Wednesday, January 22, 2014

Raffles InvitationalMathematical Olympiad

Day 1

Instructions to Contestants• This round comprises 5 problems.

• Each problem is worth 7 points for a total of 35 points.

• Indicate your answers clearly on the different answer sheets provided.

• All steps needed to justify your answers are to be clearly shown.

• Calculators, computers and other electronic devices are not allowed in thiscontest.

Page 1 of 2 Time: 4 hours and 30 minutes

Day 1Wednesday, January 22, 2014

1. Let f : N → N, where N denotes the set of positive integers, such that

• for any coprime a, b, f(ab) = f(a)f(b);• for any primes p, q, f(p + q) = f(p) + f(q).

Determine f(2014).

2. In a quadrilateral ABCD with ∠B = ∠D = 90◦, the extensions of AB andDC meet at E; and the extensions of AD and BC meet at F . A line through Bparallel to CD intersects the circumcircle ω of the triangle ABF at G distinct fromB; the line EG intersects ω at P distinct from G; and the line AP intersects CEat M . Prove that M is the midpoint of CE.

3. Determine the smallest odd integer N such that N2 is the sum of an oddnumber (greater than 1) of squares of successive integers.

4. Let k be a fixed positive integer. For all positive integers n, prove thatthere exist positive integers a1, a2, . . . , an such that n and an are coprime, and

n∑j=1

jk

aj

= 1.

5. There are 2015 points distributed regularly on the circumference of a circle.Some of the points are coloured red and the rest are coloured blue. Considerisosceles triangles formed using triples of these points. Such a triangle is calledmonochromatic if the vertices are of the same colour. What is the minimum numberof monochromatic isosceles triangles over all possible colourings?

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Page 2 of 2 Time: 4 hours and 30 minutesEach problem is worth 7 points