Let $s(n)$ denote the sum of the digits (in base ten) of a positive integer $n.$ Compute the number of positive integers $n$ at most $10^4$ that satisfy $$s(11n)=2s(n).$$

Prove that for every natural number $n$ we have $$\sum_{s=n}^{2n} 2^{2n-s}{s \choose n}= 2^{2n}.$$

Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$ (The [i]excircle[/i] of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.) [i]Proposed by Evangelos Psychas, Greece[/i]

Rick has $7$ books on his shelf: three identical red books, two identical blue books, a yellow book, and a green book. Dave accidentally knocks over the shelf and has to put the books back on in the same order. He knows that none of the red books were next to each other and that the yellow book was one of the first four books on the shelf, counting from the left. If Dave puts back the books according to the rules, but otherwise randomly, what is the probability that he puts the books back correctly?

Let $n$ be a positive integer and let $T$ denote the collection of points $(x_1, x_2, \ldots, x_n) \in \mathbb R^n$ for which there exists a permutation $\sigma$ of $1, 2, \ldots , n$ such that $x_{\sigma(i)} - x_{\sigma(i+1) } \geq 1$ for each $i=1, 2, \ldots , n.$ Prove that there is a real number $d$ satisfying the following condition: For every $(a_1, a_2, \ldots, a_n) \in \mathbb R^n$ there exist points $(b_1, \ldots, b_n)$ and $(c_1,\ldots, c_n)$ in $T$ such that, for each $i = 1, . . . , n,$ \[a_i=\frac 12 (b_i+c_i) , \quad |a_i - b_i|  \leq d, \quad \text{and} \quad |a_i - c_i| \leq d.\]

Let $ABC$ be a triangle and $D$ be a point on segment $AC$. The circumscribed circle of the triangle $BDC$ cuts $AB$ again at $E$ and the circumference circle of the triangle $ABD$ cuts $BC$ again at $F$. Prove that $AE = CF$ if and only if $BD$ is the interior bisector of $\angle ABC$.

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. The experiment is repeated with a new pendulum with a rod of length $4L$, using the same angle $\theta_0$, and the period of motion is found to be $T$. Which of the following statements is correct? $\textbf{(A) } T = 2T_0 \text{ regardless of the value of } \theta_0\\ \textbf{(B) } T > 2T_0 \text{ with } T \approx 2T_0 \text{ if } \theta_0 \ll 1\\ \textbf{(C) } T < 2T_0 \text{ with } T \approx 2T_0 \text{ if } \theta_0 \ll 1\\ \textbf{(D) } T < 2T_0 \text{ with some values of } \theta_0 \text{ and } T > 2T_0 \text{ for other values of } \theta_0\\ \textbf{(E) } T \text{ and } T_0 \text{ are not defined because the motion is not periodic unless } \theta_0 \ll 1$

Find the graph of the function $y=|2^{|x|}-1|$.

Let $p$ be a prime number. Positive integers numbers $a$ and $b$ are such $\frac{p}{a}+\frac{p}{b}=1$ and $a+b$ is divisible by $p$. What values can an expression $\frac{a+b}{p}$ take? [i]Yu.A.Karpenko[/i]

What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? $ \text{(A)}\ 30\qquad\text{(B)}\ 45\qquad\text{(C)}\ 60\qquad\text{(D)}\ 75\qquad\text{(E)}\ 90 $

Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),f_2(x),\ldots,f_k(x)$ such that \[p(x)=\sum_{j=1}^k(f_j(x))^2.\]

a) Determine if there are positive integers $a, b$ and $c$, not necessarily distinct, such that $a+b+c=2020$ and $2^a+2^b+2^c$ it's a perfect square. b) Determine if there are positive integers $a, b$ and $c$, not necessarily distinct, such that $a+b+c=2020$ and $3^a+3^b+3^c$ it's a perfect square.

Find all positive integer solutions $ (a,b,c,n)$ of the equation: $ 2^n\equal{}a!\plus{}b!\plus{}c!$.

Before district play, the Unicorns had won $45\%$ of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all? $\textbf{(A)}\ 48 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 52 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 60$

Kostya drives a car from a village to a city, driving along three roads. Moreover, on each of these roads, he drives at a constant speed. Is it possible that a third of the distance traveled was completed earlier than a third of the time, half of the distance traveled later than half of the time, and two-thirds of the distance was earlier than two-thirds of the time?

There are $15$ people attending math team: $12$ students and $3$ captains. One of the captains brings $33$ identical snacks. A nonnegative number of names (students and/or captains) are written on the NO SNACK LIST. At the end of math team, all students each get n snacks, and all captains get $n +1$ snacks, unless the person’s name is written on the board. After everyone’s snacks are distributed, there are none left. Find the number of possible integer values of $n$.

Positive integers $a,b,c$ exist such that $a+b+c+1$, $a^2+b^2+c^2 +1$, $a^3+b^3+c^3+1,$ and $a^4+b^4+c^4+7459$ are all multiples of $p$ for some prime $p$. Find the sum of all possible values of $p$ less than $1000$.

In a complete graph with $2025$ vertices, each edge has one of the colors $r_1$, $r_2$, or $r_3$. For each $i = 1,2,3$, if the $2025$ vertices can be divided into $a_i$ groups such that any two vertices connected by an edge of color $r_i$ are in different groups, find the minimum possible value of $a_1 + a_2 + a_3$.

Compute the largest value of $r$ such that three non-overlapping circles of radius $r$ can be inscribed in a unit square.

Show that for every polynomial $f(x)$ with integer coefficients, there exists a integer $C$ such that the set $\{n \in Z :$ the sum of digits of $f(n)$ is $C\}$ is not finite.

Some cities in country are connected with oneway road. It is known that every closed cyclic route, that don`t break traffic laws, consists of even roads. Prove that king of city can place military bases in some cities such that there are not roads between these cities, but for every city without base we can go from city with base by no more than $1$ road. [hide=PS]I think it should be one more condition, like there is cycle that connect all cities [/hide]

Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find \[ s_1 + s_2 + \cdots + s_n \]

Find the Jordan normal form of the operator $e^{d/dt}$ in the space of quasi-polynomials $\{e^{\lambda t}p(t)\}$ where the degree of the polynomial $p$ is less than $5$, and of the operator $\text{ad}_A$, $B\mapsto [A, B]$, in the space of $n\times n$ matrices $B$, where $A$ is a diagonal matrix.

Find all positive real numbers $p$ such that $\sqrt{a^2 + pb^2} +\sqrt{b^2 + pa^2} \ge a + b + (p - 1) \sqrt{ab}$ holds for any pair of positive real numbers $a, b$.

Is there $14$ consecutive positive integers such that each of these numbers is divisible by one of the prime numbers $p$ where $2\leq p \leq 11$.