Some cells of a $10\times10$ board are marked so that each cell has an even number of neighboring (i.e. sharing a side) marked cells. Find the maximum possible number of marked cells.

Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair, one number is divided by another.

The temperatures $f^o F$ and $c^o C$ are equal when $f = \frac95 c + 32$. What temperature is the same in both $^o F$ and $^o C$?

Prove that for some positive integer $n$ there exist positive integers $a$,$b$ and $c$ such that $a^2-n=xy$, $b^2-n=yz$ and $c^2-n=xz$ where $x,y$ and $z$ - some pairwise different positive integers.

Let $C: y=x^2+ax+b$ be a parabola passing through the point $(1,\ -1)$. Find the minimum volume of the figure enclosed by $C$ and the $x$ axis by a rotation about the $x$ axis. Proposed by kunny

$(USS 3)$ $(a)$ Prove that if $0 \le a_0 \le a_1 \le a_2,$ then $(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)$ $(b)$ Formulate and prove the analogous result for polynomials of third degree.

Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ .

Let $ABC$ be an acute triangle with $AC\neq BC$, $M$ the midpoint of side $AB$, $H$ is the orthocenter of $\triangle ABC$, $D$ on $BC$ is the foot of the altitude from $A$ and $E$ on $AC$ is the foot of the perpendicular from $B$. Prove that the lines $AB, DE$ and the perpendicular to $MH$ through $C$ are concurrent.

A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k=1,2,3,\ldots.$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? $\textbf{(A) } \frac{1}{4} \qquad\textbf{(B) } \frac{2}{7} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{3}{7}$

Kevin and Yang are playing a game. Yang has $2017 + \tbinom{2017}{2}$ cards with their front sides face down on the table. The cards are constructed as follows: [list] [*] For each $1 \le n \le 2017$, there is a blue card with $n$ written on the back, and a fraction $\tfrac{a_n}{b_n}$ written on the front, where $\gcd(a_n, b_n) = 1$ and $a_n, b_n > 0$. [*] For each $1 \le i < j \le 2017$, there is a red card with $(i, j)$ written on the back, and a fraction $\tfrac{a_i+a_j}{b_i+b_j}$ written on the front. [/list] It is given no two cards have equal fractions. In a turn Kevin can pick any two cards and Yang tells Kevin which card has the larger fraction on the front. Show that, in fewer than $10000$ turns, Kevin can determine which red card has the largest fraction out of all of the red cards.

Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$ (1) Find the minimum of $f_{2020}$. (2) Find the minimum of $f_{2020} \cdot f_{2021}$.

Define $ n_a!$ for $ n$ and $ a$ positive to be \[ n_a ! \equal{} n (n\minus{}a)(n\minus{}2a)(n\minus{}3a)...(n\minus{}ka)\] where $ k$ is the greatest integer for which $ n>ka$. Then the quotient $ 72_8!/18_2!$ is equal to $ \textbf{(A)}\ 4^5 \qquad \textbf{(B)}\ 4^6 \qquad \textbf{(C)}\ 4^8 \qquad \textbf{(D)}\ 4^9 \qquad \textbf{(E)}\ 4^{12}$

Let $n, k$ be positive integers with $k \geq 3$. The edges of of a complete graph $K_n$ are colored in $k$ colors, such that for any color $i$ and any two vertices, there exists a path between them, consisting only of edges in color $i$. Prove that there exist three vertices $A, B, C$ of $K_n$, such that $AB, BC$ and $CA$ are all distinctly colored.

Consider the sequence $x_n = 2^n-n$, $n = 0,1 ,2 ,...$. Find all integers $m \ge 0$ such that $s_m = x_0 + x_1 + x_2 + ... + x_m$ is a power of $2$.

In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.

Ara and Shea were once the same height. Since then Shea has grown $20\%$ while Ara has grow half as many inches as Shea. Shea is now $60$ inches tall. How tall, in inches, is Ara now? $\text{(A)}\ 48 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 52 \qquad \text{(D)}\ 54 \qquad \text{(E)}\ 55$

There are $20$ cities in Wonderland. The company Wonderland Airways (WA) established $18$ air routes between them. Any of the routes is closed and passes (with landing) through some $5$ different cities. Each city belongs to at least three different routes, for no two cities there exist more than one routes, which allow to fly from one to another without landing. Prove that one can fly from any city of Wonderland to any other one by airplanes of WA. (V. Kaskevich)

High schoolers chew a lot of gum. At the supermarket, $15$ packs of $14$ sticks of gum costs $\$10$. If $1400$ high schoolers chew $3$ sticks of gum per day, find the total number of dollars spent by these high schoolers on gum per week.

Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.

Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ with the property that $$ f\left( p(x)\right) =p\left( f(x)\right) ,\quad\forall x\in\mathbb{Q} , $$ for all integer polynomials $ p. $

If $\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1$, then $x$ equals: $ \textbf{(A)}\ 1/b^2 \qquad\textbf{(B)}\ 1/b \qquad\textbf{(C)}\ b^2 \qquad\textbf{(D)}\ b \qquad\textbf{(E)}\ \sqrt{b} $

An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic.

Determine the smallest integer $j$ such that it is possible to fill the fields of the table $10\times 10$ with numbers from $1$ to $100$ so that every $10$ consecutive numbers lie in some of the $j\times j$ squares of the table. Czech Republic

Denote by $\omega (n)$ the number of prime divisors of the natural number $n$ (without multiplicities). Let $$F(x)=\max_{n\leq x} \omega (n) \,\,\,\,\,\,\,\,\,\,\,\,\, G(x)=\max_{n\leq x} \left( \omega (n) + \omega (n^2+1)\right)$$ Prove that $G(x)-F(x)\to \infty$ as $x\to\infty$. (translated by Miklós Maróti)

If $a\pm bi~(b\neq 0)$ are imaginary roots of the equation $x^3+qx+r=0$ where $a,~b,~q,$ and $r$ are real numbers, then $q$ in terms of $a$ and $b$ is $\textbf{(A) }a^2+b^2\qquad\textbf{(B) }2a^2-b^2\qquad\textbf{(C) }b^2-a^2\qquad\textbf{(D) }b^2-2a^2\qquad \textbf{(E) }b^2-3a^2$