MathDB

2.4

Part of 2016 Saudi Arabia Pre-TST

Problems(2)

circle tangent to 3 circles wanted, concurrency and collinearity also

Source: 2016 Saudi Arabia Pre-TST Level 4 2.4

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 that external tangent to (O)(O) at A′A' and also tangent to the lines AB,ACAB,AC at Ab,AcA_b,A_c respectively. Define the circles (O2),(O3)(O_2), (O_3) and the points B′,C′,Bc,Ba,Ca,CbB', C', B_c , B_a, C_a, C_b similarly. 1. Denote J as the radical center of (O1),(O2),(O3)(O_1), (O_2), (O_3) and suppose that JA′JA' intersects (O1)(O_1) at the second point X,JB′X, JB' intersects (O2)(O_2) at the second point Y , JC' intersects (O3)(O_3) at the second point ZZ. Prove that the circle (XYZ)(X Y Z) is tangent to (O1),(O2),(O3)(O_1), (O_2), (O_3). 2. Prove that AA′,BB′,CC′AA', BB', CC' are concurrent at the point MM and 33 points I,M,OI,M,O are collinear.
geometrytangent circlescollinearconcurrentconcurrency
products, (a + i) | b(b + 2016), (a + i) \nmid b, (a + i)\mid (b + 2016)

Source: 2016 Saudi Arabia Pre-TST Level 4+ 2.4

9/13/2020
Let nn be a given positive integer. Prove that there are infinitely many pairs of positive integers (a,b)(a, b) with a,b>na, b > n such that ∏i=12015(a+i)∣b(b+2016),∏i=12015(a+i)∤b,∏i=12015(a+i)∣(b+2016)\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016).
number theoryProductdividesdivisible