MathDB

1

Part of 1997 Iran MO (3rd Round)

Problems(3)

All functions f such that f(1-x)=1-f(f(x))

Source: Iran Third Round MO 1997, Exam 1, P1

6/30/2012
Find all strictly ascending functions ff such that for all x∈Rx\in \mathbb R, f(1−x)=1−f(f(x)).f(1-x)=1-f(f(x)).
functionalgebra unsolvedalgebra
P(a)P(b)=-(a-b)^2 => Prove that P(a)+P(b)=0

Source: Iran Third Round MO 1997, Exam 2, P1

6/30/2012
Let PP be a polynomial with integer coefficients. There exist integers aa and bb such that P(a)⋅P(b)=−(a−b)2P(a) \cdot P(b)=-(a-b)^2. Prove that P(a)+P(b)=0P(a)+P(b)=0.
algebrapolynomialalgebra proposed
Prove that a=x and b=x^x - [Iran Third Round 1998]

Source: Iran Third Round MO 1997, Exam 3, P1

10/18/2005
Suppose that a,b,xa, b, x are positive integers such that xa+b=abbx^{a+b}=a^bb Prove that a=xa=x and b=xxb=x^x.
number theory proposednumber theory