MathDB

2

Part of 2009 IMO Shortlist

Problems(3)

IMO Shortlist 2009 - Problem A2

Source:

7/5/2010
Let aa, bb, cc be positive real numbers such that 1a+1b+1c=a+b+c\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c. Prove that: 1(2a+b+c)2+1(a+2b+c)2+1(a+b+2c)2≤316.\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}. Proposed by Juhan Aru, Estonia
IMO Shortlistthree variable inequalityInequalityinequalitiesIMO 2009
IMO Shortlist 2009 - Problem C2

Source:

7/5/2010
For any integer n≥2n\geq 2, let N(n)N(n) be the maxima number of triples (ai,bi,ci)(a_i, b_i, c_i), i=1,…,N(n)i=1, \ldots, N(n), consisting of nonnegative integers aia_i, bib_i and cic_i such that the following two conditions are satisfied: [*] ai+bi+ci=na_i+b_i+c_i=n for all i=1,…,N(n)i=1, \ldots, N(n), [*] If i≠ji\neq j then ai≠aja_i\neq a_j, bi≠bjb_i\neq b_j and ci≠cjc_i\neq c_j Determine N(n)N(n) for all n≥2n\geq 2.
Proposed by Dan Schwarz, Romania
combinatoricsExtremal combinatoricsIMO Shortlist
IMO Shortlist 2009 - Problem N2

Source:

7/5/2010
A positive integer NN is called balanced, if N=1N=1 or if NN can be written as a product of an even number of not necessarily distinct primes. Given positive integers aa and bb, consider the polynomial PP defined by P(x)=(x+a)(x+b)P(x)=(x+a)(x+b). (a) Prove that there exist distinct positive integers aa and bb such that all the number P(1)P(1), P(2)P(2),…\ldots, P(50)P(50) are balanced. (b) Prove that if P(n)P(n) is balanced for all positive integers nn, then a=ba=b.
Proposed by Jorge Tipe, Peru
algebrapolynomialnumber theoryIMO Shortlist