IrMO 1988

A pyramid with a square base, and all its edges of length $2$ , is joined to a regular tetrahedron, whose edges are also of length $2$ , by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid.

Let $x_1, . . . , x_n$ be $n$ integers, and let $p$ be a positive integer, with $p < n$ . Put $$S_1 = x_1 + x_2 + . . . + x_p$$ $$T_1 = x_{p+1} + x_{p+2} + . . . + x_n$$ $$S_2 = x_2 + x_3 + . . . + x_{p+1}$$ $$T_2 = x_{p+2} + x_{p+3} + . . . + x_n + x_1$$ $$...$$ $$S_n=x_n+x_1+...+x_{p-1}$$ $$T_n=x_p+x_{p+1}+...+x_{n-1}$$ For $a = 0, 1, 2, 3$ , and $b = 0, 1, 2, 3$ , let $m(a, b)$ be the number of numbers $i$ , $1 \leq i \leq n$ , such that $S_i$ leaves remainder $a$ on division by $4$ and $T_i$ leaves remainder $b$ on division by $4$ . Show that $m(1, 3)$ and $m(3, 1)$ leave the same remainder when divided by $4$ if, and only if, $m(2, 2)$ is even.

$ABC$ is a triangle inscribed in a circle, and $E$ is the mid-point of the arc subtended by $BC$ on the side remote from $A$ . If through $E$ a diameter $ED$ is drawn, show that the measure of the angle $DEA$ is half the magnitude of the difference of the measures of the angles at $B$ and $C$ .

Problem: A mathematical moron is given the values b; c; A for a triangle ABC and is required to find a. He does this by using the cosine rule $a^2 = b^2 + c^2 - 2bccosA$ and misapplying the low of the logarithm to this to get $log a^2 = log b^2 + log c^2 - log(2bc cos A)$ He proceeds to evaluate the right-hand side correctly, takes the anti-logarithms and gets the correct answer. What can be said about the triangle ABC?

The triangles $ABG$ and $AEF$ are in the same plane. Between them the following conditions hold: (a) $E$ is the mid-point of $AB$ ; (b) points $A,G$ and $F$ are on the same line; (c) there is a point $C$ at which $BG$ and $EF$ intersect; (d) $|CE|=1$ and $|AC|=|AE|=|FG|$ . Show that if $|AG|=x$ , then $|AB|=x^3$ .

A city has a system of bus routes laid out in such a way that (a) there are exactly $11$ bus stops on each route; (b) it is possible to travel between any two bus stops without changing routes; (c) any two bus routes have exactly one bus stop in common. What is the number of bus routes in the city?