MathDB

7

Part of Mathley 2014-15

Problems(2)

(con)cyclic wanted, 2 circles internally tangent to a third one, tangents

Source: Mathley 2014.1 p7

8/19/2020
The circles γ\gamma and δ\delta are internally tangent to the circle ω\omega at AA and BB. From AA, draw two tangent lines ℓ1,ℓ2\ell_1, \ell_2 to δ\delta, . From BB draw two tangent lines t1,t2t_1, t_2 to γ\gamma . Let ℓ1\ell_1 intersect t1t_1 at XX and ℓ2\ell_2 intersect t2t_2 at YY . Prove that the quadrilateral AXBYAX BY is cyclic.
Nguyen Van Linh, High School of Natural Sciences, Hanoi National University
Concycliccyclic quadrilateraltangent circlesTangentssummer programgeometry
(p^{2q}+q^{2p})/(p^3-pq+q^3)= r where p,q,r primes

Source: Mathley 2014.3 p7

8/18/2020
Find all primes p,q,rp,q, r such that p2q+q2pp3−pq+q3=r\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r.
Titu Andreescu, Mathematics Department, College of Texas, USA
number theoryprimesDiophantine equationdiophantine