Show that the set $\{1,2,....,2^n\}$ can be partitioned in two classes, none of which contains an arithmetic progression of length $2n$.

Let $ABC$ be a triangle whose angles measure $A$, $B$, $C$, respectively. Suppose $\tan A$, $\tan B$, $\tan C$ form a geometric sequence in that order. If $1\le \tan A+\tan B+\tan C\le 2015$, find the number of possible integer values for $\tan B$. (The values of $\tan A$ and $\tan C$ need not be integers.) [i] Proposed by Justin Stevens [/i]

The graphs of the lines $$y=x+2, \quad y=3x+4, \quad y=5x+6,\quad y=7x+8,\quad y=9x+10,\quad y=11x+12$$ are drawn. These six lines divide the plane into several regions. Compute the number of regions the plane is divided into.

Every non-negative integer is coloured white or red, so that: • there are at least a white number and a red number; • the sum of a white number and a red number is white; • the product of a white number and a red number is red. Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.

Four coplanar regular polygons share a common vertex but have no interior points in common. Each polygon is adjacent to two of the other polygons, and each pair of adjacent polygons has a common side length of $1$. How many possible perimeters are there for all such configurations? $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) more than }5$

There are $2024$ points of general position marked on the coordinate plane (i.e., points among which there are no three lying on the same straight line). Is there a polynomial of two variables $f(x,y)$ a) of degree $2025$; b) of degree $2024$ such that it equals to zero exactly at these marked points? [i]Proposed by Navid Safaei[/i]

Let $m$ and $n$ be positive integers. A child walks the Cartesian plane taking a few steps. The child begins its journey at the point $(0, n)$ and ends at the point $(m, 0)$ in such a way that: $\bullet$ Each step has length $1$ and is parallel to either the $X$ or $Y$ axis. $\bullet$ For each point $(x, y)$ of its path it is true that $x\ge 0$ and $y\ge 0$. For each step of the child, the distance between the child and the axis to which said step is parallel is calculated. If the step causes the child to be further from the point $(0, 0)$ than before, we consider that distance as positive, otherwise, we consider that distance as negative. Prove that at the end of the boy's journey, the sum of all the distances is $0$.

Find all composite positive integers $a{}$ for which there exists a positive integer $b\geqslant a$ with the same number of divisors as $a{}$ with the following property: if $a_1<\cdots<a_n$ and $b_1<\cdots<b_n$ are the proper divisors of $a{}$ and $b{}$ respectively, then $a_i+b_i, 1\leqslant i\leqslant n$ are the proper divisors of some positive integer $c.{}$

Let $k$ and $m$ be positive integers. Show that \[S(m, k)=\sum_{n=1}^{\infty}\frac{1}{n(mn+k)}\] is rational if and only if $m$ divides $k$.

Let $k$ be a positive integer. Define $n_k$ to be the number with decimal representation $70...01$ where there are exactly $k$ zeroes. Prove the following assertions: a) None of the numbers $n_k$ is divisible by $13$. b) Infinitely many of the numbers $n_k$ are divisible by $17$.

In a right triangle, the length of one side is a prime and the lengths of the other side and the hypotenuse are integral. The ratio of the triangle perimeter and the incircle diameter is also an integer. Find all possible side lengths of the triangle.

In a triangle $ABC$, $M$ and $N$ are the respective midpoints of the sides $AC$ and $AB$, and $P$ is the point of intersection of $BM$ and $CN$. Prove that, if it is possible to inscribe a circle in the quadrilateral $AMPN$, then the triangle $ABC$ is isosceles.

Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$ a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple. b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple. [i]There are exactly $n$ factors in the product on the left hand side.[/i]

Find max. numbers $A$ wich is true ineq.: $\frac{x}{\sqrt{y^{2}+z^{2}}}+\frac{y}{\sqrt{x^{2}+z^{2}}}+\frac{z}{\sqrt{x^{2}+y^{2}}}\geq A$ $x,y,z$ are positve reals numberes! :wink:

What is $\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}$? $\textbf{(A) }-1\qquad\textbf{(B) }\frac5{36}\qquad\textbf{(C) }\frac7{12}\qquad\textbf{(D) }\frac{49}{20}\qquad\textbf{(E) }\frac{43}3$

A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.

Write down $f(n)$ for the greatest odd divisor of $n \in N_0$. (a) Determine $f (n + 1) + f (n + 2) + ... + f(2n)$. (b) Determine $f(1) + f(2) + f(3) + ... + f(2n)$.

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$. Can number $7$ on the right hand side be replaced with a smaller constant?

The side lengths of a scalene triangle are roots of the polynomial $$x^3-20x^2+131x-281.3.$$ Find the square of the area of the triangle.

Given a triangle $ABC$ with incenter $I$. The incircle of triangle $ABC$ is tangent to $BC$ at $D$. Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$, respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$, respectively. Prove that $AD$ is the Euler line of triangle $IKL$. [i]Proposed by Le Viet An, Vietnam[/i]

Let $ ABC$ be an acute angled triangle with $ \angle B > \angle C$. Let $ D$ be the foot of the altitude from $ A$ on $ BC$, and let $ E$ be the foot of the perpendicular from $ D$ on $ AC$. Let $ F$ be a point on the segment $ (DE)$. Show that the lines $ AF$ and $ BF$ are perpendicular if and only if $ EF\cdot DC \equal{} BD \cdot DE$.

In an acute-angled triangle $ABC$, $a,b,c$ are sides, $m_a,m_b,m_c$ the corresponding medians, $R$ the circumradius and $r$ the inradius. Prove the inequality $$\frac{a^2+b^2}{a+b}\cdot\frac{b^2+c^2}{b+c}\cdot\frac{a^2+c^2}{a+c}\ge16R^2r\frac{m_a}a\cdot\frac{m_b}b\cdot\frac{m_c}c.$$

Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold. (Here we denote $XY$ the length of the line segment $XY$.)

X-wich has a vibrant club-life. For every pair of inhabitants there is exactly one club to which they both belong. For every pair of clubs there is exactly one person who is a member of both. No club has fewer than $3$ members, and at least one club has $17$ members. How many people live in X-wich?