1.) In the plane let us consider a set $S$ consisting of $n \geq 3$ distinct points satisfying the following three conditions: [b]I.[/b] The distance between any two points $\in S$ is not greater than 1. [b]II.[/b] For every point $A \in S$, there are exactly two “neighbor” points, i.e. two points $X, Y \in S$ for which $AX = AY = 1$. [b]III. [/b] For arbitrary two points $A, B \in S$, let $A', A''$ be the two neighbors of $A, B', B''$ the two neighbors of $B$, then $A'AA'' = B'BB''$. Is there such a set $S$ if $n = 1991$? If $n = 2000$ ? Explain your answer.

Rombus ABCD with acute angle $B$ is given. $O$ is a circumcenter of $ABC$. Point $P$ lies on line $OC$ beyond $C$. $PD$ intersect the line that goes through $O$ and parallel to $AB$ at $Q$. Prove that $\angle AQO=\angle PBC$.

Determine all integer numbers $n\ge 3$ such that the regular $n$-gon can be decomposed into isosceles triangles by non-intersecting diagonals.

Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron? (S. Markelov)

Given a triangle on the plane, construct inside the triangle the point $P$ for which the centroid of the triangle formed by the three projections of $P$ onto the sides of the triangle happens to be $P$.

Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

The pentagon $ABCDE$ which is inscribed in a circle with $AB < DE$ is the base of a pyramid with apex $S$. If the longest side from $S$ is $SA$, prove that $BS > CS$.

[b]8.1 [/b] Construct a quadrilateral using side lengths and distances between the midpoints of the diagonals. [b]8.2[/b] It is known that $a,b$ and $\sqrt{a}+\sqrt{b} $ are rational numbers. Prove that then $\sqrt{a}$, $\sqrt{b} $ are rational. [b]8.3 / 9.2[/b] Solve equation $x^3 - [x]=3$ [b]8.4[/b] Prove that if in a triangle the angle bisector of the vertex, bisects the angle between the median and the altitude, then the triangle either isosceles or right. . [b]8.5[/b] Given $n$ numbers $x_1, x_2, . . . , x_n$, each of which is equal to $+1$ or $-1$. At the same time $$x_1x_2 + x_2x_3 + . . . + x_{n-1}x_n + x_nx_1 = 0 .$$ Prove that $n$ is divisible by $4$. [b]8.6[/b] There are $n$ points marked on the circle, and it is known that for of any two, one of the arcs connecting them has a measure less than $120^0$.Prove that all points lie on an arc of size $120^0$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].

Consider the power series expansion \[\dfrac{1}{1-2x-x^2}=\sum_{n=0}^\infty a_nx^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2+a_{n+1}^2=a_m.\]

A chord $AB$ is drawn in a circle. The line $\ell$ is parallel to $AB$ and does not intersect the circle. Let $C$ be a certain point on the circle (points $C$ located on one side of $AB$ are considered). Lines $CA$ and $CB$ intersect $\ell$ at points $D$ and $E$. Prove that there exists a fixed point $F$ of the plane, not lying on line $\ell$ , such that $\angle DFE$ is constant.

Find all positive integers $n$ such that a positive integer power of $2n^2+4n-1$ equals to a positive integer power of $3n+4$.

Prove that there exists an integer polynomial $P(X)$ such that $P(n)+4^n \equiv 0 \pmod {27}$. for all $n \geq 0$.

Determine all positive integers $n$ for which there exists an integer $m$ such that $2^{n}-1$ divides $m^{2}+9$.

Solve the system of equations $(n > 2)$ \[\begin{array}{c}\ \sqrt{x_1}+\sqrt{x_2+x_3+\cdots+x_n}=\sqrt{x_2}+\sqrt{x_3+x_4+\cdots+x_n+x_1}=\cdots=\sqrt{x_n}+\sqrt{x_1+x_2+\cdots+x_{n-1}} \end{array}, \] \[x_1-x_2=1.\]

Let $x$ be the positive integer satisfying $5^2 +28^2 +39^2 = 24^2 +35^2 + x^2$. Find $x$.

Find the equation of the smallest sphere which is tangent to both of the lines $$\begin{pmatrix} x\\y\\z \end{pmatrix} =\begin{pmatrix} t+1\\ 2t+4\\ -3t +5 \end{pmatrix},\;\;\;\begin{pmatrix} x\\y\\z \end{pmatrix} =\begin{pmatrix} 4t-12\\ -t+8\\ t+17 \end{pmatrix}.$$

Let $a,b,c,d$ be the four roots of the polynomial \[x^4+3x^3-x^2+x-2.\] Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{2}$ and $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}=-\frac{3}{4}$, the value of \[\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that: $$f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q$$ holds for all $p,q\in\mathbb{P}$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square. [img]https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png[/img]

A circle centered at a point $F$ and a parabola with focus $F$ have two common points. Prove that there exist four points $A, B, C, D$ on the circle such that the lines $AB, BC, CD$ and $DA$ touch the parabola.

Let $A$ be a symmetric $m\times m$ matrix over the two-element field all of whose diagonal entries are zero. Prove that for every positive integer $n$ each column of the matrix $A^n$ has a zero entry.

Consider an arbitrary equilateral triangle $KLM$, whose vertices $K, L$ and $M$ lie on the sides $AB, BC$ and $CD$, respectively, of a given square $ABCD$. Find the locus of the midpoints of the sides $KL$ of all such triangles $KLM$.

There are $n$ coins lying in a circle. Each coin has two sides, $+$ and $-$. A $flop$ means to flip every coin that has two different neighbors simultaneously, while leaving the others alone. For instance, $++-+$, after one $flop$, becomes $+---$. For $n$ coins, let us define $M$ to be a $perfect$ $number$ if for any initial arrangement of the coins, the arrangement of the coins after $m$ $flops$ is exactly the same as the initial one. (a) When $n=1024$, find a perfect number $M$. (b) Find all $n$ for which a perfect number $M$ exist.

Consider the recurrence relation $$a_{n+3}=\frac{a_{n+2}a_{n+1}-2}{a_n}$$ with initial condition $(a_0,a_1,a_2)=(1,2,5)$. Let $b_n=a_{2n}$ for nonnegative integral $n$. It turns out that $b_{n+2}+xb_{n+1}+yb_n=0$ for some pair of real numbers $(x,y)$. Compute $(x,y)$.

Let triangle $ABC$ be such that its circumradius is $R = 1.$ Let $r$ be the inradius of $ABC$ and let $p$ be the inradius of the orthic triangle $A'B'C'$ of triangle $ABC.$ Prove that \[ p \leq 1 - \frac{1}{3 \cdot (1+r)^2}. \] [hide="Similar Problem posted by Pascual2005"] Let $ABC$ be a triangle with circumradius $R$ and inradius $r$. If $p$ is the inradius of the orthic triangle of triangle $ABC$, show that $\frac{p}{R} \leq 1 - \frac{\left(1+\frac{r}{R}\right)^2}{3}$. [i]Note.[/i] The orthic triangle of triangle $ABC$ is defined as the triangle whose vertices are the feet of the altitudes of triangle $ABC$. [b]SOLUTION 1 by mecrazywong:[/b] $p=2R\cos A\cos B\cos C,1+\frac{r}{R}=1+4\sin A/2\sin B/2\sin C/2=\cos A+\cos B+\cos C$. Thus, the ineqaulity is equivalent to $6\cos A\cos B\cos C+(\cos A+\cos B+\cos C)^2\le3$. But this is easy since $\cos A+\cos B+\cos C\le3/2,\cos A\cos B\cos C\le1/8$. [b]SOLUTION 2 by Virgil Nicula:[/b] I note the inradius $r'$ of a orthic triangle. Must prove the inequality $\frac{r'}{R}\le 1-\frac 13\left( 1+\frac rR\right)^2.$ From the wellknown relations $r'=2R\cos A\cos B\cos C$ and $\cos A\cos B\cos C\le \frac 18$ results $\frac{r'}{R}\le \frac 14.$ But $\frac 14\le 1-\frac 13\left( 1+\frac rR\right)^2\Longleftrightarrow \frac 13\left( 1+\frac rR\right)^2\le \frac 34\Longleftrightarrow$ $\left(1+\frac rR\right)^2\le \left(\frac 32\right)^2\Longleftrightarrow 1+\frac rR\le \frac 32\Longleftrightarrow \frac rR\le \frac 12\Longleftrightarrow 2r\le R$ (true). Therefore, $\frac{r'}{R}\le \frac 14\le 1-\frac 13\left( 1+\frac rR\right)^2\Longrightarrow \frac{r'}{R}\le 1-\frac 13\left( 1+\frac rR\right)^2.$ [b]SOLUTION 3 by darij grinberg:[/b] I know this is not quite an ML reference, but the problem was discussed in Hyacinthos messages #6951, #6978, #6981, #6982, #6985, #6986 (particularly the last message). [/hide]