MathDB

1

Part of 1993 Irish Math Olympiad

Problems(2)

System of eq. of 3 degree polynomials

Source: determine sum of solution pair.

7/26/2007
The following is known about the reals α \alpha and β \beta α3−3α2+5α−17=0 \alpha^{3}-3\alpha^{2}+5\alpha-17=0 and β3−3β2+5β+11=0 \beta^{3}-3\beta^{2}+5\beta+11=0 Determine α+β \alpha+\beta
algebrapolynomialalgebra proposed
integer coordinates

Source: Ireland 1993

6/29/2009
Show that among any five points P1,...,P5 P_1,...,P_5 with integer coordinates in the plane, there exists at least one pair (Pi,Pj) (P_i,P_j), with i \not\equal{} j such that the segment PiPj P_i P_j contains a point Q Q with integer coordinates other than Pi,Pj P_i, P_j.
analytic geometrymodular arithmeticgeometry unsolvedgeometry