MathDB

1

Part of 2006 IberoAmerican

Problems(2)

Isosceles triangle

Source: 21-st Iberoamerican Mathematical Olympiad

4/16/2007
In a scalene triangle ABCABC with ∠A=90∘,\angle A = 90^\circ, the tangent line at AA to its circumcircle meets line BCBC at MM and the incircle touches ACAC at SS and ABAB at R.R. The lines RSRS and BCBC intersect at N,N, while the lines AMAM and SRSR intersect at U.U. Prove that the triangle UMNUMN is isosceles.
geometrycircumcircleincentergeometry proposed
Problem 4

Source: 21-st Iberoamerican Mathematical Olympiad

5/6/2007
Find all pairs (a, b)(a,\, b) of positive integers such that 2a−12a-1 and 2b+12b+1 are coprime and a+ba+b divides 4ab+1.4ab+1.
number theory unsolvednumber theory