Time Allowed: Three and a half hours

The use of rulers and compasses is allowed, but calculators and protractors are forbidden

1 Find all two-digit positive integers N for which the sum of the digits of 10^N-N is divisible by 170.

2 Circle S lies inside circle T and touches it at A. From a point P (distinct from A) on T, chords PQ and PR of T are drawn touching S at X and Y respectively. Show that angle QAR is twice the size of angle XAY.

 

3 A tetromino is a figure made up of four unit squares connected by common edges.

(i) If we do not distinguish between the possible rotations of a tetromino within its plane, prove that there are seven distinct tetrominoes.

(ii) Prove or disprove the statement: It is possible to pack all seven distinct tetrominoes into a 4x7 rectangle without overlapping.

 

4 The sequence (a_n) is defined by a_n = n + {n} where n is a positive integer and {x} denotes the nearest integer to x, where halves are rounded up if necessary. Determine the smallest integer k for which the terms a_k, a_k+1, ..., a_k+2000 form a sequence of 2001 consecutive integers.

 

5 A triangle has sides of length a, b, c and its circumcircle has radius R. Prove that the triangle is right-angled if and only if a²+b²+c²=8R².