Question bank IMO 1963

(CZS) Determine all real solutions of the equation $\sqrt{x^2-p}+2\sqrt{x^2-1}=x$, where $p$ is a real number.

(USS) Find the locus of points in space that are vertices of right angles of which one ray passes through a given point and the other intersects a given segment.

(HUN) Prove that if all the angles of a convex $n$-gon are equal and the lengths of consecutive edges $a_1,\ldots,a_n$ satisfy $a_1\ge a_2\ge\cdots\ge a_n$, then $a_1=a_2=\cdots=a_n$.

(USS) Find all solutions $x_1,\ldots,x_5$ to the system of equations
$$\begin{cases}x_5+x_2=yx_1,\\x_1+x_3=yx_2,\\x_2+x_4=yx_3,\\x_3+x_5=yx_4,\\x_4+x_1=yx_5,\end{cases}$$
where $y$ is a real parameter.

(GDR) Prove that $\cos\frac\pi7-\cos\frac{2\pi}7+\cos\frac{3\pi}7=\frac12$.

(HUN) Five students $A$, $B$, $C$, $D$, and $E$ have taken part in a certain competition. Before the competition, two persons $X$ and $Y$ tried to guess the rankings. $X$ thought that the ranking would be $A,B,C,D,E$; and $Y$ thought that the ranking would be $D,A,E,C,B$. At the end, it was revealed that $X$ didn't guess correctly any rankings of the participants, and moreover, didn't guess any of the orderings of pairs of consecutive participants. On the other hand, $Y$ guessed the correct rankings of two participants and the correct ordering of two pairs of consecutive participants. Determine the rankings of the competition.