CMO 1970

Find all number triples $(x,y,z)$ such that when any of these numbers is added to the product of the other two, the result is 2.

Given a triangle $ABC$ with angle $A$ obtuse and with altitudes of length $h$ and $k$ as shown in the diagram, prove that $a+h\ge b+k$ . Find under what conditions $a+h=b+k$ .

$A$ $B$ $C$ $D$ $E$ $h$ $k$

A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either 1 pound or 2 pounds, and there is at least one of each weight. Prove that there are two balls having different weights and different colours.

a) Find all positive integers with initial digit 6 such that the integer formed by deleting 6 is $1/25$ of the original integer. b) Show that there is no integer such that the deletion of the first digit produces a result that is $1/35$ of the original integer.

A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths $a$ , $b$ , $c$ and $d$ of the sides of the quadrilateral satisfy the inequalities $$2\le a^2+b^2+c^2+d^2\le 4.$$