Question bank VMO 1981

Prove that a triangle is right-angled if and only if the sum of the sines of its three angles equals the sum of their cosines plus $1$.

Let $f(p)=p^{12}-p^{11}+3p^{10}-11p^3-p^2+23p+30$ and $g(p)=p^3+2p+m$. Determine all integer values of $m$ for which there exists a polynomial $h(p)$ such that $f(p)=g(p)h(p)$ for every real $p$.

Let $M,N$ be two points outside a plane $R$. Determine the position of a point $A$ on $R$ for which $AM/AN$ is minimal.

Solve over the real numbers:
$$\begin{cases}x^2+y^2+z^2+t^2=50,\\x^2-y^2+z^2-t^2=-24,\\xz=yt,\\x-y+z+t=0.\end{cases}$$

Let $t_1,\ldots,t_n$ be real numbers with $t_k\in[p,q]$ for $k=1,\ldots,n$, where $0\lt p\le q$ are given. Define
$$A=\frac1n(t_1+\cdots+t_n),\qquad B=\frac1n(t_1^2+\cdots+t_n^2).$$
Prove that $\dfrac{A^2}{B}\ge\dfrac{4pq}{(p+q)^2}$ and determine the necessary and sufficient condition for equality.

Two circles with centers $O_1,O_2$ and unequal radii are externally tangent at $A$. Let $M$ be inside the circle centered at $O_2$, with $M$ not on line $O_1O_2$. Find a line $d$ through $M$ such that the circle $ABC$ is tangent to $O_1O_2$, where $B$ is an intersection of $d$ with the circle centered at $O_1$, and $C$ is an intersection of $d$ with the circle centered at $O_2$.