MathDB

3

Part of 2015 Azerbaijan IMO TST

Problems(2)

Azerbaijan IMO TST

Source:

9/15/2015
Let nn and kk be two positive integers such that n>kn>k. Prove that the equation xn+yn=zkx^n+y^n=z^k has a solution in positive integers if and only if the equation xn+yn=zn−kx^n+y^n=z^{n-k} has a solution in positive integers.
number theory
2015 Azerbaijan IMO TST

Source: 2015 Azerbaijan IMO TST

5/29/2015
Consider a trapezoid ABCDABCD with BC∣∣ADBC||AD and BC<ADBC<AD. Let the lines ABAB and CDCD meet at XX. Let ω1\omega_1 be the incircle of the triangle XBCXBC, and let ω2\omega_2 be the excircle of the triangle XADXAD which is tangent to the segment ADAD . Denote by aa and dd the lines tangent to ω1\omega_1 , distinct from ABAB and CDCD, and passing through AA and DD, respectively. Denote by bb and cc the lines tangent to ω2\omega_2 , distinct from ABAB and CDCD, passing through BB and CC respectively. Assume that the lines a,b,ca,b,c and dd are distinct. Prove that they form a parallelogram.
geometrytrapezoid