(a) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $11n$ is twice the sum of the digits of $n$. (b) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $4n + 3$ is equal to the sum of the digits of $n$. (c) Prove that for any natural number $n$, it is possible to find $n$ consecutive numbers such that none of them is prime.

Let $L$ be the number of times the letter $L$ appeared on the Speed Round, $M$ be the number of times the letter $M$ appeared on the Speed Round, and $T$ be the number of times the letter $T$ appeared on the Speed Round. Find the value of $LMT$.

Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.

A circle has a chord of length $10$, and the distance from the center of the circle to the chord is $5$. What is the area of the circle? $\textbf{(A) }25\pi\qquad\textbf{(B) }50\pi\qquad\textbf{(C) }75\pi\qquad\textbf{(D) }100\pi\qquad\textbf{(E) }125\pi$

When Meena turned 16 years old, her parents gave her a cake with $n$ candles, where $n$ has exactly 16 different positive integer divisors. What is the smallest possible value of $n$?

Three positive reals $x,y,z$ are given so that $xy=\frac{z-x+1}{y}=\frac{z+1}2.$ Prove that one of the numbers is the arithmetic mean of the other two.

Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$. for instance, $a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

A set $G$ with elements $u,v,w...$ is a Group if the following conditions are fulfilled: $(\text{i})$ There is a binary operation $\circ$ defined on $G$ such that $\forall \{u,v\}\in G$ there is a $w\in G$ with $u\circ v = w$. $(\text{ii})$ This operation is associative; i.e. $(u\circ v)\circ w = u\circ (v\circ w)$ $\forall\{u,v,w\}\in G$. $(\text{iii})$ $\forall \{u,v\}\in G$, there exists an element $x\in G$ such that $u\circ x = v$, and an element $y\in G$ such that $y\circ u = v$. Let $K$ be a set of all real numbers greater than $1$. On $K$ is defined an operation by $ a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}$. Prove that $K$ is a Group.

Let $ \prod_{n=1}^{1996}{(1+nx^{3^n})}= 1+ a_{1}x^{k_{1}}+ a_{2}x^{k_{2}}+...+ a_{m}x^{k_{m}}$ where $a_{1}, a_{1}, . . . , a_{m}$ are nonzero and $k_{1} < k_{2} <...< k_{m}$. Find $a_{1996}$.

[b]p1.[/b] A triangle $ABC$ is drawn in the plane. A point $D$ is chosen inside the triangle. Show that the sum of distances $AD+BD+CD$ is less than the perimeter of the triangle. [b]p2.[/b] In a triangle $ABC$ the bisector of the angle $C$ intersects the side $AB$ at $M$, and the bisector of the angle $A$ intersects $CM$ at the point $T$. Suppose that the segments $CM$ and $AT$ divided the triangle $ABC$ into three isosceles triangles. Find the angles of the triangle $ABC$. [b]p3.[/b] You are given $100$ weights of masses $1, 2, 3,..., 99, 100$. Can one distribute them into $10$ piles having the following property: the heavier the pile, the fewer weights it contains? [b]p4.[/b] Each cell of a $10\times 10$ table contains a number. In each line the greatest number (or one of the largest, if more than one) is underscored, and in each column the smallest (or one of the smallest) is also underscored. It turned out that all of the underscored numbers are underscored exactly twice. Prove that all numbers stored in the table are equal to each other. [b]p5.[/b] Two stores have warehouses in which wheat is stored. There are $16$ more tons of wheat in the first warehouse than in the second. Every night exactly at midnight the owner of each store steals from his rival, taking a quarter of the wheat in his rival's warehouse and dragging it to his own. After $10$ days, the thieves are caught. Which warehouse has more wheat at this point and by how much? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A strange calculator has only two buttons with positive itegers, each of them consisting of two digits. It displays the number 1 at the beginning. Whenever a button with number $N$ is pressed, the calculator replaces the displayed number $X$ with the number $X\cdot N$ or $X+N$. Multiplication and addition alternate, multiplication is the first. (For example,if the number 10 is on the 1st button, the number 20 is on the 2nd button, and we consecutively press the 1st, 2nd, 1st and 1st button, we get the results $1\cdot 10=10$, $10+20=30$, $30\cdot 10=300$, and $300+10=310$.) Decide whether there exist particular values of the two-digit nubers on the buttons such that one can display infinitely many numbers (without cleaning the display, i.e. you must keep going and get infinitel many numbers) ending with (a) $2015$, (b) $5813$. [i]Proposed by Michal Rolínek and Peter Novotný[/i]

Let $n>2$ be a positive integer. Consider all numbers $S$ of the form \begin{align*} S= a_1 a_2 + a_2 a_3 + \ldots + a_{k-1} a_k \end{align*} with $k>1$ and $a_i$ begin positive integers such that $a_1+a_2+ \ldots + a_k=n$. Determine all the numbers that can be represented in the given form.

Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip? $\textbf{(A)} \text{ 60} \qquad \textbf{(B)} \text{ 72} \qquad \textbf{(C)} \text{ 92} \qquad \textbf{(D)} \text{ 96} \qquad \textbf{(E)} \text{ 120}$

Given a natural number $n>4$ and $2n+4$ cards numbered with $1, 2, \dots, 2n+4$. On the card with number $m$ a real number $a_m$ is written such that $\lfloor a_{m}\rfloor=m$. Prove that it's possible to choose $4$ cards in such a way that the sum of the numbers on the first two cards differs from the sum of the numbers on the two remaining cards by less than $$\frac{1}{n-\sqrt{\frac{n}{2}}}$$.

Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\).

On the plane circles $k$ and $\ell$ are intersected at points $C$ and $D$, where circle $k$ passes through the center $L$ of circle $\ell$. The straight line passing through point $D$ intersects circles $k$ and $\ell$ for the second time at points $A$ and $B$ respectively in such a way that $D$ is the interior point of segment $AB$. Show that $AB = AC$.

A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The lines $AB$ and $CD$ meet at $O$. A circle $\omega_1$ is tangent to side $BC$ at $K$ and to the extensions of sides $AB$ and $CD$, and a circle $\omega_2$ is tangent to side $AD$ at $L$ and to the extensions of sides $AB$ and $CD$. Suppose that points $O$, $K$, $L$ lie on a line. Prove that the midpoints of $BC$ and $AD$ and the center of $\omega$ also lie on a line.

Let $S = \{1, 22, 333, \dots , 999999999\}$. For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$?

Given a table $4\times 4$. a) Find, how $7$ stars can be put in its fields in such a way, that erasing of two arbitrary lines and two columns will always leave at list one of the stars. b) Prove that if there are less than $7$ stars, You can always find two columns and two rows, such, that if you erase them, no star will remain in the table.

On the parliament of Sikinia, for any two deputies, there is third deputy, which knows exactly one of the two. Every deputy belongs to one of the two ruling parties. Every day, he president tells a certain group of deputies to change the party that they belong, and all the deputies which which know at least one of the deputies of the group has to change their party too. Prove that, the president can reach any configuration of deputies between two parties.(The president himself isn't a member of the parliament of Sikinia). [I]Proposed by S. Berlov[/i]

Students from three middle schools worked on a summer project. *Seven students from Allen school worked for $3$ days. *Four students from Balboa school worked for $5$ days. *Five students from Carver school worked for $9$ days. The total amount paid for the students' work was $ \$774$. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether? $\text{(A)}\ 9.00\text{ dollars} \qquad \text{(B)}\ 48.38\text{ dollars} \qquad \text{(C)}\ 180.00\text{ dollars} \qquad \text{(D)}\ 193.50\text{ dollars} \qquad \text{(E)}\ 258.00\text{ dollars}$

Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$. Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$. [i]Proposed by Greece[/i]

A sequence $\{x_n \}=x_0, x_1, x_2, \cdots $ satisfies $x_0=a(1\le a \le 2019, a \in \mathbb{R})$, and $$x_{n+1}=\begin{cases}1+1009x_n &\ (x_n \le 2) \\ 2021-x_n &\ (2<x_n \le 1010) \\ 3031-2x_n &\ (1010<x\le 1011) \\ 2020-x_n &\ (1011<x_n) \end{cases}$$ for each non-negative integer $n$. If there exist some integer $k>1$ such that $x_k=a$, call such minimum $k$ a [i] fundamental period[/i] of $\{x_n \}$. Find all integers which can be a fundamental period of some seqeunce; and for such minimal odd period $k(>1)$, find all values of $x_0=a$ such that the fundamental period of $\{x_n \}$ equals $k$.

Determine the greatest possible value of the constant $c$ that satisfies the following condition: for any convex heptagon the sum of the lengthes of all it’s diagonals is greater than $cP$, where $P$ is the perimeter of the heptagon. (I. Zhuk)