MathDB

1

Part of 2006 Switzerland Team Selection Test

Problems(4)

In a triangle abc...

Source: Swiss Imo Selection 2006

5/25/2006
In the triangle A,B,CA,B,C, let DD be the middle of BCBC and EE the projection of CC on ADAD. Suppose ∠ACE=∠ABC\angle ACE = \angle ABC. Show that the triangle ABCABC is isosceles or rectangle.
geometrytrigonometrytrig identitiesLaw of Sinesgeometry proposed
Let 1=d_1<d_2<...<d_k=n be the positiv dividers of n

Source: Swiss Imo Selection 2006

5/25/2006
Let nn be natural number and 1=d1<d2<…<dk=n1=d_1<d_2<\ldots <d_k=n be the positive divisors of nn.
Find all nn such that 2n=d52+d62−12n = d_5^2+ d_6^2 -1.
number theory proposednumber theory
Find the value of a^2b+b^2c+c^2a

Source: Swiss Imo Selection 2006

5/25/2006
The three roots of P(x)=x3−2x2−x+1P(x) = x^3 - 2x^2 - x + 1 are a>b>c∈Ra>b>c \in \mathbb{R}. Find the value of a2b+b2c+c2aa^2b+b^2c+c^2a. :D
invariantfunctionalgebra proposedalgebra
Swiss imo selection 06 inequality

Source:

5/25/2006
Let a,b,c∈R+a,b,c \in \mathbb{R^+} and 1a+1b+1c=1\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1. Show ab+c+bc+a+ca+b≥abc+a+b+c\sqrt{ab+c} + \sqrt{bc+a} + \sqrt{ca+b} \ge \sqrt{abc} + \sqrt{a} + \sqrt{b} + \sqrt{c}. :D
inequalitiesinequalities proposed