International Mathematical Olympiad · 1963

Problems

Problem 1For which real values of $p$ does the equation $\sqrt{x^2-p}+2\sqrt{x^2-1}=x$ have real roots? What are the roots?Solutions: 1
Problem 2Given a point $A$ and a segment $BC$ , determine the locus of all points $P$ in space for which $\angle APX=90^\circ$ for some point $X$ on the segment $BC$ .Solutions: 1
Problem 3In an $n$ -gon all interior angles are equal, and the lengths of consecutive sides satisfy $a_1\ge a_2\ge\cdots\ge a_n$ . Prove that $a_1=a_2=\cdots=a_n$ .Solutions: 1
Problem 4Find all real solutions $x_1,\ldots,x_5$ of $x_i+x_{i+2}=yx_{i+1}$ for $i=1,\ldots,5$ , where subscripts are reduced modulo $5$ .Solutions: 1
Problem 5Prove that $\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7}=\frac12$ .Solutions: 1
Problem 6Five students, $A,B,C,D,E$ , took part in a contest. One prediction was the order $ABCDE$ . No contestant finished in the predicted position, and no two contestants predicted consecutively actually did so. A second prediction was $DAECB$ . Exactly two contestants finished in their predicted positions, and two disjoint pairs predicted consecutively actually did so. Determine the actual order.Solutions: 1