MathDB

2

Part of 2007 Indonesia TST

Problems(4)

find n

Source: Indonesia IMO 2007 TST, Stage 2, Test 2, Problem 2

11/15/2009
Let a,b,c a,b,c be non-zero real numbers satisfying \dfrac{1}{a}\plus{}\dfrac{1}{b}\plus{}\dfrac{1}{c}\equal{}\dfrac{1}{a\plus{}b\plus{}c}. Find all integers n n such that \dfrac{1}{a^n}\plus{}\dfrac{1}{b^n}\plus{}\dfrac{1}{c^n}\equal{}\dfrac{1}{a^n\plus{}b^n\plus{}c^n}.
algebra proposedalgebra
solving equation, with parameter

Source: Indonesia IMO 2007 TST, Stage 2, Test 1, Problem 2

11/15/2009
Solve the equation x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x} where a a is a real parameter.
parameterizationalgebra proposedalgebra
geometry iff again

Source: Indonesia IMO 2007 TST, Stage 2, Test 4, Problem 2

11/15/2009
Let ABCD ABCD be a convex quadrtilateral such that AB AB is not parallel with CD CD. Let Γ1 \Gamma_1 be a circle that passes through A A and B B and is tangent to CD CD at P P. Also, let Γ2 \Gamma_2 be a circle that passes through C C and D D and is tangent to AB AB at Q Q. Let the circles Γ1 \Gamma_1 and Γ2 \Gamma_2 intersect at E E and F F. Prove that EF EF passes through the midpoint of PQ PQ iff BC∥AD BC \parallel AD.
geometrygeometry proposed
easy functional equation

Source: Indonesia IMO 2007 TST, Stage 2, Test 5, Problem 2

11/15/2009
Find all functions f:R→R f: \mathbb{R} \rightarrow \mathbb{R} satisfying f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy for all real numbers xx and yy.
functionalgebra proposedalgebra