(AUS 1) The localities $P_1,P_2,\ldots,P_{1983}$ are served by ten international airlines $A_1,A_2,\ldots,A_{10}$. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.
(BEL 1) Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m\ge1$ is superabundant (P. Erdős, 1944) if $\forall k\in\{1,2,\ldots,m-1\},\ \frac{\sigma(m)}m>\frac{\sigma(k)}k$. Prove that there exists an infinity of superabundant numbers.
(BEL 3) IMO4. We say that a set $E$ of points of the Euclidian plane is “Pythagorean” if for any partition of $E$ into two sets $A$ and $B$, at least one of the sets contains the vertices of a right-angled triangle. Decide whether the following sets are Pythagorean:
(a) a circle;
(b) an equilateral triangle (that is, the set of three vertices and the points of the three edges).
(BEL 5) On the sides of the triangle $ABC$, three similar isosceles triangles $ABP$ $(AP=PB)$, $AQC$ $(AQ=QC)$, and $BRC$ $(BR=RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.
(BRA 1) Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A=(a_j)$ and $B=(b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k\lt b_k$ and $a_i=b_i$ for $i\lt k$. Find the terms of the first sequence of the set under this ordering.
(CAN 2) Suppose that $\{x_1,x_2,\ldots,x_n\}$ are positive integers for which $x_1+x_2+\cdots+x_n=2(n+1)$. Show that there exists an integer $r$ with $0\le r\le n-1$ for which the following $n-1$ inequalities hold:
$$\begin{aligned}x_{r+1}+\cdots+x_{r+i}&\le2i+1&&\forall i,\ 1\le i\le n-r;\\x_{r+1}+\cdots+x_n+x_1+\cdots+x_i&\le2(n-r+i)+1&&\forall i,\ 1\le i\le r-1.\end{aligned}$$
Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r$.
(CAN 5) Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0=0$ and
$$a_{n+1}=(a_n+1)a+(a+1)a_n+2\sqrt{a(a+1)a_n(a_n+1)}\qquad(n=1,2\ldots).$$
Show that for each positive integer $n$, $a_n$ is a positive integer.
(SPA 2) In a test, $3n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_1(t),N_2(t),N_3(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each $t$ during the test,
$$|N_i(t)-N_j(t)|\lt 2,\qquad i\ne j,\qquad i,j=1,2,\ldots\ .$$
(FIN 1) Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that
$$\left|P(x)-\frac pq\right|\lt \frac1{q^2}$$
for all $x\in I$.
(FIN 2′) Let $f:[0,1]\to\mathbb R$ be continuous and satisfy:
$$\begin{aligned}bf(2x)&=f(x),&&0\le x\le1/2;\\f(x)&=b+(1-b)f(2x-1),&&1/2\le x\le1,\end{aligned}$$
where $b=\frac{1+c}{2+c}$, $c>0$. Show that $0\lt f(x)-x\lt c$ for every $x$, $0\lt x\lt 1$.
(GBR 4) IMO1. Find all functions $f$ defined on the positive real numbers and taking positive real values that satisfy the following conditions:
(i) $f(xf(y))=yf(x)$ for all positive real $x,y$.
(ii) $f(x)\to0$ as $x\to+\infty$.
(LUX 2) Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$?
(POL 2) IMO5. Prove or disprove: From the interval $[1,\ldots,30000]$ one can select a set of $1000$ integers containing no arithmetic triple (three consecutive numbers of an arithmetic progression).
(POL 3) Decide whether there exists a set $M$ of natural numbers satisfying the following conditions:
(i) For any natural number $m>1$ there are $a,b\in M$ such that $a+b=m$.
(ii) If $a,b,c,d\in M$, $a,b,c,d>10$ and $a+b=c+d$, then $a=c$ or $a=d$.
(GDR 1) Let $F(n)$ be the set of polynomials $P(x)=a_0+a_1x+\cdots+a_nx^n$, with $a_0,a_1,\ldots,a_n\in\mathbb R$ and $0\le a_0=a_n\le a_1=a_{n-1}\le\cdots\le a_{[n/2]}=a_{[(n+1)/2]}$. Prove that if $f\in F(m)$ and $g\in F(n)$, then $fg\in F(m+n)$.
(GDR 3) Let $P_1,P_2,\ldots,P_n$ be distinct points of the plane, $n\ge2$. Prove that
$$\max_{1\le i\lt j\le n}P_iP_j>\frac{\sqrt3}{2}(\sqrt n-1)\min_{1\le i\lt j\le n}P_iP_j.$$
(FRG 3) IMO3. Let $a,b,c$ be positive integers satisfying $(a,b)=(b,c)=(c,a)=1$. Show that $2abc-ab-bc-ca$ is the largest integer not representable as
$$xbc+yca+zab$$
with nonnegative integers $x,y,z$.
(ROM 1) Let $(F_n)_{n\ge1}$ be the Fibonacci sequence $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$ $(n\ge1)$, and $P(x)$ the polynomial of degree $990$ satisfying
$$P(k)=F_k,\qquad k=992,\ldots,1982.$$
Prove that $P(1983)=F_{1983}-1$.
(ROM 3) Solve the system of equations
$$\begin{aligned}x_1|x_1|&=x_2|x_2|+(x_1-a)|x_1-a|,\\x_2|x_2|&=x_3|x_3|+(x_2-a)|x_2-a|,\\&\cdots\\x_n|x_n|&=x_1|x_1|+(x_n-a)|x_n-a|,\end{aligned}$$
in the set of real numbers, where $a>0$.