MathDB

1

Part of 2000 District Olympiad (Hunedoara)

Problems(4)

Romanian District Olympiad 2000, Grade IX, Problem 1

Source:

9/24/2018
a) Show that n2≥2x+3x3+⋯+nxnn−1−x, \frac{n}{2}\ge \frac{2\sqrt{x} +3\sqrt[3]{x}+\cdots +n\sqrt[n]{x}}{n-1} -x, for all non-negative reals x x and integers n≥2. n\ge 2.
b) If x,y,z∈(0,∞), x,y,z\in (0,\infty ) , then prove the inequality ∑cycx(2x+y+z)2+4≤3/16 \sum_{\text{cyc}} \frac{x}{(2x+y+z)^2+4} \le 3/16
inequalitiesromaniaInequalityalgebra
equation and system of equations

Source: Romanian District Olympiad 2000, Grade X, Problem 1

9/24/2018
a) Solve the system {3y−4x=11log⁡4x+log⁡3y=3/2 \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right.
b) Solve the equation   9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4.
equationssystem of equationsnumber theoryalgebra
matrix equation

Source: Romanian District Olympiad 2000, Grade XI, Problem 1

9/24/2018
Solve in the set of 2×2 2\times 2 integer matrices the equation X2−4⋅X+4⋅(1emsp;00emsp;1)=(7emsp;812emsp;31). X^2-4\cdot X+4\cdot\left(\begin{matrix}1  0\\0  1\end{matrix}\right) =\left(\begin{matrix}7  8\\12  31\end{matrix}\right) .
linear algebramatrixcontestsromaniaalgebra
binary operation on contest

Source: Romanian District Olympiad 2000, Grade XII, Problem 1

9/25/2018
Define the operator " ∗ * " on R \mathbb{R} as x∗y=x+y+xy. x*y=x+y+xy.
a) Show that R∖{−1}, \mathbb{R}\setminus\{ -1\} , along with the operator above, is isomorphic with R∖{0}, \mathbb{R}\setminus\{ 0\} , with the usual multiplication.
b) Determine all finite semigroups of R \mathbb{R} under " ∗. *. " Which of them are groups?
c) Prove that if H⊂∗R H\subset_{*}\mathbb{R} is a bounded semigroup, then H⊂[−2,0]. H\subset [-2, 0].
Binary operationsemigroupsGroupssuperior algebragroup theory