MathDB

5

Part of 2019 239 Open Mathematical Olympiad

Problems(2)

Tangent circles

Source: 239 2019 S5

7/31/2020
Circle Γ\Gamma touches the circumcircle of triangle ABCABC at point RR, and it touches the sides ABAB and ACAC at points PP and QQ, respectively. Rays PQPQ and BCBC intersect at point XX. The tangent line at point RR to the circle Γ\Gamma meets the segment QXQX at point YY. The line segment AXAX intersects the circumcircle of triangle APQAPQ at point ZZ. Prove that the circumscribed circles of triangles ABCABC and XYZXY Z are tangent.
geometry
n! good sets

Source: 239 2019 J5

7/31/2020
We call an ordered set of distinct natural numbers good if for any two numbers in it, the larger one is divided by the smaller one. Prove that the number (n+1)!–1(n + 1)! – 1 can be represented as x1+2x2+…+nxnx_1 + 2x_2 + \ldots + nx_n, where {x1,x2,…,xn}\{ x_1, x_2, \ldots , x_n \} is a good set, by at least n!n! ways.
combinatoricsinduction