International Mathematical Olympiad · 1960

Problems

Problem 1Determine all three-digit numbers $N$ that are divisible by $11$ , such that $N/11$ equals the sum of the squares of the digits of $N$ .Solutions: 2
Problem 2For which real numbers $x$ does the inequality $\dfrac{4x^2}{\left(1-\sqrt{1+2x}\right)^2} < 2x+9$ hold?Solutions: 1
Problem 3In a given right triangle $ABC$ , the hypotenuse $BC$ has length $a$ and is divided into $n$ equal parts, where $n$ is odd. The central part subtends an angle $\alpha$ at $A$ . If $h$ is the perpendicular distance from $A$ to $BC$ , prove that $\tan\alpha=\dfrac{4nh}{a(n^2-1)}$ .Solutions: 1
Problem 4Construct a triangle $ABC$ given the lengths of the altitudes from $A$ and $B$ and the length of the median from $A$ .Solutions: 1
Problem 5The cube $ABCDA'B'C'D'$ has $A$ above $A'$ , $B$ above $B'$ , and so on. Let $X$ be any point of the face diagonal $AC$ and $Y$ any point of $B'D'$ . (a) Find the locus of the midpoint of $XY$ . (b) Find the locus of the point $Z$ on $XY$ such that $ZY=2XZ$ .Solutions: 1
Problem 6A cone of revolution has an inscribed sphere tangent to its base and sloping surface. A cylinder is circumscribed about the sphere, with its base in the base of the cone. If the volumes of the cone and cylinder are $V_1$ and $V_2$ , respectively: (a) prove that $V_1\ne V_2$ ; (b) find the smallest possible value of $V_1/V_2$ , and in this case construct the half-angle of the cone.Solutions: 1
Problem 7In the isosceles trapezoid $ABCD$ with $AB\parallel DC$ and $BC=AD$ , let $AB=a$ , $CD=c$ , and let the perpendicular distance from $A$ to $CD$ be $h$ . Show how to construct all points $X$ on the axis of symmetry such that $\angle BXC=\angle AXD=90^\circ$ . Find the distance of each such $X$ from $AB$ and from $CD$ , and give the condition for such points to exist.Solutions: 1