MathDB

3

Part of 2012 Danube Mathematical Competition

Problems(2)

equal angles in a right triangle, danube junior geometry 2012

Source: Danube 2012 Junior P3

9/6/2018
Let ABCABC be a triangle with ∠BAC=90o\angle BAC = 90^o. Angle bisector of the ∠CBA\angle CBA intersects the segment (AB)(AB) at point EE. If there exists D∈(CE)D \in (CE) so that ∠DAC=∠BDE=xo\angle DAC = \angle BDE =x^o , calculate xx.
geometryequal angles
Danube 2012 #3

Source:

12/28/2015
Let pp and q,p<q,q, p < q, be two primes such that 1+p+p2+...+pm1 + p + p^2+...+p^m is a power of qq for some positive integer mm, and 1+q+q2+...+qn1 + q + q^2+...+q^n is a power of pp for some positive integer nn. Show that p=2p = 2 and q=2t−1q = 2^t-1 where tt is prime.
number theory