Consider a game in the integer points of the real line, where an Angel tries to escape from a Devil. A positive integer $k$ is chosen, and the Angel and the Devil take turns playing. Initially, no point is blocked. The Angel, in point $A$, can move to any point $P$ such that $|AP| \le k$, as long as $P$ is not blocked. The Devil may block an arbitrary point. The Angel loses if it cannot move and wins if it does not lose in finitely many turns. Let $f(k)$ denote the least number of rounds the Devil takes to win. Prove that $$0.5 k \log_2 (k) (1 + o(1)) \le f(k) \le k \log_2(k) (1 +o(1)).$$ Note: $a(x) = b(x) (1+o(1))$ if $\lim_{x \to \infty} \frac{b(x)}{a(x)} = 1$.

Point $P$ lies inside the triangle $ABC$. Points $K, L, M$ are symmetrics of point $P$ wrt the midpoints of the sides $BC, CA, AB$. Prove that the straight $AK, BL, CM$ intersect at one point.

Let $I$ be the incenter of a non-equilateral triangle $ABC$ and $T_1$, $T_2$, and $T_3$ be the tangency points of the incircle with the sides $BC$, $CA$ and $AB$, respectively. Prove that the orthocenter of triangle $T_1T_2T_3$ lies on the line $OI$, where $O$ is the circumcenter of triangle $ABC$. [i]Proposed by Georgi Ganchev[/i]

Let $p$ be an odd prime. Prove that the number $$\left\lfloor \left(\sqrt{5}+2\right)^{p}-2^{p+1}\right\rfloor$$ is divisible by $20p$.

Let $m$ and $n$ be positive integers such that $m>n$ and $m \equiv n \pmod{2}$. If $(m^2-n^2+1) \mid n^2-1$, then prove that $m^2-n^2+1$ is a perfect square. (proposed by G. Batzaya, folklore)

Let $a_n$ be a sequence given by $a_1 = 18$, and $a_n = a_{n-1}^2+6a_{n-1}$, for $n>1$. Prove that this sequence contains no perfect powers.

The numbers $12,14,37,65$ are one of the solutions of the equation $xy-xz+yt=182$. What number corresponds to which letter?

You are given a $5\times 6$ checkerboard with squares alternately shaded black and white. The bottom- left square is white. Each square has side length $1$ unit. You can normally travel on this board at a speed of $2$ units per second, but while you travel through the interior (not the boundary) of a black square, you are slowed down to $1$ unit per second. What is the shortest time it takes to travel from the bottom-left corner to the top-right corner of the board?

Tnag is repeating the phrase "sigma sigma on the wall'' an infinite number times. Between each word, there is exactly one second of pause. Adam has heard the phrase so many times that he has come up with a game using two numbers $x$ and $y$: Start with a score of 0. [list] [*] At a random time, Adam will hear the word $a$ (each of the 5 words are equally likely to be heard). [*] Then [list] [*] if $a$ is "sigma'', Adam will multiply his score by $x$, and [*] if $a$ is any of the other words, Adam will add $y$ to his score. [/list] [/list] Let $f(x,y)$ be Adam's expected score after infinitely many steps. Find \[ \sum_{n=2}^{\infty}f\left(\frac{1}{n}, \frac{1}{n^2}\right). \]

Let $a, b, c, d > 0$ satisfying $abcd = 1$. Prove that $$\frac{1}{a + b + 2}+\frac{1}{b + c + 2}+\frac{1}{c + d + 2}+\frac{1}{d + a + 2} \le 1$$

[color=darkred]Determine all functions $ f : [0; \plus{} \infty) \rightarrow [0; \plus{} \infty)$, such that \[ f(x \plus{} y \minus{} z) \plus{} f(2\sqrt {xz}) \plus{} f(2\sqrt {yz}) \equal{} f(x \plus{} y \plus{} z)\] for all $ x,y,z \in [0; \plus{} \infty)$, for which $ x \plus{} y\ge z$.[/color]

If $A$ and $B$ are numbers such that the polynomial $x^{2017} + Ax + B$ is divisible by $(x + 1)^2$, what is the value of $B$?

Diagonals $AC,BD$ of cyclic quadrilateral $ABCD$ intersect at $P$.Point $Q$ is on$BC$ (between$B$ and $C$) such that $PQ \perp AC$.Prove that the line passes through the circumcenters of triangles $APD$ and $BQD$ is parallel to $AD$.(A.Kuznetsov)

A wooden beam $EFGH$ $ABCD$ is with three cuts in $8$ smaller ones sawn beams. Each cut is parallel to one of the three pair of opposit sides. Each pair of saw cuts is shown perpendicular to each other. The smaller bars at the corners $A, C, F$ and $H$ have a capacity of $9, 12, 8, 24$ respectively.(The proportions in the picture are not correct!!). Calculate content of the entire bar. [asy] unitsize (0.5 cm); pair A, B, C, D, E, F, G, H; pair x, y, z; x = (1,0.5); y = (-0.8,0.8); z = (0,1); B = (0,0); C = 5*x; A = 3*y; F = 4*z; E = A + F; G = C + F; H = A + C + F; fill(y--3*y--(3*y + z)--(y + z)--cycle, gray(0.8)); fill(2*x--5*x--(5*x + z)--(2*x + z)--cycle, gray(0.8)); fill((y + z)--(y + 4*z)--(y + 4*z + 2*x)--(4*z + 2*x)--(2*x + z)--z--cycle, gray(0.8)); fill((2*x + y + 4*z)--(2*x + 3*y + 4*z)--(5*x + 3*y + 4*z)--(5*x + y + 4*z)--cycle, gray(0.8)); draw(B--C--G--H--E--A--cycle); draw(B--F); draw(E--F); draw(G--F); draw(y--(y + 4*z)--(y + 4*z + 5*x)); draw(2*x--(2*x + 4*z)--(2*x + 4*z + 3*y)); draw((3*y + z)--z--(5*x + z)); label("$A$", A, SW); label("$B$", B, S); label("$C$", C, SE); label("$E$", E, NW); label("$F$", F, SE); label("$G$", G, NE); label("$H$", H, N); [/asy]

[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one. [b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$. [b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that (a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color; (b) there are no two neighboring blue chips. [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a board $4\times 4$ the numbers from $1$ to $16$ are written, one in each box. Andres and Pablo choose four numbers each. Andrés chooses the biggest of each row and Pablo, the biggest of each column. The same number can be chosen by both. Then they are removed from the board all chosen numbers. What is the greatest value that the sum of the numbers can have what are left on the board?