MathDB

6

Part of 2013 JBMO Shortlist

Problems(2)

Concurrent perpendiculars in a rectangle

Source: JBMO Shortlist 2013, G6

6/11/2017
Let PP and QQ be the midpoints of the sides BCBC and CDCD, respectively in a rectangle ABCDABCD. Let KK and MM be the intersections of the line PDPD with the lines QBQB and QAQA, respectively, and let NN be the intersection of the lines PAPA and QBQB. Let XX, YY and ZZ be the midpoints of the segments ANAN, KNKN and AMAM, respectively. Let ℓ1\ell_1 be the line passing through XX and perpendicular to MKMK, ℓ2\ell_2 be the line passing through YY and perpendicular to AMAM and ℓ3\ell_3 the line passing through ZZ and perpendicular to KNKN. Prove that the lines ℓ1\ell_1, ℓ2\ell_2 and ℓ3\ell_3 are concurrent.
geometryrectangleperpendicular linesconcurrencymidpoints
solve in integers: x^2-y^2=z and 3xy+(x-y)z=z^2

Source: JBMO Shortlist 2013 NT6

4/24/2019
Solve in integers the system of equations: x2−y2=zx^2-y^2=z 3xy+(x−y)z=z23xy+(x-y)z=z^2
number theoryDiophantine EquationsIntegers