MathDB

1

Part of 2013 District Olympiad

Problems(6)

x^2 + y^2 + z^2 = 16(x + y + z) diophantine

Source: 2013 Romania District VIII p1

9/1/2024
Find all triples of integers (x,y,z)(x, y, z) such that x2+y2+z2=16(x+y+z).x^2 + y^2 + z^2 = 16(x + y + z).
number theoryDiophantine equationdiophantine
2013 integer solutions for radical equation 2013 Romania District VII p1

Source:

9/1/2024
Prove that the equation 1x+1006+12012−x+1006=2x+2012−x\frac{1}{\sqrt{x} +\sqrt{1006}}+\frac{1}{\sqrt{2012 -x} +\sqrt{1006}}=\frac{2}{\sqrt{x} +\sqrt{2012 -x}} has 20132013 integer solutions.
algebradiophantine
System

Source: Romania District Olympiad 2013,grade IX(problem 1)

3/14/2013
a) Prove that, whatever the real number x would be, the following inequality takes place x4−x3−x+1≥0.{{x}^{4}}-{{x}^{3}}-x+1\ge 0. b) Solve the following system in the set of real numbers: x1+x2+x3=3,x13+x23+x33=x14+x24+x34{{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}. The Mathematical Gazette
inequalitiesalgebra proposedalgebra
equation

Source: Romania District Olympiad 2013,grade X(problem 1)

3/14/2013
Let a,b∈Ra,b\in \mathbb{R} and z∈C\Rz\in \mathbb{C}\backslash \mathbb{R} so that ∣a−b∣=∣a+b−2z∣\left| a-b \right|=\left| a+b-2z \right|. a) Prove that the equation ∣z−a∣x+∣zˉ−b∣x=∣a−b∣x{{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}={{\left| a-b \right|}^{x}}, with the unknown number x∈Rx\in \mathbb{R}, has a unique solution. b) Solve the following inequation ∣z−a∣x+∣zˉ−b∣x≤∣a−b∣x{{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}\le {{\left| a-b \right|}^{x}}, with the unknown number x∈Rx\in \mathbb{R}. The Mathematical Gazette
algebra proposedalgebra
Limit sequence

Source: Romania District Olympiad 2013,grade XI(Problem 1)

3/10/2013
Let (an)n≥1{{\left( {{a}_{n}} \right)}_{n\ge 1}} an increasing sequence and bounded.Calculate lim⁡n→∞ (2an−a1−a2)(2an−a2−a3)...(2an−an−2−an−1)(2an−an−1−a1).\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).
limitinequalitiescalculuscalculus computations
Limit

Source: Romania District Olympiad 2013 ,grade 12

3/10/2013
Calculate: lim⁡n→∞ ∫01exndx\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{{{e}^{{{x}^{n}}}}dx}
limitintegrationcalculuscalculus computations