# 1963 IMO Problems

Problems of the 5th IMO 1963 Poland.

## Day I

### Problem 1

Find all real roots of the equation

$\sqrt{x^2-p}+2\sqrt{x^2-1}=x$,

where $p$ is a real parameter.

### Problem 2

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are the vertices of right angles with one side passing through $A$, and the other side intersecting the segment $BC$.

### Problem 3

In an $n$-gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation

$a_1\ge a_2\ge \cdots \ge a_n$.

Prove that $a_1=a_2=\cdots = a_n$.

## Day II

### Problem 4

Find all solutions $x_1,x_2,x_3,x_4,x_5$ of the system $\begin{eqnarray*} x_5+x_2&=&yx_1\\ x_1+x_3&=&yx_2\\ x_2+x_4&=&yx_3\\ x_3+x_5&=&yx_4\\ x_4+x_1&=&yx_5,\end{eqnarray*}$ where $y$ is a parameter.

### Problem 5

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$.

### Problem 6

Five students, $A,B,C,D,E$, took part in a contest. One prediction was that the contestants would finish in the order $ABCDE$. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.