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Romania District MO 2022 Grade 9 P4

Source: Romania District MO 2022 Grade 9

3/28/2022

We call a set of

6

points in the plane splittable if we if can denote its elements by

A,B,C,D,E

and

F

in such a way that

\triangle ABC

and

\triangle DEF

have the same centroid.
[*]Construct a splittable set. [*]Show that any set of

7

points has a subset of

6

points which is not splittable.

romaniageometryvector

Romania District MO 2022 Grade 10 P4

Source: Romania District MO 2022 Grade 10

3/28/2022

Find all functions

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

which satisfy the following relationship for all

x,y\in\mathbb{R}:

f(f(y-x)-xf(y))+f(x)=y\cdot(1-f(x)).

Romania District MO 2022 Grade 11 P4

Source: Romania District MO 2022 Grade 11

3/27/2022

Let

A\in\mathcal{M}_n(\mathbb{C})

where

n\geq 2.

Prove that if

m=|\{\text{rank}(A^k)-\text{rank}(A^{k+1})":k\in\mathbb{N}^*\}|

then

n+1\geq m(m+1)/2.

matrixrankcollege contestsromania

Romania District MO 2022 Grade 12 P4

Source: Romania District MO 2022 Grade 12

3/27/2022

Let

I\subseteq \mathbb{R}

be an open interval and

f:I\to\mathbb{R}

a strictly monotonous function. Prove that for all

c\in I

there exist

a,b\in I

such that

c\in (a,b)

and

\int_a^bf(x) \ dx=f(c)\cdot (b-a).

Integralfunctionromaniacollege contests

P4