Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 Federal Competition For Advanced Students, Part 2
Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$. The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$, respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$. Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$. Describe the locus of the intersection point.
1999 Federal Competition For Advanced Students, Part 2
Find all pairs $(x, y)$ of real numbers such that

$$y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999$$

where $f(x)=[x]$ is the floor function.
1999 Federal Competition For Advanced Students, Part 2
Ninety-nine points are given on one of the diagonals of a unit square. Prove that there is at most one vertex of the square such that the average squared distance from a given point to the vertex is less than or equal to $1/2$.
1999 Federal Competition For Advanced Students, Part 2
Given a real number $A$ and an integer $n$ with $2 \leq n \leq 19$, find all polynomials $P(x)$ with real coefficients such that $P(P(P(x))) = Ax^n +19x+99$.
1999 Federal Competition For Advanced Students, Part 2
Two players $A$ and $B$ play the following game. An even number of cells are placed on a circle. $A$ begins and $A$ and $B$ play alternately, where each move consists of choosing a free cell and writing either $O$ or $M$ in it. The player after whose move the word $OMO$ (OMO = Osterreichische Mathematik Olympiade) occurs for the first time in three successive cells wins the game. If no such word occurs, then the game is a draw. Prove that if player $B$ plays correctly, then player $A$ cannot win.
2000 Federal Competition For Advanced Students, Part 2
The sequence an is defined by $a_0 = 4, a_1 = 1$ and the recurrence formula $a_{n+1} = a_n + 6a_{n-1}$. The sequence $b_n$ is given by

$$b_n=\sum_{k=0}^n \binom nk a_k.$$

Find the coefficients $\alpha,\beta$  so that $b_n$ satisfies the recurrence formula $b_{n+1} = \alpha b_n + \beta b_{n-1}$. Find the explicit form of $b_n$.
2000 Federal Competition For Advanced Students, Part 2
A trapezoid $ABCD$ with $AB \parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chose on the arc $ADCB$ in the order $A-P -Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. Show that points $P,Q,X, Y$ lie on a circle.
2000 Federal Competition For Advanced Students, Part 2
Find all real solutions to the equation

$$| | | | | | |x^2 -x - 1| - 3| - 5| - 7| - 9| - 11| - 13| = x^2 - 2x - 48.$$
2000 Federal Competition For Advanced Students, Part 2
In a non-equilateral acute-angled triangle $ABC$ with $\angle C = 60^\circ$, $U$ is the circumcenter, $H$ the orthocenter and $D$ the intersection of $AH$ and $BC$. Prove that the Euler line $HU$ bisects the angle $BHD$.
2000 Federal Competition For Advanced Students, Part 2
Find all pairs of integers $(m, n)$ such that

$$ \left| (m^2 + 2000m+ 999999)- (3n^3 + 9n^2 + 27n) \right|= 1.$$
2000 Federal Competition For Advanced Students, Part 2
Find all functions $f : \mathbb R \to \mathbb R$ such that for all reals $x, y, z$ it holds that

$$f(x + f(y + z)) + f(f(x + y) + z) = 2y.$$
2001 Federal Competition For Advanced Students, Part 2
Prove that $\frac{1}{25} \sum_{k=0}^{2001} \left[ \frac{2^k}{25}\right]$ is a positive integer.
2001 Federal Competition For Advanced Students, Part 2
Determine all triples of positive real numbers $(x, y, z)$ such that

$$x+y+z=6,$$$$\frac 1x + \frac 1y + \frac 1z = 2 - \frac{4}{xyz}.$$
2001 Federal Competition For Advanced Students, Part 2
A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$. A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that C lies between the lines $c = AB$ and $C'$. Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$. Prove that the three lines, joining the midpoints of pairs of parallel sides of the two triangles, have a common point.
2001 Federal Competition For Advanced Students, Part 2
Find all functions $f :\mathbb R \to \mathbb R$ such that for all real $x, y$

$$f(f(x)^2 + f(y)) = xf(x) + y.$$
2001 Federal Competition For Advanced Students, Part 2
Determine all integers $m$ for which all solutions of the equation $3x^3-3x^2+m = 0$ are rational.
2001 Federal Competition For Advanced Students, Part 2
Let be given a semicircle with the diameter $AB$, and points $C,D$ on it such that $AC = CD$. The tangent at $C$ intersects the line $BD$ at $E$. The line $AE$ intersects the arc of the semicircle at $F$. Prove that $CF < FD$.
2002 Federal Competition For Advanced Students, Part 2
Consider all possible rectangles that can be drawn on a $8 \times 8$ chessboard, covering only whole cells. Calculate the sum of their areas.



What formula is obtained if “$8 \times 8$” is replaced with “$a \times b$”, where $a, b$ are positive integers?
2002 Federal Competition For Advanced Students, Part 2
Let $b$ be a positive integer. Find all $2002$−tuples $(a_1, a_2,\ldots , a_{2002})$, of natural numbers such that

$$\sum_{j=1}^{2002} a_j^{a_j}=2002b^b.$$
2002 Federal Competition For Advanced Students, Part 2
Let $ABCD$ and $AEFG$ be two similar cyclic quadrilaterals (with the vertices denoted counterclockwise). Their circumcircles intersect again at point $P$. Prove that $P$ lies on line $BE$.
2002 Federal Competition For Advanced Students, Part 2
Find all polynomials $P(x)$ of the smallest possible degree with the following properties:



(i) The leading coefficient is $200$;

(ii) The coefficient at the smallest non-vanishing power is $2$;

(iii) The sum of all the coefficients is $4$;

(iv) $P(-1) = 0, P(2) = 6, P(3) = 8$.
2002 Federal Competition For Advanced Students, Part 2
In the net drawn below, in how many ways can one reach the point $3n+1$ starting from the point $1$ so that the labels of the points on the way increase?



[asy]

import graph; size(12cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=12.32,ymin=-10.66,ymax=6.3; draw((1,2)--(xmax,0*xmax+2)); draw((1,0)--(xmax,0*xmax+0)); draw((0,1)--(1,2)); draw((1,0)--(0,1)); draw((1,2)--(3,0)); draw((1,0)--(3,2)); draw((3,2)--(5,0)); draw((3,0)--(5,2)); draw((5,2)--(7,0)); draw((5,0)--(7,2)); draw((7,2)--(9,0)); draw((7,0)--(9,2)); 

dot((1,0),linewidth(1pt)+ds); label("2",(0.96,-0.5),NE*lsf); dot((0,1),linewidth(1pt)+ds); label("1",(-0.42,0.9),NE*lsf); dot((1,2),linewidth(1pt)+ds); label("3",(0.98,2.2),NE*lsf); dot((2,1),linewidth(1pt)+ds); label("4",(1.92,1.32),NE*lsf); dot((3,2),linewidth(1pt)+ds); label("6",(2.94,2.2),NE*lsf); dot((4,1),linewidth(1pt)+ds); label("7",(3.94,1.32),NE*lsf); dot((6,1),linewidth(1pt)+ds); label("10",(5.84,1.32),NE*lsf); dot((3,0),linewidth(1pt)+ds); label("5",(2.98,-0.46),NE*lsf); dot((5,2),linewidth(1pt)+ds); label("9",(4.92,2.24),NE*lsf); dot((5,0),linewidth(1pt)+ds); label("8",(4.94,-0.42),NE*lsf); dot((8,1),linewidth(1pt)+ds); label("13",(7.88,1.34),NE*lsf); dot((7,2),linewidth(1pt)+ds); label("12",(6.8,2.26),NE*lsf); dot((7,0),linewidth(1pt)+ds); label("11",(6.88,-0.38),NE*lsf); 

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 

[/asy]
2002 Federal Competition For Advanced Students, Part 2
Let $H$ be the orthocenter of an acute-angled triangle $ABC$. Show that the triangles $ABH,BCH$ and $CAH$ have the same perimeter if and only if the triangle $ABC$ is equilateral.
2003 Federal Competition For Advanced Students, Part 2
Consider the polynomial $P(n) = n^3 -n^2 -5n+ 2$. Determine all integers $n$ for which $P(n)^2$ is a square of a prime.



Remark.I'm not sure if the statement of this problem is correct, because if $P(n)^2$ be a square of a prime, then $P(n)$ should be that prime, and I don't think the problem means that.
2003 Federal Competition For Advanced Students, Part 2
Let $a, b, c$ be nonzero real numbers for which there exist $\alpha, \beta, \gamma \in\{-1, 1\}$ with $\alpha a + \beta b + \gamma c = 0$. What is the smallest possible value of

$$\left( \frac{a^3+b^3+c^3}{abc}\right)^2 ?$$
2003 Federal Competition For Advanced Students, Part 2
For every lattice point $(x, y)$ with $x, y$ non-negative integers, a square of side $\frac{0.9}{2^x5^y}$ with center at the point $(x, y)$ is constructed. Compute the area of the union of all these squares.
2003 Federal Competition For Advanced Students, Part 2
Prove that, for any integer $g > 2$, there is a unique three-digit number $\overline{abc}_g$ in base $g$ whose representation in some base $h = g \pm 1$ is $\overline{cba}_h$.
2003 Federal Competition For Advanced Students, Part 2
We are given sufficiently many stones of the forms of a rectangle $2\times 1$ and square $1\times 1$. Let $n > 3$ be a natural number. In how many ways can one tile a rectangle $3 \times n$ using these stones, so that no two $2 \times 1$ rectangles have a common point, and each of them has the longer side parallel to the shorter side of the big rectangle?
2003 Federal Competition For Advanced Students, Part 2
Let $ABC$ be an acute-angled triangle. The circle $k$ with diameter $AB$ intersects $AC$ and $BC$ again at $P$ and $Q$, respectively. The tangents to $k$ at $A$ and $Q$ meet at $R$, and the tangents at $B$ and $P$ meet at $S$. Show that $C$ lies on the line $RS$.
2004 Federal Competition For Advanced Students, P2
Prove without using advanced (differential) calculus:

(a) For any real numbers a,b,c,d it holds that $a^6+b^6+c^6+d^6-6abcd \ge  -2$.

When does equality hold?

(b) For which natural numbers $k$ does some inequality of the form $a^k +b^k +c^k +d^k -kabcd \ge M_k$ hold for all real $a,b,c,d$? For each such $k$,
2004 Federal Competition For Advanced Students, P2
Show that every set $ \{p_1,p_2,\dots,p_k\}$ of prime numbers fulfils the following: The sum of all unit fractions (that are fractions of the type $ \frac{1}{n}$), whose denominators are exactly the $ k$ given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again.

How big is this sum if $ \frac{1}{2004}$ is among this summands?

Show that for every set $ \{p_1,p_2,\dots,p_k\}$ containing $ k$ prime numbers ($ k>2$) is the sum smaller than $ \frac{1}{N}$ with $ N=2\cdot 3^{k-2}(k-2)!$
2004 Federal Competition For Advanced Students, P2
A trapezoid $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle $k$. Let $k_a$ and $k_c$ respectively be the circles with diameters $AB$ and $CD$. Compute the area of the region which is inside the circle $k$, but outside the circles $k_a$ and $k_c$.
2004 Federal Competition For Advanced Students, P2
Show that there is an infinite sequence $a_1,a_2,...$ of natural numbers such that $a^2_1+a^2_2+ ...+a^2_N$ is a perfect square for all $N$. Give a recurrent formula for one such sequence.
2004 Federal Competition For Advanced Students, P2
Solve the following system of equations in real numbers: $\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} \\
b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} \\
c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} \\
d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}$
2004 Federal Competition For Advanced Students, P2
Over the sides of an equilateral triangle with area $ 1$ are triangles with the opposite angle $ 60^{\circ}$ to each side drawn outside of the triangle. The new corners are $ P$, $ Q$ and $ R$. (and the new triangles $ APB$, $ BQC$ and $ ARC$)

1)What is the highest possible area of the triangle $ PQR$?

2)What is the highest possible area of the triangle whose vertexes are the midpoints of the inscribed circles of the triangles $ APB$, $ BQC$ and $ ARC$?
2005 Federal Competition For Advanced Students, Part 2
Find all triples $(a,b,c)$ of natural numbers, such that $LCM(a,b,c)=a+b+c$
2005 Federal Competition For Advanced Students, Part 2
Prove that for all positive reals $a,b,c,d$, we have $\frac{a+b+c+d}{abcd}\leq \frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}+\frac{1}{d^{3}}$
2005 Federal Competition For Advanced Students, Part 2
Triangle $DEF$ is acute. Circle $c_1$ is drawn with $DF$ as its diameter and circle $c_2$ is drawn with $DE$ as its diameter. Points $Y$ and $Z$ are on $DF$ and $DE$ respectively so that $EY$ and $FZ$ are altitudes of triangle $DEF$ . $EY$ intersects $c_1$ at $P$, and $FZ$ intersects $c_2$ at $Q$. $EY$ extended intersects $c_1$ at $R$, and $FZ$  extended intersects $c_2$ at $S$. Prove that $P$, $Q$, $R$, and $S$ are concyclic points.
2005 Federal Competition For Advanced Students, Part 2
The function $f : (0,...2005) \rightarrow N$ has the properties that $f(2x+1)=f(2x)$, $f(3x+1)=f(3x)$ and $f(5x+1)=f(5x)$ with $x \in (0,1,2,...,2005)$. How many different values can the function assume?
2005 Federal Competition For Advanced Students, Part 2
Find all real $a,b,c,d,e,f$ that satisfy the system

$4a = (b + c + d + e)^4$

$4b = (c + d + e + f)^4$

$4c = (d + e + f + a)^4$

$4d = (e + f + a + b)^4$

$4e = (f + a + b + c)^4$

$4f = (a + b + c + d)^4$
2005 Federal Competition For Advanced Students, Part 2
Let $Q$ be a point inside a cube. Prove that there are infinitely many lines $l$ so that $AQ=BQ$ where $A$ and $B$ are the two points of intersection of $l$ and the surface of the cube.
2006 Federal Competition For Advanced Students, Part 2
Let $ N$ be a positive integer. How many non-negative integers $ n \le N$ are there that have an integer multiple, that only uses the digits $ 2$ and $ 6$ in decimal representation?
2006 Federal Competition For Advanced Students, Part 2
Let $ a,b,c$ be positive real numbers. Show that

$ 3(a + b + c) \ge 8 \sqrt [3]{abc} + \sqrt [3]{\frac {a^3 + b^3 + c^3}{3} }.$
2006 Federal Competition For Advanced Students, Part 2
The triangle $ ABC$ is given. On the extension of the side $ AB$ we construct the point $ R$ with $ BR = BC$, where $ AR > BR$ and on the extension of the side $ AC$ we construct the point $ S$ with $ CS = CB$, where $ AS > CS$. Let $ A_1$ be the point of intersection of the diagonals of the quadrilateral $ BRSC$.

Analogous we construct the point $ T$ on the extension of the side $ BC$, where $ CT = CA$ and $ BT > CT$ and on the extension of the side $ BA$ we construct the point $ U$ with $ AU = AC$, where $ BU > AU$. Let $ B_1$ be the point of intersection of the diagonals of the quadrilateral $ CTUA$.

Likewise we construct the point $ V$ on the extension of the side $ CA$, where $ AV = AB$ and $ CV > AV$ and on the extension of the side $ CB$ we construct the point $ W$ with $ BW = BA$ and $ CW > BW$. Let $ C_1$ be the point of intersection of the diagonals of the quadrilateral $ AVWB$.

Show that the area of the hexagon $ AC_1BA_1CB_1$ is equal to the sum of the areas of the triangles $ ABC$ and $ A_1B_1C_1$.
2006 Federal Competition For Advanced Students, Part 2
For which rational $ x$ is the number $ 1 + 105 \cdot 2^x$ the square of a rational number?
2006 Federal Competition For Advanced Students, Part 2
Find all monotonous functions $ f: \mathbb{R} \to \mathbb{R}$ that satisfy the following functional equation:

$$f(f(x)) = f( - f(x)) = f(x)^2.$$
2006 Federal Competition For Advanced Students, Part 2
Let $ A$ be an integer not equal to $ 0$. Solve the following system of equations in $ \mathbb{Z}^3$.

$ x + y^2 + z^3 = A$

$ \frac {1}{x} + \frac {1}{y^2} + \frac {1}{z^3} = \frac {1}{A}$

$ xy^2z^3 = A^2$
2007 Federal Competition For Advanced Students, Part 2
For which non-negative integers $ a<2007$ the congruence	$ x^2+a \equiv 0 \mod 2007$ has got exactly two different non-negative integer solutions?

That means, that there exist exactly two different non-negative integers $ u$ and $ v$ less than $ 2007$, such that $ u^2+a$ and $ v^2+a$ are both divisible by $ 2007$.
2007 Federal Competition For Advanced Students, Part 2
Find all tuples $ (x_1,x_2,x_3,x_4,x_5,x_6)$ of non-negative integers, such that the following system of equations holds:

$ x_1x_2(1-x_3)=x_4x_5 \\
x_2x_3(1-x_4)=x_5x_6 \\
x_3x_4(1-x_5)=x_6x_1 \\
x_4x_5(1-x_6)=x_1x_2 \\
x_5x_6(1-x_1)=x_2x_3 \\
x_6x_1(1-x_2)=x_3x_4$
2007 Federal Competition For Advanced Students, Part 2
Determine all rhombuses $ ABCD$ with the given length $ 2a$ of ist sides by giving the angle $ \alpha = \angle BAD$, such that there exists a circle which cuts each side of the rhombus in a chord of length $ a$.
2007 Federal Competition For Advanced Students, Part 2
Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?
2007 Federal Competition For Advanced Students, Part 2
38th Austrian Mathematical Olympiad 2007, round 3 problem 5



Given is a convex $ n$-gon with a triangulation, that is a partition into triangles through diagonals that don’t cut each other. Show that it’s always possible to mark the $ n$ corners with the digits of the number $ 2007$ such that every quadrilateral consisting of $ 2$ neighbored (along an edge) triangles has  got $ 9$ as the sum of the numbers on its $ 4$ corners.
2007 Federal Competition For Advanced Students, Part 2
The triangle $ ABC$ with the circumcircle $ k(U,r)$ is given. On the extension of the radii $ UA$ a point $ P$ is chosen. The reflection of the line $ PB$ on the line $ BA$ is called $ g$. Likewise the reflection of the line $ PC$ on the line $ CA$ is called $ h$. The intersection of $ g$ and $ h$ is called $ Q$.

Find the geometric location of all possible intersections $ Q$, while $ P$ passes through the extension of the radii $ UA$.
2008 Federal Competition For Advanced Students, Part 2
Prove the inequality

$$ \sqrt {a^{1 - a}b^{1 - b}c^{1 - c}} \le \frac {1}{3}

$$

holds for all positive real numbers $ a$, $ b$ and $ c$ with $ a + b + c = 1$.
2008 Federal Competition For Advanced Students, Part 2
(a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) = \frac{2008}{d}$ holds for all positive divisors of $ 2008$?

(b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) = \frac{n}{d}$ holds for all positive divisors of $ n$?
2008 Federal Competition For Advanced Students, Part 2
We are given a line $ g$ with four successive points $ P$, $ Q$, $ R$, $ S$, reading from left to right. Describe a straightedge and compass construction yielding a square $ ABCD$ such that $ P$ lies on the line $ AD$, $ Q$ on the line $ BC$, $ R$ on the line $ AB$ and $ S$ on the line $ CD$.
2008 Federal Competition For Advanced Students, Part 2
Determine all functions $ f$ mapping the set of positive integers to the set of non-negative integers satisfying the following conditions:

(1) $ f(mn) = f(m)+f(n)$,

(2) $ f(2008) = 0$, and

(3) $ f(n) = 0$ for all $ n \equiv 39\pmod {2008}$.
2008 Federal Competition For Advanced Students, Part 2
Which positive integers are missing in the sequence $ \left\{a_n\right\}$, with $ a_n = n + \left[\sqrt n\right] +\left[\sqrt [3]n\right]$ for all $ n \ge 1$? ($ \left[x\right]$ denotes the largest integer less than or equal to $ x$, i.e. $ g$ with $ g \le x < g + 1$.)
2008 Federal Competition For Advanced Students, Part 2
We are given a square $ ABCD$. Let $ P$ be a point not equal to a corner of the square or to its center $ M$. For any such $ P$, we let $ E$ denote the common point of the lines $ PD$ and $ AC$, if such a point exists. Furthermore, we let $ F$ denote the common point of the lines $ PC$ and $ BD$, if such a point exists. All such points $ P$, for which $ E$ and $ F$ exist are called acceptable points. Determine the set of all acceptable points, for which the line $ EF$ is parallel to $ AD$.
2009 Federal Competition For Advanced Students, P2
If $x,y,K,m \in N$, let us define:



$a_m= \underset{k \, twos}{2^{2^{,,,{^{2}}}}}$, $A_{km} (x)= \underset{k \, twos}{ 2^{2^{,,,^{x^{a_m}}}}}$, $B_k(y)=  \underset{m \, fours}{4^{4^{4^{,,,^{4^y}}}}}$,



Determine all pairs $(x,y)$ of non-negative integers, dependent on $k>0$, such that $A_{km} (x)=B_k(y)$
2009 Federal Competition For Advanced Students, P2
(i) For positive integers $a<b$, let $M(a,b)=\frac{\Sigma^{b}_{k=a}\sqrt{k^2+3k+3}}{b-a+1}$.

Calculate $[M(a,b)]$

(ii) Calculate $N(a,b)=\frac{\Sigma^{b}_{k=a}[\sqrt{k^2+3k+3}]}{b-a+1}$.
2009 Federal Competition For Advanced Students, P2
Let $P$ be the point in the interior of $\vartriangle ABC$. Let $D$ be the intersection of the lines $AP$ and $BC$ and let $A'$ be the point such that $\overrightarrow{AD} = \overrightarrow{DA'}$. The points $B'$ and $C'$ are defined in the similar way. Determine all points $P$ for which the triangles $A'BC, AB'C$, and $ABC'$ are congruent to $\vartriangle ABC$.
2009 Federal Competition For Advanced Students, P2
Let $ a$ be a positive integer. Consider the sequence $ (a_n)$ defined as $ a_0=a$

and $ a_n=a_{n-1}+40^{n!}$ for $ n > 0$. Prove that the sequence $ (a_n)$ has infinitely

many numbers divisible by $ 2009$.
2009 Federal Competition For Advanced Students, P2
Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k = p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$. Furthermore let $ P = P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$.



How many different values are taken by $ a_j$ if all the numbers $ a_j (1 \leq j \leq n)$ and $ P$ are prime?
2009 Federal Competition For Advanced Students, P2
The quadrilateral PQRS whose vertices are the midpoints of the sides AB, BC,

CD, DA, respectively of a quadrilateral ABCD is called the midpoint quadrilateral

of ABCD.

Determine all circumscribed quadrilaterals whose mid-point quadrilaterals are

squares.

.
2010 Federal Competition For Advanced Students, P2
Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)}

{(x - y)^5 + (y - z)^5 + (z - x)^5} \ge  3$

holds for all triples of distinct integers $x, y, z$. When does equality hold?
2010 Federal Competition For Advanced Students, P2
Determine all triples $(x, y, z)$ of positive integers $x > y > z > 0$, such that $x^2 = y \cdot 2^z + 1$
2010 Federal Competition For Advanced Students, P2
On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact.

A regular hexagon with its vertices on the circle is drawn on a circular billiard table.

A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges.

Describe a periodical track of this ball with exactly four points at the rails.

With how many different directions of impact can the ball be brought onto such a track?
2010 Federal Competition For Advanced Students, P2
Consider the part of a lattice given by the corners $(0, 0), (n, 0), (n, 2)$ and $(0, 2)$.

From a lattice point $(a, b)$ one can move to $(a + 1, b)$ or to $(a + 1, b + 1)$ or to $(a, b - 1$), provided that the second point is also contained in the part of the lattice.

How many ways are there to move from $(0, 0)$ to $(n, 2)$ considering these rules?
2010 Federal Competition For Advanced Students, P2
Two decompositions of a square into three rectangles are called substantially different, if reordering the rectangles does not change one into the other.

How many substantially different decompositions of a $2010  \times 2010$ square in three rectangles with integer side lengths are there such that the area of one rectangle is equal to the arithmetic mean of the areas of the other rectangles?
2010 Federal Competition For Advanced Students, P2
A diagonal of a convex hexagon is called long if it decomposes the hexagon into two quadrangles.

Each pair of long diagonals decomposes the hexagon into two triangles and two quadrangles.

Given is a hexagon with the property, that for each decomposition by two long diagonals the resulting triangles are both isosceles with the side of the hexagon as base.

Show that the hexagon has a circumcircle.