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?

Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and $$\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1})$$ $\forall n=1,2,...$ Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$

$ N $ grasshoppers are lined up in a row. At any time, one grasshopper is allowed to jump over exactly two grasshoppers standing to the right or left of him. At what $ n $ can grasshoppers rearrange themselves in reverse order?

$(a)$ Find a polynomial with integer coefficients of the smallest degree having $\sqrt{2} + \sqrt[3]{3}$ as a root. $(b)$ Solve $1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}$.

Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages $ 40$ miles per hour, he arrives at his workplace three minutes late. When he averages $ 60$ miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 58$

Weights of $1$ g, $2$ g,$ ...$ , $200$ g are placed on the two pans of a balance such that on each pan there are $100$ weights and the balance is in equilibrium. Prove that one can swap $50$ weights from one pan with $50$ weights from the other pan such that the balance remains in equilibrium. Kvant Magazine

In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2$ . (here $r$ is the radius of the circle inscribed in the triangle $ABC$). (Mykola Moroz)

If $\frac{a}{3}=b$ and $\frac{b}{4}=c$, what is the value of $\frac{ab}{c^2}$? $\text{(A) }12\qquad\text{(B) }36\qquad\text{(C) }48\qquad\text{(D) }60\qquad\text{(E) }144$

Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system $\begin{cases} \frac{1}{x}+\frac{4}{y}+\frac{9}{z}=3 \\ x + y + z \le 12 \end{cases}$

Sofia has $5$ pieces of paper on a table. He takes some of the pieces, cuts each one into $5$ little pieces, and puts them back on the table. She repeats this procedure several times until she gets tired. Could Sofia end up with $2010$ pieces on the table?

Given a positive integer $k$ and an integer $a\equiv 3 \pmod{8}$, show that $a^m+a+2$ is divisible by $2^k$ for some positive integer $m$.

In the $(x,y)-$plane, if $R$ is the set of points inside and on a convex polygon, let $D(x,y)$ be the distance from $(x,y)$ to the nearest point of $R.$ (a) Show that there exists constants $a,b,c,$ independent of $R$, such that $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-D(x,y)} dxdy =a+bL+cA,$$ where $L$ is the perimeter of $R$ and $A$ is the area of $R.$ (b) Find the values of $a,b$ and $c.$

Determine the positive integers $n \ge 3$ such that, for every integer $m \ge 0$, there exist integers $a_1, a_2,..., a_n$ such that $a_1 + a_2 +...+ a_n = 0$ and $a_1a_2 + a_2a_3 + ...+a_{n-1}a_n + a_na_1 = -m$ Alexandru Mihalcu

Let $f(x) = ax^2 + bx+ c$ and $g(x) = cx^2 + bx + a$. If $|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1$, prove that for $|x| \leq 1$, [b](a)[/b] $|f(x)| \leq 5/4$, [b](b)[/b] $|g(x)| \leq 2$.

Jacek has $n$ cards numbered consecutively with the numbers $1,. . . , n$, which he places in a row on the table, in any order he chooses. Jacek will remove cards from the table in the sequence consistent with the numbering of cards: first they will remove the card number $1$, then the card number $2$, and so on. Before Jacek starts taking the cards, Pie will color each one of cards in red, blue or yellow. Prove that Pie can color the cards in such a way that when Jacek takes them off, it will be fulfilled at every moment the following condition: between any two cards of the same suit there is at least one card of a different color.

Let $P$ be an arbitrary point in the plane of triangle $ABC$. Lines $PA, PB, PC$ meets the perpendicular bisectors of $BC,CA,AB$ at $O_a,O_b,O_c$ respectively. Let $(O_a)$ be the circle with center $O_a$ passing through two points $B,C$, two circles $(O_b), (O_c)$ are defined in the same manner. Two circles $(O_b), (O_c)$ meets at $A_1$, distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Let $Q$ be an arbitrary point in the plane of $ABC$ and $QB,QC$ meets $(O_c)$ and $(O_b)$ at $A_2,A_3$ distinct from $B,C$. Similarly, we have points $B_2,B_3,C_2,C_3$. Let $(K_a), (K_b), (K_c)$ be the circumcircles of triangles $A_1A_2A_3, B_1B_2B_3, C_1C_2C_3$. Prove that (a) three circles $(K_a), (K_b), (K_c)$ have a common point. (b) two triangles $K_aK_bK_c, ABC$ are similar. Trần Quang Hùng

Consider the sequence $(a_n)_{n\ge 1}$ such that $a_1=1$ and $a_{n+1}=\sqrt{a_n+n^2}$, $\forall n\ge 1$. $\textbf{(a)}$ Prove that there is exactly one rational number among the numbers $a_1,a_2,a_3,\dots$. $\textbf{(b)}$ Consider the sequence $(S_n)_{n\ge 1}$ such that $$S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}.$$ Prove that there exists an integer $N$ such that $S_n>0.9$, $\forall n>N$. [i] (Stefan Obadă)[/i]

Find the number of collections of $16$ distinct subsets of $\{1, 2, 3, 4, 5\}$ with the property that for any two subsets $X$ and $Y$ in the collection, $X\cap Y \neq \emptyset$.

In a square $ABCD$ with side $k$, let $P$ and $Q$ in $BC$ and $DC$ respectively, where $PC = 3PB$ and $QD = 2QC$. Let $M$ be the point of intersection of the lines $AQ$ and $PD$, determine the area of $QMD$ in function of $k$

Let $ABC$ be an isosceles triangle with $AB=AC=20$ . Let $P$ be a point inside the triangle $ABC$ such that the sum of the distances of $P$ to $AB$ and $AC$ is $1$ . Describe the locus of all such points inside triangle $ABC$.

A polyhedron has faces that all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces?

Show that if $ \left| ax^2+bx+c\right|\le 1, $ for all $ x\in [-1,1], $ then $ |a|+|b|+|c|\le 4. $

An urn contains balls of $k$ different colors; there are $n_i$ balls of $i-th$ color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls $m$ balls of the same color appear. Find the greatest number of selections.

The board depicts the triangle $ABC$, the altitude $AH$ and the angle bisector $AL$ which intersectthe inscribed circle in the triangle at the points $M$ and $N, P$ and $Q$, respectively. After that, the figure was erased, leaving only the points $H, M$ and $Q$. Restore the triangle $ABC$. (Bogdan Rublev)

In an $n\times n$ array all the fields are colored with a different color. In one move one can choose a row, move all the fields one place to the right, and move the last field (from the right) to the leftmost field of the row; or one can choose a column, move all the fields one place downwards, and move the field at the bottom of the column to the top field of the same column. For what values of $n$ is it possible to reach any arrangement of the $n^2$ fields using these kinds of steps? [i]Proposed by Ádám Schweitzer[/i]