Navvye/Olympiad-Problems-CSV huggingface dataset:
2008 Romanian Masters In Mathematics
Let $ ABC$ be an equilateral triangle and $ P$ in its interior. The distances from $ P$ to the triangle's sides are denoted by $ a^2, b^2,c^2$ respectively, where $ a,b,c>0$. Find the locus of the points $ P$ for which $ a,b,c$ can be the sides of a non-degenerate triangle.
2008 Romanian Masters In Mathematics
Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f=u+v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.
2008 Romanian Masters In Mathematics
Let $ a>1$ be a positive integer. Prove that every non-zero positive integer $ N$ has a multiple in the sequence $ (a_n)_{n\ge1}$, $ a_n=\left\lfloor\frac{a^n}n\right\rfloor$.
2008 Romanian Masters In Mathematics
Consider a square of sidelength $ n$ and $ (n+1)^2$ interior points. Prove that we can choose $ 3$ of these points so that they determine a triangle (eventually degenerated) of area at most $ \frac12$.
2020 Silk Road
Given a strictly increasing infinite sequence of natural numbers $ a_1, $ $ a_2, $ $ a_3, $ $ \ldots $. It is known that $ a_n \leq n + 2020 $ and the number $ n ^ 3 a_n - 1 $ is divisible by $ a_ {n + 1} $ for all natural numbers $ n $. Prove that $ a_n = n $ for all natural numbers $ n $.
2020 Silk Road
The triangle $ ABC $ is inscribed in the circle $ \omega $. Points $ K, L, M $ are marked on the sides $ AB, BC, CA $, respectively, and $ CM \cdot CL = AM \cdot BL $. Ray $ LK $ intersects line $ AC $ at point $ P $. The common chord of the circle $ \omega $ and the circumscribed circle of the triangle $ KMP $ meets the segment $ AM $ at the point $ S $. Prove that $ SK \parallel BC $.
2020 Silk Road
A polynomial $ Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0 $ with real coefficients is called  powerful  if the equality $ | k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n | $, and  non-increasing , if $ k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n $.

Let for the polynomial $ P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0 $ with nonzero real coefficients, where $ a_d> 0 $, the polynomial $ P (x) (x-1) ^ t (x + 1) ^ s $ is  powerful  for some non-negative integers $ s $ and $ t $ ($ s + t> 0 $). Prove that at least one of the polynomials $ P (x) $ and $ (- 1) ^ d P (-x) $ is  nonincreasing.
2020 Silk Road
Prove that for any natural number $ m $ there exists a natural number $ n $ such that any $ n $ distinct points on the plane can be partitioned into $ m $ non-empty sets whose convex hulls  have a common point.

The  convex hull  of a finite set $ X $ of points on the plane is the set of points lying inside or on the boundary of at least one convex polygon with vertices in $ X $, including degenerate ones, that is, the segment and the point are considered convex polygons. No three vertices of a convex polygon are collinear. The polygon contains its border.
2019 Silk Road
The altitudes of the acute-angled non-isosceles triangle $ ABC $ intersect at the point $ H $. On the segment $ C_1H $, where $ CC_1 $ is the altitude of the triangle, the point $ K $ is marked. Points $ L $ and $ M $ are the bases of perpendiculars from point $ K $ to straight lines $ AC $ and $ BC $, respectively. The lines $ AM $ and $ BL $ intersect at $ N $. Prove that $ \angle ANK = \angle HNL $.
2019 Silk Road
Let $ a_1, $ $ a_2, $ $ \ldots, $ $ a_ {99} $ be positive real numbers such that $ ia_j + ja_i \ge i + j $ for all $ 1 \le i <j \le 99. $

Prove , that $ (a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!$ .
2019 Silk Road
Find all pairs of $ (a, n) $ natural numbers such that $ \varphi (a ^ n + n) = 2 ^ n. $

($ \varphi (n) $ is the Euler function, that is, the number of integers from $1$ up to $ n $, relative prime to $ n $)
2019 Silk Road
The sequence $ \{a_n \} $ is defined as follows: $ a_0 = 1 $ and $ {a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}} $  for  $ n \ge 1. $

Prove that among $ a_1, a_2, \ldots, a_ {10 ^ 6} $ there are at least $500$ even numbers.

(Here, $ [x] $ is the largest integer not exceeding $ x $.)
2018 Silk Road
In an acute-angled triangle  $ABC$ on the sides $AB$, $BC$, $AC$   the points $H$, $L$, $K$ so that $CH \perp AB$, $HL \parallel AC$, $HK \parallel BC$. Let $P$ and $Q$ feet of altitudes of a triangle $HBL$, drawn from the vertices  $H$ and $B$ respectively. Prove that the feet of the altitudes of the triangle $AKH$, drawn from the vertices $A$ and   $H$ lie on the line $PQ$.
2018 Silk Road
Find all functions $f:\ \mathbb{R}\rightarrow\mathbb{R}$  such that for any real number $x$  the equalities are true:

$f\left(x+1\right)=1+f(x)$ and  $f\left(x^4-x^2\right)=f^4(x)-f^2(x).$



source
2018 Silk Road
Given the natural $n$. We shall call word  sequence from $n$ letters of the alphabet, and distance   $\rho(A, B)$  between words   $A=a_1a_2\dots a_n$ and $B=b_1b_2\dots b_n$ , the number of digits in which they differ (that is, the number of such $i$, for which $a_i\ne b_i$). We will say that the word  $C$ lies  between words  $A$ and   $B$ , if $\rho (A,B)=\rho(A,C)+\rho(C,B)$. What is the largest number of words  you can choose so that among any three, there is a word lying between the other two?
2018 Silk Road
Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer.



Here $\tau (n)$ denotes the number of positive divisor of $n$.
2017 Silk Road
On an infinite white checkered sheet, a square $Q$ of size $12$ × $12$ is selected. Petya wants to paint some (not necessarily all!) cells of the square with seven colors of the rainbow (each cell is just one color) so that no two of the $288$ three-cell rectangles whose centers lie in $Q$ are the same color. Will he succeed in doing this?

(Two three-celled rectangles are painted the same if one of them can be moved and possibly rotated so that each cell of it is overlaid on the cell of the second rectangle having the same color.)



(Bogdanov. I)
2017 Silk Road
The quadrilateral $ABCD$ is inscribed in the circle ω. The diagonals $AC$ and $BD$ intersect at the point $O$. On the segments $AO$ and $DO$, the points $E$ and $F$ are chosen, respectively. The straight line $EF$ intersects ω at the points $E_1$ and $F_1$. The circumscribed circles of the triangles $ADE$ and $BCF$ intersect the segment $EF$ at the points $E_2$ and $F_2$ respectively (assume that all the points $E, F, E_1, F_1, E_2$ and $F_2$ are different). Prove that $E_1E_2 = F_1F_2$.



$(N. Sedrakyan)$
2017 Silk Road
Prove that among any $42$ numbers from the interval $[1,10^6]$, you can choose four numbers so that for any permutation $(a, b, c, d)$ of these numbers, the inequality

$$25 (ab + cd) (ad + bc) \ge 16 (ac + bd)^ 2$$holds.
2017 Silk Road
Prove that for each prime $ P =9k+1$ ,exist natural n such that $P|n^3-3n+1$.
2016 Silk Road
Let $a,b$ and $c$ be real numbers such that $| (a-b) (b-c) (c-a) | = 1$. Find the smallest value of the expression $| a | + | b | + | c |$. (K.Satylhanov )
2016 Silk Road
Around the acute-angled triangle  $ABC$ ($AC>CB$)  a circle is circumscribed, and the point $N$ is midpoint of the arc $ACB$ of this circle. Let the points $A_1$  and  $B_1$ be the bases of perpendiculars on the straight line $NC$, drawn from points  $A$ and $B$  respectively (segment $NC$  lies inside the segment $A_1B_1$). Altitude $A_1A_2$ of triangle $A_1AC$  and altitude $B_1B_2$ of triangle $B_1BC$  intersect at a point $K$ . Prove that $\angle A_1KN=\angle B_1KM$, where $M$ is midpoint of the segment $A_2B_2$ .
2016 Silk Road
Given natural numbers $a,b$ and function  $f: \mathbb{N} \to \mathbb{N} $  such that for any natural  number  $n, f\left( n+a \right)$ is divided by $f\left( {\left[ {\sqrt n } \right] + b} \right)$. Prove that for any natural $n$ exist $n$ pairwise distinct and pairwise relatively prime natural numbers ${{a}_{1}}$, ${{a}_{2}}$, $\ldots$, ${{a}_{n}}$ such that the number   $f\left( {{a}_{i+1}} \right)$ is divided by $f\left( {{a}_{i}} \right)$  for each $i=1,2, \dots ,n-1$ .



(Here $[x]$ is the integer part of number $x$, that is, the largest integer not exceeding $x$.)
2016 Silk Road
Let $P(n)$   be the number of ways to split a natural number $n$  to the sum of powers of two, when the order does not matter. For example $P(5) = 4$, as $5=4+1=2+2+1=2+1+1+1=1+1+1+1+1$. Prove that for any natural the identity $P(n) + (-1)^{a_1} P(n-1) + (-1)^{a_2} P(n-2) + \ldots + (-1)^{a_{n-1}}

P(1) + (-1)^{a_n} = 0,$ is true, where  $a_k$ is the number of units in the binary number record $k$ .



source
2015 Silk Road
Prove that there is no positive real numbers $a,b,c,d$ such that both of the following equations hold.$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6 , \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}=32$$.
2015 Silk Road
Let  $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and  $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be  two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers  $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .
2015 Silk Road
Let  $B_n$ be the set of all sequences of length $n$, consisting of zeros and ones. For every two sequences  $a,b \in B_n$ (not necessarily different) we define strings  $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$

 \varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1). 

$$. Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find  $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $.

.
2015 Silk Road
Let O be a circumcenter of an acute-angled triangle ABC. Consider two circles ω and Ω inscribed in the angle BAC in such way that ω is tangent from the outside to the arc BOC of a circle circumscribed about the triangle BOC; and the circle Ω is tangent internally to a circumcircle of triangle ABC. Prove that the radius of Ω is twice the radius ω.
2015 Silk Road
Given positive real numbers $a,b,c,d$ such that



$ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6 \quad \text{and} \quad \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}=36.$



Prove the inequality



${{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}>ab+ac+ad+bc+bd+cd.$
2014 Silk Road
What is the maximum number of coins can be arranged in cells of the table $n \times n$ (each cell is not more of the one coin) so that any coin was not simultaneously below and to the right than any other?
2014 Silk Road
Let $w$ be the circumcircle of non-isosceles acute triangle $ABC$. Tangent lines to $w$ in $A$ and $B$ intersect at point $S$. Let M be the midpoint of $AB$, and $H$ be the orthocenter of triangle $ABC$. The line $HA$ intersects lines $CM$ and $CS$ at points $M_a$ and $S_a$, respectively. The points $M_b$ and $S_b$ are defined analogously. Prove that $M_aS_b$ and $M_bS_a$ are the altitudes of triangle $M_aM_bH$.
2014 Silk Road
$ a,b,c\ge 0,\ \ \ a^3+b^3+c^3+abc=4 $ Prove that

$a^3b+b^3c+c^3b \le 3$
2014 Silk Road
Find all $ f:N\rightarrow N$, such that $\forall m,n\in N $

$ 2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n) $
2013 Silk Road
Determine all pairs of positive integers $m, n,$ satisfying the equality $(2^{m}+1;2^n+1)=2^{(m;n)}+1$ , where $(a;b)$  is the greatest common divisor
2013 Silk Road
Circle with center  $I$, inscribed in a triangle $ABC$ , touches the sides $BC$ and  $AC$ at  points  $A_1$ and $B_1$ respectively. On rays $A_1I$  and $B_1I$, respectively,  let be the points  $A_2$ and  $B_2$ such that $IA_2=IB_2=R$, where $R$is the radius of the circumscribed circle of the triangle $ABC$. Prove that:

a) $AA_2 = BB_2 = OI$ where $O$ is the center of the circumscribed circle of the triangle $ABC$,

b) lines  $AA_2$ and $BB_2$ intersect on the circumcircle of the triangle $ABC$.
2013 Silk Road
Find all non-decreasing functions $ f\,:\,\mathbb{N}\to\mathbb{N} $, such that $f(f(m)f(n)+m)=f(mf(n))+f(m)$
2013 Silk Road
In the film there is $n$ roles. For each  $i$ ($1 \le i \le n$), the role of number $i$ can play $a_i$ a person, and one person can play only one role. Every day a casting is held, in which participate people for $n$ roles, and from each role only one person. Let $p$ be a prime number such that $p \ge a_1, \ldots, a_n, n$. Prove that you can have $p^k$ castings such that if we take any $k$ people who are tried in different roles, they together participated in some casting ($k$ is a natural number not exceeding $n$ ).
2012 Silk Road
Trapezium $ABCD$, where $BC||AD$, is inscribed in a circle,  $E$ is middle of the arc  $AD$  of this circle not containing point $C$ . Let $F$ be the base of the perpendicular dropped from   $E$ on the line tangent to the circle at the point $C$ . Prove that $BC=2CF$.
2012 Silk Road
In each cell of the table $4 \times 4$, in which the lines are labeled with numbers $1,2,3,4$, and columns with  letters $a,b,c,d$,  one number is written: $0$ or $1$ . Such a table is called valid  if there are exactly two units in each of its rows and in each column. Determine the number of valid  tables.
2012 Silk Road
Let $n > 1$ be an integer.

Determine the greatest common divisor of the set of numbers  $\left\{ \left( \begin{matrix}

   2n  \\
   2i+1  \\
\end{matrix} \right):0 \le i \le n-1 \right\}$

i.e. the largest positive integer, dividing  $\left( \begin{matrix}

   2n  \\
   2i+1  \\
\end{matrix} \right)$ without remainder for every $i = 0, 1, ..., n–1$ .

(Here $\left( \begin{matrix}

   m  \\
   l  \\
\end{matrix} \right)=\text{C}_{m}^{l}=\frac{m\text{!}}{l\text{!}\left( m-l \right)\text{!}}$ is binomial coefficient.)
2012 Silk Road
Prove that for any positive integer $n$, the arithmetic mean of $\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},\ldots ,\sqrt[n]{n}$ lies in $\left[ 1,1+\frac{2\sqrt{2}}{\sqrt{n}} \right]$ .
2011 Silk Road
Determine the smallest possible value of $| A_{1} \cup A_{2} \cup A_{3} \cup A_{4} \cup A_{5} |$, where $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ sets simultaneously satisfying the following conditions:

$(i)$ $| A_{i}\cap A_{j} | = 1$ for all $1\leq i < j\leq 5$, i.e. any two distinct sets contain exactly one element in common;

$(ii)$ $A_{i}\cap A_{j} \cap A_{k}\cap A_{l} =\varnothing$ for all $1\leq i<j<k<l\leq 5$, i.e. any four different sets contain no common element.

Where $| S |$ means the number of elements of $S$.
2011 Silk Road
Given an isosceles triangle $ABC$ with base $AB$. Point $K$ is taken on the extenstion of the side $AC$ (beyond the point $C$ ) so that  $\angle KBC = \angle ABC$. Denote  $S$ the  intersection point of angle - bisectors of $\angle BKC$ and $\angle ACB$. Lines  $AB$ and  $KS$ intersect at  point $L$,  lines $BS$ and  $CL$  intersect at point $M$ . Prove that line $KM$ passes through the middle of the segment $BC$.
2011 Silk Road
For all $a,b,c\in \bb{R}^+ $ such that $a+b+c=1$ and $ ( \frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} )(a-bc)(b-ac)(c-ab)\le M \cdot abc$. Find min $M$
2011 Silk Road
Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .
2010 Silk Road
In a convex quadrilateral  it is known $ABCD$ that  $\angle ADB + \angle ACB = \angle CAB + \angle DBA  = 30^{\circ}$  and $AD = BC$. Prove that from the lengths $DB$, $CA$ and  $DC$, you can make a right triangle.
2010 Silk Road
Let $N = 2010!+1$. Prove that

a) $N$ is not divisible by $4021$;

b) $N$ is not divisible by $2027,2029,2039$;

c)$ N$ has a prime divisor greater than $2050$.
2010 Silk Road
For positive real numbers $a, b, c, d,$ satisfying the following conditions:

$a(c^2 - 1)=b(b^2+c^2)$ and $d \leq 1$, prove that : $d(a \sqrt{1-d^2} + b^2 \sqrt{1+d^2}) \leq \frac{(a+b)c}{2}$
2010 Silk Road
In country there are two capitals ($A$ and $B$) and finite number of towns.

Some towns (or town with one of capital) connected with roads (one-way). (between every two towns or capital and town there are arbitrary number of roads) such that exist at least one way from $A$ to $B$.

Given, that any two ways from $A$ to $B$ have at least one common road.

Prove, that exist one road, such that all ways from $A$ to $B$ pass through this road.
2009 Silk Road
Prove that, abc≤1 and a,b,c>0 $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq 1+ \frac{6}{a+b+c} $$
2009 Silk Road
Bisectors of triangle ABC of an angles A and C intersect with BC and AB at points A1 and C1 respectively. Lines AA1 and CC1 intersect circumcircle of triangle ABC at points A2 and C2 respectively. K is intersection point of C1A2 and A1C2. I is incenter of ABC. Prove that the line KI divides AC into two equal parts.
2009 Silk Road
A tourist going to visit the Complant, found that:

a) in this country $1024$ cities, numbered by integers from $0$ to $1023$ ,

b) two cities with numbers $m$ and $n$ are connected by a straight line if and only if the binary entries of numbers $m$ and $n$ they differ exactly in one digit,

c) during the stay of a tourist in that country $8$ roads will be closed for scheduled repairs.

Prove that a tourist can make a closed route along the existing roads of Complant, passing through each of its cities exactly once.
2009 Silk Road
Prove that for any prime number $p$  there are infinitely many fours $(x, y, z, t)$ pairwise distinct natural numbers such that the number $(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)$ is a perfect square.
2008 Silk Road
Suppose $ a,c,d \in N$ and $ d|a^2b+c$ and $ d\geq a+c$

Prove that $ d\geq a+\sqrt[2b] {a}$
2008 Silk Road
In a triangle $ABC$ $A_0$,$B_0$ and $C_0$ are the midpoints of the sides $BC$,$CA$ and    $AB$.$A_1$,$B_1$,$C_1$ are the midpoints of the broken lines $BAC,CAB,ABC$.Show that  $A_0A_1,B_0B_1,C_0C_1$ are concurrent.
2008 Silk Road
Let $ G$ be a graph with $ 2n$ vertexes and $ 2n(n-1)$ edges.If we color some edge to red,then vertexes,which are connected by this edge,must be colored to red too. But not necessary that all edges from the red vertex are red.

Prove that it is possible to color some vertexes and edges in $ G$,such that all red vertexes has exactly $ n$ red edges.
2008 Silk Road
Find all polynomials $ P\in\mathbb{R}[x]$ such that for all $ r\in\mathbb{Q}$,there exist

$ d\in\mathbb{Q}$ such that $ P(d)=r$
2007 Silk Road
On the board are written $2 , 3 , 5 ,... , 2003$ , that is, all the prime numbers of the interval $[2,2007]$ . The operation of simplification  is the replacement of two numbers $a , b$ by a maximal prime number not exceeding  $\sqrt{a^2-a b+b^2}$  . First, the student erases the number $q, 2<q<2003$, then applies the simplification  operation to the remaining numbers until one number remains. Find the maximum possible and minimum possible values of the number obtained in the end. How do these values depend on the number $q$?
2007 Silk Road
Let $\omega$ be the incircle  of triangle $ABC$ touches $BC$  at point $K$ . Draw a circle passing through points $B$ and $C$ , and touching  $\omega$ at the point $S$ . Prove that $S K$ passes through the center of the exscribed circle of triangle $A B C$ , tangent to side $B C$ .
2007 Silk Road
Find the max. value of $ M$,such that for all $ a,b,c>0$:

$ a^{3}+b^{3}+c^{3}-3abc\geq M(|a-b|^{3}+|a-c|^{3}+|c-b|^{3})$
2007 Silk Road
The set of polynomials $f_1, f_2, \ldots, f_n$  with real coefficients is called special , if for any different  $i,j,k \in \{ 1,2, \ldots, n\}$ polynomial  $\dfrac{2}{3}f_i + f_j + f_k$  has no real roots, but for any different  $p,q,r,s \in \{ 1,2, \ldots, n\}$ of a polynomial $f_p + f_q + f_r + f_s$ there is a real root.

a) Give an example of a special  set of four polynomials whose sum is not a zero polynomial.

b) Is there a special  set of five polynomials?
2006 Silk Road
Found all functions $f: \mathbb{R} \to \mathbb{R}$, such that for any $x,y \in \mathbb{R}$,

$$f(x^2+xy+f(y))=f^2(x)+xf(y)+y.$$
2006 Silk Road
For positive $a,b,c$, such that $abc=1$ prove the inequality:

$4(\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}) \leq 3(2+a+b+c+\frac{1}{a}+\frac{1}{b}+ \frac{1}{c})^{\frac{2}{3}}$.
2006 Silk Road
A subset $S$ of the set $M=\{1,2,.....,p-1\}$,where $p$ is a prime number of the kind

$12n+11$,is essential,if the product ${\Pi}_s$ of all elements of the subset

is not less than the product $\bar{{\Pi}_s}$ of all other elements of the set.The

difference $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$ is called the deviation

of the subset $S$.Define the least possible remainder of division by $p$ of the deviation of an essential subset,containing $\frac{p-1}{2}$ elements.
2006 Silk Road
A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain

parallel lines and it doesn't contain three lines with a common point.We say that

the line $l_1\in L$ is  bounding the line $l_2\in L$,if all intersection points

of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$.

Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following

2 conditions are satisfied simultaneously:

1) The line $l$ is bounding the line $l'$;

2) the line $l'$ is not bounding the line $l$.
2005 Silk Road
Let $n \geq 2$ be natural number.

Prove, that $(1^{n-1}+2^{n-1}+....+(n-1)^{n-1})+1$ divided by $n$ iff for any prime divisor $p$ of $n$ $p| \frac{n}{p}-1 $ and $(p-1)| \frac{n}{p}-1$.
2005 Silk Road
Find all $(m,n) \in \mathbb{Z}^2$ that we can color each unit square of $m \times n$ with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even.
2005 Silk Road
Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$

are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$
2005 Silk Road
Suppose $\{a(n) \}_{n=1}^{\infty}$ is a sequence that:

$$ a(n) =a(a(n-1))+a(n-a(n-1)) \  \  \  \forall \  n \geq 3$$

and $a(1)=a(2)=1$.



Prove that for each $n \geq 1$ , $a(2n) \leq 2a(n)$.
2004 Silk Road
Find all $ f: \mathbb{R} \to \mathbb{R}$, such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all real $x,y$.
2004 Silk Road
find all primes $p$, for which exist natural numbers, such that $p=m^2+n^2$ and $p|(m^3+n^3-4)$.
2004 Silk Road
In-circle of $ABC$ with center $I$ touch $AB$ and $AC$ at $P$ and $Q$ respectively. $BI$ and $CI$ intersect $PQ$ at $K$ and $L$ respectively. Prove, that circumcircle of $ILK$ touch incircle of $ABC$ iff $|AB|+|AC|=3|BC|$.
2004 Silk Road
Natural $n \geq 2$ is given. Group of people calls $n-compact$, if for any men from group, we can found $n$ people (without he), each two of there are familiar.

Find maximum $N$ such that for any $n-compact$ group, consisting $N$ people contains subgroup from $n+1$ people, each of two of there are familiar.
2003 Silk Road
Let $a_1, a_2, ....., a_{2003}$ be sequence of reals number.

Call $a_k$ $leading$ element, if at least one of expression $a_k; a_k+a_{k+1}; a_k+a_{k+1}+a_{k+2}; ....; a_k+a{k+1}+a_{k+2}+....+a_{2003}$ is positive.

Prove, that if exist at least one $leading$ element, then sum of all $leading$'s elements is positive.



Official solution here
2003 Silk Road
Let $s=\frac{AB+BC+AC}{2}$ be half-perimeter of triangle $ABC$. Let $L$ and      $N$be a point's on ray's $AB$ and $CB$, for which $AL=CN=s$. Let $K$ is point, symmetric of point $B$ by circumcenter of $ABC$. Prove, that perpendicular from $K$ to $NL$ passes through incenter of $ABC$.



Solution for problem here