MathDB

Romania NMO 2023 Grade 8 P4

Source: Romania National Olympiad 2023

4/14/2023

Let

ABCD

be a tetrahedron and

M

and

N

be the midpoints of

AC

and

BD

, respectively. Show that for every point

P \in (MN)

with

P \neq M

and

P \neq N

, there exist unique points

X

and

Y

on segments

AB

and

CD

, respectively, such that

X,P,Y

are collinear.

3D geometrygeometrytetrahedron

Romania NMO 2023 Grade 5 P4

Source: Romania National Olympiad 2023

4/14/2023

We say that a number

n \ge 2

has the property

(P)

if, in its prime factorization, at least one of the factors has an exponent

3

.
a) Determine the smallest number

N

with the property that, no matter how we choose

N

consecutive natural numbers, at least one of them has the property

(P).

b) Determine the smallest

15

consecutive numbers

a_1, a_2, \ldots, a_{15}

that do not have the property

(P),

such that the sum of the numbers

5 a_1, 5 a_2, \ldots, 5 a_{15}

is a number with the property

(P).

algebranumber theory

Romania NMO 2023 Grade 6 P4

Source: Romania National Olympiad 2023

4/14/2023

Let

ABC

be a triangle with

\angle BAC = 90^{\circ}

and

\angle ACB = 54^{\circ}.

We construct bisector

BD (D \in AC)

of angle

ABC

and consider point

E \in (BD)

such that

DE = DC.

Show that

BE = 2 \cdot AD.

geometryangles

Romania NMO 2023 Grade 7 P4

Source: Romania National Olympiad 2023

4/14/2023

a) Show that there exist irrational numbers

a

,

b

, and

c

such that the numbers

a+b\cdot c

,

b+a\cdot c

, and

c+a\cdot b

are rational numbers.
b) Show that if

a

,

b

, and

c

are real numbers such that

a+b+c=1

, and the numbers

a+b\cdot c

,

b+a\cdot c

, and

c+a\cdot b

are rational and non-zero, then

a

,

b

, and

c

are rational numbers.

irrational numberalgebra

Romania NMO 2023 Grade 10 P4

Source: Romania National Olympiad 2023

4/14/2023

In an art museum,

n

paintings are exhibited, where

n \geq 33.

In total,

15

colors are used for these paintings such that any two paintings have at least one common color, and no two paintings have exactly the same colors. Determine all possible values of

n \geq 33

such that regardless of how we color the paintings with the given properties, we can choose four distinct paintings, which we can label as

T_1, T_2, T_3,

and

T_4,

such that any color that is used in both

T_1

and

T_2

can also be found in either

T_3

or

T_4

.

combinatoricsColoring

Romania NMO 2023 Grade 9 P4

Source: Romania National Olympiad 2023

4/14/2023

Let

r

and

s

be real numbers in the interval

[1, \infty)

such that for all positive integers

a

and

b

with

a \mid b \implies \left\lfloor ar \right\rfloor

divides

\left\lfloor bs \right\rfloor

.
a) Prove that

\frac{s}{r}

is a natural number.
b) Show that both

r

and

s

are natural numbers.
Here,

\lfloor x \rfloor

denotes the greatest integer that is less than or equal to

x

.

floor functionnumber theory

Romania NMO 2023 Grade 11 P4

Source: Romania National Olympiad 2023

4/14/2023

We consider a function

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

for which there exist a differentiable function

g : \mathbb{R} \rightarrow \mathbb{R}

and exist a sequence

(a_n)_{n \geq 1}

of real positive numbers, convergent to

0,

such that

g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}.

a) Give an example of such a function f that is not differentiable at any point

x \in \mathbb{R}.

b) Show that if

f

is continuous on

\mathbb{R}

, then

f

is differentiable on

\mathbb{R}.

real analysisdifferentiable functionContinous function

Romania NMO 2023 Grade 12 P4

Source: Romania National Olympiad 2023

4/14/2023

Let

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

a non-decreasing function,

f \in C^1,

for which

f(0) = 0.

Let

g:[0,1] \rightarrow \mathbb{R}

a function defined by

g(x) = f(x) + (x - 1) f'(x), \forall x \in [0,1].

a) Show that

\int_{0}^{1} g(x) \text{dx} = 0.

b) Prove that for all functions

\phi :[0,1] \rightarrow [0,1],

convex and differentiable with

\phi(0) = 0

and

\phi(1) = 1,

the inequality holds

\int_{0}^{1} g( \phi(t)) \text{dt} \leq 0.

real analysisintegrationmonotone functionsdifferentiable function

4

Problems(8)