MathDB

2

Part of 2009 Tuymaada Olympiad

Problems(3)

P(x) is a quadratic trinomial

Source: Tuymaada 2009, Junior League, First Day, Problem 2

7/19/2009
P(x) P(x) is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence P(1) P(1), P(2) P(2), P(3) P(3), …? \dots? Proposed by A. Golovanov
quadraticsalgebra unsolvedalgebra
Perpendicular to bases drawn through P meets line BQ at K

Source: Tuymaada 2009, Junior League, Second Day, Problem 2

7/19/2009
M M is the midpoint of base BC BC in a trapezoid ABCD ABCD. A point P P is chosen on the base AD AD. The line PM PM meets the line CD CD at a point Q Q such that C C lies between Q Q and D D. The perpendicular to the bases drawn through P P meets the line BQ BQ at K K. Prove that \angle QBC \equal{} \angle KDA. Proposed by S. Berlov
geometrytrapezoidgeometric transformationhomothetyangle bisectorgeometry unsolved
A necklace consists of 100 blue and several red beads

Source: Tuymaada 2009, Senior League, First Day, Problem 2

7/19/2009
A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace? Proposed by A. Golovanov
ceiling functionratiocombinatorics unsolvedcombinatorics