Question bank IMO Shortlist 1981

(BEL), IMO 4. (a) For which values of $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers?

(b) For which values of $n>2$ is there a unique set having the stated property?

(BUL) A sphere $S$ is tangent to the edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at the points $E,F,G,H$ respectively. The points $E,F,G,H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $AC$, then it is also tangent to the edge $BD$.

(CAN) Find the minimum value of
$$\max(a+b+c,b+c+d,c+d+e,d+e+f,e+f+g)$$
subject to the constraints

(i) $a,b,c,d,e,f,g\ge0$,

(ii) $a+b+c+d+e+f+g=1$.

(CAN) Let $\{f_n\}$ be the Fibonacci sequence $\{1,1,2,3,5,\ldots\}$.

(a) Find all pairs $(a,b)$ of real numbers such that for each $n$, $af_n+bf_{n+1}$ is a member of the sequence.

(b) Find all pairs $(u,v)$ of positive real numbers such that for each $n$, $uf_n^2+vf_{n+1}^2$ is a member of the sequence.

(COL) A cube is assembled with 27 white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.

(CUB) Let $P(z)$ and $Q(z)$ be complex-variable polynomials, with degree not less than 1. Let
$$P_k=\{z\in\mathbb C\mid P(z)=k\},\qquad Q_k=\{z\in\mathbb C\mid Q(z)=k\}.$$
Let also $P_0=Q_0$ and $P_1=Q_1$. Prove that $P(z)\equiv Q(z)$.

(FIN), IMO 6. Assume that $f(x,y)$ is defined for all positive integers $x$ and $y$, and that the following equations are satisfied:
$$f(0,y)=y+1,\qquad f(x+1,0)=f(x,1),\qquad f(x+1,y+1)=f(x,f(x+1,y)).$$
Determine $f(2,2)$, $f(3,3)$ and $f(4,4)$.

Alternative version: Determine $f(4,1981)$.

(FRG), IMO 2. Let $f(n,r)$ be the arithmetic mean of the minima of all $r$-subsets of the set $\{1,2,\ldots,n\}$. Prove that $f(n,r)=\frac{n+1}{r+1}$.

(FRG) A sequence $(a_n)$ is defined by means of the recursion
$$a_1=1,\qquad a_{n+1}=\frac{1+4a_n+\sqrt{1+24a_n}}{16}.$$
Find an explicit formula for $a_n$.

(FRA) Determine the smallest natural number $n$ having the following property: For every integer $p$, $p\ge n$, it is possible to subdivide (partition) a given square into $p$ squares (not necessarily equal).

(NET) On a semicircle with unit radius four consecutive chords $AB,BC,CD,DE$ with lengths $a,b,c,d$, respectively, are given. Prove that
$$a^2+b^2+c^2+d^2+abc+bcd\lt 4.$$

(NET), IMO 3. Determine the maximum value of $m^2+n^2$ where $m$ and $n$ are integers satisfying $m,n\in\{1,2,\ldots,100\}$ and $(n^2-mn-m^2)^2=1$.

(ROM) Let $P$ be a polynomial of degree $n$ satisfying
$$P(k)=\binom{n+1}{k}^{-1}\qquad(k=0,1,\ldots,n).$$
Determine $P(n+1)$.

(ROM) Prove that a convex pentagon (a five-sided polygon) $ABCDE$ with equal sides and for which the interior angles satisfy the condition $\angle A\ge\angle B\ge\angle C\ge\angle D\ge\angle E$ is a regular pentagon.

(GBR), IMO 1. Find the point $P$ inside the triangle $ABC$ for which
$$\frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF}$$
is minimal, where $PD,PE,PF$ are the perpendiculars from $P$ to $BC,CA,AB$ respectively.

(GBR) A sequence of real numbers $u_1,u_2,u_3,\ldots$ is determined by $u_1$ and the following recurrence relation for $n\ge1$:
$$4u_{n+1}=\sqrt[3]{64u_n+15}.$$
Describe, with proof, the behavior of $u_n$ as $n\to\infty$.

(USS), IMO 5. Three equal circles touch the sides of a triangle and have one common point $O$. Show that the center of the circle inscribed in and of the circle circumscribed about the triangle $ABC$ and the point $O$ are collinear.

(USS) Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.

(YUG) A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\frac{2S}{9}$.