MathDB

3

Part of 2009 District Olympiad

Problems(6)

a^2 + b^2 =p^2-q^2 2009 Romania District VII p3

Source:

8/16/2024
Let aa and bb be non-negative integers. Prove that the number a2+b2a^2 + b^2 is the difference of two perfect squares if and only if abab is even.
number theoryPerfect Squares
angle between planes wanted, regular quadrilateral prism related

Source: 2009 Romania District VIII P3

5/19/2020
Consider the regular quadrilateral prism ABCDA′B′C′D′ABCDA'B'C 'D', in which AB=a,AA′=a22AB = a,AA' = \frac{a \sqrt {2}}{2}, and MM is the midpoint of B′C′B' C'. Let FF be the foot of the perpendicular from BB on line MCMC, Let determine the measure of the angle between the planes (BDF)(BDF) and (HBS)(HBS).
geometry3D geometryprismplanesangle
Two inequalities resembling a little to Arquady’s preffered inequality-naming

Source: Romanian District Olympiad 2009, Grade IX, Problem 3

10/7/2018
a) For a,b≥0 a,b\ge 0 and x,y>0, x,y>0, show that: a3x2+b3y2≥(a+b)3(x+y)2. \frac{a^3}{x^2} +\frac{b^3}{y^2}\ge \frac{(a+b)^3}{(x+y)^2} .
b) For a,b,c≥0 a,b,c\ge 0 and x,y,z>0 x,y,z>0 under the condition a+b+c=x+y+z, a+b+c=x+y+z, prove that: a3x2+b3y2+c3z2≥a+b+c. \frac{a^3}{x^2} +\frac{b^3}{y^2} +\frac{c^3}{z^2} \ge a+b+c.
inequalities
Some coinciding solutions of equations

Source: Romanian District Olympiad 2009, Grade X, Problem 3

10/8/2018
Let A A be the set of real solutions of the equation 3x=x+2, 3^x=x+2, and let be the set B B of real solutions of the equation log⁡3(x+2)+log⁡2(3x−x)=3x−1. \log_3 (x+2) +\log_2 \left( 3^x-x \right) =3^x-1 . Prove the validity of the following subpoints:
a) A⊂B. A\subset B. b) B⊄Q∧B⊄R∖Q. B\not\subset\mathbb{Q} \wedge B\not\subset \mathbb{R}\setminus\mathbb{Q} .
equationsalgebralogarithms
Romania District Olympiad 2009 - Grade XI

Source:

4/10/2011
Let (xn)n≥1(x_n)_{n\ge 1} a sequence defined by x1=2, xn+1=xn+1n, (∀)n∈N∗x_1=2,\ x_{n+1}=\sqrt{x_n+\frac{1}{n}},\ (\forall)n\in \mathbb{N}^*. Prove that lim⁡n→∞xn=1\lim_{n\to \infty} x_n=1 and evaluate lim⁡n→∞xnn\lim_{n\to \infty} x_n^n.
limitinductionreal analysisreal analysis unsolved
MVT-like integral problem involving id.f

Source: Romanian District Olympiad 2009, Grade XII, Problem 3

10/8/2018
Let f:[0,1]⟶R f:[0,1]\longrightarrow\mathbb{R} be a continuous function such that ∫01(x−1)f(x)dx=0. \int_0^1 (x-1)f(x)dx =0. Show that:
a) There exists a∈(0,1) a\in (0,1) such that ∫0axf(x)dx=0. \int_0^a xf(x)dx =0.
b) There exists b∈(0,1) b\in (0,1) so that ∫0bxf(x)dx=bf(b). \int_0^b xf(x)dx=bf(b).
functionintegrationcalculus