Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 Belarus Team Selection Test
Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$.



Proposed by Estonia
2015 Belarus Team Selection Test
Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has $$|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.$$ Determine all possible values of $P(0)$.



Proposed by Belgium
2015 Belarus Team Selection Test
Let $n \ge 2$ be an integer, and let $A_n$ be the set $$A_n = \{2^n  - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.$$ Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .



Proposed by Serbia
2015 Belarus Team Selection Test
Define the function $f:(0,1)\to (0,1)$ by $$\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \  x < \frac 12\\ x^2 & \text{if}\ \  x \ge \frac 12 \end{array} \right.$$ Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that $$(a_n - a_{n-1})(b_n-b_{n-1})<0.$$



Proposed by Denmark
2015 Belarus Team Selection Test
Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot  CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle.



Proposed by Jack Edward Smith, UK
2015 Belarus Team Selection Test
We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ .



Proposed by Abbas Mehrabian, Iran
2015 Belarus Team Selection Test
Given a cyclic $ABCD$ with $AB=AD$. Points $M$ and $N$ are marked on the sides $CD$ and $BC$, respectively, so that $DM+BN=MN$. Prove that the circumcenter of the triangle $AMN$ belongs to the segment $AC$.



N.Sedrakian
2015 Belarus Team Selection Test
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by $$a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,$$ are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)



Proposed by Hong Kong
2015 Belarus Team Selection Test
Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite.

Prove that exist distinct  primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$.



Tuymaada Olympiad 2004, C.A.Grimm. USA
2015 Belarus Team Selection Test
In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.
2015 Belarus Team Selection Test
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying $$f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014$$ for all integers $m$ and $n$.



Proposed by Netherlands
2011 Belarus Team Selection Test
Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$.



I.Voronovich
2011 Belarus Team Selection Test
Points $L$ and $H$ are marked on the sides $AB$ of an acute-angled triangle ABC so that $CL$ is a bisector and $CH$ is an altitude. Let $P,Q$ be the feet of the perpendiculars from $L$ to $AC$ and $BC$ respectively. Prove that $AP \cdot BH = BQ \cdot AH$.



I. Gorodnin
2011 Belarus Team Selection Test
Any natural number $n, n\ge 3$ can be presented in different ways as a sum several summands (not necessarily different).  Find the greatest possible value of these summands.



Folklore
2011 Belarus Team Selection Test
Given a $n \times n$ square table. Exactly one beetle sits in each cell of the table. At $12.00$ all beetles creeps to some neighbouring cell (two cells are neighbouring if they have the common side). Find the greatest number of cells which can become empty (i.e. without beetles) if

a) $n=8$

b) $n=9$



Problem Committee of BMO 2011
2011 Belarus Team Selection Test
Is it possible to arrange the numbers $1,2,...,2011$ over the circle in some order so that among any $25$ successive numbers at least $8$ numbers are multiplies of $5$ or $7$ (or both $5$ and $7$) ?



I. Gorodnin
2011 Belarus Team Selection Test
Two different points $X,Y$ are marked on the side $AB$ of a triangle $ABC$ so that  $\frac{AX \cdot BX}{CX^2}=\frac{AY \cdot BY}{CY^2}$ . Prove that $\angle ACX=\angle BCY$.



I.Zhuk
2011 Belarus Team Selection Test
Find all functions $f: R \to R ,g: R \to R$ satisfying the following equality $f(f(x+y))=xf(y)+g(x)$ for all real $x$ and $y$.



I. Gorodnin
2011 Belarus Team Selection Test
Given nonzero real numbers a,b,c with $a+b+c=a^2+b^2+c^2=a^3+b^3+c^3$. ($*$)

a) Find $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}\right)(a+b+c-2)$

b) Do there exist pairwise different nonzero $a,b,c$ satisfying ($*$)?



D. Bazylev
2011 Belarus Team Selection Test
Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with

$a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$).

Prove that

a) $(a-1)\vdots p_i$ for some $i=1,..,n$

b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$?



I. Bliznets
2011 Belarus Team Selection Test
The external angle bisector of the angle $A$ of an acute-angled triangle $ABC$ meets the circumcircle of $\vartriangle ABC$ at point $T$. The perpendicular from the orthocenter $H$ of $\vartriangle ABC$ to the line $TA$ meets the line $BC$ at point $P$. The line $TP$ meets the circumcircce of  $\vartriangle ABC$ at point $D$. Prove that $AB^2+DC^2=AC^2+BD^2$



A. Voidelevich
2011 Belarus Team Selection Test
In a concert, 20 singers will perform. For each singer, there is a (possibly empty) set of other singers such that he wishes to perform later than all the singers from that set. Can it happen that there are exactly 2010 orders of the singers such that all their wishes are satisfied?



Proposed by Gerhard Wöginger, Austria
2011 Belarus Team Selection Test
Let $A$ be the sum of all $10$ distinct products of the sides of a convex pentagon, $S$ be the area of the pentagon.

a) Prove thas $S \le \frac15 A$.

b) Does there exist a constant $c<\frac15$ such that $S \le cA$ ?



I.Voronovich
2011 Belarus Team Selection Test
Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that

$$ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.$$



Proposed by Daniel Brown, Canada
2011 Belarus Team Selection Test
Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$



(a) Find a pair $(a,b)$ which is 51-good, but not very good.

(b) Show that all 2010-good pairs are very good.





Proposed by Okan Tekman, Turkey
2011 Belarus Team Selection Test
Let $g(n)$ be the number of all $n$-digit natural numbers each consisting only of digits $0,1,2,3$ (but not nessesarily all of them) such that the sum of no two neighbouring digits equals $2$. Determine whether $g(2010)$ and $g(2011)$ are divisible by $11$.



I.Kozlov
2011 Belarus Team Selection Test
Positive real $a,b,c$ satisfy the condition $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=1+\frac{1}{6}\left( \frac{a}{c}+\frac{b}{a}+\frac{c}{b} \right)$$Prove that $$\frac{a^3bc}{b+c}+\frac{b^3ca}{a+c}+\frac{c^3ab}{a+b}\ge \frac{1}{6}(ab+bc+ca)^2$$I.Voronovich
2011 Belarus Team Selection Test
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$



Proposed by Christopher Bradley, United Kingdom
2011 Belarus Team Selection Test
$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola.



I.Voronovich
2011 Belarus Team Selection Test
Do they exist natural numbers $m,x,y$ such that

$$m^2 +25 \vdots (2011^x-1007^y) ?$$S. Finskii
2011 Belarus Team Selection Test
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that



(i) no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);

(ii) each row and each column contains exactly 25 kings.



Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)



Proposed by Sergei Berlov, Russia
2011 Belarus Team Selection Test
In an acute-angled triangle $ABC$, the orthocenter is $H$. $I_H$ is the incenter of $\vartriangle BHC$. The bisector of $\angle BAC$ intersects the perpendicular from $I_H$ to the side $BC$ at point $K$. Let $F$ be the foot of the perpendicular from $K$ to $AB$. Prove that $2KF+BC=BH +HC$



A. Voidelevich
2011 Belarus Team Selection Test
Find all pairs $(m,n)$ of nonnegative integers for which $$m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).$$



Proposed by Angelo Di Pasquale, Australia
2011 Belarus Team Selection Test
Find all functions $f:R\to R$ such that for all real $x,y$ with $y\ne 0$ $$f(x-f(x/y))=xf(1-f(1/y))$$and

a) $f(1-f(1))\ne 0$

b) $ f(1-f(1))= 0$



S. Kuzmich, I.Voronovich
2011 Belarus Team Selection Test
Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square.



N. Sedrakian , I.Voronovich
2011 Belarus Team Selection Test
Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively,

$$\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.$$if

a)   $X_1, \ldots , X_n$ are the midpoints of the corressponding sides,

b)  $X_1, \ldots , X_n$ are the feet of the corressponding altitudes,

c)  $X_1, \ldots , X_n$ are arbitrary  points on the corressponding lines.



Modified version of IMO 2010 SL G3 (it was question c)
2011 Belarus Team Selection Test
Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$





Proposed by Sergei Berlov, Ilya Bogdanov, Russia
2009 Belarus Team Selection Test
Two equal circles $S_1$ and $S_2$ meet at two different points. The line $\ell$ intersects $S_1$ at points $A,C$ and $S_2$ at points $B,D$ respectively  (the order on $\ell$: $A,B,C,D$) . Define circles $\Gamma_1$ and $\Gamma_2$ as follows: both $\Gamma_1$ and $\Gamma_2$ touch $S_1$ internally and $S_2$ externally, both $\Gamma_1$ and $\Gamma_2$ line $\ell$, $\Gamma_1$ and $\Gamma_2$ lie in the different halfplanes relatively to line $\ell$. Suppose that $\Gamma_1$ and $\Gamma_2$ touch each other. Prove that $AB=CD$.



I. Voronovich
2009 Belarus Team Selection Test
Find all $n \in N$ for which the value of the expression $x^n+y^n+z^n$ is constant for all $x,y,z \in R$ such that $x+y+z=0$ and $xyz=1$.



D. Bazylev
2009 Belarus Team Selection Test
Points $T,P,H$ lie on the side $BC,AC,AB$ respectively of triangle $ABC$, so that $BP$ and $AT$ are angle bisectors and $CH$ is an altitude of $ABC$. Given that the midpoint of $CH$ belongs to the segment $PT,$ find the value of $\cos A + \cos B$



I. Voronovich
2009 Belarus Team Selection Test
Let $x,y,z$ be integer numbers satisfying the equality $yx^2+(y^2-z^2)x+y(y-z)^2=0$

a) Prove that number $xy$ is a perfect square.

b) Prove that there are infinitely many triples $(x,y,z)$ satisfying the equality.



I.Voronovich
2009 Belarus Team Selection Test
Prove that any positive real numbers a,b,c satisfy the inequlaity $$\frac{1}{(a+b)b}+\frac{1}{(b+c)c}+\frac{1}{(c+a)a}\ge \frac{9}{2(ab+bc+ca)}$$

I.Voronovich
2009 Belarus Team Selection Test
a) Prove that there is not an infinte sequence  $(x_n)$, $n=1,2,...$ of positive real numbers satisfying the relation

$x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}}$, $\forall n \in N$ (*)

b) Do there exist sequences satisfying (*) and containing arbitrary many terms?



I.Voronovich
2009 Belarus Team Selection Test
Given trapezoid $ABCD$ ($AD\parallel BC$) with $AD \perp AB$ and $T=AC\cap BD$.

A circle centered at point $O$ is inscribed in the trapezoid and touches the side $CD$ at point $Q$.

Let $P$ be the intersection point (different from $Q$) of the side $CD$ and the circle passing through $T,Q$ and $O$. Prove that $TP \parallel AD$.



I. Voronovich
2009 Belarus Team Selection Test
Given a graph with $n$ ($n\ge 4$) vertices . It is known that for any two vertices $A$ and $B$ there exists a vertex which is connected by edges both with $A$ and $B$. Find the smallest possible numbers of edges in the graph.



E. Barabanov
2009 Belarus Team Selection Test
Find all functions $f: R \to R$ and $g:R \to R$ such that $f(x-f(y))=xf(y)-yf(x)+g(x)$ for all real numbers $x,y$.



I.Voronovich
2009 Belarus Team Selection Test
Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE = \angle QDE$ and $ \angle PBE = \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic.



Proposed by John Cuya, Peru
2009 Belarus Team Selection Test
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations $$ a^n + pb = b^n + pc = c^n + pa$$ then $a = b = c$.



Proposed by Angelo Di Pasquale, Australia
2009 Belarus Team Selection Test
Prove that there exist many natural numbers n so that both roots of the quadratic equation $x^2+(2-3n^2)x+(n^2-1)^2=0$ are perfect squares.



S. Kuzmich
2009 Belarus Team Selection Test
In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$.



Proposed by Davood Vakili, Iran
2009 Belarus Team Selection Test
Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$,

$K=AN \cap BM$, $L=CM \cap DN$. Find the smallest possible $c\in R$ such that $S(MKNL)<c \cdot S(ABCD)$ for any convex quadrilateral $ABCD$.



I. Voronovich
2009 Belarus Team Selection Test
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i + a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.



Proposed by Mohsen Jamaali, Iran
2009 Belarus Team Selection Test
Find all real numbers $a$ for which there exists a function  $f: R \to R$ asuch that $x + f(y) =a(y + f(x))$ for all real numbers $x,y\in R$.



I.Voronovich
2009 Belarus Team Selection Test
In a triangle $ABC, AM$ is a median, $BK$ is a bisectrix, $L=AM\cap BK$. It is known that $BC=a, AB=c, a>c$.

Given that the circumcenter of triangle $ABC$ lies on the line $CL$, find $AC$



I. Voronovich
2009 Belarus Team Selection Test
In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-friends if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-clique if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements.



Proposed by Jorge Tipe, Peru
2009 Belarus Team Selection Test
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i + 1}) > a_{i - 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.



Proposed by Morteza Saghafian, Iran
2009 Belarus Team Selection Test
Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE = \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$.



Proposed by Charles Leytem, Luxembourg
2009 Belarus Team Selection Test
Let $ S = \{x_1, x_2, \ldots, x_{k + l}\}$ be a $ (k + l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called nice if

$$ \left |\frac {1}{k}\sum_{x_i\in A} x_i - \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k + l}{2kl}$$

Prove that the number of nice subsets is at least $ \dfrac{2}{k + l}\dbinom{k + l}{k}$.



Proposed by Andrey Badzyan, Russia
2009 Belarus Team Selection Test
On R a binary algebraic operation ''*'' is defined which satisfies the following two conditions:

i)  for all $a,b \in R$, there exists a unique $x \in R$ such that $x *a=b$ (write $x=b/a$)

ii)  $(a*b)*c= (a*c)* (b*c)$ for all $a,b,c \in R$

a) Is this operation necesarily commutative (i.e. $a*b=b*a$  for all $a,b \in R$) ?

b) Prove that $(a/b)/c = (a/c) / (b/c)$ and $(a/b)*c = (a*c) / (b*c)$  for all $a,b,c \in R$.



A. Mirotin
2009 Belarus Team Selection Test
Does there exist a convex pentagon $A_1A_2A_3A_4A_5$ and a point $X$ inside it such that $XA_i=A_{i+2}A_{i+3}$ for all $i=1,...,5$ (all indices are considered modulo $5$) ?



I. Voronovich
2009 Belarus Team Selection Test
a) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \in N$?

b) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \in N$?



I. Voronovich
2000 Belarus Team Selection Test
Find the minimal number of cells on a $5\times 7$ board that must be painted so that any cell which is not painted has exactly one neighboring (having a common side) painted cell.
2000 Belarus Team Selection Test
Let $P$ be a point inside a triangle $ABC$ with $\angle C = 90^o$ such that $AP = AC$, and let $M$ be the midpoint of $AB$ and $CH$ be the altitude. Prove that $PM$ bisects $\angle BPH$ if and only if $\angle A = 60^o$.
2000 Belarus Team Selection Test
Does there exist a function $f : N\to N$ such that $f ( f (n-1)) = f (n+1)- f (n)$ for all $n \ge 2$?
2000 Belarus Team Selection Test
A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$.

Prove that $\ell \le  \sin \frac{2\pi}{5}$

.
2000 Belarus Team Selection Test
All vertices of a convex polyhedron are endpoints of exactly four edges. Find the minimal possible number of triangular faces of the polyhedron.
2000 Belarus Team Selection Test
Real numbers $a$, $b$, $c$ satisfy the equation $$2a^3-b^3+2c^3-6a^2b+3ab^2-3ac^2-3bc^2+6abc=0$$. If $a<b$, find which of the numbers $b$, $c$ is larger.
2000 Belarus Team Selection Test
In the Cartesian plane, two integer points $(a_1, b_1$ and $(a_2, b_2)$ are connected if $(a_2, b_2)$ is one of the points $(-a_1, b_1 \pm 1)$, $(a_1 \pm 1,-b_1)$. Show that there exists an infinite sequence of integer points in which every integer point occurs, and every two consecutive points are connected.
2000 Belarus Team Selection Test
In a triangle ABC with $AC = b \ne BC = a$, points $E,F$ are taken on the sides $AC,BC$ respectively such that $AE = BF =\frac{ab}{a+b}$. Let $M$ and $N$ be the midpoints of $AB$ and $EF$ respectively, and $P$ be the intersection point of the segment $EF$ with the bisector of $\angle ACB$. Find the ratio of the area of $CPMN$ to that of $ABC$.
2000 Belarus Team Selection Test
In a triangle $ABC$, let $a = BC, b = AC$ and let $m_a,m_b$ be the corresponding medians.

Find all real numbers $k$ for which the equality $m_a+ka = m_b +kb$ implies that $a = b$.
2000 Belarus Team Selection Test
(a) Prove that $\{n\sqrt3\} >\frac{1}{n\sqrt3}$ for any positive integer $n$.

(b) Is there a constant $c > 1$ such that $\{n\sqrt3\} >\frac{c}{n\sqrt3}$ for all $n \in N$?
2000 Belarus Team Selection Test
Each edge of a graph with $15$ vertices is colored either red or blue in such a way that no three vertices are pairwise connected with edges of the same color. Determine the largest possible number of edges in the graph.
2000 Belarus Team Selection Test
Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$
2000 Belarus Team Selection Test
Let ABC be a triangle and $M$ be an interior point. Prove that



$$ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.$$
2000 Belarus Team Selection Test
Prove that for every real number $M$ there exists an infinite arithmetic progression such that:



- each term is a positive integer and the common difference is not divisible by 10

-  the sum of the digits of each term (in decimal representation) exceeds $M$.
2000 Belarus Team Selection Test
Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$).

If  $BC = a, AM = m_a, AL = l_a$, prove the inequalities:

(a) $a\tan \frac{a}{2} \le  2m_a  \le  a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$  and   $a\tan \frac{a}{2}  \ge  2m_a \ge  a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$

(b) $2l_a \le  a\cot \frac{a}{2}  $.
2000 Belarus Team Selection Test
Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.
2000 Belarus Team Selection Test
Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.
2000 Belarus Team Selection Test
Find the smallest natural number $n$ for which it is possible to partition the set $M = \{1,2, ... ,40\}$ into n subsets $M_1, . . . ,M_n$ so that none of the $M_i$ contains elements $a,b,c$ (not necessarily distinct) with $a+b = c$.
2000 Belarus Team Selection Test
A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le  A_1 \le A_0$.

Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.