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2

Part of 2000 District Olympiad (Hunedoara)

Problems(3)

Romanian District Olympiad 2000, Grade IX, Problem 2

Source:

9/24/2018
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.
equationsFloornumber theoryalgebraOlympiaditerations
matrix determinant problem

Source: Romanian District Olympiad 2000, Grade XI, Problem 2

9/24/2018
Calculate the determinant of the n×n n\times n complex matrix (aji)1≤j≤n1≤i≤n \left(a_j^i\right)_{1\le j\le n}^{1\le i\le n} defined by aji={1+x2,emsp;i=jx,emsp;∣i−j∣=10,emsp;∣i−j∣≥2, a_j^i=\left\{\begin{matrix} 1+x^2,  i=j\\x,  |i-j|=1\\0,  |i-j|\ge 2\end{matrix}\right. , where n n is a natural number greater than 2. 2.
linear algebramatrixdeterminant
basic complex algebra

Source: Romanian District Olympiad, Grade X, Problem 2

9/24/2018
Let z1,z2,z3∈C z_1,z_2,z_3\in\mathbb{C} such that \text{(i)}  \left|z_1\right| = \left|z_2\right| = \left|z_3\right| = 1 \text{(ii)}  z_1+z_2+z_3\neq 0 \text{(iii)}  z_1^2 +z_2^2+z_3^2 =0.
Show that ∣z13+z23+z33∣=1. \left| z_1^3+z_2^3+z_3^3\right| = 1.
complex numbersalgebra