MathDB

6

Part of 2024 Kazakhstan National Olympiad

Problems(2)

Sequence of positive integers, are they all equal 2?

Source: Kazakhstan National Olympiad 2024 (9 grade -- P6), (10-11 grade -- P5)

3/21/2024
An integer m≥3m\ge 3 and an infinite sequence of positive integers (an)n≥1(a_n)_{n\ge 1} satisfies the equation an+2=2an+1m−1+anm−1m−an+1.a_{n+2} = 2\sqrt[m]{a_{n+1}^{m-1} + a_n^{m-1}} - a_{n+1}. for all n≥1n\ge 1. Prove that a1<2ma_1 < 2^m.
algebra
Perpendiculars to the harmonic lines are also harmonic lines

Source: Kazakhstan National Olympiad 2024 (10-11 grade), P6

3/21/2024
The circle ω\omega with center at point II inscribed in an triangle ABCABC (AB≠ACAB\neq AC) touches the sides BCBC, CACA and ABAB at points DD, EE and FF, respectively. The circumcircles of triangles ABCABC and AEFAEF intersect secondary at point K.K. The lines EFEF and AKAK intersect at point XX and intersects the line BCBC at points YY and ZZ, respectively. The tangent lines to ω\omega, other than BCBC, passing through points YY and ZZ touch ω\omega at points PP and QQ, respectively. Let the lines APAP and KQKQ intersect at the point RR. Prove that if MM is a midpoint of segment YZ,YZ, then IR⊥XMIR\perp XM.
geometry