MathDB

1

JBMO Shortlist 2023 N6

p

satisfying the following conditions:
(i)

\frac{p+1}{2}

is a prime number. (ii) There are at least three distinct positive integers

n

for which

\frac{p^2+n}{p+n^2}

is an integer.
Version 2. Let

p \neq 5

be a prime number such that

\frac{p+1}{2}

is also a prime. Suppose there exist positive integers

a <b

such that

\frac{p^2+a}{p+a^2}

and

\frac{p^2+b}{p+b^2}

are integers. Show that

b=(a-1)^2+1

.

N5

1

JBMO Shortlist 2023 N5

Find the largest positive integer

k

such that we can find a set

A \subseteq \{1,2, \ldots, 100 \}

with

k

elements such that, for any

a,b \in A

,

a

divides

b

if and only if

s(a)

divides

s(b)

, where

s(k)

denotes the sum of the digits of

k

.

N4

1

JBMO Shortlist 2023 N4

The triangle

ABC

is sectioned by

AD,BE

and

CF

(where

D \in (BC), E \in (CA)

and

F \in (AB)

) in seven disjoint polygons named regions. In each one of the nine vertices of these regions we write a digit, such that each nonzero digit appears exactly once. We assign to each side of a region the lowest common multiple of the digits at its ends, and to each region the greatest common divisor of the numbers assigned to its sides.
Find the largest possible value of the product of the numbers assigned to the regions.

N3

1

JBMO Shortlist 2023 N3

Let

A

be a subset of

\{2,3, \ldots, 28 \}

such that if

a \in A

, then the residue obtained when we divide

a^2

by

29

also belongs to

A

.
Find the minimum possible value of

|A|

.

N2

1

JBMO Shortlist 2023 N2

A positive integer is called Tiranian if it can be written as

x^2+6xy+y^2

, where

x

and

y

are (not necessarily distinct) positive integers. The integer

36^{2023}

is written as the sum of

k

Tiranian integers. What is the smallest possible value of

k

?
Proposed by Miroslav Marinov, Bulgaria

G7

1

JBMO Shortlist 2023 G7

Let

D

and

E

be arbitrary points on the sides

BC

and

AC

of triangle

ABC

, respectively. The circumcircle of

\triangle ADC

meets for the second time the circumcircle of

\triangle BCE

at point

F

. Line

FE

meets line

AD

at point

G

, while line

FD

meets line

BE

at point

H

. Prove that lines

CF, AH

and

BG

pass through the same point.

G5

1

JBMO Shortlist 2023 G5

Let

D,E,F

be the points of tangency of the incircle of a given triangle

ABC

with sides

BC, CA, AB,

respectively. Denote by

I

the incenter of

ABC

, by

M

the midpoint of

BC

and by

G

the foot of the perpendicular from

M

to line

EF

. Prove that the line

ID

is tangent to the circumcircle of the triangle

MGI

.

G4

1

JBMO Shortlist 2023 G4

Let

ABCD

be a cyclic quadrilateral, for which

B

and

C

are acute angles.

M

and

N

are the projections of the vertex

B

on the lines

AC

and

AD

, respectively,

P

and

T

are the projections of the vertex

D

on the lines

AB

and

AC

respectively,

Q

and

S

are the intersections of the pairs of lines

MN

and

CD

, and

PT

and

BC

, respectively. Prove the following statements:
a)

NS \parallel PQ \parallel AC

; b)

NP=SQ

; c)

NPQS

is a rectangle if, and only if,

AC

is a diamteter of the circumscribed circle of quadrilateral

ABCD

.

G3

1

JBMO Shortlist 2023 G3

Let

A,B,C,D

and

E

be five points lying in this order on a circle, such that

AD=BC

. The lines

AD

and

BC

meet at a point

F

. The circumcircles of the triangles

CEF

and

ABF

meet again at the point

P

.
Prove that the circumcircles of triangles

BDF

and

BEP

are tangent to each other.

G2

1

JBMO Shortlist 2023 G2

Let

ABC

be a triangle with

AB<AC

and

\omega

be its circumcircle. The tangent line to

\omega

at

A

intersects line

BC

at

D

and let

E

be a point on

\omega

such that

BE

is parallel to

AD

.

DE

intersects segment

AB

and

\omega

at

F

and

G

, respectively. The circumcircle of

BGF

intersects

BE

at

N

. The line

NF

intersects lines

AD

and

EA

at

S

and

T

, respectively. Prove that

DGST

is cyclic.

G1

1

JBMO Shortlist 2023 G1

Let

ABC

be a triangle with circumcentre

O

and circumcircle

\Omega

.

\Gamma

is the circle passing through

O,B

and tangent to

AB

at

B

. Let

\Gamma

intersect

\Omega

a second time at

P \neq B

. The circle passing through

P,C

and tangent to

AC

at

C

intersects with

\Gamma

at

M

. Prove that

|MP|=|MC|

.

C5

1

JBMO Shortlist 2023 C5

Consider an increasing sequence of real numbers

a_1<a_2<\ldots<a_{2023}

such that all pairwise sums of the elements in the sequence are different. For such a sequence, denote by

M

the number of pairs

(a_i,a_j)

such that

a_i<a_j

and

a_i+a_j<a_2+a_{2022}

. Find the minimal and the maximal possible value of

M

.

C4

1

JBMO Shortlist 2023 C4

Anna and Bob are playing the following game: The number

2

is initially written on the blackboard. With Anna playing first, they alternately double the number currently written on the blackboard or square it.
The person who first writes on the blackboard a number greater than

2023^{10}

is the winner. Determine which player has a winning strategy.

C2

1

JBMO Shortlist 2023 C2

There are

n

blocks placed on the unit squares of a

n \times n

chessboard such that there is exactly one block in each row and each column. Find the maximum value

k

, in terms of

n

, such that however the blocks are arranged, we can place

k

rooks on the board without any two of them threatening each other.
(Two rooks are not threatening each other if there is a block lying between them.)

C1

1

JBMO Shortlist 2023 C1

Given is a square board with dimensions

2023 \times 2023

, in which each unit cell is colored blue or red. There are exactly

1012

rows in which the majority of cells are blue, and exactly

1012

columns in which the majority of cells are red.
What is the maximal possible side length of the largest monochromatic square?

A7

1

JBMO Shortlist 2023 A7

Let

a_1,a_2,a_3,\ldots,a_{250}

be real numbers such that

a_1=2

and

a_{n+1}=a_n+\frac{1}{a_n^2}

for every

n=1,2, \ldots, 249

. Let

x

be the greatest integer which is less than

\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_{250}}

How many digits does

x

have?
Proposed by Miroslav Marinov, Bulgaria

A6

1

JBMO Shortlist 2023 A6

Find the maximum constant

C

such that, whenever

\{a_n \}_{n=1}^{\infty}

is a sequence of positive real numbers satisfying

a_{n+1}-a_n=a_n(a_n+1)(a_n+2)

, we have

\frac{a_{2023}-a_{2020}}{a_{2022}-a_{2021}}>C.

A5

1

JBMO Shortlist 2023 A5

Let

a \geq b \geq 1 \geq c \geq 0

be real numbers such that

a+b+c=3

. Show that

3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}

A4

1

JBMO Shortlist 2023 A4

Let

a,b,c,d

be positive real numbers with

abcd=1

. Prove that

\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2

A2

1

JBMO Shortlist 2023 A2

For positive real numbers

x,y,z

with

xy+yz+zx=1

, prove that

\frac{2}{xyz}+9xyz \geq 7(x+y+z)

A1

1

JBMO Shortlist 2023 A1

Prove that for all positive real numbers

a,b,c,d

,

\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}

and determine when equality occurs.

2023 JBMO Shortlist

Subcontests

JBMO Shortlist 2023 N6

JBMO Shortlist 2023 N5

JBMO Shortlist 2023 N4

JBMO Shortlist 2023 N3

JBMO Shortlist 2023 N2

JBMO Shortlist 2023 G7

JBMO Shortlist 2023 G5

JBMO Shortlist 2023 G4

JBMO Shortlist 2023 G3

JBMO Shortlist 2023 G2

JBMO Shortlist 2023 G1

JBMO Shortlist 2023 C5

JBMO Shortlist 2023 C4

JBMO Shortlist 2023 C2

JBMO Shortlist 2023 C1

JBMO Shortlist 2023 A7

JBMO Shortlist 2023 A6

JBMO Shortlist 2023 A5

JBMO Shortlist 2023 A4

JBMO Shortlist 2023 A2

JBMO Shortlist 2023 A1