Navvye/Olympiad-Problems-CSV huggingface dataset:
2010 China Second Round Olympiad
let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let

$A_k=\frac{\sum_{i=1}^{k}a_i}{k}$

prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$.
2010 China Second Round Olympiad
the code system of a new 'MO lock' is a regular $n$-gon,each vertex labelled a number $0$ or $1$ and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide.

find the number of all possible codes(in terms of $n$).
2009 China Second Round Olympiad
Let $\omega$ be the circumcircle of acute triangle $ABC$ where $\angle A<\angle B$ and $M,N$ be the midpoints of minor arcs $BC,AC$ of $\omega$ respectively. The line $PC$ is parallel to $MN$, intersecting $\omega$ at $P$ (different from $C$). Let $I$ be the incentre of $ABC$ and let $PI$ intersect $\omega$ again at the point $T$.

1) Prove that $MP\cdot MT=NP\cdot NT$;

2) Let $Q$ be an arbitrary point on minor arc $AB$ and $I,J$ be the incentres of triangles $AQC,BCQ$. Prove that $Q,I,J,T$ are concyclic.
2009 China Second Round Olympiad
Let $n$ be a positive integer. Prove that

$$-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}$$
2009 China Second Round Olympiad
Let $k,l$ be two given integers. Prove that there exist infinite many integers $m\ge k$ such that $\gcd\left(\binom{m}{k},l\right)=1$.
2009 China Second Round Olympiad
Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$. The following conditions are given:



Every row consists of distinct numbers;

$\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$;

$x_{17}=x_{28}=x_{39}=0$;

$x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$.

The first three columns of $P$ satisfy the following property $(R)$: for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$, $1\le k\le 9$, there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$.



Prove that:

a) the elements $u_1,u_2,u_3$ come from three different columns;

b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$, where $l\ge 4$, satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$, then this column uniquely exists.
2008 China Second Round Olympiad
Given a convex quadrilateral with $\angle B+\angle D<180$.Let $P$ be an arbitrary point on the plane,define

$f(P)=PA*BC+PD*CA+PC*AB$.

(1)Prove that $P,A,B,C$ are concyclic when $f(P)$ attains its minimum.

(2)Suppose that $E$ is a point on the minor arc $AB$ of the circumcircle $O$ of $ABC$,such that$AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB$.Knowing that $DA,DC$ are tangent to circle $O$,$AC=\sqrt 2$,find the minimum of $f(P)$.
2008 China Second Round Olympiad
Let $f(x)$ be a periodic function with periods $T$ and $1$($0<T<1$).Prove that:

(1)If $T$ is rational,then there exists a prime $p$ such that $\frac{1}{p}$ is also a period of $f$;

(2)If $T$ is irrational,then there exists a strictly decreasing infinite sequence {$a_n$},with $1>a_n>0$ for all positive integer $n$,such that all $a_n$ are periods of $f$.
2008 China Second Round Olympiad
For all $k=1,2,\ldots,2008$,$a_k>0$.Prove that iff $\sum_{k=1}^{2008}a_k>1$,there exists a function $f:N\rightarrow R$ satisfying

(1)$0=f(0)<f(1)<f(2)<\ldots$;

(2)$f(n)$ has a finite limit when $n$ approaches infinity;

(3)$f(n)-f({n-1})=\sum_{k=1}^{2008}a_kf({n+k})-\sum_{k=0}^{2007}a_{k+1}f({n+k})$,for all $n=1,2,3,\ldots$.
2007 China Second Round Olympiad
In an acute triangle $ABC$, $AB<AC$. $AD$ is the altitude dropped onto $BC$ and $P$ is a point on $AD$. Let $PE\perp AC$ at $E$, $PF\perp AB$ at $F$ and let $J,K$ be the circumcentres of triangles $BDF, CDE$ respectively. Prove that $J,K,E,F$ are concyclic if and only if $P$ is the orthocentre of triangle $ABC$.
2007 China Second Round Olympiad
In a $7\times 8$ chessboard, $56$ stones are placed in the squares. Now we have to remove some of the stones such that after the operation, there are no five adjacent stones horizontally, vertically or diagonally. Find the minimal number of stones that have to be removed.
2007 China Second Round Olympiad
For positive integers $k,m$, where $1\le k\le 5$, define the function $f(m,k)$ as

$$f(m,k)=\sum_{i=1}^{5}\left[m\sqrt{\frac{k+1}{i+1}}\right]$$

where $[x]$ denotes the greatest integer not exceeding $x$. Prove that for any positive integer $n$, there exist positive integers $k,m$, where $1\le k\le 5$, such that $f(m,k)=n$.
2006 China Second Round Olympiad
An ellipse with foci $B_0,B_1$ intersects $AB_i$ at $C_i$ $(i=0,1)$. Let $P_0$ be a point on ray $AB_0$. $Q_0$ is a point on ray $C_1B_0$ such that $B_0P_0=B_0Q_0$; $P_1$ is on ray $B_1A$ such that $C_1Q_0=C_1P_1$; $Q_1$ is on ray $B_1C_0$ such that $B_1P_1=B_1Q_1$; $P_2$ is on ray $AB_0$ such that $C_0Q_1=C_0Q_2$. Prove that $P_0=P_2$ and that the four points $P_0,Q_0,Q_1,P_1$ are concyclic.
2006 China Second Round Olympiad
Let $x,y$ be real numbers. Define a sequence $\{a_n \}$ through the recursive formula

$$ a_0=x,a_1=y,a_{n+1}=\frac{a_na_{n-1}+1}{a_n+a_{n-1}},$$

Find $a_n$.
2006 China Second Round Olympiad
Solve the system of equations in real numbers:

$$ \begin{cases} x-y+z-w=2 \\ x^2-y^2+z^2-w^2=6 \\ x^3-y^3+z^3-w^3=20 \\ x^4-y^4+z^4-w^4=66 \end{cases} $$
2006 China Second Round Olympiad
Let $\triangle ABC$ be a given triangle. If $|BA-tBC| \ge |AC|$ for any $t \in \mathbb{R}$, then $\triangle ABC$ is

$ \textbf{(A)}\ \text{an acute triangle}\qquad\textbf{(B)}\ \text{an obtuse triangle}\qquad\textbf{(C)}\ \text{a right triangle}\qquad\textbf{(D)}\ \text{not known}\qquad$
2006 China Second Round Olympiad
Suppose $log_x (2x^2+x-1)>log_x 2-1$. Then the range of $x$ is

${ \textbf{(A)}\ \frac{1}{2}<x<1\qquad\textbf{(B)}\  x>\frac{1}{2}   \text{and}  x \not= 1\qquad\textbf{(C)}\ x>1\qquad\textbf{(D)}}\ 0<x<1\qquad $
2006 China Second Round Olympiad
Suppose $A = {x|5x-a \le 0}$, $B = {x|6x-b > 0}$, $a,b \in \mathbb{N}$, and $A \cap B \cap \mathbb{N} = {2,3,4}$. The number of such pairs $(a,b)$ is

${ \textbf{(A)}\ 20\qquad\textbf{(B)}\  25\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}} 42\qquad $
2006 China Second Round Olympiad
Given a right triangular prism $A_1B_1C_1 - ABC$ with $\angle BAC = \frac{\pi}{2}$ and $AB = AC = AA_1$, let $G$, $E$ be the midpoints of $A_1B_1$, $CC_1$ respectively, and $D$, $F$ be variable points lying on segments $AC$, $AB$ (not including endpoints) respectively. If $GD \bot EF$, the range of the length of $DF$ is

${ \textbf{(A)}\ [\frac{1}{\sqrt{5}}, 1)\qquad\textbf{(B)}\  [\frac{1}{5}, 2)\qquad\textbf{(C)}\ [1, \sqrt{2})\qquad\textbf{(D)}} [\frac{1}{\sqrt{2}}, \sqrt{2})\qquad $
2006 China Second Round Olympiad
Suppose $f(x) = x^3 + \log_2(x + \sqrt{x^2+1})$. For any $a,b \in \mathbb{R}$, to satisfy $f(a) + f(b) \ge 0$, the condition $a + b \ge 0$ is



$ \textbf{(A)}\ \text{necessary and sufficient}\qquad\textbf{(B)}\ \text{not necessary but sufficient}\qquad\textbf{(C)}\ \text{necessary but not sufficient}\qquad$

$\textbf{(D)}\ \text{neither necessary nor sufficient}\qquad$
2006 China Second Round Olympiad
Let $S$ be the set of all those 2007 place decimal integers $\overline{2s_1a_2a_3 \ldots a_{2006}}$ which contain odd number of digit $9$ in each sequence $a_1, a_2, a_3, \ldots, a_{2006}$. The cardinal number of $S$ is



${ \textbf{(A)}\ \frac{1}{2}(10^{2006}+8^{2006})\qquad\textbf{(B)}\ \frac{1}{2}(10^{2006}-8^{2006})\qquad\textbf{(C)}\ 10^{2006}+8^{2006}\qquad

\textbf{(D)}}\ 10^{2006}-8^{2006}\qquad $
2006 China Second Round Olympiad
Let $f(x)=\sin^4x-\sin x\cos x+cos^4 x$. Find the range of $f(x)$.
2006 China Second Round Olympiad
Let complex number $z = (a+\cos\theta)+(2a-\sin \theta)i$. Find the range of real number $a$ if $|z|\ge 2$ for any $\theta\in \mathbb{R}$.
2006 China Second Round Olympiad
Suppose points $F_1, F_2$ are the left and right foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{4}=1$ respectively, and point $P$ is on line $l:$, $x-\sqrt{3} y+8+2\sqrt{3}=0$. Find the value of ratio $\frac{|PF_1|}{|PF_2|}$ when $\angle F_1PF_2$ reaches its maximum value.
2006 China Second Round Olympiad
Suppose four solid iron balls are placed in a cylinder with the radius of 1 cm, such that every two of the four balls are tangent to each other, and the two balls in the lower layer are tangent to the cylinder base. Now put water into the cylinder. Find, in $\text{cm}^2$, the volume of water needed to submerge all the balls.
2006 China Second Round Olympiad
Find the number of real solutions to the equation $(x^{2006}+1)(1+x^2+x^4+\ldots +x^{2004})=2006x^{2005}$
2006 China Second Round Olympiad
Suppose there are 8 white balls and 2 red balls in a packet. Each time one ball is drawn and replaced by a white one. Find the probability that the last red ball is drawn in the fourth draw.
2006 China Second Round Olympiad
Given an integer $n\ge 2$, define $M_0 (x_0, y_0)$ to be an intersection point of the parabola $y^2=nx-1$ and the line $y=x$. Prove that for any positive integer $m$, there exists an integer $k\ge 2$ such that $(x^m_0, y^m_0)$ is an intersection point of $y^2=mx-1$ and the line $y=x$.
2006 China Second Round Olympiad
Let $2006$ be expressed as the sum of five positive integers $x_1, x_2, x_3, x_4, x_5$, and $S=\sum_{1\le i<j\le 5}x_ix_j$.

$ \textbf{(A)}$ What value of $x_1, x_2, x_3, x_4, x_5$ maximizes $S$?

$ \textbf{(A)}$ Find, with proof, the value of $x_1, x_2, x_3, x_4, x_5$ which minimizes of $S$ if $|x_i-x_j|\le 2$ for any $1\le i$, $j\le 5$.
2006 China Second Round Olympiad
Suppose $f(x)=x^2+a$. Define $f^1(x)=f(x)$, $f^n(x)=f(f^{n-1}(x))$, $n=2, 3, \cdots$, and let $M=\{a\in\mathbb{R}| |f^n(0)|\le 2, \text{for any  } n\in\mathbb{N}\} $. Prove that $M=[-2, \frac{1}{4}]$.
2005 China Second Round Olympiad
In $\triangle ABC$, $AB>AC$, $l$ is a tangent line of the circumscribed circle of $\triangle ABC$, passing through $A$. The circle, centered at $A$ with radius $AC$, intersects $AB$ at $D$, and line $l$ at $E, F$. Prove that lines $DE, DF$ pass through the incenter and an excenter of $\triangle ABC$ respectively.
2005 China Second Round Olympiad
Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function $$ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. $$
2005 China Second Round Olympiad
For each positive integer, define a function $$ f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. $$ Find the value of $\sum_{k=1}^{200} f(k)$.
2004 China Second Round Olympiad
In an acute triangle $ABC$, point $H$ is the intersection point of altitude $CE$ to $AB$ and altitude $BD$ to $AC$. A circle with $DE$ as its diameter intersects $AB$ and $AC$ at $F$ and $G$, respectively. $FG$ and $AH$ intersect at point $K$. If $BC=25$, $BD=20$, and $BE=7$, find the length of $AK$.
2004 China Second Round Olympiad
In a planar rectangular coordinate system, a sequence of points ${A_n}$ on the positive half of the y-axis and a sequence of points ${B_n}$ on the curve $y=\sqrt{2x}$ $(x\ge0)$ satisfy the condition $|OA_n|=|OB_n|=\frac{1}{n}$. The x-intercept of line $A_nB_n$ is $a_n$, and the x-coordinate of point $B_n$ is $b_n$, $n\in\mathbb{N}$. Prove that

(1) $a_n>a_{n+1}>4$, $n\in\mathbb{N}$;

(2) There is $n_0\in\mathbb{N}$, such that for any $n>n_0$, $\frac{b_2}{b_1}+\frac{b_3}{b_2}+\ldots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}<n-2004$.
2004 China Second Round Olympiad
For integer $n\ge 4$, find the minimal integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set ${m, m+1, \ldots, m+n+1}$ there are at least $3$ relatively prime elements.
2003 China Second Round Olympiad
From point $P$ outside a circle draw two tangents to the circle touching at $A$ and $B$. Draw a secant line intersecting the circle at points $C$ and $D$, with $C$ between $P$ and $D$. Choose point $Q$ on the chord $CD$ such that $\angle DAQ=\angle PBC$. Prove that $\angle DBQ=\angle PAC$.
2003 China Second Round Olympiad
Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.
2003 China Second Round Olympiad
Let a space figure consist of $n$ vertices and $l$ lines connecting these vertices, with $n=q^2+q+1$, $l\ge q^2(q+1)^2+1$, $q\ge2$, $q\in\mathbb{N}$. Suppose the figure satisfies the following conditions: every four vertices are non-coplaner, every vertex is connected by at least one line, and there is a vertex connected by at least $p+2$ lines. Prove that there exists a space quadrilateral in the figure, i.e. a quadrilateral with four vertices $A, B, C, D$ and four lines $ AB, BC, CD, DA$ in the figure.
2002 China Second Round Olympiad
In $\triangle ABC$, $\angle A = 60$, $AB>AC$, point $O$ is the circumcenter and $H$ is the intersection point of two altitudes $BE$ and $CF$. Points $M$ and $N$ are on the line segments $BH$ and $HF$ respectively, and satisfy $BM=CN$. Determine the value of $\frac{MH+NH}{OH}$.
2002 China Second Round Olympiad
There are real numbers $a,b$ and $c$ and a positive number $\lambda$ such that $f(x)=x^3+ax^2+bx+c$ has three real roots $x_1, x_2$ and $x_3$ satisfying

$(1) x_2-x_1=\lambda$

$(2) x_3>\frac{1}{2}(x_1+x_2)$.

Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$
2002 China Second Round Olympiad
Before The World Cup tournament, the football coach of $F$ country will let seven players, $A_1, A_2, \ldots, A_7$, join three training matches (90 minutes each) in order to assess them. Suppose, at any moment during a match, one and only one of them enters the field, and the total time (which is measured in minutes) on the field for each one of $A_1, A_2, A_3$, and $A_4$ is divisible by $7$ and the total time for each of $A_5, A_6$, and $A_7$ is divisible by $13$. If there is no restriction about the number of substitutions of players during each match, then how many possible cases are there within the total time for every player on the field?
2001 China Second Round Olympiad
Let $O,H$ be the circumcenter and orthocenter of $\triangle ABC,$ respectively. Line $AH$ and $BC$ intersect at $D,$ Line $BH$ and $AC$ intersect at $E,$ Line $CH$ and $AB$ intersect at $F,$ Line $AB$ and $ED$ intersect at $M,$ $AC$ and $FD$ intersect at $N.$ Prove that

$(1)OB\perp DF,OC\perp DE;$

$(2)OH\perp MN.$
2001 China Second Round Olympiad
If nonnegative reals $x_1, x_2, \ldots, x_n$ satisfy

$$

\sum_{i=1}^n x_i^2 + 2\sum_{1 \leq k < j  \leq n} \sqrt{\frac{k}{j}}x_kx_j = 1

$$

what are the minimum and maximum values of $\sum_{i=1}^n x_i$?
2001 China Second Round Olympiad
An $m\times n(m,n\in \mathbb{N}^*)$ rectangle is divided into some smaller squares. The sides of each square are all parallel to the corresponding sides of the rectangle, and the length of each side is integer. Determine the minimum of the sum of the sides of these squares.
2000 China Second Round Olympiad
In acute-angled triangle $ABC,$ $E,F$ are on the side $BC,$ such that $\angle BAE=\angle CAF,$ and let $M,N$ be the projections of $F$ onto $AB,AC,$ respectively. The line $AE$ intersects $ \odot (ABC) $ at $D$(different from point $A$).

Prove that $S_{AMDN}=S_{\triangle ABC}.$
2000 China Second Round Olympiad
Define the sequence $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ as $a_0=1,a_1=4,a_2=49$ and for $n \geq 0$

$$ 

\begin{cases}

a_{n+1}=7a_n+6b_n-3, \\
b_{n+1}=8a_n+7b_n-4.

\end{cases}

$$Prove that for any non-negative integer $n,$ $a_n$ is a perfect square.
2000 China Second Round Olympiad
There are $n$ people, and given that any $2$ of them have contacted with each other at most once. In any group of $n-2$ of them, any one person of the group has contacted with other people in this group for $3^k$ times, where $k$ is a non-negative integer. Determine all the possible value of $n.$
1999 China Second Round Olympiad
In convex quadrilateral $ABCD, \angle BAC=\angle CAD.$ $E$ lies on segment $CD$, and $BE$ and $AC$ intersect at $F,$ $DF$ and $BC$ intersect at $G.$ Prove that $\angle GAC=\angle EAC.$
1999 China Second Round Olympiad
Let $a$,$b$,$c$ be real numbers.

Let $z_{1}$,$z_{2}$,$z_{3}$ be complex numbers such that $|z_{k}|=1$ $(k=1,2,3)$ $~$ and $~$ $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1$

Find $|az_{1}+bz_{2}+cz_{3}|$.
1999 China Second Round Olympiad
$n$ is a given positive integer, such that it’s possible to weigh out the mass of any product weighing $1,2,3,\cdots ,ng$ with a counter balance without sliding poise and $k$ counterweights, which weigh $x_ig(i=1,2,\cdots ,k),$ respectively, where $x_i\in \mathbb{N}^*$ for any $i \in \{ 1,2,\cdots ,k\}$ and $x_1\leq x_2\leq\cdots \leq x_k.$



$(1)$Let $f(n)$ be the least possible number of $k$. Find $f(n)$ in terms of $n.$

$(2)$Find all possible number of $n,$ such that sequence $x_1,x_2,\cdots ,x_{f(n)}$ is uniquely determined.
2012 Albania National Olympiad
Find all primes $p$ such that $p+2$ and $p^2+2p-8$ are also primes.
2012 Albania National Olympiad
The trinomial $f(x)$ is such that $(f(x))^3-f(x)=0$ has three real roots. Find the y-coordinate of the vertex of $f(x)$.
2012 Albania National Olympiad
Let $S_i$ be the sum of the first $i$ terms of the arithmetic sequence $a_1,a_2,a_3\ldots $. Show that the value of the expression

$$\frac{S_i}{i}(j-k) + \frac{S_j}{j}(k-i) +\ \frac{S_k}{k}(i-j)$$

does not depend on the numbers $i,j,k$ nor on the choice of the arithmetic sequence $a_1,a_2,a_3,\ldots$.
2012 Albania National Olympiad
Find all functions  $f:\mathbb{R}\to\mathbb{R}$ such that

$$f(x^3)+f(y^3)=(x+y)f(x^2)+f(y^2)- f(xy)$$

for all $x\in\mathbb{R}$.
2012 Albania National Olympiad
Let $ABC$ be a triangle where $AC\neq BC$. Let $P$ be the foot of the altitude taken from $C$ to $AB$; and let $V$ be the orthocentre, $O$ the circumcentre of $ABC$, and $D$ the point of intersection between the radius $OC$ and the side $AB$. The midpoint of $CD$ is $E$.



a) Prove that the reflection $V'$ of $V$ in $AB$ is on the circumcircle of the triangle $ABC$.



b) In what ratio does the segment $EP$ divide the segment $OV$?
2011 Albania National Olympiad
(a)  Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$.



(b) Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other  point in the graph of the function $y=ln x$.
2011 Albania National Olympiad
Find all the values that can take the last digit of a "perfect" even number. (The natural number $n$ is called "perfect" if the sum of all its natural divisors is equal twice the number itself.For example: the number $6$ is perfect ,because $1+2+3+6=2\cdot6$).
2011 Albania National Olympiad
In a convex quadrilateral $ABCD$ ,$\angle ABC$ and $\angle BCD$ are $\geq 120^o$. Prove that $|AC|$ + $|BD| \geq |AB|+|BC|+|CD|$. (With $|XY|$ we understand the length of the segment $XY$).
2011 Albania National Olympiad
The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$.



(a) Prove that for every even $n$, $a_{n}$ is divisible by $n!$.



(b) Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.
2011 Albania National Olympiad
The triangle $ABC$ acute with gravity center $M$ is such that $\angle AMB = 2 \angle ACB$. Prove that:



(a) $AB^4=AC^4+BC^4-AC^2 \cdot BC^2,$



(b) $\angle ACB \geq 60^o$.
2010 Albania National Olympiad
Let $A$ and $B$ be two fixed points of a given circle and $XY$ a diameter of this circle. Find the locus of the intersection points of lines $AX$ and $BY$ . ($BY$ is not a diameter of the circle).



Albanian National Mathematical Olympiad 2010---12 GRADE  Question 1.
2010 Albania National Olympiad
We denote $N_{2010}=\{1,2,\cdots,2010\}$

(a)How many non empty subsets does this set have?

(b)For every non empty subset of the set $N_{2010}$ we take the product of the elements of the subset. What is the sum of these products?

(c)Same question as the (b) part for the set $-N_{2010}=\{-1,-2,\cdots,-2010\}$.





Albanian National Mathematical Olympiad 2010---12 GRADE Question 2.
2010 Albania National Olympiad
(a)Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity.

(b)What is the smallest area possible of pentagons with integral coordinates.







Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.
2010 Albania National Olympiad
The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$.

(a) Prove that $f_{2010} $ is divisible by $10$.

(b) Is $f_{1005}$ divisible by $4$?





Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.
2010 Albania National Olympiad
All members of the senate were firstly divided into $S$ senate commissions . According to the rules, no commission has less that $5$ senators and every two commissions have different number of senators. After the first session the commissions were closed and new commissions were opened. Some of the senators now are not a part of any commission. It resulted also that every two senators that were in the same commission in the first session , are not any more in the same commission.

(a)Prove that at least $4S+10$ senators were left outside the commissions.

(b)Prove that this number is achievable.



Albanian National Mathematical Olympiad 2010---12 GRADE Question 5.
2021 All-Russian Olympiad
On a circle there're $1000$ marked points, each colored in one of $k$ colors. It's known that among any $5$ pairwise intersecting segments, endpoints of which are $10$ distinct marked points, there're at least $3$ segments, each of which has its endpoints colored in different colors. Determine the smallest possible value of $k$ for which it's possible.
2021 All-Russian Olympiad
Let $n$ be a natural number. An integer $a>2$ is called $n$-decomposable, if $a^n-2^n$ is divisible by all the numbers of the form $a^d+2^d$, where $d\neq n$ is a natural divisor of $n$. Find all composite $n\in \mathbb{N}$, for which there's an $n$-decomposable number.
2021 All-Russian Olympiad
On a line $n+1$ segments are marked such that one of the points of the line is contained in all of them. Prove that one can find $2$ distinct segments $I, J$ which intersect at a segment of length at least $\frac{n-1}{n}d$, where $d$ is the length of the segment $I$.
2021 All-Russian Olympiad
Given an acute triangle $ABC$, point $D$ is chosen on the side $AB$ and a point $E$ is chosen on the extension of $BC$ beyond $C$. It became known that the line through $E$ parallel to $AB$ is tangent to the circumcircle of $\triangle ADC$. Prove that one of the tangents from $E$ to the circumcircle of $\triangle BCD$ cuts the angle $\angle ABE$ in such a way that a triangle similar to $\triangle ABC$ is formed.
2021 All-Russian Olympiad
The reals $b>0$ and $a$ are such that the quadratic $x^2+ax+b$ has two distinct real roots,  exactly one of which lies in the interval $[-1;1]$. Prove that one of the roots lies in the interval $(-b;b)$.
2021 All-Russian Olympiad
Given is a non-isosceles triangle $ABC$ with $<ABC=60$, and in its interior, a point $T$ is selected such that $<ATC=<BTC=<BTA=120$. Let $M$  the intersection point of the medians in $ABC$. Let $TM$ intersect $(ATC)$ at $K$. Find $TM/MK$.
2021 All-Russian Olympiad
Given are positive integers $n>20$ and $k>1$, such that $k^2$ divides $n$. Prove that there exist

positive integers $a, b, c$, such that $n=ab+bc+ca$.