MathDB

1

Part of 2009 District Olympiad

Problems(6)

5 | 2^n + 3^m if it 5 | 2^n + 3^m 2009 Romania District VII p1

Source:

8/16/2024
Let mm and nn be positive integers such that 55 divides 2n+3m2^n + 3^m. Prove that 55 divides 2m+3n2^m + 3^n.
number theorydivides
x^2y^2 + 1 = x^2 + xy 3x3 system 2009 Romania District VIII p1

Source:

8/16/2024
Find all non-negative real numbers x,y,zx, y, z satisfying x2y2+1=x2+xyx^2y^2 + 1 = x^2 + xy, y2z2+1=y2+yzy^2z^2 + 1 = y^2 + yz and z2x2+1=z2+xzz^2x^2 + 1 = z^2 + xz.
algebrasystem of equations
Determine the proportion TB/TA+TC/TA

Source: Romanian District Olympiad 2009, Grade IX, Problem 1

10/7/2018
On the sides AB AB and AC AC of the triangle ABC ABC consider the points D, D, respectively, E, E, such that DA→+DB→+EA→+EC→=O→. \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} . If T T is the intersection of DC DC and BE, BE, determine the real number α \alpha so that: TB→+TC→=α⋅TA→. \overrightarrow{TB} +\overrightarrow{TC} =\alpha\cdot\overrightarrow{TA} .
geometryvectorial geometry
Composition of off functions (archivation post)

Source:

10/8/2018
Let f,g:R⟶R f,g:\mathbb{R}\longrightarrow\mathbb{R} be functions with the property that f\left( g(x) \right) =g\left( f(x) \right) =-x, \forall x\in\mathbb{R}
a) Show that f,g f,g are odd. b) Give a concrete example of such f,g. f,g.
function
Romania District Olympiad 2009 - Grade XI

Source:

4/10/2011
Let A,B,C∈M3(R)A,B,C\in \mathcal{M}_3(\mathbb{R}) such that det⁡A=det⁡B=det⁡C\det A=\det B=\det C and det⁡(A+iB)=det⁡(C+iA)\det(A+iB)=\det(C+iA). Prove that det⁡(A+B)=det⁡(C+A)\det (A+B)=\det (C+A).
algebrapolynomiallinear algebralinear algebra unsolved
int f <1 implies y=0 is asymptote of id.f, where f is nonincreasing

Source: Romanian District Olympiad 2009, Grade XII, Problem 1

10/8/2018
Let f:[0,∞)⟶[0,∞) f:[0,\infty )\longrightarrow [0,\infty ) a nonincreasing function that satisfies the inequality: \int_0^x f(t)dt <1, \forall x\ge 0. Prove the following affirmations:
a) ∃lim⁡x→∞∫0xf(t)dt∈R. \exists \lim_{x\to\infty} \int_0^x f(t)dt \in\mathbb{R} .
b) lim⁡x→∞xf(x)=0. \lim_{x\to\infty} xf(x) =0.
functioninequalitiesIntegralreal analysis