MathDB

Sufficient condition for a point to lie on a median

Source: Romanian National Olympiad, grade ix, p.3

8/23/2019

Let be a point

P

in the interior of a triangle

ABC.

The lines

AP,BP,CP

meet

BC,AC,

respectively,

AB

at

A_1,B_1,

respectively,

C_1.

If

\mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} ,

show that

P

lies on a median of

ABC.

\mathcal{A}

denotes area.

geometrymedianarea

variation of f(g(x)+g(y))=f(g(y))-y (same thing though)

Source: Romania National Olympiad 2015, grade x, p.3

8/23/2019

Find all functions

f,g:\mathbb{Q}\longrightarrow\mathbb{Q}

that verify the relations

\left\{\begin{matrix} f(g(x)+g(y))=f(g(x))+y \\ g(f(x)+f(y))=g(f(x))+y\end{matrix}\right. ,

for all

x,y\in\mathbb{Q} .

functionalgebra

condition for the convergence of sum (x^i/i^b)

Source: Romania National Olympiad 2015, grade xi, p. 3

8/23/2019

Let be two nonnegative real numbers

a,b

with

b>a,

and a sequence

\left( x_n \right)_{n\ge 1}

of real numbers such that the sequence

\left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1}

is bounded.
Show that the sequence

\left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1}

is convergent.

real analysisSequences

Romanian National Olympiad 2015 - Grade 12 - Problem 3

Source: Romanian National Olympiad 2015 - Grade 12 - Problem 3

8/17/2024

Let

\mathcal{C}

be the set of all twice differentiable functions

f:[0,1] \to \mathbb{R}

with at least two (not necessarily distinct) zeros and

|f''(x)| \le 1,

for all

x \in [0,1].

Find the greatest value of the integral

\int\limits_0^1 |f(x)| \mathrm{d}x

when

f

runs through the set

\mathcal{C},

as well as the functions that achieve this maximum.
Note: A differentiable function

f

has two zeros in the same point

a

if

f(a)=f'(a)=0.

calculusIntegral calculusIntegral inequality

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