MathDB

1

Part of 2000 Turkey Team Selection Test

Problems(2)

Prove that number of ordered pairs (x,y) is divisible by 3

Source: Turkey TST 2000 P1

3/12/2011
(a)(a) Prove that for every positive integer nn, the number of ordered pairs (x,y)(x, y) of integers satisfying x2−xy+y2=nx^2-xy+y^2 = n is divisible by 3.3. (b)(b) Find all ordered pairs of integers satisfying x2−xy+y2=727.x^2-xy+y^2=727.
algebrapolynomialVietanumber theory unsolvednumber theory
Triangular prism of infinite length cut by a plane

Source: Turkey TST 2000 P4

3/12/2011
Show that any triangular prism of infinite length can be cut by a plane such that the resulting intersection is an equilateral triangle.
geometry3D geometryprismgeometry unsolved