MathDB

2

Part of 2010 Irish Math Olympiad

Problems(2)

Triangle is Isosceles

Source: 2010 IrMO Paper 1 Problem 2

1/30/2018
Let ABCABC be a triangle and let PP denote the midpoint of the side BCBC. Suppose that there exist two points MM and NN interior to the side ABAB and ACAC respectively, such that ∣AD∣=∣DM∣=2∣DN∣,|AD|=|DM|=2|DN|, where DD is the intersection point of the lines MNMN and APAP. Show that ∣AC∣=∣BC∣|AC|=|BC|.
geometry
Real Roots of Polynomial

Source: 2010 IrMO Paper 2 Problem 2

1/30/2018
For each odd integer p≥3p\ge 3 find the number of real roots of the polynomial fp(x)=(x−1)(x−2)⋯(x−p+1)+1.f_p(x)=(x-1)(x-2)\cdots (x-p+1)+1.
algebrapolynomial