MathDB

1

Part of 2008 Bulgaria National Olympiad

Problems(2)

Circles

Source: Bulgarian MO 2008, Day 1, Problem 1

5/17/2008
Let ABC ABC be an acute triangle and CL CL be the angle bisector of ∠ACB \angle ACB. The point P P lies on the segment CLCL such that \angle APB\equal{}\pi\minus{}\frac{_1}{^2}\angle ACB. Let k1 k_1 and k2 k_2 be the circumcircles of the triangles APC APC and BPC BPC. BP\cap k_1\equal{}Q, AP\cap k_2\equal{}R. The tangents to k1 k_1 at Q Q and k2 k_2 at B B intersect at S S and the tangents to k1 k_1 at A A and k2 k_2 at R R intersect at T T. Prove that AS\equal{}BT.
geometrycircumcircletrigonometryangle bisectorgeometry proposed
k-th power of natural number

Source: Bulgarian Mathematical Olympiad 2008

2/22/2009
Find the smallest natural number k k for which there exists natural numbers m m and n n such that 1324 \plus{} 279m \plus{} 5^n is k k-th power of some natural number.
quadraticsnumber theory unsolvednumber theory