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\section{Third International Olympiad, 1961}
\subsection{1961/1.}
Solve the system of equations:
\begin{eqnarray*}
x+y+z &=&a \\
x^{2}+y^{2}+z^{2} &=&b^{2} \\
xy &=&z^{2}
\end{eqnarray*}
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.
\subsection{1961/2. }
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?
\subsection{1961/3.}
Solve the equation $\cos ^{n}x-\sin ^{n}x=1,$ where $n$ is a natural number.
\subsection{1961/4.}
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
\[
\frac{P_{1}P}{PQ_{1}},\frac{P_{2}P}{PQ_{2}},\frac{P_{3}P}{PQ_{3}}
\]
at least one is $\leq 2$ and at least one is $\geq 2$.
\subsection{1961/5.}
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}\leq c