IMO 1990 Shortlisted Problems with Solutions

Algebra

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

A2. Let \(x_1,\dots,x_n>0\). Prove \[ \frac{1}{1+x_1}+\cdots+\frac{1}{1+x_n}\ge\frac{n}{1+\sqrt[n]{x_1x_2\cdots x_n}}. \] S

A3. Let \(a,b,c\) be positive integers such that \((a+1)(b+1)(c+1)\) is divisible by \(abc+1\). Prove that \[ \frac{(a+1)(b+1)(c+1)}{abc+1} \] is a perfect square. S

Combinatorics

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

C2. A tile consists of 5 unit squares. For a \(5\times n\) rectangle, let \(T_n\) be the number of tilings. Show that \(T_{2n}=2\cdot3^{n-1}\) and \(T_{2n+1}=0\). S

C3. Let \(x_1,\dots,x_n\) be real numbers such that \(\lfloor x_i\rfloor\le i-1\le\lceil x_i\rceil\). Show there exists a permutation \((p_1,\dots,p_n)\) with \(p_i\le x_i\). S

Geometry

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

Number Theory

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

N2. Let \(n\) be a positive integer not divisible by 3. Show there exists a multiple of \(n\) whose digit sum is 1990. S