MathDB

1

JBMO Shortlist 2022 N6

n

for which there exists an integer multiple of

2022

such that the sum of the squares of its digits is equal to

n

.

N5

1

JBMO Shortlist 2022 N5

Find all pairs

(a, p)

of positive integers, where

p

is a prime, such that for any pair of positive integers

m

and

n

the remainder obtained when

a^{2^n}

is divided by

p^n

is non-zero and equals the remainder obtained when

a^{2^m}

is divided by

p^m

.

N4

1

JBMO Shortlist 2022 N4

Consider the sequence

u_0, u_1, u_2, ...

defined by

u_0 = 0, u_1 = 1,

and

u_n = 6u_{n - 1} + 7u_{n - 2}

for

n \ge 2

. Show that there are no non-negative integers

a, b, c, n

such that

ab(a + b)(a^2 + ab + b^2) = c^{2022} + 42 = u_n.

N2

1

JBMO Shortlist 2022 N2

Let

a < b < c < d < e

be positive integers. Prove that

\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1

where

[x, y]

is the least common multiple of

x

and

y

(e.g.,

[6, 10] = 30

). When does equality hold?

N1

1

JBMO Shortlist 2022 N1

Determine all pairs

(k, n)

of positive integers that satisfy

1! + 2! + ... + k! = 1 + 2 + ... + n.

G6

1

JBMO Shortlist 2022 G6

Let

ABC

be a right triangle with hypotenuse

BC

. The tangent to the circumcircle of triangle

ABC

at

A

intersects the line

BC

at

T

. The points

D

and

E

are chosen so that

AD = BD, AE = CE,

and

\angle CBD = \angle BCE < 90^{\circ}

. Prove that

D, E,

and

T

are collinear.
Proposed by Nikola Velov, Macedonia

G5

1

JBMO Shortlist 2022 G5

Given is an acute angled triangle

ABC

with orthocenter

H

and circumcircle

k

. Let

\omega

be the circle with diameter

AH

and

P

be the point of intersection of

\omega

and

k

other than

A

. Assume that

BP

and

CP

intersect

\omega

for the second time at points

Q

and

R

, respectively. If

D

is the foot of the altitude from

A

to

BC

and

S

is the point of the intersection of

\omega

and

QD

, prove that

HR = HS

.

G4

1

JBMO Shortlist 2022 G4

Given is an equilateral triangle

ABC

and an arbitrary point, denoted by

E

, on the line segment

BC

. Let

l

be the line through

A

parallel to

BC

and let

K

be the point on

l

such that

KE

is perpendicular to

BC

. The circle with centre

K

and radius

KE

intersects the sides

AB

and

AC

at

M

and

N

, respectively. The line perpendicular to

AB

at

M

intersects

l

at

D

, and the line perpendicular to

AC

at

N

intersects

l

at

F

. Show that the point of intersection of the angle bisectors of angles

MDA

and

NFA

belongs to the line

KE

.

G2

1

JBMO Shortlist 2022 G2

Let

ABC

be a triangle with circumcircle

k

. The points

A_1, B_1,

and

C_1

on

k

are the midpoints of arcs

\widehat{BC}

(not containing

A

),

\widehat{AC}

(not containing

B

), and

\widehat{AB}

(not containing

C

), respectively. The pairwise distinct points

A_2, B_2,

and

C_2

are chosen such that the quadrilaterals

AB_1A_2C_1, BA_1B_2C_1,

and

CA_1C_2B_1

are parallelograms. Prove that

k

and the circumcircle of triangle

A_2B_2C_2

have a common center. Comment. Point

A_2

can also be defined as the reflection of

A

with respect to the midpoint of

B_1C_1

, and analogous definitions can be used for

B_2

and

C_2

.

G1

1

JBMO Shortlist 2022 G1

Let

ABCDE

be a cyclic pentagon such that

BC = DE

and

AB

is parallel to

DE

. Let

X, Y,

and

Z

be the midpoints of

BD, CE,

and

AE

respectively. Show that

AE

is tangent to the circumcircle of the triangle

XYZ

.
Proposed by Nikola Velov, Macedonia

C6

1

JBMO Shortlist 2022 C6

Let

n \ge 2

be an integer. In each cell of a

4n \times 4n

table we write the sum of the cell row index and the cell column index. Initially, no cell is colored. A move consists of choosing two cells which are not colored and coloring one of them in red and one of them in blue. Show that, however Alex perfors

n^2

moves, Jane can afterwards perform a number of moves (eventually none) after which the sum of the numbers written in the red cells is the same as the sum of the numbers written in the blue ones.

C5

1

JBMO Shortlist 2022 C5

Let

S

be a finite set of points in the plane, such that for each

2

points

A

and

B

in

S

, the segment

AB

is a side of a regular polygon all of whose vertices are contained in

S

. Find all possible values for the number of elements of

S

.
Proposed by Viktor Simjanoski, Macedonia

C3

1

JBMO Shortlist 2022 C3

There are

200

boxes on the table. In the beginning, each of the boxes contains a positive integer (the integers are not necessarily distinct). Every minute, Alice makes one move. A move consists of the following. First, she picks a box

X

which contains a number

c

such that

c = a + b

for some numbers

a

and

b

which are contained in some other boxes. Then she picks a positive integer

k > 1

. Finally, she removes

c

from

X

and replaces it with

kc

. If she cannot make any mobes, she stops. Prove that no matter how Alice makes her moves, she won't be able to make infinitely many moves.

C2

1

JBMO Shortlist 2022 C2

Let

n \ge 2

be an integer. Alex writes the numbers

1, 2, ..., n

in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment

s

we denote by

d_s

the difference of the numbers written in its extremities and by

p_s

the number of all other drawn segments which intersect

s

in its interior. Find the greatest

n

for which Alex can write the numbers on the circle such that

p_s \le |d_s|

, for each drawn segment

s

.

C1

1

JBMO Shortlist 2022 C1

Anna and Bob, with Anna starting first, alternately color the integers of the set

S = \{1, 2, ..., 2022 \}

red or blue. At their turn each one can color any uncolored number of

S

they wish with any color they wish. The game ends when all numbers of

S

get colored. Let

N

be the number of pairs

(a, b)

, where

a

and

b

are elements of

S

, such that

a

,

b

have the same color, and

b - a = 3

. Anna wishes to maximize

N

. What is the maximum value of

N

that she can achieve regardless of how Bob plays?

A6

1

JBMO Shortlist 2022 A6

Let

a, b,

and

c

be positive real numbers such that

a^2 + b^2 + c^2 = 3

. Prove that

\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 + |(a - b)(b - c)(c - a)|.

A5

1

JBMO Shortlist 2022 A5

The numbers

2, 2, ..., 2

are written on a blackboard (the number

2

is repeated

n

times). One step consists of choosing two numbers from the blackboard, denoting them as

a

and

b

, and replacing them with

\sqrt{\frac{ab + 1}{2}}

.

(a)

If

x

is the number left on the blackboard after

n - 1

applications of the above operation, prove that

x \ge \sqrt{\frac{n + 3}{n}}

.

(b)

Prove that there are infinitely many numbers for which the equality holds and infinitely many for which the inequality is strict.

A4

1

JBMO Shortlist 2022 A4

Suppose that

a, b,

and

c

are positive real numbers such that

a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.

Find the largest possible value of the expression

\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.

A3

1

JBMO Shortlist 2022 A3

Let

a, b,

and

c

be positive real numbers such that

a + b + c = 1

. Prove the following inequality

a \sqrt[3]{\frac{b}{a}} + b \sqrt[3]{\frac{c}{b}} + c \sqrt[3]{\frac{a}{c}} \le ab + bc + ca + \frac{2}{3}.

Proposed by Anastasija Trajanova, Macedonia

A2

1

JBMO Shortlist 2022 A2

Let

x, y,

and

z

be positive real numbers such that

xy + yz + zx = 3

. Prove that

\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.

Proposed by Petar Filipovski, Macedonia

2022 JBMO Shortlist

Subcontests

JBMO Shortlist 2022 N6

JBMO Shortlist 2022 N5

JBMO Shortlist 2022 N4

JBMO Shortlist 2022 N2

JBMO Shortlist 2022 N1

JBMO Shortlist 2022 G6

JBMO Shortlist 2022 G5

JBMO Shortlist 2022 G4

JBMO Shortlist 2022 G2

JBMO Shortlist 2022 G1

JBMO Shortlist 2022 C6

JBMO Shortlist 2022 C5

JBMO Shortlist 2022 C3

JBMO Shortlist 2022 C2

JBMO Shortlist 2022 C1

JBMO Shortlist 2022 A6

JBMO Shortlist 2022 A5

JBMO Shortlist 2022 A4

JBMO Shortlist 2022 A3

JBMO Shortlist 2022 A2