MathDB

2

Part of 2002 Romania National Olympiad

Problems(6)

a,b,c just isn't cool enough

Source: Romanian MO 2002

12/8/2010
Given real numbers a,c,da,c,d show that there exists at most one function f:R→Rf:\mathbb{R}\rightarrow\mathbb{R} which satisfies: f(ax+c)+d\le x\le f(x+d)+c \text{for any}\ x\in\mathbb{R}
functionalgebra proposedalgebra
Every x can be written as a difference of two irrationals

Source: Romanian MO 2002

12/8/2010
Prove that any real number 0<x<10<x<1 can be written as a difference of two positive and less than 11 irrational numbers.
number theory proposednumber theory
(3 \sqrt 3 - 4), A ratio of

Source: RMO 2002 - Grade IX - Problem 2

2/15/2007
Let ABCABC be a right triangle where ∡A=90∘\measuredangle A = 90^\circ and M∈(AB)M\in (AB) such that AMMB=33−4\frac{AM}{MB}=3\sqrt{3}-4. It is known that the symmetric point of MMwith respect to the line GIGI lies on ACAC. Find the measure of ∡B\measuredangle B.
ratiogeometrygeometric transformationreflectioncircumcircleinradiustrigonometry
The real polynomials f and g

Source: Romanian MO 2002

12/8/2010
Find all real polynomials ff and gg, such that: (x2+x+1)⋅f(x2−x+1)=(x2−x+1)⋅g(x2+x+1),(x^2+x+1)\cdot f(x^2-x+1)=(x^2-x+1)\cdot g(x^2+x+1), for all x∈Rx\in\mathbb{R}.
algebrapolynomialalgebra proposed
f has no local extrema and limits at any point

Source:

12/8/2010
Let f:R→Rf:\mathbb{R}\rightarrow\mathbb{R} be a function that has limits at any point and has no local extrema. Show that: a)a) ff is continuous; b)b) ff is strictly monotone.
functionlimitreal analysisreal analysis unsolved
There exists x_1,x_2 for an integrable function f

Source:

12/8/2010
Let f:[0,1]→Rf:[0,1]\rightarrow\mathbb{R} be an integrable function such that: 0<∣∫01f(x) dx∣≤1.0<\left\vert \int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1. Show that there exists x1≠x2,x1,x2∈[0,1]x_1\not= x_2, x_1,x_2\in [0,1], such that: ∫x1x2f(x) dx=(x1−x2)2002\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}
functionintegrationcalculusinequalitiesreal analysisreal analysis unsolved