IMO 1961

Problem 1. Solve the system of equations:

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.

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

Problem 3. Solve the equation ${\cos }^{n}x-{\sin }^{n}x=1$, where $n$ is a natural number.

Problem 4. Consider triangle $P_1{P}_2{P}_3$ and a point $P$ within the triangle. Lines $P_{1}P,P_{2}P,P_{3}P$ intersect the opposite sides in points $Q_1,Q_2,Q_3$ respectively. Prove that, of the numbers

$\frac{P_{1}P}{P{Q}_1},\frac{P_{2}P}{P{Q}_2},\frac{P_{3}P}{P{Q}_3}$

at least one is $\le 2$ and at least one is $\ge 2$.

Problem 5. Construct triangle $ABC$ if $AC=b,AB=c$ and $\angle AMB=\omega$, where $M$ is the midpoint of segment $BC$ and $\omega <90^{\circ }$. Prove that a

solution exists if and only if

$b\tan \frac{\omega }{2}\le c<b.$

In what case does the equality hold?

Problem 6. Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane a take three arbitrary points $A^{\prime },B^{\prime },C^{\prime }$. Let $L,M,N$ be the midpoints of segments $A{A}^{\prime },B{B}^{\prime },C{C}^{\prime }$; let $G$ be the centroid of triangle $LMN$. (We will not consider positions of the points $A^{\prime },B^{\prime },C^{\prime }$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A^{\prime },B^{\prime },C^{\prime }$ range independently over the plane $\epsilon$?