MathDB

4

Part of 2014 Dutch IMO TST

Problems(2)

Prove that <EBF=<ECF

Source: Dutch IMO TST I Problem 4

7/17/2014
Let △ABC\triangle ABC be a triangle with ∣AC∣=2∣AB∣|AC|=2|AB| and let OO be its circumcenter. Let DD be the intersection of the bisector of ∠A\angle A with BCBC. Let EE be the orthogonal projection of OO to ADAD and let F≠DF\ne D be the point on ADAD satisfying ∣CD∣=∣CF∣|CD|=|CF|. Prove that ∠EBF=∠ECF\angle EBF=\angle ECF.
geometrygeometric transformationreflectionangle bisectorgeometry proposed
Pairs of primes and perfect squares

Source: Dutch IMO TST II Problem 4

7/17/2014
Determine all pairs (p,q)(p, q) of primes for which pq+1+qp+1p^{q+1}+q^{p+1} is a perfect square.
modular arithmeticnumber theory unsolvednumber theory