In the plane, let $M_0M_1\cdots M_{2n}$, $n\ge1$, be a convex polygon whose vertices occur counterclockwise on a circle $(C)$ of radius $R$. A point $A$ inside the polygon satisfies
$$\angle M_0AM_1=\angle M_1AM_2=\cdots=\angle M_{2n}AM_0=\frac{360^\circ}{2n+1}.$$
Assume $A$ is not the center. Let $B$ lie on the circle so that $AB$ is perpendicular to the diameter through $A$. Prove
$$\frac{2n+1}{\frac1{AM_0}+\frac1{AM_1}+\cdots+\frac1{AM_{2n}}}\lt AB\lt\frac{AM_0+AM_1+\cdots+AM_{2n}}{2n+1}\lt R.$$
The source calls the same circle $(O)$ in the sentence defining $B$; we use $(C)$ consistently and denote its center by $O$. Either of the two possible points $B$ gives the same length.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. The printed dates are 16–17 May 1990; grouping Q1–3 and Q4–6 into sessions is an editorial inference. Import scoring multipliers: correct 1, incorrect 0, blank 0.
Let $a,b,A,B$ be positive real numbers. Define $x_1=a$, $x_2=b$ and
$$x_{n+1}=A\sqrt[3]{x_n^2}+B\sqrt[3]{x_{n-1}^2}\qquad(n=2,3,4,\ldots).$$
Prove that $\lim_{n\to\infty}x_n$ exists and determine it.
A finite set $T$ of positive integers satisfies:
1. For any two elements of $T$, their greatest common divisor and least common multiple also belong to $T$.
2. For every $x\in T$, there is $x'\in T$ such that $\gcd(x,x')=1$ and $\operatorname{lcm}(x,x')$ is the largest element of $T$.
Write $l(T)$ for the number of elements. Find its greatest possible value subject to $l(T)\lt1990$.
In a tetrahedron, the product of the lengths of each pair of opposite edges is $l$. Let $\alpha,\beta,\gamma$ be the angles between the three pairs of opposite edges, and let $R_1,R_2,R_3,R_4$ be the circumradii of its four faces. Prove
$$\sin^2\alpha+\sin^2\beta+\sin^2\gamma\ge\frac{l}{\sqrt{R_1R_2R_3R_4}}.$$
There are $n\ge3$ pupils standing on a circle, always facing their teacher at its center. Each time the teacher blows a whistle, some two adjacent pupils exchange positions and all others stay still. Find the smallest number $M$ of whistles that allows suitable exchanges with this final property: every two pupils who were initially adjacent are again adjacent, but their left/right roles are reversed. Thus, if pupil $A$ was initially on pupil $B$'s left, $A$ is finally on $B$'s right.
Let the sequence $\{x_n\}_{n=1}^{+\infty}$, with $|x_1|\lt 1$, be defined by
$$x_{n+1}=\frac{-x_n+\sqrt{3-3x_n^2}}2.$$
(a) What condition on $x_1$ is necessary for every term of this sequence to be positive?
(b) Is this sequence periodic? Explain your answer.
Consider the $2n-1$ consecutive natural numbers $1,2,3,\ldots,2n-1$. Cross out at least $n-1$ numbers according to these rules:
- If $a$ is crossed out, then $2a$ must be crossed out.
- If $a$ and $b$ are crossed out, then $a+b$ must be crossed out.
Which numbers should be crossed out to maximize the sum of those remaining? Explain your answer.
A tetrahedron is to be cut by three planes to obtain a parallelepiped whose three faces through one vertex lie on faces of the tetrahedron.
(a) Can the parallelepiped have volume $9/10$ of the tetrahedron's volume? Explain.
(b) Determine a common point of the three cutting planes so that the parallelepiped has volume $11/50$ of the tetrahedron's volume.
In a plane, let $ABC$ be a triangle. Determine the locus of points $M$ in the plane such that
$$MA\cdot MA'=MB\cdot MB'=MC\cdot MC',$$
where $A',B',C'$ are the perpendicular projections of $M$ onto the lines $BC,CA,AB$, respectively.
Let $f(x)$ be a nonconstant polynomial with real coefficients satisfying
$$f(x)f(2x^2)=f(2x^3+x)$$
for every $x\in\mathbb R$. Prove that $f$ has no real roots.
The children in a class stand in a circle to play a candy game. The teacher gives each child an even number of candies (every child has candies, but their numbers may differ). A child gives half of their candies to the child immediately on their right. The child who has just received candies does the same if their new number is even; if it is odd, they receive one candy from the teacher before sharing in the same way. The children continue passing candies around the circle in turn. Prove that eventually there is a situation in which, if a child gives half of their candies to the teacher instead of to their neighbor, all children have equal numbers of candies.