MathDB

1

Part of 2010 Germany Team Selection Test

Problems(5)

Prove that |RP| = |PD|

Source: VAIMO 1, German Pre-TST 2010

7/16/2011
The quadrilateral ABCDABCD is a rhombus with acute angle at A.A. Points MM and NN are on segments AC‾\overline{AC} and BC‾\overline{BC} such that ∣DM∣=∣MN∣.|DM| = |MN|. Let PP be the intersection of ACAC and DNDN and let RR be the intersection of ABAB and DM.DM. Prove that ∣RP∣=∣PD∣.|RP| = |PD|.
geometryrhombusgeometry unsolved
Consider 2009 cards which are lying in sequence on a table

Source: VAIMO 4, German Pre-TST 2010

7/16/2011
Consider 2009 cards which are lying in sequence on a table. Initially, all cards have their top face white and bottom face black. The cards are enumerated from 1 to 2009. Two players, Amir and Ercole, make alternating moves, with Amir starting. Each move consists of a player choosing a card with the number kk such that k<1969k < 1969 whose top face is white, and then this player turns all cards at positions k,k+1,…,k+40.k,k+1,\ldots,k+40. The last player who can make a legal move wins.
(a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player?
Also compare shortlist 2009, combinatorics problem C1.
combinatorics unsolvedcombinatoricsgame strategy
Let k be the smallest digest then a_{n+1} = a_n + 2^k

Source: AIMO 3/1, German Pre-TST 2010

7/16/2011
A sequence (an)\left(a_n\right) with a1=1a_1 = 1 satisfies the following recursion: In the decimal expansion of ana_n (without trailing zeros) let kk be the smallest digest then an+1=an+2k.a_{n+1} = a_n + 2^k. How many digits does a9⋅102010a_{9 \cdot 10^{2010}} have in the decimal expansion?
algebra unsolvedalgebra
AIMO 5/1, German TST 2010

Source: AIMO 5/1, German TST 2010

7/16/2011
In the plane we have points P,Q,A,B,CP,Q,A,B,C such triangles APQ,QBPAPQ,QBP and PQCPQC are similar accordantly (same direction). Then let A′A' (B′,C′B',C' respectively) be the intersection of lines BPBP and CQCQ (CPCP and AQ;AQ; APAP and BQ,BQ, respectively.) Show that the points A,B,C,A′,B′,C′A,B,C,A',B',C' lie on a circle.
geometry unsolvedgeometry
Non-real roots of polynomial X^{n+1}-X^2+aX+1

Source: AIMO 6/1, German TST 2010

7/16/2011
Let a∈R.a \in \mathbb{R}. Show that for n≥2n \geq 2 every non-real root zz of polynomial Xn+1−X2+aX+1X^{n+1}-X^2+aX+1 satisfies the condition ∣z∣>1nn.|z| > \frac{1}{\sqrt[n]{n}}.
algebrapolynomialalgebra unsolved