P4

Part of 2024 District Olympiad

Problems(4)

Source: Romanian District Olympiad 2024 9.4

H{}

be the orthocenter of the triangle

ABC{}

X{}

be the midpoint of the side

BC.

The perpendicular at

H{}

HX{}

intersects the sides

(AB)

(AC)

Y{}

Z{}

respectively. Let

O{}

be the circumcenter of

ABC{}

O'

be the circumcenter of

BHC.

HY=HZ.

\overrightarrow{AY}+\overrightarrow{AZ}=2\overrightarrow{OO'}.

injectivity in 0

Source: Romanian District Olympiad, grade 10, p4

n\in\mathbb{N}\setminus\left\{0\right\}

be a positive integer. Find all the functions

f:\mathbb{R}\rightarrow \mathbb{R}

satisfying that :

f(x+y^{2n})=f(f(x))+y^{2n-1}f(y),(\forall)x,y\in\mathbb{R},

f(x)=0

has an unique solution.

algebrafunctional equation

Yet another convergence problem

Source: Romanian District Olympiad 2024 11.4

Consider the functions

f,g:\mathbb{R}\to\mathbb{R}

f{}

is continous. For any real numbers

a<b<c

there exists a sequence

(x_n)_{n\geqslant 1}

which converges to

b{}

and for which the limit of

g(x_n)

n{}

tends to infinity exists and satisfies

f(a)<\lim_{n\to\infty}g(x_n)<f(c).

[*]Give an example of a pair of such functions

f,g

g{}

is discontinous at every point. [*]Prove that if

g{}

is monotonous, then

f=g.

real analysislimit

Cute integral inequality

Source: Romanian District Olympiad 2024 12.4

f:[0,\infty)\to\mathbb{R}

be a differentiable function, with a continous derivative. Given that

f(0)=0

0\leqslant f'(x)\leqslant 1

x>0

\frac{1}{n+1}\int_0^af(t)^{2n+1}\mathrm{d}t\leqslant\left(\int_0^af(t)^n\mathrm{d}t\right)^2,

for any positive integer

n{}

and real number

a>0.

Integralinequalitiesreal analysis