IMO 1991 Shortlisted Problems with Solutions

Algebra

A1. Find all polynomials \(P(x)\in\mathbb{R}[x]\) such that \[ P(x+1)P(x-1)=P(x)^2-1 \] for all real \(x\). S

A2. Let \(x_1,\dots,x_n>0\). Prove that \[ \frac{1}{1+x_1}+\cdots+\frac{1}{1+x_n}\ge\frac{n}{1+\sqrt[n]{x_1\cdots x_n}}. \] S

Combinatorics

C1. Let \(S\) be a finite set of integers. Show that there exists \(T\subseteq S\) such that \[ \Bigl|\sum_{t\in T}t\Bigr|\le\frac{1}{2}\sum_{s\in S}|s|. \] S

C2. In a rectangle of real numbers with row and column sums all equal to 1, show that every number lies between 0 and 1. S

C3. Define a *nice* sequence of integers of length \(n\). Show that a sequence of length 1991 contains a nice subsequence of length 1991. S

Geometry

G1. In triangle \(ABC\), let \(H\) be the orthocenter and \(O\) the circumcenter. Show that \[ AH^2+BH^2+CH^2 = 9OG^2, \] where \(G\) is the centroid. S

G2. Let \(ABCD\) be a convex quadrilateral with perpendicular diagonals. Let \(P\) be the foot from \(B\) to \(AD\). Show that the circumcircle of triangle \(BPC\) has special properties. S

Number Theory

N1. Prove there are infinitely many primes \(p\) with \(p\equiv 1\pmod{4}\). S

N2. Let \(n\) be a positive integer not divisible by 3. Show that there exists a multiple of \(n\) with digit sum equal to 1991. S