MathDB

3

Part of 2011 Romania National Olympiad

Problems(6)

midpoint wanted, angle bisectors related, <BCD = <ADC >= 90 ^o

Source: 2011 Romanian NMO grade VII P3

5/18/2020
In the convex quadrilateral ABCDABCD we have that ∠BCD=∠ADC≥90o\angle BCD = \angle ADC \ge 90 ^o. The bisectors of ∠BAD\angle BAD and ∠ABC\angle ABC intersect in MM. Prove that if M∈CDM \in CD, then MM is the middle of CDCD.
geometryangle bisectormidpointequal angles
angle between lateral edge of pyramid and plane of base wanted

Source: 2011 Romanian NMO grade VIII P3

5/18/2020
Let VABCVABC be a regular triangular pyramid with base ABCABC, of center OO. Points II and HH are the center of the inscribed circle, respectively the orthocenter △VBC\vartriangle VBC. Knowing that AH=3OIAH = 3 OI, determine the measure of the angle between the lateral edge of the pyramid and the plane of the base.
pyramidangle3D geometrygeometry
Prove concurrence involving excirlce

Source: Romanian NO, grade ix, p.3

10/3/2019
Let ABC ABC be a triangle, Ia I_a be center of the A-excircle. A\text{-excircle}. This excircle intersects the lines AB,AC, AB, AC, at P, P, respectively, Q. Q. The line PQ PQ intersects the lines IaB,IaC I_aB,I_aC at D, D, respectively, E. E. Let A1 A_1 be the intersection of DC DC with BE, BE, and define, analogously, B1,C1. B_1,C_1. Show that AA1,BB1,CC1 AA_1,BB_1,CC_1 are concurrent.
geometry
Monotony of a rational cyclic function

Source: Romanian NO 2011, grade x, p. 3

10/3/2019
Let be three positive real numbers a,b,c. a,b,c. Show that the function f:R⟶R, f:\mathbb{R}\longrightarrow\mathbb{R} , f(x)=axbx+cx+bxax+cx+cxax+bx, f(x)=\frac{a^x}{b^x+c^x} +\frac{b^x}{a^x+c^x} +\frac{c^x}{a^x+b^x} , is nondecresing on the interval [0,∞) \left[ 0,\infty \right) and nonincreasing on the interval (−∞,0]. \left( -\infty ,0 \right] .
functionalgebracyclic function
Romania National Olympiad 2011 - Grade XI - problem 3

Source:

4/19/2011
Let g:R→Rg:\mathbb{R}\to\mathbb{R} be a continuous and strictly decreasing function with g(R)=(−∞,0)g(\mathbb{R})=(-\infty,0) . Prove that there are no continuous functions f:R→Rf:\mathbb{R}\to\mathbb{R} with the property that there exists a natural number k≥2k\ge 2 so that : f∘f∘…∘f⏟k times=g\underbrace{f\circ f\circ\ldots\circ f}_{k\text{ times}}=g .
functionreal analysisreal analysis unsolved
A characterization of division rings in terms of an equation concerning its elem

Source: Romanian NO 2011, grade xii, p.3

10/3/2019
The equation xn+1+x=0 x^{n+1} +x=0 admits 0 0 and 1 1 as its unique solutions in a ring of order n≥2. n\ge 2. Prove that this ring is a skew field.
abstract algebraRing Theory