IMO 1993 Shortlisted Problems with Solutions

Algebra

A1. Let \(a,b,c\) be real numbers such that \((a-1)(b-1)(c-1)=abc\). Prove that \[ (a-1)(b-1)(c-1)(a+b+c-2)=abc. \] S

A2. 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

A3. Let \(x_1,x_2,\dots,x_n\) be positive real numbers. Prove that \[ \frac{x_1}{1+x_1}+\frac{x_2}{1+x_2}+\cdots+\frac{x_n}{1+x_n} \geq \frac{n\sqrt[n]{x_1x_2\cdots x_n}}{1+\sqrt[n]{x_1x_2\cdots x_n}}. \] S

Combinatorics

C1. A tile consists of five unit squares as shown. Let \(T_n\) be the number of ways of covering a \(5\times n\) rectangle with \(n\) tiles. Prove that \(T_{2n}=2\cdot 3^{n-1}\) and \(T_{2n+1}=0\). S

C2. Show that the total number of zeros in all entries of a \(n\times n\) 0-1 matrix with exactly \(k\) ones equals \(\frac{n(n-1)}{2}\). S

C3. A sequence of real numbers \((x_1,\dots,x_n)\) satisfies: \(\lfloor x_i\rfloor\le i-1\le\lceil x_i\rceil\). Show that there exists a permutation \((p_1,\dots,p_n)\) of \(\{1,2,\dots,n\}\) such that \(p_i\le x_i\). S

Geometry

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

G2. In triangle \(ABC\), with points \(D\), \(E\), \(F\) on sides \(BC\), \(CA\), \(AB\) respectively such that \(\angle BDE=\angle CDF=\angle AEF=90^\circ\), show that points \(D,E,F\) are collinear. S

Number Theory

N1. Let \(p\) be prime. Show that for any integers \(a_1,\dots,a_p\), \[ a_1^p+a_2^p+\cdots+a_p^p-a_1-a_2-\cdots-a_p \] is divisible by \(p\). 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 1993. S

N3. Suppose that \(a_n\) is defined by \(a_0=0,a_1=1\), and \[ a_{n+2}=na_{n+1}+(n+1)a_n. \] Find \(a_{1993}\). S