TR-MO 1997

Find all pairs of integers $(x, y)$ such that $5x^{2}-6xy+7y^{2}=383$ .

Let $F$ be a point inside a convex pentagon $ABCDE$ , and let $a_{1}$ , $a_{2}$ , $a_{3}$ , $a_{4}$ , $a_{5}$ denote the distances from $F$ to the lines $AB$ , $BC$ , $CD$ , $DE$ , $EA$ , respectively. The points $F_{1}$ , $F_{2}$ , $F_{3}$ , $F_{4}$ , $F_{5}$ are chosen on the inner bisectors of the angles $A$ , $B$ , $C$ , $D$ , $E$ of the pentagon respectively, so that $AF_{1} = AF$ , $BF_{2} = BF$ , $CF_{3} = CF$ , $DF_{4} = DF$ and $EF_{5} = EF$ . If the distances from $F_{1}$ , $F_{2}$ , $F_{3}$ , $F_{4}$ , $F_{5}$ to the lines $EA$ , $AB$ , $BC$ , $CD$ , $DE$ are $b_{1}$ , $b_{2}$ , $b_{3}$ , $b_{4}$ , $b_{5}$ , respectively. Prove that $a_{1} + a_{2} + a_{3} + a_{4} + a_{5} \leq b_{1} + b_{2} + b_{3} + b_{4} + b_{5}$

Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$ .

In a triangle $ABC$ , the inner and outer bisectors of the $\angle A$ meet the line $BC$ at $D$ and $E$ , respectively. Let $d$ be a common tangent of the circumcircle $(O)$ of $\triangle ABC$ and the circle with diameter $DE$ and center $F$ . The projections of the tangency points onto $FO$ are denoted by $P$ and $Q$ , and the length of their common chord is denoted by $m$ . Prove that $PQ = m$

Let $D_{1}, D_{2}, . . . , D_{n}$ be rectangular parallelepipeds in space, with edges parallel to the $x, y, z$ axes. For each $D_{i}$ , let $x_{i} , y_{i} , z_{i}$ be the lengths of its projections onto the $x, y, z$ axes, respectively. Suppose that for all pairs $D_{i}$ , $D_{j}$ , if at least one of $x_{i} < x_{j}$ , $y_{i} < y_{j}$ , $z_{i} < z_{j}$ holds, then $x_{i} \leq x_{j}$ , $y_{i} \leq y_{j}$ , and $z_{i} < z_{j}$ . If the volume of the region $\bigcup^{n}_{i=1}{D_{i}}$ equals 1997, prove that there is a subset $\{D_{i_{1}}, D_{i_{2}}, . . . , D_{i_{m}}\}$ of the set $\{D_{1}, . . . , D_{n}\}$ such that $(i)$ if $k \not= l$ then $D_{i_{k}} \cap D_{i_{l}} = \emptyset$ , and $(ii)$ the volume of $\bigcup^{m}_{k=1}{D_{i_{k}}}$ is at least 73.