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Romanian District Olympiad 2000, Grade IX, Problem 2

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September 24, 2018
equationsFloornumber theoryalgebraOlympiaditerations

Problem Statement

a) Let a,b a,b two non-negative integers such that a2>b. a^2>b. Show that the equation ⌊x2+2ax+b⌋=x+a−1 \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 has an infinite number of solutions in the non-negative integers. Here, ⌊α⌋ \lfloor\alpha\rfloor denotes the floor of α. \alpha.
b) Find the floor of m=2+2+⋯⏟n times+2, m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , where n n is a natural number. Justify.
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