International Mathematical Olympiad · 1976

Problems

Problem 1In a convex quadrilateral of area $32$ , the sum of the lengths of two opposite sides and one diagonal is $16$ . Determine all possible lengths of the other diagonal.Solutions: 1
Problem 2Let $P_1(x) = x^2 - 2$ and $P_j(x) = P_1(P_{j-1}(x))$ for $j = 2, 3, \ldots$ . Prove that, for any positive integer $n$ , the roots of the equation $P_n(x) = x$ are all real and distinct.Solutions: 1
Problem 3A box whose shape is a rectangular parallelepiped can be completely filled with cubes of side $1$ . If one places inside it the maximum possible number of cubes, each of volume $2$ , with their sides parallel to those of the box, then exactly $40\%$ of the volume of the box is occupied. Determine the possible dimensions of all such boxes.Solutions: 1
Problem 4Determine, with proof, the greatest number which is the product of positive integers whose sum is $1976$ .Solutions: 1
Problem 5Consider the system of $p$ equations in $q=2p$ unknowns $x_1,\ldots,x_q$ , with every coefficient $a_{ij}$ in $\{-1,0,1\}$ . Prove that the system has a solution such that (a) all $x_j$ are integers; (b) at least one $x_j\ne0$ ; (c) $|x_j|\le q$ for every $j$ .Solutions: 1
Problem 6A sequence $(u_n)$ is defined by $u_0=2$ , $u_1=\frac52$ , and $u_{n+1}=u_n(u_{n-1}^2-2)-u_1$ for $n=1,2,\ldots$ . Prove that for every positive integer $n$ , $\lfloor u_n\rfloor=2^{\frac{2^n-(-1)^n}{3}}$ , where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ .Solutions: 1