MathDB

10

Part of 2022 MMATHS

Problems(2)

2022 MMATHS Team Online p10 - sequence of circumcenters, fixed sum ZA^2_i

Source:

10/1/2023
Suppose that A1A2A3A_1A_2A_3 is a triangle with A1A2=16A_1A_2 = 16 and A1A3=A2A3=10A_1A_3 = A_2A_3 = 10. For each integer n≥4n \ge 4, set An to be the circumcenter of triangle An−1An−2An−3A_{n-1}A_{n-2}A_{n-3}. There exists a unique point ZZ lying in the interiors of the circumcircles of triangles AkAk+1Ak+2A_kA_{k+1}A_{k+2} for all integers k≥1k \ge 1. If ZA12+ZA22+ZA32+ZA42ZA^2_1+ ZA^2_2+ ZA^2_3+ ZA^2_4 can be expressed as ab\frac{a}{b} for positive integers a,ba, b with gcd(a,b)=1gcd(a, b) = 1, find a+ba + b.
geometrycircumcircleCircumcenterMMATHS
2022 MMATHS Individual p10 - sum of f(d) when d|M

Source:

10/1/2023
Define a function ff on the positive integers as follows: f(n)=mf(n) = m, where mm is the least positive integer such that nn is a factor of m2m^2. Find the smallest integer MM such that M\sqrt{M} is both a product of prime numbers, of which there are at least 33, and a factor of ∑d∣Mf(d),\sum_{ d|M} f(d), the sum of f(d)f(d) for all positive integers dd that divide MM.
number theoryprime numbersMMATHS