MathDB

3

Part of 2008 Germany Team Selection Test

Problems(4)

k^3 - 36k^2 + 51k - 97 is a multiple of 3^2008.

Source: VAIMO 2008, Problem 3

1/3/2009
Prove there is an integer k k for which k^3 \minus{} 36 k^2 \plus{} 51 k \minus{} 97 is a multiple of 32008. 3^{2008.}
number theory unsolvednumber theory
2yf(x+y) + (x-y)(f(x) + f(y)) >= 0

Source: AIMO 2008, TST 5, P3, Suggested by Gunther Vogel

1/4/2009
Find all real polynomials f f with x,y∈R x,y \in \mathbb{R} such that 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0.
algebrapolynomialinequalitiesalgebra unsolved
Geometric location: |PA| * |PC| = |PB| * |PD|

Source: AIMO 2008, TST 4, P3, Suggested by Christian Reiher

1/4/2009
Let ABCD ABCD be an isosceles trapezium. Determine the geometric location of all points P P such that |PA| \cdot |PC| \equal{} |PB| \cdot |PD|.
symmetrygeometryperpendicular bisectorgeometry unsolved
f(x - f(y)) = f(x + y) + f(y)

Source: AIMO 2008, TST 6, P3

1/4/2009
Determine all functions f:R↦R f: \mathbb{R} \mapsto \mathbb{R} with x,y∈R x,y \in \mathbb{R} such that f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)
functionalgebra unsolvedalgebra