MathDB

4

Part of 2009 Belarus Team Selection Test

Problems(2)

xy is perfect square if yx^2+(y^2-z^2)x+y(y-z)^2=0 in integers x,y,z

Source: 2009 Belarus TST 1.4

11/8/2020
Let x,y,zx,y,z be integer numbers satisfying the equality yx2+(y2−z2)x+y(y−z)2=0yx^2+(y^2-z^2)x+y(y-z)^2=0 a) Prove that number xyxy is a perfect square. b) Prove that there are infinitely many triples (x,y,z)(x,y,z) satisfying the equality.
I.Voronovich
number theoryPerfect SquarediophantineDiophantine equation
min no of graphs such any 2 vertices are connected with a third vertex by edges

Source: 2009 Belarus TST 2.4

11/8/2020
Given a graph with nn (n≥4n\ge 4) vertices . It is known that for any two vertices AA and BB there exists a vertex which is connected by edges both with AA and BB. Find the smallest possible numbers of edges in the graph.
E. Barabanov
combinatoricsgraph theorygraph