(SWE 2) Two ships sail on the sea with constant speeds and fixed directions. It is known that at 9:00 the distance between them was 20 miles; at 9:35, 15 miles; and at 9:55, 13 miles. At what moment were the ships the smallest distance from each other, and what was that distance?
(ROM 5), IMO 1. Prove that there exists a unique triangle whose side lengths are consecutive natural numbers and one of whose angles is twice the measure of one of the others.
(BUL 2), IMO 3. Let $a,b,c$ be real numbers. Prove that the system of equations
$$\begin{cases}ax_1^2+bx_1+c=x_2,\\ax_2^2+bx_2+c=x_3,\\\cdots\\ax_{n-1}^2+bx_{n-1}+c=x_n,\\ax_n^2+bx_n+c=x_1.\end{cases}$$
has a unique real solution if and only if $(b-1)^2-4ac=0$.
(BUL 5) Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon $(n\ge3)$ inscribed in a circle of radius $r$. Prove the inequality
$$(n+1)h_{n+1}-nh_n>r.$$
Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n\ge3$.
(HUN 1) If $a_i$ $(i=1,2,\ldots,n)$ are distinct non-zero real numbers, prove that the equation
$$\frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x}=n$$
has at least $n-1$ real roots.
(HUN 5) Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $\frac{3\sqrt3}{8}$ times the product of the side lengths of the triangle. When does equality hold?
(ROM 2) Given an oriented line $\Delta$ and a fixed point $A$ on it, consider all trapezoids $ABCD$ one of whose bases $AB$ lies on $\Delta$, in the positive direction. Let $E,F$ be the midpoints of $AB$ and $CD$ respectively. Find the loci of vertices $B,C,D$ of trapezoids that satisfy the following:
(i) $|AB|\le a$ ($a$ fixed);
(ii) $|EF|=l$ ($l$ fixed);
(iii) the sum of squares of the nonparallel sides of the trapezoid is constant.
Remark. The constants are chosen so that such trapezoids exist.
(ROM 3) Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a,d_b,d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a,b,c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality
$$ab d_a d_b+bc d_b d_c+ca d_c d_a\le\frac{4S^2}{3}.$$
Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.
(ROM 6) Find all solutions $(x_1,x_2,\ldots,x_n)$ of the equation
$$1+\frac1{x_1}+\frac{x_1+1}{x_1x_2}+\frac{(x_1+1)(x_2+1)}{x_1x_2x_3}+\cdots+\frac{(x_1+1)\cdots(x_{n-1}+1)}{x_1x_2\cdots x_n}=0.$$
(POL 1) If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that
$$\left(1+\frac ab\right)^m+\left(1+\frac ba\right)^m\ge2^{m+1}.$$
(POL 5) Given two congruent triangles $A_1A_2A_3$ and $B_1B_2B_3$ $(A_iA_k=B_iB_k)$, prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented.
(BUL 5) A line in the plane of a triangle $ABC$ intersects the sides $AB$ and $AC$ respectively at points $X$ and $Y$ such that $BX=CY$. Find the locus of the center of the circumcircle of triangle $XAY$.
(GBR 1), IMO 6. Let $[x]$ denote the integer part of $x$, i.e., the greatest integer not exceeding $x$. If $n$ is a positive integer, express as a simple function of $n$ the sum
$$\left\lfloor\frac{n+1}{2}\right\rfloor+\left\lfloor\frac{n+2}{4}\right\rfloor+\cdots+\left\lfloor\frac{n+2^i}{2^{i+1}}\right\rfloor+\cdots.$$
(GBR 3) A polynomial $p(x)=a_0x^k+a_1x^{k-1}+\cdots+a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by $m$ for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k!a_0$ is also divisible by $m$. Also prove that if $a_0,k,m$ are nonnegative integers for which $k!a_0$ is divisible by $m$, there exists a polynomial $p(x)=a_0x^k+\cdots+a_k$ divisible by $m$.
(GBR 4) Given a point $O$ and lengths $x,y,z$, prove that there exists an equilateral triangle $ABC$ for which $OA=x,OB=y,OC=z$, if and only if $x+y\ge z,y+z\ge x,z+x\ge y$ (the points $O,A,B,C$ are coplanar).
(ITA 2) If an acute-angled triangle $ABC$ is given, construct an equilateral triangle $A'B'C'$ in space such that lines $AA',BB',CC'$ pass through a given point.
(ITA 5) We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0,1,2,\ldots$ from it we obtain points with abscisas $n=0,1,2,\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $1/5$ apart?
(CZS 1) Given $n$ $(n\ge3)$ points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1,A_2,\ldots,A_n$ in such a way that for each $i,j,k$, $1\le i\lt j\lt k\le n$, angle $A_iA_jA_k$ is greater than or equal to $120^\circ$.