MathDB

1

Part of 2012 District Olympiad

Problems(6)

\sqrt{a^2_1+a^2_2+ ...+a^2_{2012}-1} is irrational 2012 Romania District VII p1

Source:

9/1/2024
Let a1,a2,...,a2012a_1, a_2, ... , a_{2012} be odd positive integers. Prove that the number A=a12+a22+...+a20122−1A=\sqrt{a^2_1+ a^2_2+ ...+ a^2_{2012}-1} is irrational.
algebranumber theoryirrational number
a -\sqrt{ab}, b+ \sqrt{ab} rationals 2012 Romania District VIII p1

Source:

9/1/2024
Let aa and bb be distinct positive real numbers, such that a−aba -\sqrt{ab} and b−abb -\sqrt{ab} are both rational numbers. Prove that aa and bb are rational numbers.
algebrarational
|f(x)-f(y)|<=|sinx-siny| implies id+f is monotone

Source: Romanian District Olympiad 2012, Grade X, Problem 1

10/9/2018
Let f:[0,∞)⟶R f:[0,\infty )\longrightarrow\mathbb{R} a bounded and periodic function with the property that |f(x)-f(y)|\le |\sin x-\sin y|, \forall x,y\in[0,\infty ) . Show that the function [0,∞)∋x↦x+f(x) [0,\infty ) \ni x\mapsto x+f(x) is monotone.
functionalgebraPost 3 is the actual problem
Solve [x]^5+{x}^5=x^5

Source: Romanian District Olympiad 2012, Grade IX, Problem 1

10/9/2018
Solve in R \mathbb{R} the equation [x]5+{x}5=x5, [x]^5+\{ x\}^5 =x^5, where [],{} [],\{\} are the integer part, respectively, the fractional part.
equationsalgebrafractional partInteger PartFloor
sequence in which sum of terms equals product of the same terms

Source: Romanian District Olympiad 2012, Grade XI, Problem 1

10/9/2018
Consider the sequence (xn)n≥1 \left( x_n \right)_{n\ge 1} having x1>1 x_1>1 and satisfying the equation x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} , \forall n\in\mathbb{N} . Show that this sequence is convergent and find its limit.
Sequenceslimitreal analysis
Romania District Olympiad 2012 - Grade XII

Source:

3/10/2012
Let a,b,ca,b,c three positive distinct real numbers. Evaluate:
lim⁡t→∞∫0t1(x2+a2)(x2+b2)(x2+c2)dx\lim_{t\to \infty} \int_0^t \frac{1}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}dx
limitintegrationcalculusreal analysisreal analysis unsolved