MathDB

1

JBMO Shortlist 2019 G6

ABC

be a non-isosceles triangle with incenter

I

. Let

D

be a point on the segment

BC

such that the circumcircle of

BID

intersects the segment

AB

at

E\neq B

, and the circumcircle of

CID

intersects the segment

AC

at

F\neq C

. The circumcircle of

DEF

intersects

AB

and

AC

at the second points

M

and

N

respectively. Let

P

be the point of intersection of

IB

and

DE

, and let

Q

be the point of intersection of

IC

and

DF

. Prove that the three lines

EN, FM

and

PQ

are parallel.
Proposed by Saudi Arabia

G3

1

JBMO Shortlist 2019 G3

Let

ABC

be a triangle with incenter

I

. The points

D

and

E

lie on the segments

CA

and

BC

respectively, such that

CD = CE

. Let

F

be a point on the segment

CD

. Prove that the quadrilateral

ABEF

is circumscribable if and only if the quadrilateral

DIEF

is cyclic.
Proposed by Dorlir Ahmeti, Albania

G1

1

JBMO Shortlist 2019 G1

Let

ABC

be a right-angled triangle with

\angle A = 90^{\circ}

and

\angle B = 30^{\circ}

. The perpendicular at the midpoint

M

of

BC

meets the bisector

BK

of the angle

B

at the point

E

. The perpendicular bisector of

EK

meets

AB

at

D

. Prove that

KD

is perpendicular to

DE

.
Proposed by Greece

G5

1

JBMO Shortlist 2019 G5

Let

P

be a point in the interior of a triangle

ABC

. The lines

AP, BP

and

CP

intersect again the circumcircles of the triangles

PBC, PCA

and

PAB

at

D, E

and

F

respectively. Prove that

P

is the orthocenter of the triangle

DEF

if and only if

P

is the incenter of the triangle

ABC

.
Proposed by Romania

G7

1

JBMO Shortlist 2019 G7

Let

ABC

be a right-angled triangle with

\angle A = 90^{\circ}

. Let

K

be the midpoint of

BC

, and let

AKLM

be a parallelogram with centre

C

. Let

T

be the intersection of the line

AC

and the perpendicular bisector of

BM

. Let

\omega_1

be the circle with centre

C

and radius

CA

and let

\omega_2

be the circle with centre

T

and radius

TB

. Prove that one of the points of intersection of

\omega_1

and

\omega_2

is on the line

LM

.
Proposed by Greece

N7

1

JBMO Shortlist 2019 N7

Find all perfect squares

n

such that if the positive integer

a\ge 15

is some divisor

n

then

a+15

is a prime power.
Proposed by Saudi Arabia

N5

1

JBMO Shortlist 2019 N5

Find all positive integers

x, y, z

such that

45^x-6^y=2019^z

Proposed by Dorlir Ahmeti, Albania

N4

1

JBMO Shortlist 2019 N4

Find all integers

x,y

such that

x^3(y+1)+y^3(x+1)=19

.
Proposed by Bulgaria

N3

1

JBMO Shortlist 2019 N3

Find all prime numbers

p

and nonnegative integers

x\neq y

such that

x^4- y^4=p(x^3-y^3)

.
Proposed by Bulgaria

N2

1

JBMO Shortlist 2019 N2

Find all triples

(p, q, r)

of prime numbers such that all of the following numbers are integers

\frac{p^2 + 2q}{q+r}, \frac{q^2+9r}{r+p}, \frac{r^2+3p}{p+q}

Proposed by Tajikistan

C5

1

JBMO Shortlist 2019 C5

An economist and a statistician play a game on a calculator which does only one operation. The calculator displays only positive integers and it is used in the following way: Denote by

n

an integer that is shown on the calculator. A person types an integer,

m

, chosen from the set

\{ 1, 2, . . . , 99 \}

of the first

99

positive integers, and if

m\%

of the number

n

is again a positive integer, then the calculator displays

m\%

of

n

. Otherwise, the calculator shows an error message and this operation is not allowed. The game consists of doing alternatively these operations and the player that cannot do the operation looses. How many numbers from

\{1, 2, . . . , 2019\}

guarantee the winning strategy for the statistician, who plays second?
For example, if the calculator displays

1200

, the economist can type

50

, giving the number

600

on the calculator, then the statistician can type

25

giving the number

150

. Now, for instance, the economist cannot type

75

as

75\%

of

150

is not a positive integer, but can choose

40

and the game continues until one of them cannot type an allowed number
Proposed by Serbia

C4

1

JBMO Shortlist 2019 C4

We have a group of

n

kids. For each pair of kids, at least one has sent a message to the other one. For each kid

A

, among the kids to whom

A

has sent a message, exactly

25 \%

have sent a message to

A

. How many possible two-digit values of

n

are there?
Proposed by Bulgaria

C2

1

JBMO Shortlist 2019 C2

In a certain city there are

n

straight streets, such that every two streets intersect, and no three streets pass through the same intersection. The City Council wants to organize the city by designating the main and the side street on every intersection. Prove that this can be done in such way that if one goes along one of the streets, from its beginning to its end, the intersections where this street is the main street, and the ones where it is not, will apear in alternating order.
Proposed by Serbia

C1

1

JBMO Shortlist 2019 C1

Let

S

be a set of

100

positive integer numbers having the following property: “Among every four numbers of

S

, there is a number which divides each of the other three or there is a number which is equal to the sum of the other three.” Prove that the set

S

contains a number which divides all other

99

numbers of

S

.
Proposed by Tajikistan

A7

1

JBMO Shortlist 2019 A7

Show that for any positive real numbers

a, b, c

such that

a + b + c = ab + bc + ca

, the following inequality holds

3 + \sqrt[3]{\frac{a^3+1}{2}}+\sqrt[3]{\frac{b^3+1}{2}}+\sqrt[3]{\frac{c^3+1}{2}}\leq 2(a+b+c)

Proposed by Dorlir Ahmeti, Albania

A6

1

JBMO Shortlist 2019 A6

Let

a, b, c

be positive real numbers. Prove the inequality

(a^2+ac+c^2) \left( \frac{1}{a+b+c}+\frac{1}{a+c} \right)+b^2 \left( \frac{1}{b+c}+\frac{1}{a+b} \right)>a+b+c

.
Proposed by Tajikistan

A5

1

JBMO Shortlist 2019 A5

Let

a, b, c, d

be positive real numbers such that

abcd = 1

. Prove the inequality

\frac{1}{a^3 + b + c + d} +\frac{1}{a + b^3 + c + d}+\frac{1}{a + b + c^3 + d} +\frac{1}{a + b + c + d^3} \leq \frac{a+b+c+d}{4}

Proposed by Romania

A3

1

JBMO Shortlist 2019 A3

Let

A

and

B

be two non-empty subsets of

X = \{1, 2, . . . , 11 \}

with

A \cup B = X

. Let

P_A

be the product of all elements of

A

and let

P_B

be the product of all elements of

B

. Find the minimum and maximum possible value of

P_A +P_B

and find all possible equality cases.
Proposed by Greece

A1

1

JBMO Shortlist 2019 A1

Real numbers

a

and

b

satisfy

a^3+b^3-6ab=-11

. Prove that

-\frac{7}{3}<a+b<-2

. Proposed by Serbia

N6

1

Problem 5

a,b,c

are non-negative integers. Solve:

a!+5^b=7^c

Proposed by Serbia

G2

1

Geometry

Let

ABC

be a triangle with circumcircle

\omega

. Let

l_B

and

l_C

be two lines through the points

B

and

C

, respectively, such that

l_B \parallel l_C

. The second intersections of

l_B

and

l_C

with

\omega

are

D

and

E

, respectively. Assume that

D

and

E

are on the same side of

BC

as

A

. Let

DA

intersect

l_C

at

F

and let

EA

intersect

l_B

at

G

. If

O

,

O_1

and

O_2

are circumcenters of the triangles

ABC

,

ADG

and

AEF

, respectively, and

P

is the circumcenter of the triangle

OO_1O_2

, prove that

l_B \parallel OP \parallel l_C

.
Proposed by Stefan Lozanovski, Macedonia

A2

1

Simple inequality

Let

a, b, c

be positive real numbers such that

abc = \frac {2} {3}.

Prove that:

\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.

2019 JBMO Shortlist

Subcontests

JBMO Shortlist 2019 G6

JBMO Shortlist 2019 G3

JBMO Shortlist 2019 G1

JBMO Shortlist 2019 G5

JBMO Shortlist 2019 G7

JBMO Shortlist 2019 N7

JBMO Shortlist 2019 N5

JBMO Shortlist 2019 N4

JBMO Shortlist 2019 N3

JBMO Shortlist 2019 N2

JBMO Shortlist 2019 C5

JBMO Shortlist 2019 C4

JBMO Shortlist 2019 C2

JBMO Shortlist 2019 C1

JBMO Shortlist 2019 A7

JBMO Shortlist 2019 A6

JBMO Shortlist 2019 A5

JBMO Shortlist 2019 A3

JBMO Shortlist 2019 A1

Problem 5

Geometry

Simple inequality