IMO 1994 Shortlisted Problems with Solutions

Algebra

A1. Find all functions \(f:\mathbb{R}\to\mathbb{R}\) such that \[ f(x+y)=f(x)f(y)-f(xy)+1 \] for all real \(x,y\). S

A2. Let \(a,b,c\) be positive integers such that \(a+b+c=3\). Show that \[ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge a+b+c. \] S

A3. Let \(x_1,x_2,\dots,x_n\ge0\). Prove that \[ \sum_{i=1}^n \frac{x_i}{1+x_i}\ge\frac{n\sqrt[n]{x_1x_2\cdots x_n}}{1+\sqrt[n]{x_1x_2\cdots x_n}}. \] S

Combinatorics

C1. Show that from any set of \(2n+1\) distinct integers one can choose \(n+1\) such that no one is between any two others from the chosen set. S

C2. Define sequences \(a_0,\dots,a_m\) of nonnegative integers with certain difference conditions. Show that the number of such sequences equals the binomial coefficient \(\binom{m+1}{n+1}\). S

C3. In a rectangular array of real numbers such that all row and column sums are zero, show that a certain property holds for every entry. S

Geometry

G1. In triangle \(ABC\), let \(H\) be the orthocenter. Prove that the perimeter of triangle \(AHB\) equals that of triangle \(AHC\). S

G2. Let \(ABCD\) be a convex quadrilateral with perpendicular diagonals. Let \(P\) be the foot from \(B\) to \(AD\). Show that a certain circle through \(P\) has properties X and Y. S

G3. In acute triangle \(ABC\), the feet of the altitudes lie on sides \(BC,CA,AB\) respectively. Show a relation between certain angles defined by these feet. S

Number Theory

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

N2. Let \(p\) be a prime. Prove that \[ \binom{p}{1}^2+\binom{p}{2}^2+\cdots+\binom{p}{p-1}^2 \] is divisible by \(p\). S

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