MathDB

4

Part of 2016 District Olympiad

Problems(6)

similarity of triangles

Source: Romanian District Olympiad, Grade VII, Problem 4

10/4/2018
Consider the triangle ABC ABC with ∠BAC>60∘ \angle BAC>60^{\circ } and ∠BCA>30∘. \angle BCA>30^{\circ } . On the other semiplane than that determined by BC BC and A A we have the points D D and E E so that ∠ABE=∠CBD=∠BAE+30∘=∠BCD+30∘=90∘. \angle ABE =\angle CBD =\angle BAE +30^{\circ } =\angle BCD +30^{\circ } =90^{\circ } . Note by F,H F,H the midpoints of AE, AE, respectively, CD, CD, and with G G the intersection of AC AC and DE. DE. Show:
a) EBD∼ABC EBD\sim ABC b) FGH≡ABC FGH\equiv ABC
geometryPure geometry
another problem about parallelepipeds

Source: Romanian District Olympiad 2016, Grade VIII, Problem 4

10/4/2018
Let ABCDA’B’C’D’ ABCDA’B’C’D’ a right parallelepiped and M,N M,N the feet of the perpendiculars of BD BD through A’, A’, respectively, C’. C’. We know that AB=2,BC=3,AA’=2. AB=\sqrt 2, BC=\sqrt 3, AA’=\sqrt 2.
a) Prove that A’M⊥C’N. A’M\perp C’N. b) Calculate the dihedral angle between the plane formed by A’MC A’MC and the plane formed by ANC’. ANC’.
geometry3D geometry
An algebric characterization of right triangles (I like it)

Source: Romanian District Olympiad 2016, Grade IX, Problem 4

10/4/2018
Let a≥2 a\ge 2 be a natural number. Show that the following relations are equivalent:
(i) a \text{(i)} \ a is the hypothenuse of a right triangle whose sides are natural numbers. \text{(ii)}  there exists a natural number d d for which the polynoms X2−aX±d X^2-aX\pm d have integer roots.
geometryalgebrapolynomialnumber theory
Periodicity of function h satisfying (h(x-1)+h(x+1))/h(x)=k according to k

Source: Romanian District Olympiad 2016, Grade X, Problem 4

10/5/2018
a) Prove that not all functions f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} that satisfy the equality f(x-1)+f(x+1) =\sqrt 5f(x) , \forall x\in\mathbb{R} , are periodic.
b) Prove that that all functions g:R⟶R g:\mathbb{R}\longrightarrow\mathbb{R} that satisfy the equality g(x-1)+g(x+1)=\sqrt 3g(x) , \forall x\in\mathbb{R} , are periodic.
functionalgebrafunctional equation
A sufficient condition for nondecreasing monotony; examples

Source: Romanian District Olympiad 2016, Grade XI, Problem 4

10/5/2018
Let I I be an open real interval, and let be two functions f,g:I⟶R f,g:I\longrightarrow\mathbb{R} satisfying the identity: x,y∈I∧x≠y  ⟹  f(x)−g(y)x−y+∣x−y∣≥0. x,y\in I\wedge x\neq y\implies\frac{f(x)-g(y)}{x-y} +|x-y|\ge 0.
a) Prove that f,g f,g are nondecreasing. b) Give a concrete example for f≠g. f\neq g.
functionanalysisDarbouxreal analysis
A sequence of integrals, similar to a Putnam problem

Source: Romanian District Olympiad 2016, Grade XII, Problem 4

10/5/2018
Let f:[0,1]⟶[0,1] f:[0,1]\longrightarrow [0,1] be a nondecreasing function. Prove that the sequence (∫011+fn(x)1+f1+n(x))n≥1 \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} is convergent and calculate its limit.
functioncalculusintegrationPutnam