International Mathematical Olympiad  – IMO 1961  Problems

Problem 1

Solve the system of equations:\[ x+y+z=a \]\[ x^2+y^2+z^2=b^2 \]\[ xy=z^2 \]where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.

Problem 2

Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove:
\[ a^{2} + b^{2} + c^{2}\geq 4S \sqrt {3}
\]
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 $P_1P_2P_3$ 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\[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \]
at least one is $\leq 2$ and at least one is $\geq 2$

Problem 5

Construct a triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=w$, where $M$ is the midpoint of the segment $BC$ and $w<90$. Prove that a solution exists if and only if\[ b \tan{\dfrac{w}{2}} \leq c <b \]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 $\epsilon$ 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 the 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 $\epsilon$?