BG-MO 1979

Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$ .

Daniel Harrer

Points $P,Q,R,S$ are taken on respective edges $AC$ , $AB$ , $BD$ , and $CD$ of a tetrahedron $ABCD$ so that $PR$ and $QS$ intersect at point $N$ and $PS$ and $QR$ intersect at point $M$ . The line $MN$ meets the plane $ABC$ at point $L$ . Prove that the lines $AL$ , $BP$ , and $CQ$ are concurrent.

Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$ .

For each real number $k$ , denote by $f(k)$ the larger of the two roots of the quadratic equation $$(k^2+1)x^2+10kx-6(9k^2+1)=0.$$ Show that the function $f(k)$ attains a minimum and maximum and evaluate these two values.

A convex pentagon $ABCDE$ satisfies $AB=BC=CA$ and $CD=DE=EC$ . Let $S$ be the center of the equilateral triangle $ABC$ and $M$ and $N$ be the midpoints of $BD$ and $AE$ , respectively. Prove that the triangles $SME$ and $SND$ are similar.

The set $M=\{1,2,\ldots,2n\}~(n\ge2)$ is partitioned into $k$ nonintersecting subsets $M_1,M_2,\ldots,M_k$ , where $k^3+1\le n$ . Prove that there exist $k+1$ even numbers $2j_1,2j_2,\ldots,2j_{k+1}$ in $M$ that are in one and the same subset $M_j$ $(1\le j\le k)$ such that the numbers $2j_1-1,2j_2-1,\ldots,2j_{k+1}-1$ are also in one and the same subset $M_r$ $(1\le r\le k)$ .