Let \(AOB\) be a given angle less than \(180^{\circ}\) and let \(P\) be an interior point of the angular region determined by \(\angle AOB\). Show, with proof, how to construct, using only ruler and compass, a line segment \(CD\) passing through \(P\) such that \(C\) lies on the way \(OA\) and \(D\) lies on the ray \(OB\), and \(CP:PD=1:2\).

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

Find the number of integers $x$ such that the following three conditions all hold: $\bullet$ $x$ is a multiple of $5$ $\bullet$ $121 < x < 1331$ $\bullet$ When $x$ is written as an integer in base $11$ with no leading $0$s (i.e. no $0$s at the very left), its rightmost digit is strictly greater than its leftmost digit.

Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on $F$ such that for all $x,y,z$ in $F$, $(\text i)$ $x*z=y*z$ implies $x=y$ $(\text{ii})$ $x*(y*z)\ne(x*y)*z$

Let $f(x)$ be a polynomial with integer coefficients such that $f(15) f(21) f(35) - 10$ is divisible by $105$. Given $f(-34) = 2014$ and $f(0) \ge 0$, find the smallest possible value of $f(0)$. [i]Proposed by Michael Kural and Evan Chen[/i]

We have a $6 \times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?

Let $ABC$ be a triangle with $AB \neq BC$ and let $BD$ the interior bisectrix of $ \angle ABC$ with $D \in AC$ . Let $M$ be the midpoint of the arc $AC$ that contains the point $B$ in the circumcircle of the triangle $ABC$ .The circumcircle of the triangle $BDM$ intersects the segment $AB$ in $K \neq B$ . Denote by $J$ the symmetric of $A$ with respect to $K$ .If $DJ$ intersects $AM$ in $O$ then prove that $J,B,M,O$ are concyclic.

Let $A_1A_2...A_{2010}$ be a regular $2010$-gon. Find the number of obtuse triangles whose vertices are among $A_1$, $A_2$,$ ...$, $A_{2010}$.

Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and randomly presses up or down at each floor. After doing this $482$ times, the probability that Dave is on the first floor can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is the remainder when $m+n$ is divided by $1000$? $\text{(A) }136\qquad\text{(B) }294\qquad\text{(C) }508\qquad\text{(D) }692\qquad\text{(E) }803$

There are ten cards with the number $a$ on each, ten with $b$ and ten with $c$, where $a, b$ and $c$ are distinct real numbers. For every five cards, it is possible to add another five cards so that the sum of the numbers on these ten cards is $0$. Prove that one of $a, b$ and $c$ is $0$.

The numbers 1 through 9 can be arranged in the triangles labeled $a$ through $i$ illustrated below so that the numbers in each of the $2\times2$ triangles sum to the value $n$; that is \[a+b+c+d=b+e+f+g=d+g+h+i=n.\] For each possible sum $n$, show an arrangement, labeled with the sum as shown below. Prove that there are no possible arrangements for any other values of $n$. [asy] size(150); defaultpen(linewidth(0.7)+fontsize(12)); picture p = new picture; draw(p,(-3,-3^.5)/2--(3,-3^.5)/2^^(-1,0)--(1,0)^^(-1,3^.5)/2--(1,3^.5)/2); add(p); add(rotate(120)*p); add(rotate(240)*p); string[] hexlbl = {'d','c','b','f','g','h'}, trilbl = {'a','e','i'}; for(int i = 0; i < hexlbl.length; ++i) label('$'+hexlbl[i]+'$',dir(30+60*i)/3^.5); for(int i = 0; i < trilbl.length; ++i) label('$'+trilbl[i]+'$',dir(90+120*i)*2/3^.5);[/asy]

In triangle $ ABC$ we have $ a \geq b$ and $ a \geq c.$ Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis $ s_a$ to the altitude $ a_a.$ When do we have equality?

Find all primes which are sum of two primes and difference of two primes.

14 students attend the IMO training camp. Every student has at least $k$ favourite numbers. The organisers want to give each student a shirt with one of the student's favourite numbers on the back. Determine the least $k$, such that this is always possible if: $a)$ The students can be arranged in a circle such that every two students sitting next to one another have different numbers. $b)$ $7$ of the students are boys, the rest are girls, and there isn't a boy and a girl with the same number.

In the Cartesian plane consider the hyperbola \[\Gamma=\{M(x,y)\in\mathbb{R}^2 \vert \frac{x^2}{4}-y^2=1\} \] and a conic $\Gamma '$, disjoint from $\Gamma$. Let $n(\Gamma ,\Gamma ')$ be the maximal number of pairs of points $(A,A')\in\Gamma\times\Gamma '$ such that $AA'\le BB'$, for any $(B,B')$ For each $p\in\{0,1,2,4\}$, find the equation of $\Gamma'$ for which $n(\Gamma ,\Gamma ')=p$. Justify the answer.

Compute the sum of all real solutions $\alpha$ (in radians) to the equation \[ |\sin\alpha|=\left\lfloor \frac{\alpha}{20} \right\rfloor. \]

Let $a_1, a_2, a_3, \dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n > N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1, a_2, \dots, a_{n-1}$. Prove that at least one of the sequence $a_1, a_3, a_5, \dots$ and $a_2, a_4, a_6, \dots$ is eventually periodic. (An infinite sequence $b_1, b_2, b_3, \dots$ is eventually periodic if there exist positive integers $p$ and $M$ such that $b_{m+p} = b_m$ for all $m \ge M$.)

Let $f$ be a function from $\mathbb Z$ to $\mathbb R$ which is bounded from above and satisfies $f(n)\le\frac12(f(n-1)+f(n+1))$ for all $n$. Show that $f$ is constant.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Gamma$, and let $E$ be the midpoint of the diagonal $BD$. Let $I_1, I_2, I_3, I_4$ be the centers of the circles inscribed into triangles $\vartriangle ABE$, $\vartriangle ADE$, $\vartriangle BCE$, $\vartriangle CDE$, in that order. Prove that the circles $AI_1I_2$ and $CI_3I_4$ intersect $\Gamma$ at diametrically opposite points. Remark: For a circle $C$ and points $X, Y \in C$, we say that $X$ and $Y$ are diametrically opposite if $XY$ is a diameter of $C$.

From a point $A$ construct tangents to a circle centered at point $O$, intersecting the circle at $P$ and $Q$ respectively. Let $M$ be the midpoint of $PQ$. If $K$ and $L$ are points on circle $O$ such that $K, L$, and $A$ are collinear, prove $\angle MKO = \angle MLO$.

Show that a continuous function $f : \mathbb{R} \to \mathbb{R}$ is increasing if and only if \[(c - b)\int_a^b f(x)\, \text{d}x \le (b - a) \int_b^c f(x) \, \text{d}x,\] for any real numbers $a < b < c$.

From an arbitrary point $O$ inside a convex $n$-gon, perpendiculars are drawn on (extensions of the) sides of the $n$-gon. Along each perpendicular a vector is constructed, starting from $O$, directed towards the side onto which the perpendicular is drawn, and of length equal to half the length of the corresponding side. Find the sum of these vectors.

Given a graph with $n \geq 4$ vertices. It is known that for any two of vertices there is a vertex connected with none of these two vertices. Find the greatest possible number of the edges in the graph.

A triangle has sides $a, b,c$, radius of the incircle $r$ and radii of the excircles $r_a, r_b, r_c$: Prove that: a) The triangle is right-angled if and only if: $r + r_a + r_b + r_c = a + b + c$. b) The triangle is right-angled if and only if: $r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2$.

Let $a,b,c$, and $d$ be real numbers such that $$a+b = c +d+ 12$$ and $$ab + cd - 28 = bc + ad.$$ Find the minimum possible value of $a^4+b^4+c^4+d^4$.