3

Part of 1988 Irish Math Olympiad

Problems(2)

Triangle Inscribed in Circle

Source: 1988 IrMO Paper 1 Problem 3

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

E

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

C

Finding Number of Bus Routes in a City

Source: 1988 IrMO Paper 2 Problem 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?