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20-th Korean Mathematical Olympiad 2007: Final Round

This document contains the problems from the final round of the 20th Korean Mathematical Olympiad in 2007. The first day contained 3 problems, the first involving proving an equation about the lengths of line segments related to a circle tangent to sides and circumcenter of a triangle. The second asked for the number of ways to write 0s and 1s on a 4x4 board such that adjacent cells multiply to 0. The third asked for positive integer triples satisfying a quadratic equation. The second day contained 3 more problems, the first asking for pairs of primes such that their product divides an expression involving the primes, the second proving an inequality relating lengths along angle bisectors of a triangle to its perimeter, and the third involving properties of a

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292 views1 page

20-th Korean Mathematical Olympiad 2007: Final Round

This document contains the problems from the final round of the 20th Korean Mathematical Olympiad in 2007. The first day contained 3 problems, the first involving proving an equation about the lengths of line segments related to a circle tangent to sides and circumcenter of a triangle. The second asked for the number of ways to write 0s and 1s on a 4x4 board such that adjacent cells multiply to 0. The third asked for positive integer triples satisfying a quadratic equation. The second day contained 3 more problems, the first asking for pairs of primes such that their product divides an expression involving the primes, the second proving an inequality relating lengths along angle bisectors of a triangle to its perimeter, and the third involving properties of a

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20-th Korean Mathematical Olympiad 2007

Final Round
First Day – March 24, 2007

1. Let O be the circumcenter of an acute triangle ABC and let k be the circle with
center O′ that is tangent to O at A and tangent to side BC at D. Circle k meets AB
and AC again at E and F respectively. The lines OO′ and EO′ meet k again at A′
and G. Lines BO and A′ G intersect at H. Prove that DF 2 = AF · GH.
2. How many ways are there to write either 0 or 1 in each cell of a 4 × 4 board so
that the product of numbers in any two cells sharing an edge is always 0?
3. Find all triples (x, y, z) of positive integers satisfying 1 + 4x + 4y = z2 .

Second Day – March 25, 2007

4. Find all pairs (p, q) of primes such that p p + qq + 1 is divisible by pq.


5. For the vertex A of a triangle ABC, let la be the distance between the projections
on AB and AC of the intersection of the angle bisector of ∠A with side BC. Define
lb and lc analogously. If l is the perimeter of triangle ABC, prove that
la lb lc 1
3
≤ .
l 64

6. Let f : N → N be a function satisfying k f (n) ≤ f (kn) ≤ k f (n) + k − 1 for all


k, n ∈ N.
(a) Prove that f (a) + f (b) ≤ f (a + b) ≤ f (a) + f (b) + 1 for all a, b ∈ N.
(b) If f satisfies f (2007n) ≤ 2007 f (n) + 2005 for every n ∈ N, show that there
exists c ∈ N such that f (2007c) = 2007 f (c).

The IMO Compendium Group,


D. Djukić, V. Janković, I. Matić, N. Petrović
www.imomath.com

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