MathDB

4

Part of 2011 Romania National Olympiad

Problems(6)

angle chasing, <BCA = 2 <MBC, <ABC= 60 ^o

Source: 2011 Romanian NMO grade VII P4

5/18/2020
Consider △ABC\vartriangle ABC where ∠ABC=60o\angle ABC= 60 ^o. Points MM and DD are on the sides (AC)(AC), respectively (AB)(AB), such that ∠BCA=2∠MBC\angle BCA = 2 \angle MBC, and BD=MCBD = MC. Determine ∠DMB\angle DMB.
geometryanglesAngle Chasing
sum of digits is multiple of 2011

Source: 2011 Romania NMO VIII p4

8/15/2024
A positive integer will be called typical if the sum of its decimal digits is a multiple of 20112011. a) Show that there are infinitely many typical numbers, each having at least 20112011 multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?
number theorysum of digits
The parity of the floor of a power of 2 multiple of sqrt(2)

Source: Romanian NO, grade ix, p.4

10/3/2019
Let be a natural number n. n. Prove that there exists a number k∈{0,1,2,…n} k\in\{ 0,1,2,\ldots n \} such that the floor of 2n+k2 2^{n+k}\sqrt 2 is even.
number theoryinequalitiesFloor
Romania National Olympiad 2011 - Grade XI - problem 4

Source:

4/19/2011
Let A , B∈M2(C)A\, ,\, B\in\mathcal{M}_2(\mathbb{C}) so that : A2+B2=2ABA^2+B^2=2AB . a) Prove that : AB=BAAB=BA . b) Prove that : tr (A)=tr (B)\text{tr}\, (A)=\text{tr}\, (B) .
linear algebralinear algebra unsolved
Quadratic integers

Source: Romanian NO 2011, grade x, p.4

10/3/2019
a) Show that there exists exactly a sequence (xn,yn)n≥0 \left( x_n,y_n \right)_{n\ge 0} of pairs of nonnegative integers, that satisfy the property that (1+33)n=xn+yn33, \left( 1+\sqrt 33 \right)^n=x_n+y_n\sqrt 33, for all nonegative integers n. n.
b) Having in mind the sequence from a), prove that, for any natural prime p, p, at least one of the numbers yp−1,yp y_{p-1} ,y_p and yp+1 y_{p+1} are divisible by p. p.
algebra
Cond. on lateral lim. for a f to be the integral of certain f (hard)

Source: Romanian NO 2011, grade xii, p.4

10/3/2019
Let f,F:R⟶R f,F:\mathbb{R}\longrightarrow\mathbb{R} be two functions such that f f is nondecreasing, F F admits finite lateral derivates in every point of its domain, lim⁡x→y−f(x)≤lim⁡x→y−F(x)−F(y)x−y,lim⁡x→y+f(x)≥lim⁡x→y+F(x)−F(y)x−y, \lim_{x\to y^-} f(x)\le\lim_{x\to y^-}\frac{F(x)-F\left( y \right)}{x-y} ,\lim_{x\to y^+} f(x)\ge\lim_{x\to y^+}\frac{F(x)-F\left( y \right)}{x-y} , for all real numbers y, y, and F(0)=0. F(0)=0.
Prove that F(x)=∫0xf(t)dt, F(x)=\int_0^x f(t)dt, for all real numbers x. x.
functionintegrationreal analysis