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2

Part of 2005 Moldova Team Selection Test

Problems(2)

\sum1/{4-ab}

Source: Moldova TST 2005

3/21/2005
Let a a, b b, c c be positive reals such that a^4 \plus{} b^4 \plus{} c^4 \equal{} 3. Prove that \sum\frac1{4 \minus{} ab}\leq1, where the ∑ \sum sign stands for cyclic summation. Alternative formulation: For any positive reals a a, b b, c c satisfying a^4 \plus{} b^4 \plus{} c^4 \equal{} 3, prove the inequality \frac{1}{4\minus{}bc}\plus{}\frac{1}{4\minus{}ca}\plus{}\frac{1}{4\minus{}ab}\leq 1.
inequalitiesfunctioninequalities proposed
2005 divides smth

Source: Moldova TST 2005

4/9/2005
Let m∈Nm\in N and E(x,y,m)=(72x)m+(72y)m−xm−ymE(x,y,m)=(\frac{72}x)^m+(\frac{72}y)^m-x^m-y^m, where xx and yy are positive divisors of 72. a) Prove that there exist infinitely many natural numbers mm so, that 2005 divides E(3,12,m)E(3,12,m) and E(9,6,m)E(9,6,m). b) Find the smallest positive integer number m0m_0 so, that 2005 divides E(3,12,m0)E(3,12,m_0) and E(9,6,m0)E(9,6,m_0).
number theory unsolvednumber theory