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Part of 2010 Moldova Team Selection Test

Problems(2)

Diophantine equation

Source: Moldova TST 2010, day 1, problem 1

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-digit numbers such that placing to the right side of the number its successor we get a

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-digit number which is a perfect square.

modular arithmeticnumber theory proposednumber theory

A polynomial with negative roots

Source: Moldova TST 2010, day 2, problem 1

Let p\in\mathbb{R}_\plus{} and k\in\mathbb{R}_\plus{}. The polynomial F(x)\equal{}x^4\plus{}a_3x^3\plus{}a_2x^2\plus{}a_1x\plus{}k^4 with real coefficients has

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negative roots. Prove that F(p)\geq(p\plus{}k)^4

algebrapolynomialalgebra proposed