MathDB

2

Part of 2007 Germany Team Selection Test

Problems(4)

Dirichlet strikes back

Source: AIMO 2007, TST 2, P2

1/11/2009
Let n,k∈N n, k \in \mathbb{N} with 1≤k≤n2−1. 1 \leq k \leq \frac {n}{2} - 1. There are n n points given on a circle. Arbitrarily we select nk+1 nk + 1 chords among the points on the circle. Prove that of these chords there are at least k+1 k + 1 chords which pairwise do not have a point in common.
inductioncombinatorics unsolvedcombinatorics
Integer quadruples s.t. p^m q^n = (p+q)^2 + 1

Source: VAIMO 2007, P2

1/3/2009
Find all quadruple (m,n,p,q)∈Z4 (m,n,p,q) \in \mathbb{Z}^4 such that p^m q^n \equal{} (p\plus{}q)^2 \plus{} 1.
LaTeXnumber theory unsolvednumber theory
Nested fractional function equation

Source: AIMO 2007, TST 5, P2

1/11/2009
Determine all functions f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{} which satisfy f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1}   \forall x,y > 0
functionalgebrafunctional equationalgebra unsolved
Sum of magnitudes for complex roots 20x^8 + 7ix^7 - 7ix + 20

Source: AIMO 2007, TST 3, P2

1/11/2009
Determine the sum of absolute values for the complex roots of 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.
absolute valuealgebra unsolvedalgebra