MathDB

3

Part of 2007 IMO Shortlist

Problems(4)

Inequality with x^n + y^n = 1

Source: IMO Shortlist 2007, A3

7/13/2008
Let n n be a positive integer, and let x x and y y be a positive real number such that x^n \plus{} y^n \equal{} 1. Prove that \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
Author: Juhan Aru, Estonia
inequalitiesfunctionalgebracalculusIMO ShortlistHi
Numbers in set can be colored red and blue

Source: IMO Shortlist 2007, C3, AIMO 2008, TST 2, P2

7/13/2008
Find all positive integers n n for which the numbers in the set S \equal{} \{1,2, \ldots,n \} can be colored red and blue, with the following condition being satisfied: The set S×S×S S \times S \times S contains exactly 2007 2007 ordered triples (x,y,z) \left(x, y, z\right) such that: (i) the numbers x x, y y, z z are of the same color, and (ii) the number x \plus{} y \plus{} z is divisible by n n. Author: Gerhard Wöginger, Netherlands
combinatoricsmodular arithmeticcountingIMO Shortlisttriplets
Equal angles (a very old problem)

Source: ISL 2007, G3, VAIMO 2008, P5

7/13/2008
The diagonals of a trapezoid ABCD ABCD intersect at point P P. Point Q Q lies between the parallel lines BC BC and AD AD such that \angle AQD \equal{} \angle CQB, and line CD CD separates points P P and Q Q. Prove that \angle BQP \equal{} \angle DAQ.
Author: Vyacheslav Yasinskiy, Ukraine
geometrytrapezoidhomothetyIMO ShortlistDDIT
Subset Y, |Y| = 2007 and mod(a - b + c - d + e, 47) &lt;&gt; 0

Source: IMO Shortlist 2007, N3, AIMO 2008, TST 2, P3

7/13/2008
Let X X be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset Y Y of X X such that a \minus{} b \plus{} c \minus{} d \plus{} e is not divisible by 47 for any a,b,c,d,e∈Y. a,b,c,d,e \in Y. Author: Gerhard Wöginger, Netherlands
modular arithmeticnumber theoryDivisibilityExtremal combinatoricsAdditive combinatoricsIMO Shortlist