IMO 1985 Shortlisted Problems with Solutions

Algebra

A1. Find all functions \(f:\mathbb{R}\to\mathbb{R}\) satisfying \[ f(x+y)+f(x-y)=2f(x)f(y) \] for all real \(x,y\). S

A2. Let \(x_1,x_2,\dots,x_n>0\). Prove \[ \frac{1}{1+x_1}+\frac{1}{1+x_2}+\cdots+\frac{1}{1+x_n} \ge\frac{n}{1+\sqrt[n]{x_1x_2\cdots x_n}}. \] S

Combinatorics

C1. A tile made of 5 unit squares is used to tile a \(5\times n\) rectangle. Let \(T_n\) be the number of tilings. Show that \[ T_{2n}=2\cdot 3^{n-1},\quad T_{2n+1}=0. \] S

C2. 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

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

Geometry

G1. In triangle \(ABC\), let \(O\) be the circumcenter and \(H\) the orthocenter. Prove 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 invariant properties. S

Number Theory

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

N2. Let \(n\) be a positive integer not divisible by 3. Show that there exists a multiple of \(n\) whose digit sum equals 1985. S