MathDB

1

Part of 2022 Dutch IMO TST

Problems(3)

n = d^3 + m^3 , m | n, d is the smallest divisor of n greater than 1

Source: 2022 Dutch IMO TST 1.1

12/3/2022
Determine all positive integers n≥2n \ge 2 which have a positive divisor m∣nm | n satisfying n=d3+m3.n = d^3 + m^3. where dd is the smallest divisor of nn which is greater than 11.
number theoryDiophantine equationdiophantine
diophantine ab =2(1 + cd), exists a triangle with sides a - c, b - d and c + d

Source: 2022 Dutch IMO TST 3.1

12/3/2022
Find all quadruples (a,b,c,d)(a, b, c, d) of non-negative integers such that ab=2(1+cd)ab =2(1 + cd) and there exists a non-degenerate triangle with sides of length a−ca - c, b−db - d, and c+dc + d.
number theoryDiophantine equationdiophantine
F is circumcenter of DMN if B is circumcenter of FMN

Source: 2022 Dutch IMO TST 2.1

12/3/2022
Consider an acute triangle ABCABC with ∣AB∣>∣CA∣>∣BC∣|AB| > |CA| > |BC|. The vertices D,ED, E, and FF are the base points of the altitudes from A,BA, B, and CC, respectively. The line through F parallel to DEDE intersects BCBC in MM. The angular bisector of ∠MFE\angle MF E intersects DEDE in NN. Prove that FF is the circumcentre of △DMN\vartriangle DMN if and only if BB is the circumcentre of △FMN\vartriangle FMN.
geometrycircumcircle