$ABCD$ is a parallelogram; $AB=a$, $AD=1$, $\alpha$ is the size of $\angle DAB$, and the three angles of the triangle $ABD$ are acute. Prove that the four circles $K_A,K_B,K_C,K_D$, each of radius $1$, whose centers are the vertices $A,B,C,D$, cover the parallelogram if and only if $a\leq\cos\alpha+\sqrt3\sin\alpha$.
Let $k,m,n$ be positive integers such that $m+k+1$ is a prime number greater than $n+1$. Write $c_s$ for $s(s+1)$. Prove that the product $(c_{m+1}-c_k)(c_{m+2}-c_k)\cdots(c_{m+n}-c_k)$ is divisible by the product $c_1c_2\cdots c_n$.
The triangles $A_0B_0C_0$ and $A'B'C'$ have all their angles acute. Describe how to construct one of the triangles $ABC$, similar to $A'B'C'$ and circumscribing $A_0B_0C_0$ (so that $A,B,C$ correspond to $A',B',C'$, and $AB$ passes through $C_0$, $BC$ through $A_0$, and $CA$ through $B_0$). Among these triangles $ABC$ describe, and prove, how to construct the triangle with the maximum area.
Consider the sequence $(c_n)$:
$$\begin{aligned}c_1&=a_1+a_2+\cdots+a_8,\\c_2&=a_1^2+a_2^2+\cdots+a_8^2,\\&\cdots\\c_n&=a_1^n+a_2^n+\cdots+a_8^n,\\&\cdots\end{aligned}$$
where $a_1,a_2,\ldots,a_8$ are real numbers, not all equal to zero. 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 competition lasting $n$ days there are $m$ medals to be won. On the first day, one medal and $1/7$ of the remaining $m-1$ medals are won. On the second day, $2$ medals and $1/7$ of the remainder are won. And so on. On the $n$th day exactly $n$ medals are won. How many days did the competition last and what was the total number of medals?