MathDB

P2

Part of Russian TST 2021

Problems(2)

Excircle geometry

Source: Russian TST 2021, Day 7 P2

3/21/2023
The AA{}-excircle ωA\omega_A{} of the triangle ABCABC touches the side of the BCBC at point A1A_1 and the extensions of the sides ABAB and ACAC are at points C1C_1 and B1B_1 respectively. Let PP{} be the middle of the segment B1C1B_1C_1. The line A1PA_1P intersects ωA\omega_A{} a second time at point XX{}. The tangents to the circumcircle of the triangle ABCABC at point AA{} and to ωA\omega_A{} at point XX{} intersect at point RR. Prove that RP=RXRP = RX.
Number of lattice points inside triangle

Source: Russian TST 2021, Day 8 P2

3/21/2023
The natural numbers tt{} and qq{} are given. For an integer ss{}, we denote by f(s)f(s) the number of lattice points lying in the triangle with vertices (0;−t/q),(0;t/q)(0;-t/q), (0; t/q) and (t;ts/q)(t; ts/q). Suppose that qq{} divides rs−1rs-1{}. Prove that f(r)=f(s)f(r) = f(s).
number theorylattice points