BG-MO 1984

Solve the equation $5^x7^y+4=3^z$ in nonnegative integers.

The diagonals of a trapezoid $ABCD$ with bases $AB$ and $CD$ intersect in a point $O$ , and $AB/CD=k>1$ . The bisectors of the angles $AOB,BOC,COD,DOA$ intersect $AB,BC,CD,DA$ respectively at $K,L,M,N$ . The lines $KL$ and $MN$ meet at $P$ , and the lines $KN$ and $LM$ meet at $Q$ . If the areas of $ABCD$ and $OPQ$ are equal, find the value of $k$ .

Points $P_1,P_2,\ldots,P_n,Q$ are given in space $(n\ge4)$ , no four of which are in a plane. Prove that if for any three distinct points $P_\alpha,P_\beta,P_\gamma$ there is a point $P_\delta$ such that the tetrahedron $P_\alpha P_\beta P_\gamma P_\delta$ contains the point $Q$ , then $n$ is an even number.

Let $a,b,a_2,\ldots,a_{n-2}$ be real numbers with $ab\ne0$ such that all the roots of the equation $$ax^n-ax^{n-1}+a_2x^{n-2}+\ldots+a_{n-2}x^2-n^2bx+b=0$$ are positive and real. Prove that these roots are all equal.

Let $0<x_i<1$ and $x_i+y_i=1$ for $i=1,2,\ldots,n$ . Prove that $$(1-x_1x_2\cdots x_n)^m+(1-y_1^m)(1-y_2^m)\cdots(1-y_n^m)>1$$ for any natural numbers $m$ and $n$ .

Let there be given a pyramid $SABCD$ whose base $ABCD$ is a parallelogram. Let $N$ be the midpoint of $BC$ . A plane $\lambda$ intersects the lines $SC,SA,AB$ at points $P,Q,R$ respectively such that $\overline{CP}/\overline{CS}=\overline{SQ}/\overline{SA}=\overline{AR}/\overline{AB}$ . A point $M$ on the line $SD$ is such that the line $MN$ is parallel to $\lambda$ . Show that the locus of points $M$ , when $\lambda$ takes all possible positions, is a segment of the length $\frac{\sqrt5}2SD$ .