MathDB

1

n-complete multisets

Oriol has a finite collection of cards, each one with a positive integer written on it. We say the collection is

n

-complete if for any integer

k

from

1

to

n

(inclusive), he can choose some cards such that the sum of the numbers on them is exactly

k

. Suppose that Oriol's collection is

n

-complete, but it stops being

n

-complete if any card is removed from it. What is the maximum possible sum of the numbers on all the cards?
Proposed by Ariel García

#5

1

NT Functional with factorial and exponent

Let

\mathbb{N} = \{1, 2, 3, \dots \}

be the set of positive integers. Find all functions

f:\mathbb{N}\rightarrow \mathbb{N}

, such that for all positive integers

n

and prime numbers

p

:

p \mid f(n)f(p-1)!+n^{f(p)}.

Proposed by Dorlir Ahmeti

#4

1

Orthocenter is the midpoint of the altitude

Let

ABC

be an acute triangle with orthocenter

H

. Let

A_1

,

B_1

and

C_1

be the feet of the altitudes of triangle

ABC

opposite to vertices

A

,

B

, and

C

respectively. Let

B_2

and

C_2

be the midpoints of

BB_1

and

CC_1

, respectively. Let

O

be the intersection of lines

BC_2

and

CB_2

. Prove that

O

is the circumcenter of triangle

ABC

if and only if

H

is the midpoint of

AA_1

.
Proposed by Dorlir Ahmeti

#3

1

Two circles and concyclic points

Let

\Gamma_1

and

\Gamma_2

be circles intersecting at points

A

and

B

. A line through

A

intersects

\Gamma_1

and

\Gamma_2

at

C

and

D

respectively. Let

P

be the intersection of the lines tangent to

\Gamma_1

at

A

and

C

, and let

Q

be the intersection of the lines tangent to

\Gamma_2

at

A

and

D

. Let

X

be the second intersection point of the circumcircles of

BCP

and

BDQ

, and let

Y

be the intersection of lines

AB

and

PQ

. Prove that

C

,

D

,

X

and

Y

are concyclic.
Proposed by Ariel García

#2

1

Classrooms with rare capacities

Let

n

be an integer greater than

1

. A certain school has

1+2+\dots+n

students and

n

classrooms, with capacities for

1, 2, \dots, n

people, respectively. The kids play a game in

k

rounds as follows: in each round, when the bell rings, the students distribute themselves among the classrooms in such a way that they don't exceed the room capacities, and if two students shared a classroom in a previous round, they cannot do it anymore in the current round. For each

n

, determine the greatest possible value of

k

.
Proposed by Victor Domínguez

#1

1

Floor and ceiling inequality with 2020

Let

a, b

and

c

be real numbers such that

\lceil a \rceil + \lceil b \rceil + \lceil c \rceil + \lfloor a + b \rfloor + \lfloor b + c \rfloor + \lfloor c + a \rfloor = 2020

Prove that

\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor + \lceil a + b + c \rceil \ge 1346

Note:

\lfloor x \rfloor

is the greatest integer less than or equal to

x

, and

\lceil x \rceil

is the smallest integer greater than or equal to

x

. That is,

\lfloor x \rfloor

is the unique integer satisfying

\lfloor x \rfloor \le x < \lfloor x \rfloor + 1

, and

\lceil x \rceil

is the unique integer satisfying

\lceil x \rceil - 1 < x \le \lceil x \rceil

.
Proposed by Ariel García

Mexican Quarantine Mathematical Olympiad

Subcontests

n-complete multisets

NT Functional with factorial and exponent

Orthocenter is the midpoint of the altitude

Two circles and concyclic points

Classrooms with rare capacities

Floor and ceiling inequality with 2020