In an acute-angled triangle $ ABC$, $ D,E,F$ are the feet of the altitudes from $ A,B,C$, respectively, and $ P,Q,R$ are the feet of the perpendiculars from $ A,B,C$ onto $ EF,FD,DE$, respectively. Prove that the lines $ AP,BQ,CR$ are concurrent.

The bisectors of the angles $ CAB, ABC, BCA $ of the triangle $ ABC $ intersect the circle circumcribed around this triangle at points $ K, L, M $, respectively. Prove that $$ AK+BL+CM > AB+BC+CA.$$

The diagonals $AC{}$ and $BD$ of the trapezoid $ABCD$ intersect at $E{}.$ The bisector of the angle $BEC$ intersects the bases $BC$ and $AD$ at $X{}$ and $Z{}$. The perpendicular bisector of the segment $XZ$ intersects the sides $AB$ and $CD$ at $Y{}$ and $T{}$. Prove that $XYZT{}$ is a rhombus. [i]Proposed by M. Didin, I. Kukharchuk and P. Puchkov[/i]

Integers $1, 2,...,100$ are written on a circle, not necessarily in that order. Can it be that the absolute value of the dierence between any two adjacent integers is at least $30$ and at most $50$?

Peter has a deck of $1001$ cards, and with a blue pen he has written the numbers $1,2,\ldots,1001$ on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card $C$, he took $500$ consecutive cards following $C$ (clockwise order), and denoted by $f(C)$ the number of blue numbers written on those $500$ cards that are greater than the blue number written on $C$ itself. After all, he wrote this $f(C)$ number on the top side of the card $C$ with a red pen. Prove that Peter's friend Basil, who sees all the red numbers on these cards, can determine the blue number on each card.

Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$. Prove that $m^k$ is divisible by $k^m$.

Let $a<b<c<d<e$ be real numbers. Among the $10$ sums of the pairs of these numbers, the least $3$ are $32,36,37$, while the largest two are $48$ and $51$. Find all possible values of $e$

Evaluate \[\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}.\]

Determine and justify all solutions $(x,y, z)$ of the system of equations: $x^2 = y + z$ $y^2 = x + z$ $z^2 = x + y$

Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Let one of its internal tangents touches $\omega_1$ and $\omega_2$ at $A_1$ and $A_2$, respectively, and let one of its external tangents touches $\omega_1$ and $\omega_2$ at $B_1$ and $B_2$, respectively. Prove that if $A_1B_2\perp A_2B_1$, then $A_1B_2 = A_2B_1$.

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees? [i]Proposed by Seyed Reza Hosseini[/i]

A magician should determine the area of a hidden convex $ 2008$-gon $ A_{1}A_{2}\cdots A_{2008}$. In each step he chooses two points on the perimeter, whereas the chosen points can be vertices or points dividing selected sides in selected ratios. Then his helper divides the polygon into two parts by the line through these two points and announces the area of the smaller of the two parts. Show that the magician can find the area of the polygon in $ 2006$ steps.

$m$ and $n$ are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows: Starting from one of the hexagons, the Donkey moves $m$ cells in one of the $6$ directions, then it turns $60$ degrees clockwise and after that moves $n$ cells in this new direction until it reaches it's final cell. At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey? [i]Proposed by Shayan Dashmiz[/i]

Let $\ell = 1$, $M = 23$, $N = 45$, and $u = 67$. Compute the number of ordered pairs of nonnegative integers $(X, Y)$ with $X \leq M - \ell$ and $Y \leq N + u$ such that the sum \[ \sum_{k=\ell}^{u} \binom{X + k}{M}\cdot\binom{Y - k}{N} \] is divisible by $89$ (for integers $a$ and $b$, define the binomial coefficient $\tbinom{a}{b}$ to be the number of $b$-element subsets of any given $a$-element set, which is $0$ when $a < 0$, $b < 0$, or $b > a$).

In a room containing $N$ people, $N > 3$, at least one person has not shaken hands with everyone else in the room. What is the maximum number of people in the room that could have shaken hands with everyone else? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }N-1\qquad\textbf{(D) }N\qquad \textbf{(E) }\text{none of these}$

Q.A bug travels in the co-ordinate plane moving along only the lines that are parallel to the $X$ and $Y$ axes.Let $A=(-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$.How many lattice points lie on at least one of these paths. My answer ($87$)

a) Given a big square consisting of $7\times 7$ squares. You should mark the centres of $k$ points in such a way, that no quadruple of the marked points will be the vertices of a rectangle with the sides parallel to the sides of the given squares. What is the greatest $k$ such that the problem has solution? b) The same problem for $13\times 13$ square.

Let $a$ be a positive integer.Suppose that $\forall n$ ,$\exists d$, $d\not =1$, $d\equiv 1\pmod n$ ,$d\mid n^2a-1$.Prove that $a$ is a perfect square.

Let $k$ be a positive integer. Find all collection of integers $(a_1, a_2,\cdots, a_k)$ such that there exist a non-linear polynomial $P$ with integer coefficients, so that for all positive integers $n$ there exist a positive integer $m$ satisfying: $$P(n+a_1)+P(n+a_2)+...+P(n+a_k)=P(m)$$ [i]Proposed by Ivan Chan Kai Chin[/i]

A sequence $a_1, a_2,\dots, a_n$ of positive integers is [i]alagoana[/i], if for every $n$ positive integer, one have these two conditions I- $a_{n!} = a_1\cdot a_2\cdot a_3\cdots a_n$ II- The number $a_n$ is the $n$-power of a positive integer. Find all the sequence(s) [i]alagoana[/i].

Let $ABC$ be any triangle with $\angle BAC \le \angle ACB \le \angle CBA$. Let $D, E$ and $F$ be the midpoints of $BC, CA$ and $AB$, respectively, and let $\epsilon$ be a positive real number. Suppose there is an ant (represented by a point $T$ ) and two spiders (represented by points $P_1$ and $P_2$, respectively) walking on the sides $BC, CA, AB, EF, FD$ and $DE$. The ant and the spiders may vary their speeds, turn at an intersection point, stand still, or turn back at any point; moreover, they are aware of their and the others’ positions at all time. Assume that the ant’s speed does not exceed $1$ mm/s, the first spider’s speed does not exceed $\frac{\sin A}{2 \sin A+\sin B}$ mm/s, and the second spider’s speed does not exceed $\epsilon$ mm/s. Show that the spiders always have a strategy to catch the ant regardless of the starting points of the ant and the spiders. Note: the two spiders can discuss a plan before the hunt starts and after seeing all three starting points, but cannot communicate during the hunt.

Let $n$ be a positive integer. Find the greatest possible integer $m$, in terms of $n$, with the following property: a table with $m$ rows and $n$ columns can be filled with real numbers in such a manner that for any two different rows $\left[ {{a_1},{a_2},\ldots,{a_n}}\right]$ and $\left[ {{b_1},{b_2},\ldots,{b_n}} \right]$ the following holds: \[\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,...,\left| {{a_n} - {b_n}} \right|} \right) = 1\] [i]Poland (Tomasz Kobos)[/i]

A point $P$ in the interior of a regular hexagon is at distances $8,8,16$ units from three consecutive vertices of the hexagon, respectively. If $r$ is radius of the circumscribed circle of the hexagon, what is the integer closest to $r$?