ZA-SAMO 1995

Prove that there are no integers $m$ and $n$ such that $$19m^2+95mn+2000n^2=1995.$$

$ABC$ is a triangle with $\hat{A}<\hat{C}$ , and $D$ is the point on $BC$ such that $B\hat{A}D=A\hat{C}B$ . The perpendicular bisectors of $AD$ and $AC$ intersect in the point $E$ . Prove that $B\hat{A}E=90^\circ$ .

Suppose that $a_1,a_2,\dots,a_n$ are the numbers $1,2,3,\dots,n$ but written in any order. Prove that $$(a_1-1)^2+(a_2-2)^2+\cdots+(a_n-n)^2$$ is always even.

Three circles, with radii $p$ , $q$ and $r$ and centres $A$ , $B$ and $C$ respectively, touch one another externally at points $D$ , $E$ and $F$ . Prove that the ratio of the areas of $\triangle DEF$ and $\triangle ABC$ equals $$\frac{2pqr}{(p+q)(q+r)(r+p)}.$$

The convex quadrilateral $ABCD$ has area $1$ , and $AB$ is produced to $E$ , $BC$ to $F$ , $CD$ to $G$ and $DA$ to $H$ , such that $AB=BE$ , $BC=CF$ , $CD=DG$ and $DA=AH$ . Find the area of the quadrilateral $EFGH$ .

Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$ .