MathDB

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Part of 2007 All-Russian Olympiad

Problems(4)

at least one of three euqations has a real root

Source: All-Russian 2007

5/4/2007
Given reals numbers aa, bb, cc. Prove that at least one of three equations x2+(a−b)x+(b−c)=0x^{2}+(a-b)x+(b-c)=0, x2+(b−c)x+(c−a)=0x^{2}+(b-c)x+(c-a)=0, x2+(c−a)x+(a−b)=0x^{2}+(c-a)x+(a-b)=0 has a real root. O. Podlipsky
algebrapolynomialquadraticsalgebra proposed
Unitary quadratic trinomials

Source: All-Russian 2007

5/4/2007
Unitary quadratic trinomials f(x) f(x) and g(x) g(x) satisfy the following interesting condition: f(g(x)) \equal{} 0 and g(f(x)) \equal{} 0 do not have real roots. Prove that at least one of equations f(f(x)) \equal{} 0 and g(g(x)) \equal{} 0 does not have real roots too.
S. Berlov
quadraticsalgebra unsolvedalgebra
Faces of a cube $9\times 9\times 9$

Source: All-Russian 2007

5/4/2007
Faces of a cube 9×9×99\times 9\times 9 are partitioned onto unit squares. The surface of a cube is pasted over by 243243 strips 2×12\times 1 without overlapping. Prove that the number of bent strips is odd. A. Poliansky
geometry3D geometrycombinatorics proposedcombinatorics
cos and sin

Source: All-Russian 2007

5/4/2007
Prove that for k>10k>10 Nazar may replace in the following product some one cos⁡\cos by sin⁡\sin so that the new function f1(x)f_{1}(x) would satisfy inequality ∣f1(x)∣≤3⋅2−1−k|f_{1}(x)|\le 3\cdot 2^{-1-k} for all real xx. f(x)=cos⁡xcos⁡2xcos⁡3x…cos⁡2kxf(x) = \cos x \cos 2x \cos 3x \dots \cos 2^{k}x N. Agakhanov
trigonometryfunctioninequalitiesinequalities unsolved