a) Given a circle with two inscribed triangles $T_1$ and $T_2$. The vertices of $T_1$ are the midpoints of the arcs with the ends in the vertices of $T_2$. Consider a hexagon -- the intersection of $T_1$ and $T_2$. Prove that its main diagonals are parallel to $T_1$ sides and are intersecting in one point. b) The segment, that connects the midpoints of the arcs $AB$ and $AC$ of the circle circumscribed around the $ABC$ triangle, intersects $[AB]$ and $[AC]$ sides in $D$ and $K$ points. Prove that the points $A,D,K$ and $O$ -- the centre of the circle -- are the vertices of a diamond.

Let $n > 1$ be an integer. A set $S \subseteq \{ 0, 1, 2, \cdots , 4n - 1 \}$ is called ’sparse’ if for any $k \in \{ 0, 1, 2, \cdots , n - 1 \}$ the following two conditions are satisfied: $(a)$ The set $S \cap \{4k - 2, 4k - 1, 4k, 4k + 1, 4k + 2 \}$ has at most two elements; $(b)$ The set $S \cap \{ 4k +1, 4k +2, 4k +3 \}$ has at most one element. Prove that there are exactly $8 \cdot 7^{n-1}$ sparse subsets.

In $\Delta ABC$ with $AC=10$ and $BC=15$ the points $G$ and $I$ are its centroid and the center of its inscribed circle respectively. Find $AB$, if $\angle GIC=90^\circ$.

Given $n$ real numbers $a_1$, $a_2$ $\ldots$ $a_n$. ($n\geq 1$). Prove that there exists real numbers $b_1$, $b_2$ $\ldots$ $b_n$ satisfying: (a) For any $1 \leq i \leq n$, $a_i - b_i$ is a positive integer. (b)$\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}$

$16$ progamers are playing in another single elimination tournament. Each round, each of the remaining progamers plays against another and the loser is eliminated. Additionally, each time a progamer wins, he will have a ceremony to celebrate. A player's rst ceremony is ten seconds long, and afterward each ceremony is ten seconds longer than the last. What is the total length in seconds of all the ceremonies over the entire tournament?

How namy subsets with $3$ elements of set $S=\{1,2,3,...,19,20\}$ exist, such that their product is divisible by $4$.

Let $ABCD$ and $AEF G$ be unit squares such that the area of their intersection is $\frac{20}{21}$ . Given that $\angle BAE < 45^o$, $\tan \angle BAE$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.

Numbers $1,2, ..., 2n$ are partitioned into two sequences $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$. Prove that number \[W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|\] is a perfect square.

Determine the greatest number with $n$ digits in the decimal representation which is divisible by $429$ and has the sum of all digits less than or equal to $11$.

Consider a $6 \times 6$ grid of squares. Edmond chooses four of these squares uniformly at random. What is the probability that the centers of these four squares form a square?

The floor plan of a contemporary art museum is a (not necessarily convex) polygon and its walls are solid. The security guard guarding the museum has two favourite spots (points $A$ and $B$) because one can see the whole area of the museum standing at either point. Is it true that from any point of the $AB$ section one can see the whole museum?

Find the smallest real number \(a\) such that for any positive integer number \(n > 2\) and any arrangement of the numbers from 1 to \(n\) on a circle, there exists a pair of adjacent numbers whose ratio (when dividing the larger number by the smaller one) is less than \(a\). [i]Proposed by Mykhailo Shtandenko[/i]

Given real numbers $x,y,z$ which satisfies $$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$ Show that $max\{ |x|,|y|,|z|\} \le 1$.

How many integers $ N$ less than 1000 can be written as the sum of $ j$ consecutive positive odd integers from exactly 5 values of $ j\ge 1$?

A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mathematicians have no soul! It’s all numbers and equations! What is the root of your incompetence?!” Her boyfriend assumes she means the square root of himself, or the square root of i. What two answers should he give?

Find all functions $f: R\to R$ such that: \[ f(x^2)-f(y^2)=(x+y)(f(x)-f(y)), x,y \in R \]

How many kites are there such that all of its four vertices are vertices of a given regular icosagon ($20$-gon)? $ \textbf{(A)}\ 105 \qquad\textbf{(B)}\ 100 \qquad\textbf{(C)}\ 95 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 85 $

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Prove that $$2 \left( \frac{ab}{a + b} +\frac{bc}{b + c} +\frac{ca}{c+ a}\right)+ 1 \ge 6(ab + bc + ca)$$ Trần Nam Dũng

Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.

Solve the equation \[(x+1)\log^2_{3}x+4x\log_{3}x-16=0\]

Some time ago, on a radio program, a baker announced a special promotion in the purchase of two stuffed cakes. Each cake could contain up to five fillings of which had in the pastry. On the show, a lady said there were $1,048,576$ different possibilities to choose the two stuffed cakes. How many different fillings did the pastry chef have?

Let $S$ be a point outside of the plane of the parallelogram $ABCD$, such that the triangles $SAB$, $SBC$, $SCD$ and $SAD$ are equivalent. a) Prove that $ABCD$ is a rhombus. b) If the distance from $S$ to the plane $(A, B, C, D)$ is $12$, $BD = 30$ and $AC = 40$, compute the distance from the projection of the point $S$ on the plane $(A, B, C, D)$ to the plane $(S,B,C)$ .

Diameter $AB$ is drawn to a circle with radius $1$. Two straight lines $s$ and $t$ touch the circle at points $A$ and $B$, respectively. Points $P$ and $Q$ are chosen on the lines $s$ and $t$, respectively, so that the line $PQ$ touches the circle. Find the smallest possible area of the quadrangle $APQB$.

Given a set of boys and girls, we call a pair $(A,B)$ amicable if $A$ and $B$ are friends. The friendship relation is symmetric. A set of people is affectionate if it satisfy the following conditions: i) The set has the same number of boys and girls. ii) For every four different people $A,B,C,D$ if the pairs $(A,B),(B,C),(C,D)$ and $(D,A)$ are all amicable, then at least one of the pairs $(A,C)$ and $(B,D)$ is also amicable. iii) At least $\frac{1}{2013}$ of all boy-girl pairs are amicable. Let $m$ be a positive integer. Prove that there exists an integer $N(m)$ such that if a affectionate set has al least $N(m)$ people, then there exists $m$ boys that are pairwise friends or $m$ girls that are pairwise friends.

Determine all positive integers $n$ such that all positive integers less than $n$ and coprime to $n$ are powers of primes.