IMO 1967

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$ , and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only $$a\le\cos A+\sqrt3\sin A.$$

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $\frac{1}{8}.$

Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$ . Let $c_s=s(s+1)$ . Prove that

$$(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)$$

is divisible by the product $c_1c_2\ldots c_n$ .

$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$ , and $AB$ contains $C_0$ ) and similar to $A_1B_1C_1.$

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let $$c_n = \sum^8_{k=1} a^n_k$$ for $n=1,2,3,\ldots$ . Given that among the numbers of the sequence $(c_n)$ , there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?