia Team Selection Test
A trader owns horses of $3$ races, and exacly $b_j$ of each race (for $j=1,2,3$). He want to leave these horses heritage to his $3$ sons. He knowns that the boy $i$ for horse $j$ (for $i,j=1,2,3$) would pay $a_{ij}$ golds, such that for distinct $i,j$ holds holds $a_{ii}> a_{ij}$ and $a_{jj} >a_{ij}$.



Prove that there exists a natural number $n$ such that whenever it holds  $\min\{b_1,b_2,b_3\}>n$, trader can give the horses to their sons such that after getting the horses each son values his horses more than the other brother is getting, individually.
2019 Serbia Team Selection Test
Solve the equation in nonnegative integers:



$2^x=5^y+3$
2019 Serbia Team Selection Test
A figuric  is a convex polyhedron with $26^{5^{2019}}$ faces. On every face of a figuric we write down a number. When we throw two figurics (who don't necessarily have the same set of numbers on their sides) into the air, the figuric which falls on a side with the greater number wins; if this number is equal for both figurics, we repeat this process until we obtain a winner. Assume that a figuric has an equal probability of falling on any face. We say that one figuric rules over another if when throwing these figurics into the air, it has a strictly greater probability to win than the other figuric (it can be possible that given two figurics, no figuric rules over the other).

Milisav and Milojka both have a blank figuric. Milisav writes some (not necessarily distinct) positive integers on the faces of his figuric so that they sum up to $27^{5^{2019}}$. After this, Milojka also writes positive integers on the faces of her figuric so that they sum up to $27^{5^{2019}}$. Is it always possible for Milojka to create a figuric that rules over Milisav's?



Proposed by Bojan Basic
2018 Serbia Team Selection Test
Prove that there exists infinetly many natural number $n$ such that at least one of the numbers $2^{2^n}+1$ and $2018^{2^n}+1$ is not a prime.
2018 Serbia Team Selection Test
Let $n$ be a fixed positive integer and let $x_1,\ldots,x_n$ be positive real numbers. Prove that

$$x_1\left(1-x_1^2\right)+x_2\left(1-(x_1+x_2)^2\right)+\cdots+x_n\left(1-(x_1+...+x_n)^2\right)<\frac{2}{3}.$$
2018 Serbia Team Selection Test
Ana and Bob are playing the following game.



First, Bob draws triangle $ABC$ and a point $P$ inside it.

Then Ana and Bob alternate, starting with Ana, choosing three different permutations $\sigma_1$, $\sigma_2$ and $\sigma_3$ of $\{A, B, C\}$.

Finally, Ana draw a triangle $V_1V_2V_3$.



For $i=1,2,3$, let $\psi_i$ be the similarity transformation which takes $\sigma_i(A), \sigma_i(B)$ and $\sigma_i(C)$  to $V_i, V_{i+1}$ and $ X_i$ respectively (here $V_4=V_1$) where triangle $\Delta V_iV_{i+1}X_i$ lies on the outside of triangle $V_1V_2V_3$. Finally, let $Q_i=\psi_i(P)$. Ana wins if triangles $Q_1Q_2Q_3$ and $ABC$ are similar (in some order of vertices) and Bob wins otherwise. Determine who has the winning strategy.
2018 Serbia Team Selection Test
An isosceles trapezium is called right if only one pair of its sides are parallel (i.e parallelograms are not right).

A dissection of a rectangle into $n$ (can be different shapes) right isosceles trapeziums is called strict if the union of any $i,(2\leq i \leq n)$  trapeziums in the dissection do not form a right isosceles trapezium.

Prove that for any $n, n\geq 9$ there is a strict dissection of a $2017 \times 2018$ rectangle into $n$ right isosceles trapeziums.



Proposed by Bojan Basic
2018 Serbia Team Selection Test
Let $H $ be the orthocenter of $ABC $ ,$AB\neq AC $ ,and let $F $ be a point on circumcircle of $ABC $ such that $\angle AFH=90^{\circ} $.$K $ is the symmetric point of $H $ wrt $B $.Let $P $ be a point such that $\angle PHB=\angle PBC=90^{\circ} $,and $Q $ is the foot of $B $ to $CP $.Prove that $HQ $ is tangent to tge circumcircle of $FHK $.
2018 Serbia Team Selection Test
For any positive integer $n$, define

$$c_n=\min_{(z_1,z_2,...,z_n)\in\{-1,1\}^n} |z_1\cdot 1^{2018} + z_2\cdot 2^{2018} + ... + z_n\cdot n^{2018}|.$$Is the sequence $(c_n)_{n\in\mathbb{Z}^+}$ bounded?
2017 Serbia Team Selection Test
Let $ABC$ be a triangle and $D$ the midpoint of the side $BC$. Define points $E$ and $F$ on $AC$ and $B$, respectively, such that $DE=DF$ and $\angle EDF =\angle BAC$. Prove that $$DE\geq \frac {AB+AC} 4.$$
2017 Serbia Team Selection Test
Initally a pair $(x, y)$ is written on the board, such that exactly one of it's coordinates is odd. On such a pair we perform an operation to get pair $(\frac x 2, y+\frac x 2)$ if $2|x$ and $(x+\frac y 2, \frac y 2)$ if $2|y$. Prove that for every odd $n>1$ there is a even positive integer $b<n$ such that starting from the pair $(n, b)$ we will get the pair $(b, n)$ after finitely many operations.
2017 Serbia Team Selection Test
A function $f:\mathbb{N} \rightarrow \mathbb{N} $ is called nice if $f^a(b)=f(a+b-1)$, where $f^a(b)$ denotes $a$ times applied function $f$.

Let $g$ be a nice function, and an integer $A$ exists such that $g(A+2018)=g(A)+1$.

a) Prove that $g(n+2017^{2017})=g(n)$ for all $n \geq A+2$.

b) If $g(A+1) \neq g(A+1+2017^{2017})$ find $g(n)$ for $n <A$.
2017 Serbia Team Selection Test
We have an $n \times n$ square divided into unit squares. Each side of unit square is called unit segment. Some isoceles right triangles of hypotenuse $2$ are put on the square so all their vertices are also vertices of unit squares. For which $n$ it is possible that every unit segment belongs to exactly one triangle(unit segment belongs to a triangle even if it's on the border of the triangle)?
2017 Serbia Team Selection Test
Let $n \geq 2$ be a positive integer and $\{x_i\}_{i=0}^n$ a sequence such that not all of its elements are zero and there is a positive constant $C_n$ for which:

(i) $x_1+ \dots +x_n=0$, and

(ii) for each $i$ either $x_i\leq x_{i+1}$ or $x_i\leq x_{i+1} + C_n x_{i+2}$ (all indexes are assumed modulo $n$).

Prove that

a) $C_n\geq 2$, and

b) $C_n=2$ if and only $2 \mid n$.
2017 Serbia Team Selection Test
Let $k$ be a positive integer and let $n$ be the smallest number with exactly $k$ divisors. Given $n$ is a cube, is it possible that $k$ is divisible by a prime factor of the form $3j+2$?
2016 Serbia Additional Team Selection Test
Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:

$P_{n+1}=P_n(1+x)P_n(1-x)-1$.

Prove that $x^{2016}|P_{2016}(x)$.
2016 Serbia Additional Team Selection Test
Let $ABCD$ be a square with side $4$. Find, with proof, the biggest $k$ such that no matter how we place $k$ points into $ABCD$, such that they are on the interior but not on the sides, we always have a square with sidr length $1$, which is inside the square $ABCD$, such that it contains no points in its interior(they can be on the sides).
2016 Serbia Additional Team Selection Test
Let $w(x)$ be largest odd divisor of $x$. Let $a,b$ be natural numbers such that $(a,b)=1$ and

$a+w(b+1)$ and $b+w(a+1)$ are powers of two. Prove that $a+1$ and $b+1$ are powers of two.
2013 Serbia Additional Team Selection Test
We call polynomials $A(x) = a_n x^n +. . .+a_1 x+a_0$ and $B(x) = b_m x^m +. . .+b_1 x+b_0$

($a_n b_m \neq 0$) similar if the following conditions hold:

$(i)$ $n = m$;

$(ii)$ There is a permutation $\pi$ of the set $\{ 0, 1, . . . , n\} $ such that $b_i = a_{\pi (i)}$ for each $i \in {0, 1, . . . , n}$.

Let $P(x)$ and $Q(x)$ be similar polynomials with integer coefficients. Given that

$P(16) = 3^{2012}$, find the smallest possible value of $|Q(3^{2012})|$.



Proposed by Milos Milosavljevic
2013 Serbia Additional Team Selection Test
In an acute $\triangle ABC$ ($AB \neq AC$) with angle $\alpha$ at the vertex $A$, point $E$ is

the nine-point center, and $P$ a point on the segment $AE$. If $\angle ABP = \angle ACP = x$,

prove that $x = 90$° $ -2 \alpha $.



Proposed by Dusan Djukic
2013 Serbia Additional Team Selection Test
Let $p > 3$ be a given prime number. For a set $S \subseteq  \mathbb{Z}$ and $a \in  \mathbb{N}$ , define

$S_a  = \{ x \in \{ 0,1, 2,...,p-1 \}$ | $(\exists_s \in S) x \equiv_p a \cdot s \}$ .

$(a)$ How many sets $S \subseteq \{ 1, 2,...,p-1 \} $ are there for which the sequence

$S_1 , S_2 , ..., S_{p-1}$ contains exactly two distinct terms?

$(b)$ Determine all numbers $k \in  \mathbb{N}$ for which there is a set $ S \subseteq \{ 1, 2,...,p-1 \} $ such

that the sequence $S_1 , S_2 , ..., S_{p-1} $ contains exactly $k$ distinct terms.



Proposed by Milan Basic and Milos Milosavljevic
2012 Serbia Team Selection Test
Let $P(x)$ be a polynomial  of degree $2012$ with real coefficients satisfying the condition $$P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),$$ for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?
2012 Serbia Team Selection Test
Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.
2012 Serbia Team Selection Test
Let $P$ and $Q$ be points inside triangle $ABC$ satisfying $\angle PAC=\angle QAB$ and $\angle PBC=\angle QBA$.



a) Prove that feet of perpendiculars from $P$ and $Q$ on the sides of triangle $ABC$ are concyclic.



b) Let $D$ and $E$ be feet of perpendiculars from $P$ on the lines $BC$ and $AC$ and $F$ foot of perpendicular from $Q$ on $AB$. Let $M$ be intersection point of $DE$ and $AB$. Prove that $MP\bot CF$.
2009 Serbia Team Selection Test
Let $ \alpha$ and $ \beta$ be the angles of a non-isosceles triangle $ ABC$ at points $ A$ and $ B$, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in $ D$ and $ E$, respectively. Prove that the acute angle between the lines $ DE$ and $ AB$ isn't greater than $ \frac{|\alpha-\beta|}3$.
2009 Serbia Team Selection Test
Find the least number which is divisible by 2009 and its sum of digits is 2009.
2009 Serbia Team Selection Test
Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\ldots,S_n$ such that:



$ 1^\circ$ $ |S_i\cup S_j|\leq 2004$ for each two $ 1\leq i,j\le n$ and

$ 2^\circ$ $ S_i\cup S_j\cup S_k=\{1,2,\ldots,2008\}$ for each three integers $ 1\le i<j<k\le n$.
2009 Serbia Team Selection Test
Let $ x,y,z$ be positive real numbers such that $ xy + yz + zx = x + y + z$. Prove the inequality



$ \frac1{x^2 + y + 1} + \frac1{y^2 + z + 1} + \frac1{z^2 + x + 1}\le1$



When does the equality hold?
2009 Serbia Team Selection Test
Let $ k$ be the inscribed circle of non-isosceles triangle $ \triangle ABC$, which center is $ S$. Circle $ k$ touches sides $ BC,CA,AB$ in points $ P,Q,R$ respectively. Line $ QR$ intersects $ BC$ in point $ M$. Let a circle which contains points $ B$ and $ C$ touch $ k$ in point $ N$. Circumscribed circle of $ \triangle MNP$ intersects line $ AP$ in point $ L$, different from $ P$. Prove that points $ S,L$ and $ M$ are collinear.
2006 Serbia Team Selection Test
$$Problem 1 $$The set S = {1,2,3,...,2006} is partitioned into two disjoint subsets A and B

such that:

(i) 13 ∈ A;

(ii) if a ∈ A, b ∈ B, a+b ∈ S, then a+b ∈ B;

(iii) if a ∈ A, b ∈ B, ab ∈ S, then ab ∈ A.

Determine the number of elements of A
2006 Serbia Team Selection Test
$$problem 2$$:A point $P$ is taken in the interior of a right triangle$ ABC$ with $\angle C = 90$ such hat

$AP = 4, BP = 2$, and$ CP = 1$. Point $Q$ symmetric to $P$ with respect to $AC$ lies on

the circumcircle of triangle $ABC$. Find the angles of triangle $ABC$.
2006 Serbia Team Selection Test
Determine all natural numbers $n$ and $k > 1$ such that $k$ divides each of the numbers$\binom{n}{1}$,$\binom{n}{2}$,..........,$\binom{n}{n-1}$
2005 Serbia Team Selection Test
Prove that there is n rational number  $r$ such that   $cosr\pi=\frac{3}{5}$
2005 Serbia Team Selection Test
A convex angle $xOy$ and a point $M$ inside it are given in the plane. Prove that there is a unique point $P$ in the plane with the following property:

- For any line $l$ through $M$, meeting the rays $x$ and $y$ (or their extensions) at $X$ and $Y$, the angle $XPY$ is not obtuse.
2005 Serbia Team Selection Test
Find all polynomial with real coefficients such that:

P(x^2+1)=P(x)^2+1
2005 Serbia Team Selection Test
Let $T$ be the centroid of triangle $ABC$. Prove that $$ \frac 1{\sin \angle TAC} + \frac 1{\sin \angle TBC} \geq 4 $$
2005 Serbia Team Selection Test
Let $a,b,c$ be positive reals such that $abc=1$ .Prove the inequality $\frac{a}{a^2+2}+\frac{b}{b^2+2}+\frac{c}{c^2+2}\leq 1$
2005 Serbia Team Selection Test
We say that $ n$ squares in a $ n\times n$ board are scattered if no  two of them are in the same row or column.In every square of this board is witten a natural number so that the sum of numbrs in $ n$ scattered squares is always the same

and no row or no column contains two equal numbers .It turned out that the numbers on the main diagonal are arranged in the increasing order ,and that their product  is the smallest among all products of $ n$ scattered  numbers .Prove that scattered numbers with the greatest product are exactly those on the other diagonal.
2004 Serbia Team Selection Test
Let ABCD be a square and K

be a circle with diameter AB. For an

arbitrary point P on side CD, segments AP and BP meet K

again at

points M and N, respectively, and lines DM and CN meet at point Q.

Prove that Q lies on the circle K

and that AQ : QB = DP : PC.
2004 Serbia Team Selection Test
Let $a$, $b$ and $c$ be real numbers such that $abc=1$. Prove that the most two of numbers

$$2a-\frac{1}{b},\ 2b-\frac{1}{c},\ 2c-\frac{1}{a}$$

are greater than $1$.
2004 Serbia Team Selection Test
Let $P(x)$ be a polynomial of degree $n$ whose roots are $i-1, i-2,\cdot\cdot\cdot, i-n$ (where $i^2=-1$), and  let $R(x)$ and $S(x)$ be the polynomials with real coefficients such that $P(x)=R(x)+iS(x)$. Show that the polynomial $R$ has $n$ real roots. (R. Stanojevic)
2003 Serbia Team Selection Test
If $ p(x)$ is a polynomial, denote by $ p^n(x)$ the polynomial $ p(p(...(p(x))..)$, where $ p$ is iterated $ n$ times. Prove that the polynomial $ p^{2003}(x)-2p^{2002}(x)+p^{2001}(x)$ is divisible by $ p(x)-x$
2003 Serbia Team Selection Test
Let M and N be the distinct points in the plane of the triangle ABC such that AM : BM : CM = AN : BN : CN. Prove that the line MN contains the circumcenter of △ABC.
2003 Serbia Team Selection Test
Each edge and each diagonal of the convex $ n$-gon $ (n\geq 3)$ is colored in red or blue. Prove that the vertices of the $ n$-gon can be labeled as $ A_1,A_2,...,A_n$ in such a way that one of the following two conditions is satisfied:



(a) all segments $ A_1A_2,A_2A_3,...,A_{n-1}A_n,A_nA_1$ are of the same colour.

(b) there exists a number $ k, 1<k<n$ such that the segments $ A_1A_2,A_2A_3,...,A_{k-1}A_k$ are blue, and the segments $ A_kA_{k+1},...,A_{n-1}A_n,A_nA_1$ are red.
1999 Yugoslav Team Selection Test
For a natural number $n$, let $P(x)$ be the polynomial of $2n$−th degree such that:

$P(0) = 1$ and $P(k) = 2^{k-1}$ for $k = 1, 2, . . . , 2n$.

Prove that $2P(2n + 1) - P(2n + 2) = 1$.



P.S. I tried to prove it by firstly expressing this polynomial using Lagrange interpolation but get bored of computations - it seems like it can be done this way, but I'd like to see more 'clever' solution.  :)
1999 Yugoslav Team Selection Test
Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$. The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ in the line $BC$, let $X$ be the foot of the perpendicular from $A$ to $BE$, and let $Y$ be the midpoint of the segment $AX$. Let the line $BY$ intersect the circle $\omega$ again at $Z$.



Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$.



commentEdited by Orl.
1999 Yugoslav Team Selection Test
Consider the set $A_n=\{x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n\}$ of $2n$ variables. How many permutations of set $A_n$ are there for which it is possible to assign real values from the interval $(0,1)$ to the $2n$ variables so that:

(i) $x_i+y_i=1$ for each $i$;

(ii) $x_1<x_2<\ldots<x_n$;

(iii) the $2n$ terms of the permutation form a strictly increasing sequence?
1999 Yugoslav Team Selection Test
For a natural number $d$, $M_d$ denotes the set of natural numbers which are not representable as the sum of at least two consecutive terms of an arithmetic progression with the common difference d whose terms are integers. Prove that each $c\in M_3$ can be written in the form $c=ab$, where $a\in M_1$ and $b\in M_2\setminus\{2\}$.
1998 Yugoslav Team Selection Test
From a deck of playing cards, four threes, four fours and four fives are selected and put down on a table with the main side up. Players $A$ and $B$ alternately take the cards one by one and put them on the pile. Player $A$ begins. A player after whose move the sum of values of the cards on the pile is

(a) greater than 34;

(b) greater than 37;

loses the game. Which player has a winning strategy?
1998 Yugoslav Team Selection Test
In a convex quadrilateral $ABCD$, the diagonal $AC$ intersects the diagonal $BD$ at its midpoint $S$. The radii of incircles of triangles $ABS,BCS,CDS,DAS$ are $r_1,r_2,r_3,r_4$, respectively. Prove that

$$|r_1-r_2+r_3-r_4|\le\frac18|AB-BC+CD-DA|.$$
1998 Yugoslav Team Selection Test
Prove that there are no positive integers $n$ and $k\le n$ such that the numbers

$$\binom nk,\binom n{k+1},\binom n{k+2},\binom n{k+3}$$in this order form an arithmetic progression.
1997 Yugoslav Team Selection Test
Consider a regular $n$-gon $A_1A_2\ldots A_n$ with area $S$. Let us draw the lines $l_1,l_2,\ldots,l_n$ perpendicular to the plane of the $n$-gon at $A_1,A_2,\ldots,A_n$ respectively. Points $B_1,B_2,\ldots,B_n$ are selected on lines $l_1,l_2,\ldots,l_n$ respectively so that:



(i) $B_1,B_2,\ldots,B_n$ are all on the same side of the plane of the $n$-gon;

(ii) Points $B_1,B_2,\ldots,B_n$ lie on a single plane;

(iii) $A_1B_1=h_1,A_2B_2=h_2,\ldots,A_nB_n=h_n$.



Express the volume of polyhedron $A_1A_2\ldots A_nB_1B_2\ldots B_n$ as a function in $S,h_1,\ldots,h_n$.
1997 Yugoslav Team Selection Test
Given a natural number $k$, find the smallest natural number $C$ such that

$$\frac C{n+k+1}\binom{2n}{n+k}$$is an integer for every integer $n\ge k$.
1997 Yugoslav Team Selection Test
Numbers $1,2,\ldots,1997^2$ are written in the cells of a $1997\times1997$ table. It is allowed to apply the following transformations: exchange places of any two rows or any two columns, or reverse a row or column. (When a row or column is reversed, the first and last entry exchange their positions, so do the second and second last, etc.) Is it possible that, after finitely many such transformations, arbitrary two numbers exchange their positions and no other number changes its position?
1996 Yugoslav Team Selection Test
Let $\mathfrak F=\{A_1,A_2,\ldots,A_n\}$ be a collection of subsets of the set $S=\{1,2,\ldots,n\}$ satisfying the following conditions:



(a) Any two distinct sets from $\mathfrak F$ have exactly one element in common;

(b) each element of $S$ is contained in exactly $k$ of the sets in $\mathfrak F$.



Can $n$ be equal to $1996$?
1996 Yugoslav Team Selection Test
Let there be given a set of $1996$ equal circles in the plane, no two of them having common interior points. Prove that there exists a circle touching at most three other circles.
1996 Yugoslav Team Selection Test
The sequence $\{x_n\}$ is given by

$$x_n=\frac14\left(\left(2+\sqrt3\right)^{2n-1}+\left(2-\sqrt3\right)^{2n-1}\right),\qquad n\in\mathbb N.$$Prove that each $x_n$ is equal to the sum of squares of two consecutive integers.
1995 Yugoslav Team Selection Test
Determine all triples $(x,y,z)$ of positive rational numbers with $x\le y\le z$ such that $x+y+z,\frac1x+\frac1y+\frac1z$, and xyz are natural numbers.
1995 Yugoslav Team Selection Test
A natural number $n$ has exactly $1995$ units in its binary representation. Show that $n!$ is divisible by $2^{n-1995}$.
1995 Yugoslav Team Selection Test
Let $SABCD$ be a pyramid with the vertex $S$ whose all edges are equal. Points $M$ and $N$ on the edges $SA$ and $BC$ respectively are such that $MN$ is perpendicular to both $SA$ and $BC$. Find the ratios $SM:MA$ and $BN:NC$.
1992 Yugoslav Team Selection Test
Three squares $BCDE,CAFG$ and $ABHI$ are constructed outside the triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. Prove that $APQ$ is an isosceles right triangle.