$P$ is a point inside $\triangle ABC$. $AP, BP, CP$ intersects $BC, CA, AB$ at $D, E, F$, respectively. $AD$ meets $(ABC)$ again at $D_1$. $S$ is a point on $(ABC)$. Lines $AS$, $EF$ intersect at $T$, lines $TP, BC$ intersect at $K$, and $KD_1$ meets $(ABC)$ again at $X$. Prove that $S, D, X$ are colinear.

Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]

There are $128$ numbered seats arranged around a circle in a palaestra. The first person to enter the place would sit on seat number $1$. Since a contagious disease is infecting the people of the city, each person who enters the palaestra would sit on a seat whose distance is the longest to the nearest occupied seat. If there are several such seats, the newly entered person would sit on the seat with the smallest number. What is the number of the seat on which the $39$th person sits?

For which pair $(a,b)$, there is no positive real pair $(x,y)$ satisfying $x+2y < a$ and $xy > b$ ? $\textbf{(A)}\ \left (\frac{15}{7}, \frac{4}{7}\right ) \qquad\textbf{(B)}\ \left (\frac{18}{11}, \frac{1}{3}\right ) \qquad\textbf{(C)}\ \left (\frac{5}{7}, \frac{1}{16}\right ) \qquad\textbf{(D)}\ \left (\frac{6}{7}, \frac{1}{11}\right ) \qquad\textbf{(E)}\ \text{None}$

Given that $a,b,c,d,e$ are real numbers such that $a+b+c+d+e=8$, $a^2+b^2+c^2+d^2+e^2=16$. Determine the maximum value of $e$.

Let $ \triangle{XOY}$ be a right-angled triangle with $ m\angle{XOY}\equal{}90^\circ$. Let $ M$ and $ N$ be the midpoints of legs $ OX$ and $ OY$, respectively. Given that $ XN\equal{}19$ and $ YM\equal{}22$, find $ XY$. $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to $5$ instead of $2000$ below.) What is the number in the lowermost row? 1 2 3 4 5 3 5 7 9 8 12 16 20 28 4

Ana and Natalia alternately play on a $ n \times n$ board (Ana rolls first and $n> 1$). At the beginning, Ana's token is placed in the upper left corner and Natalia's in the lower right corner. A turn consists of moving the corresponding piece in any of the four directions (it is not allowed to move diagonally), without leaving the board. The winner is whoever manages to place their token on the opponent's token. Determine if either of them can secure victory after a finite number of turns.

Given a positive integer $n$, determine the largest integer $M$ satisfying $$\lfloor \sqrt{a_1}\rfloor + ... + \lfloor \sqrt{a_n} \rfloor \ge \lfloor\sqrt{ a_1 + ... + a_n +M \cdot min(a_1,..., a_n)}\rfloor $$ for all non-negative integers $a_1,...., a_n$. S. Berlov, A. Khrabrov

Let $f : \mathbb R \to \mathbb R$ be a function such that $f(1)=1$ and \[f(x+y)=f(x)+f(y)\] And for all $x \in \mathbb R / \{0\}$ we have $f\left( \frac 1x \right) = \frac{1}{f(x)}.$ Find all such functions $f.$

$f(x),g(x),h(x)$ are square trinomials with discriminant, that equals $2$. And $f(x)+g(x),f(x)+h(x),g(x)+h(x)$ are square trinomials with discriminant, that equals $1$. Prove,that $f(x)+g(x)+h(x)$ has not roots.

On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships have returned to port, how many meetings of two ships have taken place?

Given an isosceles triangle $ ABC$ with $ AB \equal{} BC$. A point $ M$ is chosen inside $ ABC$ such that $ \angle AMC \equal{} 2\angle ABC$ . A point $ K$ lies on segment $ AM$ such that $ \angle BKM \equal{}\angle ABC$. Prove that $ BK \equal{} KM\plus{}MC$.

$O$ is circumcircle and $AH$ is the altitude of $\triangle ABC$. $P$ is the point on line $OC$ such that $AP \perp OC$. Prove, that midpoint of $AB$ lies on the line $HP$.

What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common? $ \textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5 \qquad \textbf{(D) }6 \qquad \textbf{(E) }10 \qquad $

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.