MathDB

6

Part of 2018 China Northern MO

Problems(2)

Equal Angles

Source: 2018 China North Mathematical Olympiad Grade 10 Test 2 P2

7/26/2018
Let HH be the orthocenter of triangle ABCABC. Let DD and EE be points on ABAB and ACAC such that DEDE is parallel to CHCH. If the circumcircle of triangle BDHBDH passes through MM, the midpoint of DEDE, then prove that ∠ABM=∠ACM\angle ABM=\angle ACM
geometryChinacircumcircle
na_{n+1}=2(n+1)a_n-n-2

Source: 2018 China North Mathematical Olympiad Grade 11 Test 2 P2

6/24/2019
For a1=3a_1 = 3, define the sequence a1,a2,a3,…a_1, a_2, a_3, \ldots for n≥1n \geq 1 as nan+1=2(n+1)an−n−2.na_{n+1}=2(n+1)a_n-n-2. Prove that for any odd prime pp, there exist positive integer m,m, such that p∣amp|a_m and p∣am+1.p|a_{m+1}.
number theorySequence