(BUL 7) IMO 5. Let $f$ and $\varphi$ be real functions defined on the set $\mathbb R$ satisfying the functional equation
$$f(x+y)+f(x-y)=2\varphi(y)f(x),\tag{1}$$
for arbitrary real $x,y$ (give examples of such functions). Prove that if $f(x)$ is not identically $0$ and $|f(x)|\le1$ for all $x$, then $|\varphi(x)|\le1$ for all $x$.
(CZS 1) We are given $3n$ points $A_1,A_2,\ldots,A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i$.
(CZS 4) Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1+x_2+\cdots+x_n=0$. Let $m$ be the least and $M$ the greatest among them. Prove that
$$x_1^2+x_2^2+\cdots+x_n^2\le-nmM.$$
(GDR 1) Let $n_1,n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1\cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2$.
(GDR 2) Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if
$$AB^2+CD^2=BC^2+AD^2=CA^2+BD^2.$$
(GBR 1) IMO 6.
(a) A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. We define $p,q,r$ to be the signed distances of $P,Q,R$ from $\pi$ measured along a directed normal to $\pi$. Prove that
$$p^2+q^2+r^2+(q-r)^2+(r-p)^2+(p-q)^2=2a^2,$$
where $a$ is the length of an edge of the tetrahedron.
(b) Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane.
(NET 2) IMO 4. Find all solutions in positive real numbers $x_i$ $(i=1,2,3,4,5)$ of the following system of inequalities:
$$\begin{aligned}(x_1^2-x_3x_5)(x_2^2-x_3x_5)&\le0&&\text{(i)}\\ (x_2^2-x_4x_1)(x_3^2-x_4x_1)&\le0&&\text{(ii)}\\ (x_3^2-x_5x_2)(x_4^2-x_5x_2)&\le0&&\text{(iii)}\\ (x_4^2-x_1x_3)(x_5^2-x_1x_3)&\le0&&\text{(iv)}\\ (x_5^2-x_2x_4)(x_1^2-x_2x_4)&\le0&&\text{(v)}.\end{aligned}$$
(NET 6) Consider a sequence of circles $K_1,K_2,K_3,K_4,\ldots$ of radii $r_1,r_2,r_3,r_4,\ldots$, respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1,BA$, and $BC$; $K_3$ is tangent to $K_2,CA$, and $CB$; $K_4$ is tangent to $K_3,AB$, and $AC$; etc.
(a) Prove the relation
$$r_1\cot\frac A2+2\sqrt{r_1r_2}+r_2\cot\frac B2=r\left(\cot\frac A2+\cot\frac B2\right),$$
where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that
$$r_1=r\cot\frac B2\cot\frac C2\sin^2t_1.$$
(b) Prove that the sequence of circles $K_1,K_2,\ldots$ is periodic.
(USS 2) IMO 1. A set of $10$ positive integers is given such that the decimal expansion of each of them has two digits. Prove that there are two disjoint subsets of the set with equal sums of their elements.