Find a factor of $2^{33}-2^{19}-2^{17}-1$ that lies between $1000$ and $5000$.

Solve the system of equations $$ \left\{\begin{array}{l} x \log x+y \log y+z \log x=0\\ \\ \dfrac{\log x}{x}+\dfrac{\log y}{y}+\dfrac{\log z}{z}=0 \end{array} \right. $$

Let $m$ be a positive integer. Find all bounded sequences of integers $a_1, a_2, a_3,... $for which $a_n + a_{n+1} + a_{n+m }= 0$ for all $n \in N$.

Compute $1+2\cdot3^4$. [i]Proposed by Evan Chen[/i]

Let $ABC$ be an acute triangle, $D$ be the midpoint of $BC$. Bisectors of angles $ADB$ and $ADC$ intersect the circles circumscribed around the triangles $ADB$ and $ADC$ at points $E$ and $F$, respectively. Prove that $EF\perp AD$.

Compute the value of \[ \frac{d}{dx}\int_{1}^{10} x^3\,dx. \]

Given a cube with side length $ 1$, we perform six cuts as follows: one cut parallel to the $xy$-plane, two cuts parallel to the $yz$-plane, and three cuts parallel to the $xz$-plane, where the cuts are made uniformly independent of each other. What is the expected value of the volume of the largest piece?

In triangle $ABC, \angle A = 90^o$. $M$ is the midpoint of $BC$ and $H$ is the foot of the altitude from $A$ to $BC$. The line passing through $M$ and perpendicular to $AC$ meets the circumcircle of triangle $AMC$ again at $P$. If $BP$ intersects $AH$ at $K$, prove that $AK = KH$.

In the triangle $ABC$ it is known that $\angle ACB> 90 {} ^ \circ$, $\angle CBA> 45 {} ^ \circ$. On the sides $AC$ and $AB$, respectively, there are points $P$ and $T$ such that $ABC$ and $PT = BC$. The points ${{P} _ {1}}$ and ${{T} _ {1}}$ on the sides $AC$ and $AB$ are such that $AP = C {{P} _ {1}}$ and $AT = B {{T} _ {1}}$. Prove that $\angle CBA- \angle {{P} _ {1}} {{T} _ {1}} A = 45 {} ^ \circ$. (Anton Trygub)

Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$. (I.Voronovich)

Find all functions $f: R \to R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$

Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that \[ xyz(x+y+z)+2021\geqslant 2024xyz\]

Inside an $11\times 11$ square, Pablo drew a rectangle and extending its sides divided the square into $5$ rectangles, as shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/5/a/7774da7085f283b3aae74fb5ff472572571827.gif[/img] Sofía did the same, but she also managed to make the lengths of the sides of the $5$ rectangles be whole numbers between $1$ and $10$, all different. Show a figure like the one Sofia made.

There are scales and $(n+1)$ weights with the total weight $2n$. Each weight is an integer. We put all the weights in turn on the lighter side of the scales, starting from the heaviest one, and if the scales is in equilibrium -- on the left side. Prove that when all the weights will be put on the scales, they will be in equilibrium.

Given four colours and enough square plates $1\times 1$. We have to paint four edges of every plate with four different colours and combine plates, putting them with the edges of the same colour together. Describe all the pairs $m,n$, such that we can combine those plates in a $n\times m$ rectangle, that has every edge of one colour, and its four edges have different colours.

Given the polynomial $P(x)=(x^2-7x+6)^{2n}+13$ where $n$ is a positive integer. Prove that $P(x)$ can't be written as a product of $n+1$ non-constant polynomials with integer coefficients.

A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have? [asy]draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4)); draw((3,0)--(3,3)); draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed); draw((2.5,4)--(2.5,1),dashed); label("$x$",(2.75,3.5),NNE); label("$y$",(4.125,1.5),NNE); [/asy] $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

Let $A$ and $B$ be sets of positive integers with $|A|\ge 2$ and $|B|\ge 2$. Let $S$ be a set consisting of $|A|+|B|-1$ numbers of the form $ab$ where $a\in A$ and $b\in B$. Prove that there exist pairwise distinct $x,y,z\in S$ such that $x$ is a divisor of $yz$.

Let $n$ be an even integer greater than $2$. On the vertices of a regular polygon with n sides we can place red or blue chips. Two players, $A$ and $B$, play alternating turns of the next mode: each player, on their turn, chooses two vertices that have no tiles and places on one of them a red chip and in the other a blue chip. The goal of $A$ is to get three vertices consecutive with tiles of the same color. $B$'s goal is to prevent this from happening. To the beginning of the game there are no tiles in any of the vertices. Show that regardless of who starts to play, Player $B$ can always achieve his goal.

[b]p1.[/b] $17$ rooks are placed on an $8\times 8$ chess board. Prove that there must be at least one rook that is attacking at least $2$ other rooks. [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins? [b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h, i$ if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc} & & a & b & c \\ x & & & d & e \\ \hline & f & a & c & c \\ + & g & h & i & \\ \hline f & f & f & c & c \\ \end{tabular}$ [b]p4.[/b] Pinocchio rode a bicycle for $3.5$ hours. During every $1$-hour period he went exactly $5$ km. Is it true that his average speed for the trip was $5$ km/h? Explain your reasoning. [b]p5.[/b] Let $a, b, c$ be odd integers. Prove that the equation $ax^2 + bx + c = 0$ cannot have a rational solution. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x_i\ge -1$ and $\sum^n_{i=1}x_i^3=0$. Prove $\sum^n_{i=1}x_i \le \frac{n}{3}$.

suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.

Let $n$ numbers all equal to $1$ be written on a blackboard. A move consists of replacing two numbers on the board with two copies of their sum. It happens that after $h$ moves all $n$ numbers on the blackboard are equal to $m.$ Prove that $h \leq \frac{1}{2}n \log_2 m.$

Let $k$ be a positive integer and $n_1, n_2, ..., n_k$ be non-negative integers. Points $P_1, P_2, ..., P_k$ lie on a circle in such a way that at point $P_i$ there are $n_i$ stones. Leandro wishes to change the position of some of these stones in order to accomplish his objective which is to have the same number of stones at each point of the circle. He does this by repeating as many times as necessary the following operation: if there exists a point on the circle with at least $k - 1$ stones, he can choose $k -1$ of these and distribute them by giving one to each of the remaining $k - 1$ points. For which values $n_1, n_2, ..., n_k$ can Leandro accomplish his objective? In the figure below there is a configuration of stones for $k = 4$. On the right is the initial division of stones, while on the left there is the configuration obtained from the initial one by choosing $k - 1 = 3$ stones from the top point on the circle and distributing one each to the other points. [figures missing]

Let functions $f, g: \mathbb{R}^+ \to \mathbb{R}^+$ satisfy the following: \[ f(g(x)y + f(x)) = (y+2015)f(x) \] for every $x,y \in \mathbb{R}^+$. (a) Prove that $g(x) = \frac{f(x)}{2015}$ for every $x \in \mathbb{R}^+. $ (b) State an example of function that satisfy the equation above and $f(x), g(x) \ge 1$ for every $x \in \mathbb{R}^+$.