3

Part of 2006 Junior Balkan Team Selection Tests - Moldova

Problems(2)

Problem 3 (2nd jbmo tst)

Source: 2nd TST for JBMO, Moldova

Determine all 2nd degree polynomials with integer coefficients of the form

P(X)=aX^{2}+bX+c

, that satisfy:

P(a)=b

P(b)=a

a\neq b

algebrapolynomialalgebra proposed

Poblem 3 (1st jbmo tst), moldova

Source: 1st JBMO TST, Moldova

The convex polygon

A_{1}A_{2}\ldots A_{2006}

has opposite sides parallel

(A_{1}A_{2}||A_{1004}A_{1005}, \ldots)

. Prove that the diagonals

A_{1}A_{1004}, A_{2}A_{1005}, \ldots A_{1003}A_{2006}

are concurrent if and only if opposite sides are equal.

geometry proposedgeometry