(BUL 1) Let $ABC$ be a triangle with bisectors $AA_1,BB_1,CC_1$ ($A_1\in BC$, etc.) and $M$ their common point. Consider the triangles $MB_1A,MC_1A,MC_1B,MA_1B,MA_1C,MB_1C$, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then $AB=BC=CA$.
(BUL 3) Let $a_0,a_1,\ldots,a_n,a_{n+1}$ be a sequence of real numbers satisfying the following conditions:
$$a_0=a_{n+1}=0,\qquad |a_{k-1}-2a_k+a_{k+1}|\le1\quad(k=1,2,\ldots,n).$$
Prove that $|a_k|\le\frac{k(n+1-k)}2\quad(k=0,1,\ldots,n+1)$.
(CZS 3) IMO1. In a convex quadrangle with area $32\,\mathrm{cm}^2$, the sum of the lengths of two nonadjacent edges and of the length of one diagonal is equal to $16\,\mathrm{cm}$.
(a) What is the length of the other diagonal?
(b) What are the lengths of the edges of the quadrangle if the perimeter is a minimum?
(c) Is it possible to choose the edges in such a way that the perimeter is a maximum?
(GBR 1a) IMO6. For all positive integral $n$, $u_{n+1}=u_n(u_{n-1}^2-2)-u_1$, $u_0=2$, and $u_1=5/2$. Prove that
$$3\log_2[u_n]=2^n-(-1)^n,$$
where $[x]$ is the integral part of $x$.
(NET 3) IMO5. Let a set of $p$ equations be given,
$$\begin{aligned}a_{11}x_1+\cdots+a_{1q}x_q&=0,\\a_{21}x_1+\cdots+a_{2q}x_q&=0,\\&\vdots\\a_{p1}x_1+\cdots+a_{pq}x_q&=0,\end{aligned}$$
with coefficients $a_{ij}$ satisfying $a_{ij}=-1,0$, or $+1$ for all $i=1,\ldots,p$ and $j=1,\ldots,q$. Prove that if $q=2p$, there exists a solution $x_1,\ldots,x_q$ of this system such that all $x_j$ ($j=1,\ldots,q$) are integers satisfying $|x_j|\le q$ and $x_j\ne0$ for at least one value of $j$.
(NET 4) IMO3. A rectangular box can be filled completely with unit cubes. If one places cubes with volume $2$ in the box such that their edges are parallel to the edges of the box, one can fill exactly $40\%$ of the box. Determine all possible (interior) sizes of the box.
(POL 1b) Let $I=(0,1]$ be the unit interval of the real line. For a given number $a\in(0,1)$ we define a map $T:I\to I$ by the formula
$$T(x,y)=\begin{cases}x+(1-a)&\text{if }0\lt x\le a,\\x-a&\text{if }a\lt x\le1.\end{cases}$$
Show that for every interval $J\subset I$ there exists an integer $n\gt 0$ such that $T^n(J)\cap J\ne\varnothing$.
(SWE 3) Let $P$ be a polynomial with real coefficients such that $P(x)\gt 0$ if $x\gt 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x)=\frac{Q(x)}{R(x)}$ if $x\gt 0$.
(FIN 2) IMO2. Let $P_1(x)=x^2-2$, $P_j(x)=P_1(P_{j-1}(x))$, $j=2,3,\ldots$. Show that for arbitrary $n$ the roots of the equation $P_n(x)=x$ are real and different from one another.
(VIE 1) Prove that there exist infinitely many positive integers $n$ such that the decimal representation of $5^n$ contains a block of $1976$ consecutive zeros.
(VIE 2) The polynomial $1976(x+x^2+\cdots+x^n)$ is decomposed into a sum of polynomials of the form $a_1x+a_2x^2+\cdots+a_nx^n$, where $a_1,a_2,\ldots,a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.