For real $a, b$ define the sequence $x_{0}, x_{1}, x_{2}, ...$ by $x_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}$. If $b = 1$, show that the sequence converges to a finite limit for all $a$. If $b > 2$, show that the sequence diverges for some $a$.

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

Select a number $X$ from the set of all $3$-digit natural numbers uniformly at random. Let $A \in [0,1]$ be the probability that $X$ is divisible by $11$, given that it is palindromic. Let $B \in [0,1]$ be the probability that X is palindromic, given that it is divisible by $11$. Compute $B-A$. Recall that a $3$-digit number is a palindrome if it reads the same left to right as right to left. For instance, $484$ is a palindrome, but $603$ is not a palindrome.

Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$. Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]

We write pairs of integers on a blackboard. Initially, the pair $(1,2)$ is written. On a move, if $(a, b)$ is on the blackboard, we can add $(-a, -b)$ or $(-b, a+b)$. In addition, if $(a, b)$ and $(c, d)$ are written on the blackboard, we can add $(a+c, b+d)$. Can we reach $(2022, 2023)$?

Show that for any positive real numbers $a, x, y, z$ the following inequalities are true $$\frac{a+z}{a+x}\cdot x+\frac{a+x}{a+y}\cdot y+\frac{a+y}{a+z}\cdot z \leq x+y+z \leq \frac{a+y}{a+z}\cdot x+\frac{a+z}{a+x}\cdot y+\frac{a+x}{a+y}\cdot z.$$

Let $ABCD$ be a parallelogram. On the ray $(DB$ a point $E$ is given such that the ray $(AB$ is the angle bisector of $\angle CAE$. Let $F$ be the intersection of $CE$ and $AB$. Prove that $\frac{AB}{BF} - \frac{AC}{AE} = 1$

A wishing well is located at the point $(11,11)$ in the $xy$-plane. Rachelle randomly selects an integer $y$ from the set $\left\{ 0, 1, \dots, 10 \right\}$. Then she randomly selects, with replacement, two integers $a,b$ from the set $\left\{ 1,2,\dots,10 \right\}$. The probability the line through $(0,y)$ and $(a,b)$ passes through the well can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [i]Proposed by Evan Chen[/i]

[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 the blackboard, there are numbers $1, 2, \dots , 100$. At each move, Bob erases arbitrary two numbers $a$ and $b$, where $a \ge b > 0$, and writes the single number $\lfloor{a/b}\rfloor$. After $99$ such moves the blackboard will contain a single number. What is its maximum possible value? (Reminder that $\lfloor{x}\rfloor$ is the maximum integer not exceeding $x$.)

Suppose that $a_1, a_2, \dots a_{2021}$ are non-negative numbers such that $\sum_{k=1}^{2021} a_k=1$. Prove that $$ \sum_{k=1}^{2021}\sqrt[k]{a_1 a_2\dots a_k} \leq 3. $$

In a right triangle the square of the hypotenuse is equal to twice the product of the legs. One of the acute angles of the triangle is: $ \textbf{(A)}\ 15^{\circ} \qquad\textbf{(B)}\ 30^{\circ} \qquad\textbf{(C)}\ 45^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 75^{\circ} $

On an $n\times n$ chart, where $n \geq 4$, stand "$+$" signs in the cells of the main diagonal and "$-$" signs in all the other cells. You can change all the signs in one row or in one column, from $-$ to $+$ or from $+$ to $-$. Prove that you will always have $n$ or more $+$ signs after finitely many operations.

The numbers from $1$ to $2015$ are written on sheets so that if if $n-m$ is a prime, then $n$ and $m$ are on different sheets. What is the minimum number of sheets required?

What is the largest integer less than to $\sqrt[3]{(2011)^3 + 3 \times (2011)^2 + 4 \times 2011+ 5}$ ? (A) $2010$, (B) $2011$, (C) $2012$, (D) $2013$, (E) None of the above.

Let $\lfloor z \rfloor$ denote the greatest integer less than or equal to $z.$ Compute $$\sum_{j=-1000}^{1000} \left\lfloor \frac{2025}{j+0.5}\right\rfloor.$$

Given a triangle $ABC$ with [i]Lemoine[/i] point $L.$ Choose points $X,Y,Z$ on the segments $LA,LB,LC,$ respectively such that:$$\angle XBA=\angle YAB,\angle XCA=\angle ZAC.$$ Prove that: $\angle ZBC=\angle YCB.$

Let $ x$ and $ y$ be positive integers such that $ 7x^5 \equal{} 11y^{13}$. The minimum possible value of $ x$ has a prime factorization $ a^cb^d$. What is $ a \plus{} b \plus{} c \plus{} d$? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

Find the number of $10$-tuples $(a_1,a_2,\dots,a_9,a_{10})$ of integers such that $|a_1|\leq 1$ and \[a_1^2+a_2^2+a_3^2+\cdots+a_{10}^2-a_1a_2-a_2a_3-a_3a_4-\cdots-a_9a_{10}-a_{10}a_1=2.\]

Does there exist a set $S$ consisting of rational numbers with the following property: for every integer $n$ there is a unique nonempty, finite subset of $S$, whose elements sum to $n$?

In a certain kingdom, the king has decided to build $25$ new towns on $13$ uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

The triangle was divided into five triangles similar to it. Is it true that the original triangle is right-angled? (S. Markelov)

Let $a_1,a_2,a_3, \cdots $ be distinct positive integers, and $0<c<\frac{3}{2}$ . Prove that : There exist infinitely many positive integers $k$, such that $[a_k,a_{k+1}]>ck $.

$2021$ points are marked on a circle. $2021$ segments with marked endpoints are drawn. After that one counts the number of different points where some $2$ drawn segments intersect(endpoints of segments do [b]not[/b] count as intersections) Find the maximum number one can get.