(FRA) There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions:
(i) Every route includes exactly three ports;
(ii) No two routes contain the same three ports;
(iii) The series offers exactly two routes to each tourist who desires to visit two different arbitrary ports?
(CZS) IMO1. Let $x_1\ge x_2\ge\cdots\ge x_n$ and $y_1\ge y_2\ge\cdots\ge y_n$ be two $n$-tuples of numbers. Prove that
$$\sum_{i=1}^n(x_i-y_i)^2\le\sum_{i=1}^n(x_i-z_i)^2$$
is true when $z_1,z_2,\ldots,z_n$ denote $y_1,y_2,\ldots,y_n$ taken in another order.
(USA) Find the integer represented by $\left[\sum_{n=1}^{10^9}n^{-2/3}\right]$. Here $[x]$ denotes the greatest integer less than or equal to $x$ (e.g. $[\sqrt2]=1$).
(SWE) Let $a_1,a_2,\ldots,a_n,\ldots$ be a sequence of real numbers such that $0\le a_n\le1$ and $a_n-2a_{n+1}+a_{n+2}\ge0$ for $n=1,2,3,\ldots$. Prove that
$$0\le(n+1)(a_n-a_{n+1})\le2$$
for $n=1,2,3,\ldots$.
(SWE) Let $M$ be the set of all positive integers that do not contain the digit 9 (base 10). If $x_1,\ldots,x_n$ are arbitrary but distinct elements in $M$, prove that
$$\sum_{j=1}^n\frac1{x_j}\lt 80.$$
(USS) IMO4. Let $A$ be the sum of the digits of the number $16^{16}$ and $B$ the sum of the digits of the number $A$. Find the sum of the digits of the number $B$ without calculating $16^{16}$.
(GDR) Prove that from $x+y=1$ ($x,y\in\mathbb R$) it follows that
$$x^{m+1}\sum_{j=0}^n\binom{m+j}{j}y^j+y^{n+1}\sum_{i=0}^m\binom{n+i}{i}x^i=1\quad(m,n=0,1,2,\ldots).$$
(NET) IMO3. On the sides of an arbitrary triangle $ABC$, triangles $BPC$, $CQA$, and $ARB$ are externally erected such that
$$\angle PBC=\angle CAQ=45^\circ,$$
$$\angle BCP=\angle QCA=30^\circ,$$
$$\angle ABR=\angle BAR=15^\circ.$$
Prove that $\angle QRP=90^\circ$ and $QR=RP$.
(NET) Let $f(x)$ be a continuous function defined on the closed interval $0\le x\le1$. Let $G(f)$ denote the graph of $f(x)$: $G(f)=\{(x,y)\in\mathbb R^2\mid0\le x\le1,y=f(x)\}$. Let $G_a(f)$ denote the graph of the translated function $f(x-a)$ (translated over a distance $a$), defined by $G_a(f)=\{(x,y)\in\mathbb R^2\mid a\le x\le a+1,y=f(x-a)\}$. Is it possible to find for every $a$, $0\lt a\lt 1$, a continuous function $f(x)$, defined on $0\le x\le1$, such that $f(0)=f(1)=0$ and $G(f)$ and $G_a(f)$ are disjoint point sets?
(GBR) IMO6. The function $f(x,y)$ is a homogeneous polynomial of the $n$th degree in $x$ and $y$. If $f(1,0)=1$ and for all $a,b,c$,
$$f(a+b,c)+f(b+c,a)+f(c+a,b)=0,$$
prove that $f(x,y)=(x-2y)(x+y)^{n-1}$.
(GBR) IMO2. Let $a_1,a_2,a_3,\ldots$ be any infinite increasing sequence of positive integers. (For every integer $i\gt 0$, $a_{i+1}\gt a_i$.) Prove that there are infinitely many $m$ for which positive integers $x,y,h,k$ can be found such that $0\lt h\lt k\lt m$ and $a_m=xa_h+ya_k$.
(GRE) Consider on the first quadrant of the trigonometric circle the arcs $AM_1=x_1$, $AM_2=x_2$, $AM_3=x_3$, $\ldots$, $AM_\nu=x_\nu$, such that $x_1\lt x_2\lt x_3\lt \cdots\lt x_\nu$. Prove that
$$\sum_{i=0}^{\nu-1}\sin2x_i-\sum_{i=0}^{\nu-1}\sin(x_i-x_{i+1})\lt \frac\pi2+\sum_{i=0}^{\nu-1}\sin(x_i+x_{i+1}).$$
(ROM) Let $A_0,A_1,\ldots,A_n$ be points in a plane such that
(i) $A_0A_1\le\frac12A_1A_2\le\cdots\le\frac1{2^{n-1}}A_{n-1}A_n$ and
(ii) $0\lt \angle A_0A_1A_2\lt \angle A_1A_2A_3\lt \cdots\lt \angle A_{n-2}A_{n-1}A_n\lt 180^\circ$,
where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0\le k\le m-2\lt n-2$.
(USS) IMO5. Is it possible to plot 1975 points on a circle with radius 1 so that the distance between any two of them is a rational number (distances have to be measured by chords)?