(BUL 2) Consider a sequence of polynomials $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$, where $P_0(x)=2$, $P_1(x)=x$ and for every $n\ge1$ the following equality holds:
$$P_{n+1}(x)+P_{n-1}(x)=xP_n(x).$$
Prove that there exist three real numbers $a,b,c$ such that for all $n\ge1$,
$$(x^2-4)[P_n(x)^2-4]=[aP_{n+1}(x)+bP_n(x)+cP_{n-1}(x)]^2.\tag{1}$$
(BUL 5) (IMO 5) Prove that for every natural number $m\ge1$ there exists a finite set $S_m$ of points in the plane satisfying the following condition: If $A$ is any point in $S_m$, then there are exactly $m$ points in $S_m$ whose distance to $A$ equals $1$.
(GDR 1) Knowing that the system
$$\begin{cases}x+y+z=3,\\x^3+y^3+z^3=15,\\x^4+y^4+z^4=35,\end{cases}$$
has a real solution $x,y,z$ for which $x^2+y^2+z^2\lt 10$, find the value of $x^5+y^5+z^5$ for that solution.
(GBR 3) We are given two mutually tangent circles in the plane, with radii $r_1,r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.
(HUN 1) (IMO 1) Let $a,b,c,d,e$ be real numbers. Prove that the expression
$$(a-b)(a-c)(a-d)(a-e)+(b-a)(b-c)(b-d)(b-e)+(c-a)(c-b)(c-d)(c-e)$$
$$+(d-a)(d-b)(d-c)(d-e)+(e-a)(e-b)(e-c)(e-d)$$
is nonnegative.
(HUN 7) Let $n\ge2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2^n$-gon such that the following conditions are satisfied:
(1) only digits $1$ and $2$ are used;
(2) each number consists of exactly $n$ digits;
(3) different numbers are assigned to different vertices;
(4) the numbers assigned to two neighboring vertices differ at exactly one digit.
(NET 1) (IMO 4) Given a tetrahedron $ABCD$ whose all faces are acute-angled triangles, set
$$\sigma=\angle DAB+\angle BCD-\angle ABC-\angle CDA.$$
Consider all closed broken lines $XYZTX$ whose vertices $X,Y,Z,T$ lie in the interior of segments $AB,BC,CD,DA$ respectively. Prove that:
(a) if $\sigma\ne0$, then there is no broken line $XYZT$ of minimal length;
(b) if $\sigma=0$, then there are infinitely many such broken lines of minimal length. That length equals $2AC\sin(\alpha/2)$, where
$$\alpha=\angle BAC+\angle CAD+\angle DAB.$$
(NET 4) Determine whether there exist distinct real numbers $a,b,c,t$ for which:
(i) the equation $ax^2+btx+c=0$ has two distinct real roots $x_1,x_2$,
(ii) the equation $bx^2+ctx+a=0$ has two distinct real roots $x_2,x_3$,
(iii) the equation $cx^2+atx+b=0$ has two distinct real roots $x_3,x_1$.
(POL 1) Let $T_k=k-1$ for $k=1,2,3,4$ and
$$T_{2k-1}=T_{2k-2}+2^{k-2},\qquad T_{2k}=T_{2k-5}+2^k\quad(k\ge3).$$
Show that for all $k$,
$$1+T_{2n-1}=\left[\frac{12}7\,2^{n-1}\right]\qquad1+T_{2n}=\left[\frac{17}7\,2^{n-1}\right],$$
where $[x]$ denotes the greatest integer not exceeding $x$.
(POL 3) The matrix
$$\begin{pmatrix}a_{11}&\ldots&a_{1n}\\\vdots&\cdots&\vdots\\a_{n1}&\ldots&a_{nn}\end{pmatrix}$$
satisfies the inequality $\sum_{j=1}^n|a_{j1}x_1+\cdots+a_{jn}x_n|\le M$ for each choice of numbers $x_i$ equal to $\pm1$. Show that
$$|a_{11}+a_{22}+\cdots+a_{nn}|\le M.$$
(POL 6) Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB',CC'$ either are collinear or form an equilateral triangle.
(SWE 5) (IMO 6) Consider the $n\times n$ array of nonnegative integers
$$\begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\ldots&a_{nn}\end{pmatrix},$$
with the following property: If an element $a_{ij}$ is zero, then the sum of the elements of the $i$th row and the $j$th column is greater than or equal to $n$. Prove that the sum of all the elements is greater than or equal to $\frac12n^2$.
(USS 1) A broken line $A_1A_2\ldots A_n$ is drawn in a $50\times50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248$.
(USS 2) Natural numbers from $1$ to $99$ (not necessarily distinct) are written on $99$ cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by $100$. Show that all the cards contain the same number.
(USS 5) (IMO 2) Given a convex polyhedron $P_1$ with $9$ vertices $A_1,\ldots,A_9$, let us denote by $P_2,P_3,\ldots,P_9$ the images of $P_1$ under the translations mapping the vertex $A_1$ to $A_2,A_3,\ldots,A_9$, respectively. Prove that among the polyhedra $P_1,\ldots,P_9$ at least two have a common interior point.
(YUG 1) Prove the inequality
$$\frac{a_1+a_3}{a_1+a_2}+\frac{a_2+a_4}{a_2+a_3}+\frac{a_3+a_1}{a_3+a_4}+\frac{a_4+a_2}{a_4+a_1}\ge4,$$
where $a_i\gt 0$, $i=1,2,3,4$.