MathDB

2

Part of 2005 Germany Team Selection Test

Problems(6)

Train your abilities of making partial scores

Source: 1st German pre-TST 2005, 6 Dec 2004, Problem 2

12/15/2004
Let MM be a set of points in the Cartesian plane, and let (S)\left(S\right) be a set of segments (whose endpoints not necessarily have to belong to MM) such that one can walk from any point of MM to any other point of MM by travelling along segments which are in (S)\left(S\right). Find the smallest total length of the segments of (S)\left(S\right) in the cases
a.) M={(−1,0),(0,0),(1,0),(0,−1),(0,1)}M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}. b.) M={(−1,−1),(−1,0),(−1,1),(0,−1),(0,0),(0,1),(1,−1),(1,0),(1,1)}M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}.
In other words, find the Steiner trees of the set MM in the above two cases.
analytic geometrygeometry unsolvedgeometrySteiner TreesGermanyTSTTeam Selection Test
Japanese ineq

Source: Japan MO; posted by Zhaobin in Pre-Olympiad section

2/5/2005
If aa, bb, cc are positive reals such that a+b+c=1a+b+c=1, prove that
1+a1−a+1+b1−b+1+c1−c≤2(ba+cb+ac).\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).
inequalitiesfunction3-variable inequalityNesbittGermanyJapan
Excircle of APR at AP = excircle of PQB at PQ <==> S is...

Source: 4th German TST 2005, problem 2, by Arend Bayer

5/12/2005
Let ABCABC be a triangle satisfying BC<CABC < CA. Let PP be an arbitrary point on the side ABAB (different from AA and BB), and let the line CPCP meet the circumcircle of triangle ABCABC at a point SS (apart from the point CC). Let the circumcircle of triangle ASPASP meet the line CACA at a point RR (apart from AA), and let the circumcircle of triangle BPSBPS meet the line CBCB at a point QQ (apart from BB). Prove that the excircle of triangle APRAPR at the side APAP is identical with the excircle of triangle PQBPQB at the side PQPQ if and only if the point SS is the midpoint of the arc ABAB on the circumcircle of triangle ABCABC.
geometrycircumcircleangle bisectorgeometry proposed
Polynomial

Source: 17-th Iranian Mathematical Olympiad 1999/2000; 6th German TST 2005, problem 2

10/26/2003
For any positive integer n n, prove that there exists a polynomial P P of degree n n such that all coeffients of this polynomial P P are integers, and such that the numbers P(0) P\left(0\right), P(1) P\left(1\right), P(2) P\left(2\right), ..., P(n) P\left(n\right) are pairwisely distinct powers of 2 2.
algebrapolynomialfunctionIranpower of 2
How often has this already been posted on ml?

Source: 5th German TST 2005, problem 2, not from the shortlist

5/12/2005
Let n be a positive integer, and let a1a_1, a2a_2, ..., ana_n, b1b_1, b2b_2, ..., bnb_n be positive real numbers such that a1≥a2≥...≥ana_1\geq a_2\geq ...\geq a_n and b1≥a1b_1\geq a_1, b1b2≥a1a2b_1b_2\geq a_1a_2, b1b2b3≥a1a2a3b_1b_2b_3\geq a_1a_2a_3, ..., b1b2...bn≥a1a2...anb_1b_2...b_n\geq a_1a_2...a_n. Prove that b1+b2+...+bn≥a1+a2+...+anb_1+b_2+...+b_n\geq a_1+a_2+...+a_n.
inequalities proposedinequalities
Cyclic inequality a_1 (b_1 + a_2) + a_2 (b_2 + a_3) + ... &lt;1

Source: 7th German TST 2005, problem 2; Japan Mathematical Olympiad Finals 2002 , Problem 4

5/30/2005
Let n n be a positive integer such that n≥3 n\geq 3. Let a1 a_1, a2 a_2, ..., an a_n and b1 b_1, b2 b_2, ..., bn b_n be 2n 2n positive real numbers satisfying the equations a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1,   \text{and}   b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1. Prove the inequality a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.
inequalitiesn-variable inequalityGermanyTST