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Balkan 2003

This document contains 4 math problems: 1) Whether there exists a set of positive integers whose subsets of 2003 elements never sum to a multiple of 2003. 2) Showing that three points related to a triangle and its circumcircle are collinear. 3) Finding functions satisfying three properties related to addition and evaluation. 4) Relating the signs of distances between points along a diagonal of a divided rectangle to the length of the diagonal.

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Mauricio Mallma
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229 views1 page

Balkan 2003

This document contains 4 math problems: 1) Whether there exists a set of positive integers whose subsets of 2003 elements never sum to a multiple of 2003. 2) Showing that three points related to a triangle and its circumcircle are collinear. 3) Finding functions satisfying three properties related to addition and evaluation. 4) Relating the signs of distances between points along a diagonal of a divided rectangle to the length of the diagonal.

Uploaded by

Mauricio Mallma
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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20th Balkan 2003

1. Is there a set of 4004 positive integers such that the sum of each subset of
2003 elements is not divisible by 2003?

2. ABC is a triangle. The tangent to the circumcircle at A meets the line BC at


D. The perpendicular to BC at B meets the perpendicular bisector of AB at E,
and the perpendicular to BC at C meets the perpendicular bisector of AC at F.
Show that D, E, F are collinear.

3. Find all real-valued functions f(x) on the rationals such that:


(1) f(x + y) - y f(x) - x f(y) = f(x) f(y) - x - y + xy, for all x, y
(2) f(x) = 2 f(x+1) + 2 + x, for all x and
(3) f(1) + 1 > 0.

4. A rectangle ABCD has side lengths AB = m, AD = n, with m and n


relatively prime and both odd. It is divided into unit squares and the diagonal
AC intersects the sides of the unit squares at the points A1 = A, A2, A3, ... , AN
= C. Show that A1A2 - A2A3 + A3A4 - ... ± AN-1AN = AC/mn.

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