(BUL 1) Prove that all numbers in the sequence
$$\frac{107811}{3},\quad\frac{110778111}{3},\quad\frac{111077781111}{3},\quad\ldots$$
are perfect cubes.
(BUL 4) Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that:
(a) The medians of that triangle correspond to the sides of a right-angled triangle.
(b) The inequality $$5(a^2+b^2-c^2)\geq8ab$$ is valid, where $a,b,c$ are side lengths of the given triangle.
(CZS 1) Find all real solutions of the system of equations
$$\begin{aligned}x_1+x_2+\cdots+x_n&=a,\\x_1^2+x_2^2+\cdots+x_n^2&=a^2,\\&\ \vdots\\x_1^n+x_2^n+\cdots+x_n^n&=a^n.\end{aligned}$$
(CZS 2) IMO1 $ABCD$ is a parallelogram; $AB=a$, $AD=1$, $\alpha$ is the size of $\angle DAB$, and the three angles of the triangle $ABD$ are acute. Prove that the four circles $K_A,K_B,K_C,K_D$, each of radius $1$, whose centers are the vertices $A,B,C,D$, cover the parallelogram if and only if $a\leq\cos\alpha+\sqrt3\sin\alpha$.
(CZS 3) The circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k$.
(CZS 4) The square $ABCD$ is to be decomposed into $n$ triangles (nonoverlapping) all of whose angles are acute. Find the smallest integer $n$ for which there exists a solution to this problem and construct at least one decomposition for this $n$. Answer whether it is possible to ask additionally that (at least) one of these triangles has a perimeter less than an arbitrarily given positive number.
(CZS 6) Given a segment $AB$ of the length $1$, define the set $M$ of points in the following way: it contains the two points $A,B$, and also all points obtained from $A,B$ by iterating the following rule: (*) for every pair of points $X,Y$ in $M$, the set $M$ also contains the point $Z$ of the segment $XY$ for which $YZ=3XZ$.
(a) Prove that the set $M$ consists of points $X$ from the segment $AB$ for which the distance from the point $A$ is either
$$AX=\frac{3k}{4^n}\quad\lor\quad AX=\frac{3k-2}{4^n},$$
where $n,k$ are nonnegative integers.
(b) Prove that the point $X_0$ for which $AX_0=1/2=X_0B$ does not belong to the set $M$.
(GDR 1) Find whether among all quadrilaterals whose interiors lie inside a semicircle of radius $r$ there exists one (or more) with maximal area. If so, determine their shape and area.
(GDR 2) Which fraction $p/q$, where $p,q$ are positive integers less than $100$, is closest to $\sqrt2$? Find all digits after the decimal point in the decimal representation of this fraction that coincide with digits in the decimal representation of $\sqrt2$ (without using any tables).
(GDR 3) Suppose $\tan\alpha=p/q$, where $p$ and $q$ are integers and $q\ne0$. Prove that the number $\tan\beta$ for which $\tan2\beta=\tan3\alpha$ is rational only when $p^2+q^2$ is the square of an integer.
(GDR 4) Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by numbers of the form $z_{k_1,k_2}=k_1r_1+k_2r_2$, $k_1,k_2$ integers; i.e., for every real number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found such that $|x-(k_1r_1+k_2r_2)|\lt p$.
(GBR 1) IMO3 Let $k,m,n$ be positive integers such that $m+k+1$ is a prime number greater than $n+1$. Write $c_s$ for $s(s+1)$. Prove that the product $(c_{m+1}-c_k)(c_{m+2}-c_k)\cdots(c_{m+n}-c_k)$ is divisible by the product $c_1c_2\cdots c_n$.
(GBR 5) If $x$ is a positive rational number, show that $x$ can be uniquely expressed in the form
$$x=a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots,$$
where $a_1,a_2,\ldots$ are integers, $0\leq a_n\leq n-1$ for $n>1$, and the series terminates.
Show also that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6$.
(GBR 6) The $n$ points $P_1,P_2,\ldots,P_n$ are placed inside or on the boundary of a disk of radius $1$ in such a way that the minimum distance $d_n$ between any two of these points has its largest possible value $D_n$. Calculate $D_n$ for $n$ from $2$ to $7$ and justify your answer.
(HUN 1) In space, $n$ points ($n\geq3$) 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 a polygonal line in such a way.
Source footnote. The statement so formulated is false. It would be trivially true under the additional assumption that the polygonal line is closed. However, from the offered solution, which is not clear, it does not seem that the proposer had this in mind.