MathDB

4

Part of 2019 District Olympiad

Problems(6)

equilateral wanted, right isosceles (2019 Romania District VII p4)

Source:

5/22/2020
Consider the isosceles right triangleABC,∠A=90o ABC, \angle A = 90^o, and point D∈(AB)D \in (AB) such that AD=13ABAD = \frac13 AB. In the half-plane determined by the line ABAB and point CC , consider a point EE such that ∠BDE=60o\angle BDE = 60^o and ∠DBE=75o\angle DBE = 75^o. Lines BCBC and DEDE intersect at point GG, and the line passing through point GG parallel to the line ACAC intersects the line BEBE at point HH. Prove that the triangle CEHCEH is equilateral.
geometryEquilateralright triangleisosceles
[x+1/x] = [x^2+1/x^2]

Source: 2019 Romania District VIII p4

9/1/2024
Solve the equation in the set of real numbers: [x+1x]=[x2+1x2]\left[ x+\frac{1}{x} \right] = \left[ x^2+\frac{1}{x^2} \right] where [a][a], represents the integer part of the real number aa.
algebrafloor functionInteger Part
Romanian District Olympiad 2019 - Grade 9 - Problem 4

Source: Romanian District Olympiad 2019 - Grade 9 - Problem 4

3/18/2019
Find all positive integers pp for which there exists a positive integer nn such that pn+3n ∣ pn+1+3n+1.p^n+3^n~|~p^{n+1}+3^{n+1}.
number theory
Romanian District Olympiad 2019 - Grade 10 - Problem 4

Source: Romanian District Olympiad 2019 - Grade 10 - Problem 4

3/17/2019
Find the smallest positive real number λ\lambda such that for every numbers a1,a2,a3∈[0,12]a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right] and b1,b2,b3∈(0,∞)b_1,b_2,b_3 \in (0, \infty) with ∑i=13ai=∑i=13bi=1,\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1, we have b1b2b3≤λ(a1b1+a2b2+a3b3).b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3).
Inequalityalgebra
Romanian District Olympiad 2019 - Grade 11 - Problem 4

Source: Romanian District Olympiad 2019 - Grade 11 - Problem 4

3/16/2019
Let f:[0,∞)→[0,∞)f: [0, \infty) \to [0, \infty) be a continuous function with f(0)>0f(0)>0 and having the property x−y<f(y)−f(x)≤0 ∀ 0≤x<y.x-y<f(y)-f(x) \le 0~\forall~0 \le x<y. Prove that: a)a) There exists a unique α∈(0,∞)\alpha \in (0, \infty) such that (f∘f)(α)=α.(f \circ f)(\alpha)=\alpha. b)b) The sequence (xn)n≥1,(x_n)_{n \ge 1}, defined by x1≥0x_1 \ge 0 and xn+1=f(xn) ∀ n∈Nx_{n+1}=f(x_n)~\forall~n \in \mathbb{N} is convergent.
continuous functionConvergenceSequencescalculus
Romanian District Olympiad 2019 - Grade 12 - Problem 4

Source: Romanian District Olympiad 2019 - Grade 12 - Problem 4

3/16/2019
Let aa be a real number, a>1.a>1. Find the real numbers b≥1b \ge 1 such that lim⁡x→∞∫0x(1+ta)−bdt=1.\lim_{x \to \infty} \int\limits_0^x (1+t^a)^{-b} \mathrm{d}t=1.
Improper integralIntegrallimitcalculus