MathDB

4

Part of 2019 Czech-Polish-Slovak Junior Match

Problems(2)

similar triangles for juniors

Source: Czech-Polish-Slovak Match Junior 2019, individual p4 CPSJ

1/20/2020
Let kk be a circle with diameter ABAB. A point CC is chosen inside the segment ABAB and a point DD is chosen on kk such that BCDBCD is an acute-angled triangle, with circumcentre denoted by OO. Let EE be the intersection of the circle kk and the line BOBO (different from BB). Show that the triangles BCDBCD and ECAECA are similar.
geometrysimilar triangles
possible values of xy+yz+zx when x^2-yz = y^2-zx = z^2-xy = 2

Source: Czech-Polish-Slovak Match Junior 2019, team p4 CPSJ

1/20/2020
Determine all possible values of the expression xy+yz+zxxy+yz+zx with real numbers x,y,zx, y, z satisfying the conditions x2−yz=y2−zx=z2−xy=2x^2-yz = y^2-zx = z^2-xy = 2.
algebrasystem of equations