MathDB

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Part of 2018 District Olympiad

Problems(6)

\sqrt{n + [ \sqrt{n} +/2]} irrational 2018 Romania District VII p1

Source:

9/1/2024
Show that n+[n+12]\sqrt{n + \left[ \sqrt{n} +\frac12\right]} is an irrational number, for every positive integer nn.
algebrairrational number
{m/a} + {n/m} \ne 1

Source: 2018 Romania District VIII p1

9/1/2024
Prove that {mn}+{nm}≠1\left\{ \frac{m}{n}\right\}+\left\{ \frac{n}{m}\right\} \ne 1 , for any positive integers m,nm, n.
number theoryfractional part
Finding Strictly Increasing Functions

Source: Romanian District Olympiad 2018 - Grade IX - Problem 1

3/10/2018
Find all strictly increasing functions f:N→Nf : \mathbb{N} \to \mathbb{N} such that f(x)+f(y)1+f(x+y)\frac {f(x) + f(y)}{1 + f(x + y)} is a non-zero natural number, for all x,y∈Nx, y\in\mathbb{N}.
function
Logarithmic Equation

Source: Romanian District Olympiad 2018 - Grade X - Problem 1

3/10/2018
Find x∈Rx\in\mathbb{R} for which
log⁡2(x2+4)−log⁡2x+x2−4x+2=0.\log_2(x^2 + 4) - \log_2x + x^2 - 4x + 2 = 0.
logarithms
Invertible Matrices

Source: Romanian District Olympiad 2018 - Grade XI - Problem 1

3/10/2018
Show that if n≥2n\ge 2 is an integer, then there exist invertible matrices A1,A2,…,An∈M2(R)A_1, A_2, \ldots, A_n \in \mathcal{M}_2(\mathbb{R}) with non-zero entries such that:
A1−1+A2−1+…+An−1=(A1+A2+…+An)−1.A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.
[hide=Edit.] The 77777th77777^{\text{th}} topic in College Math :coolspeak:
Matriceslinear algebra
Set of Continuous Functions

Source: Romanian District Olympiad 2018 - Grade XII - Problem 1

3/10/2018
Let F\mathcal{F} be the set of continuous functions f:[0,1]→Rf : [0, 1]\to\mathbb{R} satisfying max⁡0≤x≤1∣f(x)∣=1\max_{0\le x\le 1} |f(x)| = 1 and let I:F→RI : \mathcal{F} \to \mathbb{R},
I(f)=∫01f(x) dx−f(0)+f(1).I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).
a) Show that I(f)<3I(f) < 3, for any f∈Ff \in \mathcal{F}.
b) Determine sup⁡{I(f)∣f∈F}\sup\{I(f) \mid f \in \mathcal{F}\}.
functioncontinuitysupremum