MathDB

2

Part of 2019 Tournament Of Towns

Problems(6)

Combinatorics(pretty hard,pls help)

Source: Tournament of towns

3/13/2019
Consider 2n+1 coins lying in a circle. At the beginning, all the coins are heads up. Moving clockwise, 2n+1 flips are performed: one coin is flipped, the next coin is skipped, the next coin is flipped, the next two coins are skipped, the next coin is flipped,the next three coins are skipped and so on, until finally 2n coins are skipped and the next coin is flipped.Prove that at the end of this procedure,exactly one coin is heads down.
combinatoricsInvariantscoinscombinatorics unsolvedCircular array
a^{n+1} + b^{n+1} is divisible by a^n + b^n for infi nitely n

Source: Tournament of Towns, Senior O-Level , Spring 2019 p2

5/11/2020
Consider two positive integers aa and bb such that an+1+bn+1a^{n+1} + b^{n+1} is divisible by an+bna^n + b^n for infi nitely many positive integers nn. Is it necessarily true that a=ba = b?
(Boris Frenkin)
number theorySum of powersDivisibility
position of orthocenter of OXY does not depend on the choice of P on circle

Source: Tournament of Towns, Junior O-Level , Fall 2019 p2

4/19/2020
Let ω\omega be a circle with the center OO and AA and CC be two different points on ω\omega. For any third point PP of the circle let XX and YY be the midpoints of the segments APAP and CPCP. Finally, let HH be the orthocenter (the point of intersection of the altitudes) of the triangle OXYOXY . Prove that the position of the point H does not depend on the choice of PP.
(Artemiy Sokolov)
geometryFixed pointfixedorthocentercircle
S/(AB + AC)> S_1/(A_1B_1 + A_1C_1), area triangle inequality

Source: Tournament of Towns, Junior A-Level , Fall 2019 p2

4/19/2020
Two acute triangles ABCABC and A1B1C1A_1B_1C_1 are such that B1B_1 and C1C_1 lie on BCBC, and A1A_1 lies inside the triangle ABCABC. Let SS and S1S_1 be the areas of those triangles respectively. Prove that SAB+AC>S1A1B1+A1C1\frac{S}{AB + AC}> \frac{S_1}{A_1B_1 + A_1C_1}
(Nairi Sedrakyan, Ilya Bogdanov)
geometrygeometric inequalityarea of a triangletriangle inequalityinequalities
Easy geom to solve

Source: International Mathematical Tournament of towns

12/2/2019
Given a convex pentagon ABCDEABCDE such that AEAE is parallel to CDCD and AB=BCAB=BC. Angle bisectors of angles AA and CC intersect at KK. Prove that BKBK and AEAE are parallel.
geometryangle bisector
P is orthocenter of acute ABC, 3 equal chords given

Source: Tournament of Towns, Senior A-Level , Fall 2019 p2

4/18/2020
Let ABCABC be an acute triangle. Suppose the points A′,B′,C′A',B',C' lie on its sides BC,AC,ABBC,AC,AB respectively and the segments AA′,BB′,CC′AA',BB',CC' intersect in a common point PP inside the triangle. For each of those segments let us consider the circle such that the segment is its diameter, and the chord of this circle that contains the point PP and is perpendicular to this diameter. All three these chords occurred to have the same length. Prove that PP is the orthocenter of the triangle ABCABC.
(Grigory Galperin)
orthocentergeometryChordsequal segments