MathDB

8

Part of 2022 Tuymaada Olympiad

Problems(2)

Tuymaada 2022

Source: Tuymaada 2022 Junior P-8

9/12/2022
Eight poles stand along the road. A sparrow starts at the first pole and once in a minute flies to a neighboring pole. Let a(n)a(n) be the number of ways to reach the last pole in 2n+12n + 1 flights (we assume a(m)=0a(m) = 0 for m<3m < 3). Prove that for all n≥4n \ge 4 a(n)−7a(n−1)+15a(n−2)−10a(n−3)+a(n−4)=0.a(n) - 7a(n-1)+ 15a(n-2) - 10a(n-3) +a(n-4)=0. (T. Amdeberhan, F. Petrov )
number theorycombinatoricscounting
Tuymaada 2022

Source: Tuymaada 2022 Senior P-8

9/12/2022
In an acute triangle △ABC\triangle ABC the points Cm,Am,BmC_m, A_m, B_m are the midpoints of AB,BC,CAAB, BC, CA respectively. Inside the triangle △ABC\triangle ABC a point PP is chosen so that ∠PCB=∠BmBC\angle PCB = \angle B_mBC and ∠PAB=∠ABBm.\angle PAB = \angle ABB_m. A line passing through PP and perpendicular to ACAC meets the median BBmBB_m at E.E. Prove that EE lies on the circumcircle of the triangle △AmBmCm.\triangle A_mB_mC_m. (K. Ivanov )
geometrymediansines