MathDB

1

Part of 2008 Germany Team Selection Test

Problems(4)

Show that there is a digit unequal to 2

Source: VAIMO 2008, Problem 1

1/3/2009
Show that there is a digit unequal to 2 in the decimal represesentation of 33 \sqrt [3]{3} between the 1000000 1000000-th und 3141592 3141592-th position after decimal point.
floor functionmodular arithmeticinequalities unsolvedinequalities
Find Q for sequence with non-negative reals elements

Source: AIMO 2008, TST 4, P1, Suggested by Christian Reiher

1/4/2009
Determine Q∈R Q \in \mathbb{R} which is so big that a sequence with non-negative reals elements a1,a2,… a_1 ,a_2, \ldots which satisfies the following two conditions: (i) ∀m,n≥1 \forall m,n \geq 1 we have a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right) (ii) ∀k≥0 \forall k \geq 0 we have a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}} such that for each sequence element we have the inequality an≤Q. a_n \leq Q.
inequalitiesalgebra unsolvedalgebra
Find S_1024 for sequence of sets of natural numbers

Source: AIMO 2008, TST 5, P1, Suggested by Gunther Vogel

1/4/2009
A sequence (Sn),n≥1 (S_n), n \geq 1 of sets of natural numbers with S1={1},S2={2} S_1 = \{1\}, S_2 = \{2\} and Sn+1={k∈N∣k−1∈Sn XOR k∈Sn−1}.{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}. Determine S1024. S_{1024}.
inductionLaTeXalgebra unsolvedalgebra
ABC similar to some permutation of T_aT_bT_c

Source: AIMO 2008, TST 6, P1

1/4/2009
Let ABC ABC be an acute triangle, and Ma M_a, Mb M_b, Mc M_c be the midpoints of the sides a a, b b, c c. The perpendicular bisectors of a a, b b, c c (passing through Ma M_a, Mb M_b, Mc M_c) intersect the boundary of the triangle again in points Ta T_a, Tb T_b, Tc T_c. Show that if the set of points {A,B,C} \left\{A,B,C\right\} can be mapped to the set {Ta,Tb,Tc} \left\{T_a, T_b, T_c\right\} via a similitude transformation, then two feet of the altitudes of triangle ABC ABC divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?
ratiogeometry unsolvedgeometry