Let there be a regular hexagon with sidelength $1$. Find the greatest integer $n\geq2$ for which there exist $n{}$ points inside or on the sides of the hexagon such that the distance between every two points is no less than $\sqrt{2}$.

Let $A$ be the subset of $\{1, 2, ..., 16\}$ that has $6$ elements. Prove that there exist $2$ subsets of $A$ that are disjoint, and the sum of their elements are the same.

Can you draw some diagonals in a convex $2014$-gon so that they do not intersect, the whole $2014$-gon is divided into triangles and each vertex belongs to an odd number of these triangles?

You are given a quadrilateral $ABCD$. It is known that $\angle BAC = 30^o$, $\angle D = 150^o$ and, in addition, $AB = BD$. Prove that $AC$ is the bisector of angle $C$.

Let $ABC$ be an acute triangle. Let $D$ and $E$ be the feet of the altitudes of the triangle $ABC$ from $A$ and $B$, respectively. Let $F$ be the reflection of the point $A$ over $BC$. Let $G$ be a point such that the quadrilateral $ABCG$ is a parallelogram. Show that the circumcircles of triangles $BCF$ , $ACG$ and $CDE$ are concurrent on a point different from $C$.

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$. [i]Remark[/i]. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.

Let $x,y$ be positive real numbers such that $x+y=1$. Prove that $\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.

Show that for every real function f in 1-period $L^2(0, 1)$ there exist three functions $g_1, g_2, g_3$ with the same properties and constants $c_0, c_1, c_2, c_3$ satisfying $$f(x)=c_0+\sum_{i=1}^3(g_i(x+c_i)-g_i(x))$$

Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}. \]

Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.

let $F_q$ be a finite field with char ≠ 2, and let $V = F_q \times F_q$ be the 2-dimensional vector space over $F_q$. Let L ⊂ V be a subset containing lines in all directions. The order of a point in V is the number of lines in L that pass through the point. Prove that L contains at least q lines having a third-order point.

Prove that the equation $$3x(x-3y)=y^2+z^2$$doesn't have any integer solutions except $x=0,y=0,z=0$.

Let $a$, $b$, $c$ be positive real numbers with $abc \leq a+b+c$. Show that \[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \] [i]Cristinel Mortici, Romania[/i]

$ \lim_{n\to\infty} n2^n\int_1^n \frac{dx}{\left( 1+x^2\right)^n} $ [i][i]Bogdan Enescu[/i][/i]

[b]8.1[/b] On the median drawn from the vertex of the triangle to the base, point $A$ is taken. The sum of the distances from $A$ to the sides of the triangle is equal to $s$. Find the distances from $A$ to the sides if the lengths of the sides are equal to $x$ and $y$. [b]8.2[/b] Fraction $0, abc...$ is composed according to the following rule: $a$ and $c$ are arbitrary digits, and each next digit is equal to the remainder of the sum of the previous two digits when divided by $10$. Prove that this fraction is purely periodic. [b]8.3[/b] Two convex polygons with $m$ and $n$ sides are drawn on the plane ($m>n$). What is the greatest possible number of parts, they can break the plane? [b]8.4 [/b]The sum of three integers that are perfect squares is divisible by $9$. Prove that among them, there are two numbers whose difference is divisible by $9$. [b]8.5 / 9.5[/b] Given $k+2$ integers. Prove that among them there are two integers such that either their sum or their difference is divisible by $2k$. [b]8.6[/b] A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points sides of an angle and a given circle. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

Determine, with proof, whether a square can be dissected into finitely many (not necessarily congruent) triangles, each of which has interior angles $30^\circ, 75^\circ,$ and $75^\circ.$

Through a random point $C_1$ from the edge $DC$ of the regular tetrahedron $ABCD$ is drawn a plane, parallel to the plane $ABC$. The plane constructed intersects the edges $DA$ and $DB$ at the points $A_1,B_1$ respectively. Let the point $H$ is the midpoint of the altitude through the vertex $D$ of the tetrahedron $DA_1B_1C_1$ and $M$ is the center of gravity (barycenter) of the triangle $ABC_1$. Prove that the measure of the angle $HMC$ doesn’t depend on the position of the point $C_1$. [i](Ivan Tonov)[/i]

Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x = 0$ of $(1 + x)^{\alpha}$ . Evaluate $$\int_{0}^{1} \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\, dy.$$

For any positive integer $a$, denote $f(a)=|\{b\in\mathbb{N}| b|a$ $\text{and}$ $b\mod{10}\in\{1,9\}\}|$ and $g(a)=|\{b\in\mathbb{N}| b|a$ $\text{and}$ $b\mod{10}\in\{3,7\}\}|$. Prove that $f(a)\geq g(a)\forall a\in\mathbb{N}$.

Two chords are drawn in a circle of radius equal to unit, $AB$ and $AC$ of equal length. a) Describe how you can construct a third chord $DE$ that is divided into three equal parts by the intersections with $AB$ and $AC$. b) If $AB = AC =\sqrt2$, what are the lengths of the two segments that the chord $DE$ determines in $AB$?

In Mexico, there live $n$ Mexicans, some of whom know each other. They decided to play a game. On the first day, each Mexican wrote a non-negative integer on their forehead. On each following day, they changed their number according to the following rule: On day $i+1$, each Mexican writes on their forehead the smallest non-negative integer that did not appear on the forehead of any of their acquaintances on day $i$. It is known that on some day every Mexican wrote the same number as on the previous day, after which they decided to stop the game. Determine the maximum number of days this game could have lasted. [i]Proposed by Pavle Martinović[/i]

On the semicircle with diameter $AB$ and center $O$ point $D$ is marked. Points $E$ and $F$ are the midpoints of minor arcs $AD$ and $BD$ respectively. It turned out that the line connecting orthocenters of $ADF$ and $BDE$ passes through $O$ Find $\angle AOD$

Consider a set $ A$ of positive integers such that the least element of $ A$ equals $ 1001$ and the product of all elements of $ A$ is a perfect square. What is the least possible value of the greatest element of $ A$?

The sequence $a_n$ is defined by the following conditions: $a_1=1$, and for any $n\in \mathbb N$, the number $a_{n+1}$ is obtained from $a_n$ by adding three if $n$ is a member of this sequence, and two if it is not. Prove that $a_n<(1+\sqrt 2)n$ for all $n$. [i]Proposed by Mikhail Ivanov[/i]