Question bank VMO 1980

Let $\alpha_1,\ldots,\alpha_n\in[0^\circ,180^\circ]$ and suppose $\sum_{k=1}^n(1+\cos\alpha_k)$ is an odd integer. Prove that $\sum_{k=1}^n\sin\alpha_k\ge1$.

Let $T$ be the sum of $k$ positive numbers $m_1,\ldots,m_k$. Prove the inequality printed in the source:
$$\sum_{k=1}^{n}\left(m_k+\frac1{m_k}\right)^2\ge k\left(\frac{k}{T}+\frac{T}{k}\right)^2.$$
Editorial transcription note: the source uses $n$ as the upper limit and reuses $k$ as the summation index, although only $k$ numbers are defined. This inconsistency is retained above; the solution explicitly treats the natural $k$-term interpretation.

Let $P$ lie inside triangle $A_1A_2A_3$. The line $PA_i$ meets the opposite side at $B_i$. Let $C_i$ be the midpoint of $A_iB_i$, and $D_i$ the midpoint of $PB_i$. Compare the areas of triangles $C_1C_2C_3$ and $D_1D_2D_3$.

Prove that for every tetrahedron there exist two planes in space such that the ratio of the areas of its orthogonal projections onto these planes is at least $\sqrt2$.

Can the equation $z^3-2z^2-2z+m=0$ have three distinct rational roots? Why?

Let $n\gt 1$ be an integer and $p\gt 0$ a real number. Find the maximum of $\sum_{k=1}^{n-1}x_kx_{k+1}$ over all nonnegative real $x_1,\ldots,x_n$ satisfying $\sum_{k=1}^n x_k=p$.