IMO 1959–1966 - Longlisted Problems (individual years unknown)

(CZS) We are given $n>3$ points in the plane, no three of which lie on a line. Does there necessarily exist a circle that passes through at least three of the given points and contains none of the other given points in its interior?

(GDR) Given $n$ positive real numbers $a_1,a_2,\ldots,a_n$ such that $a_1a_2\cdots a_n=1$, prove that
$$(1+a_1)(1+a_2)\cdots(1+a_n)\ge2^n.$$

(BUL) A regular triangular prism has height $h$ and a base of side length $a$. Both bases have small holes in the centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once and leaves through the hole in the bottom. Find the angle at which the light enters and the length of its path inside the prism.

(POL) Five points in the plane are given, no three of which are collinear. Show that some four of them form a convex quadrilateral.

(USS) Prove the inequality
$$\tan\frac{\pi\sin x}{4\sin\alpha}+\tan\frac{\pi\cos x}{4\cos\alpha}>1$$
for any $x,\alpha$ with $0\le x\le\pi/2$ and $\pi/6\lt y\lt \pi/3$.

(USS) A convex planar polygon $\mathcal M$ with perimeter $l$ and area $S$ is given. Let $M(R)$ be the set of all points in space that lie a distance at most $R$ from a point of $\mathcal M$. Show that the volume $V(R)$ of this set equals
$$V(R)=\frac43\pi R^3+\frac\pi2 lR^2+2SR.$$

(USS) For which arrangements of two infinite circular cylinders does their intersection lie in a plane?

(USS) We are given a bag of sugar, a two-pan balance, and a weight of 1 gram. How do we obtain 1 kilogram of sugar in the smallest possible number of weighings?

(ROM) Find $x$ such that
$$\frac{\sin3x\cos(60^\circ-4x)+1}{\sin(60^\circ-7x)-\cos(30^\circ+x)+m}=0,$$
where $m$ is a fixed real number.

(GDR) How many real solutions are there to the equation $x=1964\sin x-189$?

(CZS) Does there exist an integer $z$ that can be written in two different ways as $z=x!+y!$, where $x,y$ are natural numbers with $x\le y$?

(BUL) Find digits $x,y,z$ such that the equality $$\sqrt{\underbrace{\overline{xx\cdots x}}_{2n}-\underbrace{\overline{yy\cdots y}}_n}=\underbrace{\overline{zz\cdots z}}_n$$ holds for at least two values of $n\in\mathbb N$, and in that case find all $n$ for which this equality is true.

(YUG) Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Prove the inequality $$\binom n2\sum_{i<j}\frac1{a_i a_j}\ge4\left(\sum_{i<j}\frac1{a_i+a_j}\right)^2$$ and find the conditions on the numbers $a_i$ for equality to hold.

(POL) Compute the largest number of regions into which one can divide a disk by joining $n$ points on its circumference.

(POL) Points $A,B,C,D$ lie on a circle such that $AB$ is a diameter and $CD$ is not. If the tangents at $C$ and $D$ meet at $P$ while $AC$ and $BD$ meet at $Q$, show that $PQ$ is perpendicular to $AB$.

(CZS) We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XYZ$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q$.

(ROM) Suppose $ABCD$ and $A^\prime B^\prime C^\prime D^\prime$ are two parallelograms arbitrarily arranged in space, and let points $M,N,P,Q$ divide the segments $AA^\prime,BB^\prime,CC^\prime,DD^\prime$ respectively in equal ratios.

(a) Show that $MNPQ$ is a parallelogram;

(b) Find the locus of $MNPQ$ as $M$ varies along the segment $AA^\prime$.

(HUN) Solve the equation $\frac1{\sin x}+\frac1{\cos x}=\frac1p$, where $p$ is a real parameter. Discuss for which values of $p$ the equation has at least one real solution and determine the number of solutions in $[0,2\pi)$ for a given $p$.

(HUN) Construct a triangle given the three exradii.

(HUN) We are given three equal rectangles with the same center in three mutually perpendicular planes, with the long sides also mutually perpendicular. Consider the polyhedron with vertices at the vertices of these rectangles.

(a) Find the volume of this polyhedron;

(b) can this polyhedron be regular, and under what conditions?