IMO 1979 - Shortlisted problems

(BEL 1) Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).

(BEL 4) From a bag containing $5$ pairs of socks, each pair a different color, a random sample of $4$ single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are $4$ socks of different colors held outside the bag. What is the probability of the latter alternative?

(BUL 1) Find all polynomials $f(x)$ with real coefficients for which
$$f(x)f(2x^2)=f(2x^3+x).$$

(BUL 3) IMO2. A pentagonal prism $A_1A_2\ldots A_5B_1B_2\ldots B_5$ is given. The edges, the diagonals of the lateral walls and the internal diagonals of the prism are each colored either red or green in such a way that no triangle whose vertices are vertices of the prism has its three edges of the same color. Prove that all edges of the bases are of the same color.

(CZS 2) Let $n\ge2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j,k)$ of integers, $1\le j\lt k\le n$, with the following property: If $(j,k)\in M$, then $(k,m)\notin M$ for any $m$.

(CZS 4) Find the real values of $p$ for which the equation
$$\sqrt{2p+1-x^2}+\sqrt{3x+p+4}=\sqrt{x^2+9x+3p+9}$$
in $x$ has exactly two real distinct roots ($\sqrt t$ means the positive square root of $t$).

(FRG 1) IMO1. Given that $1-\frac12+\frac13-\frac14+\cdots-\frac1{1318}+\frac1{1319}=\frac pq$, where $p$ and $q$ are natural numbers having no common factor, prove that $p$ is divisible by $1979$.

(FRG 3) For all rational $x$ satisfying $0\le x\lt 1$, $f$ is defined by
$$f(x)=\begin{cases}f(2x)/4,&0\le x\lt 1/2,\\3/4+f(2x-1)/4,&1/2\le x\lt 1.\end{cases}$$
Given that $x=0.b_1b_2b_3\ldots$ is the binary representation of $x$, find $f(x)$.

(FRG 4) IMO6. Let $S$ and $F$ be two opposite vertices of a regular octagon. A counter starts at $S$ and each second is moved to one of the two neighboring vertices of the octagon. The direction is determined by the toss of a coin. The process ends when the counter reaches $F$. We define $a_n$ to be the number of distinct paths of duration $n$ seconds that the counter may take to reach $F$ from $S$. Prove that for $n=1,2,3,\ldots$,
$$a_{2n-1}=0,\qquad a_{2n}=\frac1{\sqrt2}(x^{n-1}-y^{n-1}),\qquad x=2+\sqrt2,\quad y=2-\sqrt2.$$

(FIN 3) Show that for any vectors $a,b$ in Euclidean space,
$$|a\times b|^3\le\frac{3\sqrt3}{8}|a|^2|b|^2|a-b|^2.$$
Remark. Here $\times$ denotes the vector product.

(GDR 1) Given real numbers $x_1,x_2,\ldots,x_n$ ($n\ge2$), with $x_i\ge1/n$ ($i=1,2,\ldots,n$) and with $x_1^2+x_2^2+\cdots+x_n^2=1$, find whether the product $P=x_1x_2x_3\cdots x_n$ has a greatest and/or least value and if so, give these values.

(GDR 3) Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an S-family over $R$ if and only if it satisfies the following three conditions:
(i) For no two sets $X,Y$ in $F$ is $X\subseteq Y$;
(ii) For any three sets $X,Y,Z$ in $F$, $X\cup Y\cup Z\ne R$;
(iii) $\bigcup_{X\in F}X=R$.
We define $|F|$ to be the number of elements of $F$ (i.e., the number of subsets of $R$ belonging to $F$). Determine, if it exists, $h=\max|F|$, the maximum being taken over all S-families over $R$.

(GRE 1) Show that $\frac{20}{60}\lt \sin20^\circ\lt \frac{21}{60}$.

(GRE 5) Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

(ISR 2) IMO5. The nonnegative real numbers $x_1,x_2,x_3,x_4,x_5,a$ satisfy the following relations:
$$\sum_{i=1}^5ix_i=a,\qquad\sum_{i=1}^5i^3x_i=a^2,\qquad\sum_{i=1}^5i^5x_i=a^3.$$
What are the possible values of $a$?

(ISR 4) Let $K$ denote the set $\{a,b,c,d,e\}$. $F$ is a collection of $16$ different subsets of $K$, and it is known that any three members of $F$ have at least one element in common. Show that all $16$ members of $F$ have exactly one element in common.

(NET 1) Inside an equilateral triangle $ABC$ one constructs points $P,Q,R$ such that
$$\angle QAB=\angle PBA=15^\circ,\qquad\angle RBC=\angle QCB=20^\circ,\qquad\angle PCA=\angle RAC=25^\circ.$$
Determine the angles of triangle $PQR$.

(POL 1) Let $m$ positive integers $a_1,\ldots,a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1,\ldots,b_n$ such that all sums of distinct $b_k$'s are distinct and all $a_i$ ($i\le m$) occur among them.

(ROM 1) Consider the sequences $(a_n),(b_n)$ defined by
$$a_1=3,\qquad b_1=100,\qquad a_{n+1}=3^{a_n},\qquad b_{n+1}=100^{b_n}.$$
Find the smallest integer $m$ for which $b_m\gt a_{100}$.

(SWE 2) Given the integer $n\gt 1$ and the real number $a\gt 0$, determine the maximum of $\sum_{i=1}^{n-1}x_ix_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a$.