Show that the polynomial $x^6 -x^5 +x^4 -x^3 +x^2 -x+\frac34$ has no real zeroes.

Let $\Delta ABC$ be a triangle and $P$ be a point in its interior. Prove that \[ \frac{[BPC]}{PA^2}+\frac{[CPA]}{PB^2}+\frac{[APB]}{PC^2} \ge \frac{[ABC]}{R^2} \] where $R$ is the radius of the circumcircle of $\Delta ABC$, and $[XYZ]$ is the area of $\Delta XYZ$.

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that for all $x, y\in\mathbb{R}^+$, \[ \frac{f(x)}{y^2} - \frac{f(y)}{x^2} \le \left(\frac{1}{x}-\frac{1}{y}\right)^2\] ($\mathbb{R}^+$ denotes the set of positive real numbers.) [i](Proposed by Ivan Chan Guan Yu)[/i]

Determine all positive integers $a$ for which $a^a$ is divisible by $20^{19}$.

Let $n$ be a positive integer. Alice and Bob play the following game. First, Alice picks $n + 1$ subsets $A_1,...,A_{n+1}$ of $\{1,... ,2^n\}$ each of size $2^{n-1}$. Second, Bob picks $n + 1$ arbitrary integers $a_1,...,a_{n+1}$. Finally, Alice picks an integer $t$. Bob wins if there exists an integer $1 \le i \le n + 1$ and $s \in A_i$ such that $s + a_i \equiv t$ (mod $2^n$). Otherwise, Alice wins. Find all values of $n$ where Alice has a winning strategy.

What is the largest number of horses that you can put on a chessboard without there being two horses that can beat each other? a. Describe an arrangement with that maximum number. b. Prove that a larger number is not possible. (A chessboard consists of $8 \times 8$ spaces and a horse jumps from one field to another field according to the line "two squares vertically and one squared horizontally" or "one square vertically and two squares horizontally") [asy] unitsize (0.5 cm); int i, j; for (i = 0; i <= 7; ++i) { for (j = 0; j <= 7; ++j) { if ((i + j) % 2 == 0) { if ((i - 2)^2 + (j - 3)^2 == 5) { fill(shift((i,j))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); } else { fill(shift((i,j))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); } } }} for (i = 0; i <= 8; ++i) { draw((i,0)--(i,8)); draw((0,i)--(8,i)); } label("$a$", (0.5,-0.5), fontsize(10)); label("$b$", (1.5,-0.5), fontsize(10)); label("$c$", (2.5,-0.5), fontsize(10)); label("$d$", (3.5,-0.5), fontsize(10)); label("$e$", (4.5,-0.5), fontsize(10)); label("$f$", (5.5,-0.5), fontsize(10)); label("$g$", (6.5,-0.5), fontsize(10)); label("$h$", (7.5,-0.5), fontsize(10)); label("$1$", (-0.5,0.5), fontsize(10)); label("$2$", (-0.5,1.5), fontsize(10)); label("$3$", (-0.5,2.5), fontsize(10)); label("$4$", (-0.5,3.5), fontsize(10)); label("$5$", (-0.5,4.5), fontsize(10)); label("$6$", (-0.5,5.5), fontsize(10)); label("$7$", (-0.5,6.5), fontsize(10)); label("$8$", (-0.5,7.5), fontsize(10)); label("$P$", (2.5,3.5), fontsize(10)); [/asy]

Show that there do not exist more than $27$ half-lines (or rays) emanating from the origin in the $3$-dimensional space, such that the angle between each pair of rays is $\geq \frac{\pi}{4}$.

Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers, and let $b_1, b_2, \ldots, b_m$ be $m$ positive integers such that $a_1 a_2 \cdots a_n = b_1 b_2 \cdots b_m$. Prove that a rectangular table with $n$ rows and $m$ columns can be filled with positive integer entries in such a way that * the product of the entries in the $i$-th row is $a_i$ (for each $i \in \left\{1,2,\ldots,n\right\}$); * the product of the entries in the $j$-th row is $b_j$ (for each $i \in \left\{1,2,\ldots,m\right\}$).

I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day.

$234$ viewers came to the cinema. Determine for which$ n \ge 4$ the viewers could be can be arranged in $n$ rows so that every viewer in $i$-th row gets to know just $j$ viewers in $j$-th row for any $i, j \in \{1, 2,... , n\}, i\ne j$. (The relationship of acquaintance is mutual.) (Tomáš Jurík)

A regular tetrahedron is sectioned with a plane after a rhombus. Prove that the rhombus is square.

Two players play a game on an $8$ by $8$ chessboard according to the following rules. The first player places a knight on the board. Then each player in turn moves the knight , but cannot place it on a square where it has been before. The player who has no move loses. Who wins in an errorless game , the first player or the second one? (The knight moves are the normal ones. ) (V . Zudilin , year 12 student , Beltsy (Moldova))

Find all arcs $\theta$ such that $\frac{1}{\sin ^2 \theta}, \frac{1}{\cos ^2 \theta} $ are integer numbers and roots of equation $$x^2-ax+a=0.$$

A set $D$ of positive integers is called [i]indifferent[/i] if there are at least two integers in the set, and for any two distinct elements $x,y\in D$, their positive difference $|x-y|$ is also in $D$. Let $M(x)$ be the smallest size of an indifferent set whose largest element is $x$. Compute the sum $M(2)+M(3)+\dots+M(100)$. [i]Proposed by Yannick Yao[/i]

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ is tangent to the line $CD$, and the circle with diameter $CD$ is tangent to the line $AB$. Prove that the two intersection points of these circles and the point $AC \cap BD$ are collinear.

Find all triplets $(a,b,c)$ of complex numbers such that the equation \[x^4-ax^3-bx+c=0\] has $a,b,c$ as roots.

The sequence $(p_n)$ is defined as follows: $p_1=2$ and for all $n$ greater than or equal to $2$, $p_n$ is the largest prime divisor of the expression $p_1p_2p_3\ldots p_{n-1}+1$. Prove that every $p_n$ is different from $5$.

What is the largest possible value of $|...| |a_1 - a_2| - a_3| - ... - a_{1990}|$, where $a_1, a_2, ... , a_{1990}$ is a permutation of $1, 2, 3, ... , 1990$?

[b]a)[/b] Among $9$ apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making $2$ weighing in a two-course balance? [b]b)[/b] Find the least necessary number of weighing that must be done to cover a false currency between $27$ coins if all the others are true.

Let $(I),(O)$ be the incircle, and, respectiely, circumcircle of $ABC$. $(I)$ touches $BC,CA,AB$ in $D,E,F$ respectively. We are also given three circles $\omega_a,\omega_b,\omega_c$, tangent to $(I),(O)$ in $D,K$ (for $\omega_a$), $E,M$ (for $\omega_b$), and $F,N$ (for $\omega_c$). [b]a)[/b] Show that $DK,EM,FN$ are concurrent in a point $P$; [b]b)[/b] Show that the orthocenter of $DEF$ lies on $OP$.

Let $ ABC$ is a triangle, let $ H$ is orthocenter of $ \triangle ABC$, let $ M$ is midpoint of $ BC$. Let $ (d)$ is a line perpendicular with $ HM$ at point $ H$. Let $ (d)$ meet $ AB, AC$ at $ E, F$ respectively. Prove that $ HE \equal{}HF$.

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$

The fraction halfway between $ \frac{1}{5}$ and $ \frac{1}{3}$ (on the number line) is \[ \textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{2}{15} \qquad \textbf{(C)}\ \frac{4}{15} \qquad \textbf{(D)}\ \frac{53}{200} \qquad \textbf{(E)}\ \frac{8}{15} \]

Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $D$ be the midpoint of side $BC$ and $BE,CZ$ be the altitudes of the triangle $ABC$. Line $ZE$ intersects line $BC$ at point $O$. (i) Find all the angles of the triangle $ZDE$ in terms of angle $\angle A$ of the triangle $ABC$ (ii) Find the angle $\angle BOZ$ in terms of angles $\angle B$ and $\angle C$ of the triangle $ABC$

Given a quadrangle $ABCD$ and a point $M$ inside it such that $ABMD$ is a parallelogram. $ \angle CBM = \angle CDM$. Prove that the $ \angle ACD = \angle BCM$.