MathDB

2.3

Part of 2016 Saudi Arabia Pre-TST

Problems(2)

u^n - v^n are integers for infinite many n

Source: 2016 Saudi Arabia Pre-TST Level 4 2.3

9/13/2020
Let uu and vv be positive rational numbers with u≠vu \ne v. Assume that there are infinitely many positive integers nn with the property that un−vnu^n - v^n are integers. Prove that uu and vv are integers.
number theorypositive integers
circle (I) is inscribed in hexagon with 6 vertices A_b,A_c , B_c , B_a, C_a, C_b

Source: 2016 Saudi Arabia Pre-TST Level 4+ 2.3

9/13/2020
Let ABCABC be a non isosceles triangle with circumcircle (O)(O) and incircle (I)(I). Denote (O1)(O_1) as the circle internal tangent to (O)(O) at A1A_1 and also tangent to segments AB,ACAB,AC at Ab,AcA_b,A_c respectively. Define the circles (O2),(O3)(O_2), (O_3) and the points B1,C1,Bc,Ba,Ca,CbB_1, C_1, B_c , B_a, C_a, C_b similarly. 1. Prove that AA1,BB1,CC1AA_1, BB_1, CC_1 are concurrent at the point MM and 33 points I,M,OI,M,O are collinear. 2. Prove that the circle (I)(I) is inscribed in the hexagon with 66 vertices Ab,Ac,Bc,Ba,Ca,CbA_b,A_c , B_c , B_a, C_a, C_b.
geometryinscribedconcurrencycollinear