International Competitions 1961

Solve the system of equations: $$\begin{aligned} x + y + z &= a \\ x^2 + y^2 + z^2 &= b^2 \\ xy &= z^2 \end{aligned}$$ where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x, y, z$ (the solutions of the system) are distinct positive numbers.

Let $a, b, c$ be the sides of a triangle, and $T$ its area. Prove: $a^2 + b^2 + c^2 \geq 4\sqrt{3}T$. In what case does equality hold?

Solve the equation $\cos^n x - \sin^n x = 1$, where $n$ is a natural number.

Consider triangle $P_1P_2P_3$ and a point $P$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers $$\frac{P_1P}{PQ_1}, \frac{P_2P}{PQ_2}, \frac{P_3P}{PQ_3}$$ at least one is $\leq 2$ and at least one is $\geq 2$.

Construct triangle $ABC$ if $AC = b$, $AB = c$ and $\measuredangle AMB = \omega$, where $M$ is the midpoint of segment $BC$ and $\omega < 90°$. Prove that a solution exists if and only if $$b \tan \frac{\omega}{2} \leq c < b.$$ In what case does the equality hold?

Consider a plane $\varepsilon$ and three non-collinear points $A, B, C$ on the same side of $\varepsilon$; suppose the plane determined by these three points is not parallel to $\varepsilon$. In plane $\varepsilon$ take three arbitrary points $A', B', C'$. Let $L, M, N$ be the midpoints of segments $AA', BB', CC'$; let $G$ be the centroid of triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L, M, N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\varepsilon$?