MathDB

3

Part of 2010 IFYM, Sozopol

Problems(5)

Prove HE = HF

Source:

2/23/2009
Let ABC ABC is a triangle, let H H is orthocenter of △ABC \triangle ABC, let M M is midpoint of BC BC. Let (d) (d) is a line perpendicular with HM HM at point H H. Let (d) (d) meet AB,AC AB, AC at E,F E, F respectively. Prove that HE \equal{}HF.
geometrycircumcirclecongruent trianglesangle bisectorgeometry unsolved
Problem 3 of Second round - Constructing two points on given circles

Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade

12/14/2019
Two circles are intersecting in points PP and QQ. Construct two points AA and BB on these circles so that P∈ABP\in AB and the product AP.PBAP.PB is maximal.
geometryconstructionconstructive geometry
Problem 3 of Third round - Least value of leading coefficient

Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade

12/14/2019
Let a,b,ca,b,c be integers, a>0a>0 and the equation ax2−bx+c=0ax^2-bx+c=0 has two distinct real roots in the interval (0,1)(0,1). Find the least possible value of aa.
algebra
Problem 3 of Fourth round

Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade

12/15/2019
Through vertex CC of ΔABC\Delta ABC are constructed lines l1l_1 and l2l_2 which are symmetrical about the angle bisector CLcCL_c. Prove that the projections of AA and BB on lines l1l_1 and l2l_2 lie on one circle.
geometrysymmetry
P(X) is divisible by Q(X) iff b=1

Source: Romanian MO 2001

1/14/2011
Let n≥2n\ge 2 be an even integer and a,ba,b real numbers such that bn=3a+1b^n=3a+1. Show that the polynomial P(X)=(X2+X+1)n−Xn−aP(X)=(X^2+X+1)^n-X^n-a is divisible by Q(X)=X3+X2+X+bQ(X)=X^3+X^2+X+b if and only if b=1b=1.
algebrapolynomialalgebra proposed