MathDB

1

JBMO Shortlist 2021 C6

m \times n

table consisting of

mn

unit cells. Alice and Bob play the following game: Alice goes first and the one who moves colors one of the empty cells with one of the given three colors. Alice wins if there is a figure, such as the ones below, having three different colors. Otherwise Bob is the winner. Determine the winner for all cases of

m

and

n

where

m, n \ge 3

.
Proposed by Toghrul Abbasov, Azerbaijan

C4

1

JBMO Shortlist 2021 C4

Alice and Bob play a game together as a team on a

100 \times 100

board with all unit squares initially white. Alice sets up the game by coloring exactly

k

of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least

10

red squares and color all of the remaining squares in it red. What is the smallest

k

such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?
Proposed by Nikola Velov, Macedonia

C3

1

JBMO Shortlist 2021 C3

We have a set of

343

closed jars, each containing blue, yellow and red marbles with the number of marbles from each color being at least

1

and at most

7

. No two jars have exactly the same contents. Initially all jars are with the caps up. To flip a jar will mean to change its position from cap-up to cap-down or vice versa. It is allowed to choose a triple of positive integers

(b; y; r) \in \{1; 2; ...; 7\}^3

and flip all the jars whose number of blue, yellow and red marbles differ by not more than

1

from

b, y, r

, respectively. After

n

moves all the jars turned out to be with the caps down. Find the number of all possible values of

n

, if

n \le 2021

.

C2

1

JBMO Shortlist 2021 C2

Let

n

be a positive integer. We are given a

3n \times 3n

board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a

2 \times 2

square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all

n

for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.
Proposed by Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina

C1

1

JBMO Shortlist 2021 C1

In Mathcity, there are infinitely many buses and infinitely many stations. The stations are indexed by the powers of

2: 1, 2, 4, 8, 16, ...

Each bus goes by finitely many stations, and the bus number is the sum of all the stations it goes by. For simplifications, the mayor of Mathcity wishes that the bus numbers form an arithmetic progression with common difference

r

and whose first term is the favourite number of the mayor. For which positive integers

r

is it always possible that, no matter the favourite number of the mayor, given any

m

stations, there is a bus going by all of them?
Proposed by Savinien Kreczman and Martin Rakovsky, France

N7

1

JBMO Shortlist 2021 N7

Alice chooses a prime number

p > 2

and then Bob chooses a positive integer

n_0

. Alice, in the first move, chooses an integer

n_1 > n_0

and calculates the expression

s_1 = n_0^{n_1} + n_1^{n_0}

; then Bob, in the second move, chooses an integer

n_2 > n_1

and calculates the expression

s_2 = n_1^{n_2} + n_2^{n_1}

; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number

n_k

such that

p

divides

s_k(s_1 + 2s_2 + · · · + ks_k)

. Who has a winning strategy?
Proposed by Borche Joshevski, Macedonia

N6

1

JBMO Shortlist 2021 N6

Given a positive integer

n \ge 2

, we define

f(n)

to be the sum of all remainders obtained by dividing

n

by all positive integers less than

n

. For example dividing

5

with

1, 2, 3

and

4

we have remainders equal to

0, 1, 2

and

1

respectively. Therefore

f(5) = 0 + 1 + 2 + 1 = 4

. Find all positive integers

n \ge 3

such that

f(n) = f(n - 1) + (n - 2)

.

N5

1

JBMO Shortlist 2021 N5

Find all pairs of integers

(x, y)

such that

x^2 + 5y^2 = 2021y

.

N4

1

JBMO Shortlist 2021 N4

Dragos, the early ruler of Moldavia, and Maria the Oracle play the following game. Firstly, Maria chooses a set

S

of prime numbers. Then Dragos gives an infinite sequence

x_1, x_2, ...

of distinct positive integers. Then Maria picks a positive integer

M

and a prime number

p

from her set

S

. Finally, Dragos picks a positive integer

N

and the game ends. Dragos wins if and only if for all integers

n \ge N

the number

x_n

is divisible by

p^M

; otherwise, Maria wins. Who has a winning strategy if the set S must be:

\hspace{5px}

a) finite;

\hspace{5px}

b) infinite?
Proposed by Boris Stanković, Bosnia and Herzegovina

N2

1

JBMO Shortlist 2021 N2

The real numbers

x, y

and

z

are such that

x^2 + y^2 + z^2 = 1

. a) Determine the smallest and the largest possible values of

xy + yz - xz

. b) Prove that there does not exist a triple

(x, y, z)

of rational numbers, which attains any of the two values in a).

N1

1

JBMO Shortlist 2021 N1

Find all positive integers

a, b, c

such that

ab + 1

,

bc + 1

, and

ca + 1

are all equal to factorials of some positive integers.
Proposed by Nikola Velov, Macedonia

G5

1

JBMO Shortlist 2021 G5

Let

ABC

be an acute scalene triangle with circumcircle

\omega

. Let

P

and

Q

be interior points of the sides

AB

and

AC

, respectively, such that

PQ

is parallel to

BC

. Let

L

be a point on

\omega

such that

AL

is parallel to

BC

. The segments

BQ

and

CP

intersect at

S

. The line

LS

intersects

\omega

at

K

. Prove that

\angle BKP = \angle CKQ

.
Proposed by Ervin Macić, Bosnia and Herzegovina

G4

1

JBMO Shortlist 2021 G4

Let

ABCD

be a convex quadrilateral with

\angle B = \angle D = 90^{\circ}

. Let

E

be the point of intersection of

BC

with

AD

and let

M

be the midpoint of

AE

. On the extension of

CD

, beyond the point

D

, we pick a point

Z

such that

MZ = \frac{AE}{2}

. Let

U

and

V

be the projections of

A

and

E

respectively on

BZ

. The circumcircle of the triangle

DUV

meets again

AE

at the point

L

. If

I

is the point of intersection of

BZ

with

AE

, prove that the lines

BL

and

CI

intersect on the line

AZ

.

G3

1

JBMO Shortlist 2021 G3

Let

ABC

be an acute triangle with circumcircle

\omega

and circumcenter

O

. The perpendicular from

A

to

BC

intersects

BC

and

\omega

at

D

and

E

, respectively. Let

F

be a point on the segment

AE

, such that

2 \cdot FD = AE

. Let

l

be the perpendicular to

OF

through

F

. Prove that

l

, the tangent to

\omega

at

E

, and the line

BC

are concurrent.
Proposed by Stefan Lozanovski, Macedonia

G2

1

JBMO Shortlist 2021 G2

Let

P

be an interior point of the isosceles triangle

ABC

with

\hat{A} = 90^{\circ}

. If

\widehat{PAB} + \widehat{PBC} + \widehat{PCA} = 90^{\circ},

prove that

AP \perp BC

.
Proposed by Mehmet Akif Yıldız, Turkey

A3

1

JBMO Shortlist 2021 A3

Let

n

be a positive integer. A finite set of integers is called

n

-divided if there are exactly

n

ways to partition this set into two subsets with equal sums. For example, the set

\{1, 3, 4, 5, 6, 7\}

is

2

-divided because the only ways to partition it into two subsets with equal sums is by dividing it into

\{1, 3, 4, 5\}

and

\{6, 7\}

, or

\{1, 5, 7\}

and

\{3, 4, 6\}

. Find all the integers

n > 0

for which there exists a

n

-divided set.
Proposed by Martin Rakovsky, France

A2

1

JBMO Shortlist 2021 A2

Let

n > 3

be a positive integer. Find all integers

k

such that

1 \le k \le n

and for which the following property holds: If

x_1, . . . , x_n

are

n

real numbers such that

x_i + x_{i + 1} + ... + x_{i + k - 1} = 0

for all integers

i > 1

(indexes are taken modulo

n

), then

x_1 = . . . = x_n = 0

.
Proposed by Vincent Jugé and Théo Lenoir, France

2021 JBMO Shortlist

Subcontests

JBMO Shortlist 2021 C6

JBMO Shortlist 2021 C4

JBMO Shortlist 2021 C3

JBMO Shortlist 2021 C2

JBMO Shortlist 2021 C1

JBMO Shortlist 2021 N7

JBMO Shortlist 2021 N6

JBMO Shortlist 2021 N5

JBMO Shortlist 2021 N4

JBMO Shortlist 2021 N2

JBMO Shortlist 2021 N1

JBMO Shortlist 2021 G5

JBMO Shortlist 2021 G4

JBMO Shortlist 2021 G3

JBMO Shortlist 2021 G2

JBMO Shortlist 2021 A3

JBMO Shortlist 2021 A2