Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 Switzerland Team Selection Test
Let $(a,b)$ be a pair of natural numbers. Henning and Paul play the following game. At the beginning there are two piles of $a$ and $b$ coins respectively. We say that $(a,b)$ is the starting position of the game. Henning and Paul play with the following rules:

$\bullet$ They take turns alternatively where Henning begins.

$\bullet$ In every step each player either takes a positive integer number of coins from one of the two piles or takes same natural number of coins from both piles.

$\bullet$  The player how take the last coin wins.

Let $A$ be the set of all positive integers like $a$ for which there exists a positive integer $b<a$ such that Paul has a wining strategy for the starting position $(a,b)$. Order the elements of $A$ to construct a sequence $a_1<a_2<a_3<\dots$

$(a)$ Prove that $A$ has infinity many elements.

$(b)$ Prove that the sequence defined by $m_k:=a_{k+1}-a_{k}$ will never become periodic. (This means the sequence $m_{k_0+k}$ will not be periodic for any choice of $k_0$)
2019 Switzerland Team Selection Test
Prove that for all positive integers $n$ there are positive integers $a,b$ such that $$n\mid 4a^2+9b^2-1.$$
2019 Switzerland Team Selection Test
Let $k,n,r$ be positive integers and $r<n$. Quirin owns $kn+r$ black and $kn+r$ white socks. He want to clean his cloths closet such there does not exist $2n$ consecutive socks $n$ of which black and the other $n$ white. Prove that

he can clean his closet in the desired manner if and only if $r\geq k$ and $n>k+r$.
2019 Switzerland Team Selection Test
Let $ABC$ be an acute triangle with $AB<AC$. $E,F$ are foots of the altitudes drawn from $B,C$ respectively. Let $M$ be the midpoint of segment $BC$. The tangent at $A$ to the circumcircle of $ABC$ cuts $BC$ in $P$ and $EF$ cuts the parallel to $BC$ from $A$ at $Q$. Prove that $PQ$ is perpendicular to $AM$.
2019 Switzerland Team Selection Test
Let $n \geq 5$ be an integer. A shop sells balls in $n$ different colors. Each of $n + 1 $ children bought three balls with different colors, but no two children bought exactly the same color combination. Show that there are at least two children who bought exactly one ball of the same color.
2019 Switzerland Team Selection Test
Let $n $ be a positive integer. Determine whether there exists a positive real number $\epsilon >0$ (depending on $n$) such that for all positive real numbers $x_1,x_2,\dots ,x_n$, the inequality $$\sqrt[n]{x_1x_2\dots x_n}\leq (1-\epsilon)\frac{x_1+x_2+\dots+x_n}{n}+\epsilon \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\dots +\frac{1}{x_n}},$$holds.
2016 Switzerland Team Selection Test
Let $n$ be a natural number. Two numbers  are called  "unsociable" if their greatest common divisor is $1$. The numbers $\{1,2,...,2n\}$ are partitioned into $n$ pairs. What is the minimum number of "unsociable" pairs that are formed?
2016 Switzerland Team Selection Test
Find all polynomial functions with real coefficients for which $$(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)$$for all real $x$
2016 Switzerland Team Selection Test
Find all integers $n \geq 1$ such that for all $x_1,...,x_n \in \mathbb{R}$ the following inequality is satisfied

$$\left(\frac{x_1^n+...+x_n^n}{n}-x_1....x_n\right)\left(x_1+...+x_n\right) \geq 0$$
2016 Switzerland Team Selection Test
Find all positive integers $n$ such that $$\sum_{d|n, 1\leq d <n}d^2=5(n+1)$$
2016 Switzerland Team Selection Test
Let $ABC$ be a triangle with $AB  \neq AC$ and let $M$ be the middle of $BC$. The bisector of $\angle BAC$ intersects the line $BC$ in $Q$. Let $H$ be the foot of $A$ on $BC$. The perpendicular to $AQ$ passing through $A$ intersects the line $BC$ in $S$. Show that $MH \times QS=AB \times AC$.
2016 Switzerland Team Selection Test
Find all functions $f : \mathbb{R} \mapsto  \mathbb{R} $ such that

$$ \left(f(x)+y\right)\left(f(x-y)+1\right)=f\left(f(xf(x+1))-yf(y-1)\right)$$for all $x,y \in \mathbb{R}$
2016 Switzerland Team Selection Test
Let $ABC$ be a non-rectangle triangle with $M$ the middle of $BC$. Let $D$ be a point on the line $AB$ such that $CA=CD$ and let $E$ be a point on the line $BC$ such that $EB=ED$. The parallel to $ED$ passing through $A$ intersects the line $MD$ at the point $I$ and the line $AM$ intersects the line $ED$ at the point $J$. Show that the points $C, I$ and $J$ are aligned.
2016 Switzerland Team Selection Test
In an EGMO exam, there are three exercises, each of which can yield a number of points between $0$ and $7$. Show that, among the $49$ participants, one can always find two such that the first in each of the three tasks was at least as good as the other.
2015 Switzerland Team Selection Test
What is the maximum number of 1 × 1 boxes that can be colored black in a n × n chessboard so that any 2 × 2 square contains a maximum of 2 black boxes?
2015 Switzerland Team Selection Test
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that

$$ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. $$
2015 Switzerland Team Selection Test
Let $ABC$ be a triangle with $AB> AC$. Let $D$ be a point on $AB$ such that $DB = DC$ and $M$ the middle of $AC$. The parallel to $BC$ passing through $D$ intersects the line $BM$ in $K$. Show that $\angle KCD = \angle DAC$.
2015 Switzerland Team Selection Test
Find all relatively prime integers $a,b$  such that $$a^2+a=b^3+b$$
2015 Switzerland Team Selection Test
Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$
2015 Switzerland Team Selection Test
Find all finite and non-empty sets $A$ of functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that for all $f_1, f_2 \in A$, there exists $g \in A$ such that for all $x, y \in \mathbb{R}$

$$f_1 \left(f_2 (y)-x\right)+2x=g(x+y)$$
2015 Switzerland Team Selection Test
Let $n \geq 2$ be a positive integer. At the center of a circular garden is a guard tower. On the outskirt of the garden there are $n$ garden dwarfs regularly spaced. In the tower are attentive supervisors. Each supervisor controls a portion of the garden delimited by two dwarfs.

We say that the supervisor $A$ controls the supervisor $B$ if the region of $B$ is contained in that of $A$.

Among the supervisors there are two groups: the apprentices and the teachers. Each apprentice is controlled by exactly one teachers, and controls no one, while the teachers are not controlled by anyone.

The entire garden has the following maintenance costs:

- One apprentice costs 1 gold per year.

- One teacher costs 2 gold per year.

- A garden dwarf costs 2 gold per year.

Show that the garden dwarfs cost at least as much as the supervisors.
2015 Switzerland Team Selection Test
Let $ABCD$ be a parallelogram. Suppose that there exists a point $P$ in the interior of the parallelogram which is on the perpendicular bisector of $AB$ and such that $\angle PBA = \angle ADP$

Show that $\angle CPD = 2 \angle BAP$
2015 Switzerland Team Selection Test
In Thailand there are $n$ cities. Each pair of cities is connected by a one-way street which can be borrowed, depending on its type, only by bike or by car. Show that there is a city from which you can reach any other city, either by bike or by car.

Remark : It is not necessary to use the same means of transport for each city
2006 Switzerland Team Selection Test
In the triangle $A,B,C$, let $D$ be the middle of $BC$ and $E$ the projection of $C$ on $AD$. Suppose $\angle ACE = \angle ABC$. Show that the triangle $ABC$ is isosceles or rectangle.
2006 Switzerland Team Selection Test
Let $n\ge5$ be an integer. Find the biggest integer $k$ such that there always exists a $n$-gon with exactly $k$ interior right angles. (Find $k$ in terms of $n$).
2006 Switzerland Team Selection Test
Let $n$ be natural number. Each of the numbers $\in\{1,2,\ldots ,n\}$ is coloured in black or white. When we choose a number, we flip it's colour and the colour of all the numbers which have at least one common divider with the chosen number. At the beginning all the numbers were coloured white. For which $n$ are all the numbers black after a finite number of changes?
2006 Switzerland Team Selection Test
Let $n$ be natural number and $1=d_1<d_2<\ldots <d_k=n$ be the positive divisors of $n$.



Find all $n$ such that $2n = d_5^2+ d_6^2 -1$.
2006 Switzerland Team Selection Test
Let $D$ be inside $\triangle ABC$ and $E$ on $AD$ different to $D$. Let $\omega_1$ and $\omega_2$ be the circumscribed circles of $\triangle BDE$ and $\triangle CDE$ respectively. $\omega_1$ and $\omega_2$ intersect $BC$ in the interior points $F$ and $G$ respectively. Let $X$ be the intersection between $DG$ and $AB$ and $Y$ the intersection between $DF$ and $AC$. Show that $XY$ is $\|$ to $BC$.
2006 Switzerland Team Selection Test
Find all the functions $f : \mathbb{R} \to \mathbb{R}$ satisfying for all $x,y \in \mathbb{R}$ $f(f(x)-y^2) = f(x)^2 - 2f(x)y^2 + f(f(y))$.
2006 Switzerland Team Selection Test
The three roots of $P(x) = x^3 - 2x^2 - x + 1$ are $a>b>c \in \mathbb{R}$. Find the value of $a^2b+b^2c+c^2a$. :D
2006 Switzerland Team Selection Test
We place randomly the numbers $1,2, \dots ,2006$ around a circle. A move consists of changing two neighbouring numbers. After a limited numbers of moves all the numbers are diametrically opposite to their starting position. Show that we changed at least once two numbers which had the sum $2007$.
2006 Switzerland Team Selection Test
Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.
2006 Switzerland Team Selection Test
Let $a,b,c \in \mathbb{R^+}$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show $\sqrt{ab+c} + \sqrt{bc+a} + \sqrt{ca+b} \ge \sqrt{abc} + \sqrt{a} + \sqrt{b} + \sqrt{c}$. :D
2006 Switzerland Team Selection Test
Find all naturals $k$ such that $3^k+5^k$ is the power of a natural number with the exponent $\ge 2$.
2006 Switzerland Team Selection Test
An airport contains 25 terminals which are two on two connected by tunnels. There is exactly 50 main tunnels which can be traversed in the two directions, the others are with single direction. A group of four terminals is called good if of each terminal of the four we can arrive to the 3 others by using only the tunnels connecting them. Find the maximum number of good groups.
2004 Switzerland Team Selection Test
Let $S$ be the set of all n-tuples $(X_1,...,X_n)$ of subsets of the set $\{1,2,..,1000\}$, not necessarily different and not necessarily nonempty. For $a = (X_1,...,X_n)$ denote by $E(a)$ the number of elements of $X_1\cup ... \cup X_n$. Find an explicit formula for the sum $\sum_{a\in S} E(a)$
2004 Switzerland Team Selection Test
Find the largest natural number $n$ for which $4^{995} +4^{1500} +4^n$ is a square.
2004 Switzerland Team Selection Test
Second Test, May 16



Let $a$, $b$, and $c$ be positive real numbers such that $abc = 1$. Prove that

$\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca}\le 1$ .

When does equality hold?
2004 Switzerland Team Selection Test
A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.
2004 Switzerland Team Selection Test
Find all finite sequences $(x_0, x_1, \ldots,x_n)$ such that for every $j$, $0 \leq j \leq n$, $x_j$ equals the number of times $j$ appears in the sequence.
2004 Switzerland Team Selection Test
The real numbers $a,b,c,d$ satisfy the equations:

$a =\sqrt{45-\sqrt{21-a}}, b =\sqrt{45+\sqrt{21-b}}, c =\sqrt{45-\sqrt{21+c}}, d=\sqrt{45+\sqrt{21+d}}$.

Prove that $abcd = 2004$.
2004 Switzerland Team Selection Test
Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows:

$$x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}$$Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .



Proposed by Marcin Kuczma, Poland
2004 Switzerland Team Selection Test
Let $A_{1}, ..., A_{n}$ be different subsets of an $n$-element set $X$. Show that there exists $x\in X$ such that the sets

$A_{1}-\{x\}, A_{2}-\{x\}, ..., A_{n}-\{x\}$ are all different.
2004 Switzerland Team Selection Test
In an acute-angled triangle $ABC$ the altitudes $AU,BV,CW$ intersect at $H$.

Points $X,Y,Z$, different from $H$, are taken on segments $AU,BV$, and $CW$, respectively.

(a) Prove that if $X,Y,Z$ and $H$ lie on a circle, then the sum of the areas of triangles $ABZ, AYC, XBC$ equals the area of $ABC$.

(b) Prove the converse of (a).
2004 Switzerland Team Selection Test
Find all injective functions $f : R \to R$ such that for all real $x \ne y$ , $f\left(\frac{x+y}{x-y}\right) = \frac{f(x)+ f(y)}{f(x)- f(y)}$
2004 Switzerland Team Selection Test
Find all natural numbers which can be written in the form $\frac{(a+b+c)^2}{abc}$ , where $a,b,c \in N$.
2003 Switzerland Team Selection Test
Real numbers $x,y,a$ satisfy the equations $x+y = x^3 +y^3 = x^5 +y^5 = a$.

Find all possible values of $a$.
2003 Switzerland Team Selection Test
In an acute-angled triangle $ABC, E$ and $F$ are the feet of the altitudes from $B$ and $C$, and $G$ and $H$ are the projections of $B$ and $C$ on $EF$, respectively. Prove that $HE = FG$.
2003 Switzerland Team Selection Test
Find the largest real number $ C_1 $ and the smallest real number $ C_2 $, such that, for all reals $ a,b,c,d,e $, we have $$ C_1 < \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a} < C_2 $$
2003 Switzerland Team Selection Test
Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.
2003 Switzerland Team Selection Test
There are $n$ pieces on the squares of a $5 \times 9$ board, at most one on each square at any time during the game. A move in the game consists of simultaneously moving each piece to a neighboring square by side, under the restriction that a piece having been moved horizontally in the previous move must be moved vertically and vice versa. Find the greatest value of $n$ for which there exists an initial position starting at which the game can be continued until the end of the world.
2003 Switzerland Team Selection Test
Let $ a,b,c $ be positive real numbers satisfying $ a+b+c=2 $. Prove the inequality $$ \frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca} \ge \frac{27}{13} $$
2003 Switzerland Team Selection Test
Find all polynomials $Q(x)= ax^2+bx+c$ with integer coefficients for which there exist three different prime numbers $p_1, p_2, p_3$ such that $|Q(p_1)| = |Q(p_2)| = |Q(p_3)| = 11$.
2003 Switzerland Team Selection Test
Let $A_1A_2A_3$ be a triangle and $\omega_1$ be a circle passing through $A_1$ and $A_2$.

Suppose that there are circles $\omega_2,...,\omega_7$ such that:

(a) $\omega_k$ passes through $A_k$ and $A_{k+1}$ for $k = 2,3,...,7$, where $A_i = A_{i+3}$,

(b) $\omega_k$ and $\omega_{k+1}$ are externally tangent for $k = 1,2,...,6$.

Prove that $\omega_1 = \omega_7$.
2003 Switzerland Team Selection Test
Given integers $0 < a_1 < a_2 <... < a_{101} < 5050$, prove that one can always choose for different numbers $a_k,a_l,a_m,a_n$ such that $5050 | a_k +a_l -a_m -a_n$
2003 Switzerland Team Selection Test
Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.
2002 Switzerland Team Selection Test
In space are given $24$ points, no three of which are collinear. Suppose that there are exactly $2002$ planes determined by three of these points. Prove that there is a plane containing at least six points.
2002 Switzerland Team Selection Test
A point$ O$ inside a parallelogram $ABCD$ satisfies $\angle AOB + \angle COD = \pi$. Prove that $\angle CBO = \angle CDO$.
2002 Switzerland Team Selection Test
Let $d_1,d_2,d_3,d_4$ be the four smallest divisors of a positive integer $n$ (having at least four divisors). Find all $n$ such that $d_1^2+d_2^2+d_3^2+d_4^2 = n$.
2002 Switzerland Team Selection Test
A $7 \times 7$ square is divided into unit squares by lines parallel to its sides. Some Swiss crosses (obtained by removing corner unit squares from a square of side $3$) are to be put on the large square, with the edges along division lines. Find the smallest number of unit squares that need to be marked in such a way that every cross covers at least one marked square.
2002 Switzerland Team Selection Test
Find all $f: R\rightarrow R$ such that

(i) The set $\{\frac{f(x)}{x}| x\in R-\{0\}\}$ is finite

(ii) $f(x-1-f(x)) = f(x)-1-x$ for all $x$
2002 Switzerland Team Selection Test
A sequence $x_1,x_2,x_3,...$ has the following properties:

(a) $1 = x_1 < x_2 < x_3 < ...$

(b) $x_{n+1} \le 2n$ for all $n \in N$.

Prove that for each positive integer $k$ there exist indices $i$ and $j$ such that $k =x_i -x_j$.
2002 Switzerland Team Selection Test
In a group of $n$ people, every weekend someone organizes a party in which he invites all of his acquaintances. Those who meet at a party become acquainted. After each of the $n$ people has organized a party, there still are two people not knowing each other. Show that these two will never get to know each other at such a party.
2002 Switzerland Team Selection Test
For each real number $a$ and integer $n \ge 1$ prove the inequality $a^n +\frac{1}{a^n} -2 \ge n^2 \left(a +\frac{1}{a} -2\right)$ and find the cases of equality.
2002 Switzerland Team Selection Test
Given an integer $m\ge 2$, find the smallest integer $k > m$ such that for any partition of the set $\{m,m + 1,..,k\}$ into two classes $A$ and $B$ at least one of the classes contains three numbers $a,b,c$ (not necessarily distinct) such that $a^b = c$.
2001 Switzerland Team Selection Test
The $2001 \times 2001$ trees in a park form a square grid. What is the largest number of trees that can be cut so that no tree stump can be seen from any other? (Each tree has zero width.)
2001 Switzerland Team Selection Test
If $a,b$, and $c$ are the sides of a triangle, prove the inequality $\sqrt{a+b-c}+\sqrt{c+a-b}+\sqrt{b+c-a } \le \sqrt{a}+\sqrt{b}+\sqrt{c}$.

When does equality occur?
2001 Switzerland Team Selection Test
In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.
2001 Switzerland Team Selection Test
For a natural number $n \ge 2$, consider all representations of $n$ as a sum of its distinct divisors, $n = t_1 + t_2 + ... + t_k, t_i| n$. Two such representations differing only in order of the summands are considered the same (for example, $20 = 10+5+4+1$ and $20 = 5+1+10+4$). Let $a(n)$ be the number of different representations of $n$ in this form. Prove or disprove: There exists M such that $a(n) \le M$ for all $n \ge 2$.
2001 Switzerland Team Selection Test
Let $a_1 < a_2 < ... < a_n$ be a sequence of natural numbers such that for $i < j$ the decimal representation of $a_i$ does not occur as the leftmost digits of the decimal representation of $a_j$ . (For example, $137$ and $13729$ cannot both occur in the sequence.) Prove that $\sum_{i=1}^n \frac{1}{a_i} \le 1+\frac12 +\frac13 +...+\frac19$

.
2001 Switzerland Team Selection Test
A function $f : [0,1] \to R$ has the following properties:

(a) $f(x) \ge 0$ for $0 < x < 1$,

(b) $f(1) = 1$,

(c) $f(x+y) \ge f(x)+ f(y) $ whenever $x,y,x+y \in [0,1]$.

Prove that $f(x) \le 2x$ for all $x \in [0,1]$.
2001 Switzerland Team Selection Test
Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circle $S$ through $A,B$, and $O$ intersects $AC$ and $BC$ again at points $P$ and $Q$ respectively. Prove that $CO \perp PQ$.
2001 Switzerland Team Selection Test
Find two smallest natural numbers $n$ for which each of the fractions

$\frac{68}{n+70},\frac{69}{n+71},\frac{70}{n+72},...,\frac{133}{n+135}$ is irreducible.
2001 Switzerland Team Selection Test
In Geneva there are $16$ secret agents, each of whom is watching one or more other agents. It is known that if agent $A$ is watching agent $B$, then $B$ is not watching $A$. Moreover, any $10$ agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any $11$ agents can also be so ordered.
2001 Switzerland Team Selection Test
Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.
2000 Switzerland Team Selection Test
A convex quadrilateral $ABCD$ is inscribed in a circle. Show that the line connecting the midpoints of the arcs $AB$ and $CD$ and the line connecting the midpoints of the arcs $BC$ and $DA$ are perpendicular.
2000 Switzerland Team Selection Test
Real numbers $a_1,a_2,...,a_{16}$ satisfy the conditions $\sum_{i=1}^{16}a_i = 100$ and $\sum_{i=1}^{16}a_i^2 = 1000$ .

What is the greatest possible value of $a_16$?
2000 Switzerland Team Selection Test
An equilateral triangle of side $1$ is covered by five congruent equilateral triangles of side $s < 1$ with sides parallel to those of the larger triangle. Show that some four of these smaller triangles also cover the large triangle.
2000 Switzerland Team Selection Test
Let $q(n)$ denote the sum of the digits of a natural number $n$. Determine $q(q(q(2000^{2000})))$.
2000 Switzerland Team Selection Test
Consider all words of length $n$ consisting of the letters $I,O,M$.

How many such words are there, which contain no two consecutive $M$’s?
2000 Switzerland Team Selection Test
Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$.

Can number $7$ on the right hand side be replaced with a smaller constant?
2000 Switzerland Team Selection Test
Show that the equation $14x^2 +15y^2 = 7^{2000}$ has no integer solutions.
2000 Switzerland Team Selection Test
Let $f(x) = \frac{4^x}{4^x+2}$ for $x > 0$. Evaluate $\sum_{k=1}^{1920}f\left(\frac{k}{1921}\right)$
2000 Switzerland Team Selection Test
Two given circles $k_1$ and $k_2$ intersect at points $P$ and $Q$.

Construct a segment $AB$ through $P$ with the endpoints at $k_1$ and $k_2$ for which $AP \cdot PB$ is maximal.
2000 Switzerland Team Selection Test
At $n$ distinct points of a circular race course there are $n$ cars ready to start. Each car moves at a constant speed and covers the circle in an hour. On hearing the initial signal, each of them selects a direction and starts moving immediately. If two cars meet, both of them change directions and go on without loss of speed. Show that at a certain moment each car will be at its starting point.
2000 Switzerland Team Selection Test
The vertices of a regular $2n$-gon ($n \ge 3$) are labelled with the numbers $1,2,...,2n$ so that the sum of the numbers at any two adjacent vertices equals the sum of the numbers at the vertices diametrically opposite to them. Show that this is only possible if $n$ is odd.
2000 Switzerland Team Selection Test
Find all functions $f : R \to R$ such that for all real $x,y$, $f(f(x)+y) = f(x^2 -y)+4y f(x)$
2000 Switzerland Team Selection Test
The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. Let $P$ be an internal point of triangle $ABC$ such that the incircle of triangle $ABP$ touches $AB$ at $D$ and the sides $AP$ and $BP$ at $Q$ and $R$. Prove that the points $E,F,R,Q$ lie on a circle.
2000 Switzerland Team Selection Test
The polynomial $P$ of degree $n$ satisfies $P(k) = \frac{k}{k +1}$ for $k = 0,1,2,...,n$. Find $P(n+1)$.
2000 Switzerland Team Selection Test
Let $S = \{P_1,P_2,...,P_{2000}\}$ be a set of $2000$ points in the interior of a circle of radius $1$, one of which at its center. For $i = 1,2,...,2000$ denote by $x_i$ the distance from $P_i$ to the closest point $P_j \ne P_i$. Prove that $x_1^2 +x_2^2  +...+x_{2000}^2<9$ .