MathDB

1

JBMO 2018 P3, the shortlist version, min k for the nxn system

k > 1, n > 2018

be positive integers, and let

n

be odd. The nonzero rational numbers

x_1,x_2,\ldots,x_n

are not all equal and satisfy

x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}

Find: a) the product

x_1 x_2 \ldots x_n

as a function of

k

and

n

b) the least value of

k

, such that there exist

n,x_1,x_2,\ldots,x_n

satisfying the given conditions.

G6

1

MP \cdot XY \ge 2 \cdot QX \cdot PY, related to a chord

Let

XY

be a chord of a circle

\Omega

, with center

O

, which is not a diameter. Let

P, Q

be two distinct points inside the segment

XY

, where

Q

lies between

P

and

X

. Let

\ell

the perpendicular line drawn from

P

to the diameter which passes through

Q

. Let

M

be the intersection point of

\ell

and

\Omega

, which is closer to

P

. Prove that

MP \cdot XY \ge 2 \cdot QX \cdot PY

G5

1

strange area condition given (GCE)=1/2 (a^3/b+ab) inside a rectangle

Given a rectangle

ABCD

such that

AB = b > 2a = BC

, let

E

be the midpoint of

AD

. On a line parallel to

AB

through point

E

, a point

G

is chosen such that the area of

GCE

is

(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)

Point

H

is the foot of the perpendicular from

E

to

GD

and a point

I

is taken on the diagonal

AC

such that the triangles

ACE

and

AEI

are similar. The lines

BH

and

IE

intersect at

K

and the lines

CA

and

EH

intersect at

J

. Prove that

KJ \perp AB

.

G4

1

sum R^4/P_i^2 >= 16, geometric inequality with areas P_i, by incenter I

Let

ABC

be a triangle with side-lengths

a, b, c

, inscribed in a circle with radius

R

and let

I

be ir's incenter. Let

P_1, P_2

and

P_3

be the areas of the triangles

ABI, BCI

and

CAI

, respectively. Prove that

\frac{R^4}{P_1^2}+\frac{R^4}{P_2^2}+\frac{R^4}{P_3^2}\ge 16

G2

1

< NA'T = < ADT wanted, starting with a right triangle, symmetric, projections

Let

ABC

be a right angled triangle with

\angle A = 90^o

and

AD

its altitude. We draw parallel lines from

D

to the vertical sides of the triangle and we call

E, Z

their points of intersection with

AB

and

AC

respectively. The parallel line from

C

to

EZ

intersects the line

AB

at the point

N

. Let

A'

be the symmetric of

A

with respect to the line

EZ

and

I, K

the projections of

A'

onto

AB

and

AC

respectively. If

T

is the point of intersection of the lines

IK

and

DE

, prove that

\angle NA'T = \angle ADT

.

G1

1

MN bisects segment CH

Let

H

be the orthocentre of an acute triangle

ABC

with

BC > AC

, inscribed in a circle

\Gamma

. The circle with centre

C

and radius

CB

intersects

\Gamma

at the point

D

, which is on the arc

AB

not containing

C

. The circle with centre

C

and radius

CA

intersects the segment

CD

at the point

K

. The line parallel to

BD

through

K

, intersects

AB

at point

L

. If

M

is the midpoint of

AB

and

N

is the foot of the perpendicular from

H

to

CL

, prove that the line

MN

bisects the segment

CH

.

C3

1

Alice and Bob play, 8x8 table, white red black, minimum n for victory

The cells of a

8 \times 8

table are initially white. Alice and Bob play a game. First Alice paints

n

of the fields in red. Then Bob chooses

4

rows and

4

columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of

n

such that Alice can win the game no matter how Bob plays.

C1

1

if x \in S => (x - 1) or (x+1) \in S, S has at least 4 elements

A set

S

is called neighbouring if it has the following two properties: a)

S

has exactly four elements b) for every element

x

of

S

, at least one of the numbers

x - 1

or

x+1

belongs to

S

. Find the number of all neighbouring subsets of the set

\{1,2,... ,n\}

.

A5

1

0<= a,b,c,d<=1<=x,y,z,t and a+b+c+d +x+y+z+t=8 show sum of squares<=28

Let a

,b,c,d

and

x,y,z,t

be real numbers such that

0\le a,b,c,d \le 1

,

x,y,z,t \ge 1

and

a+b+c+d +x+y+z+t=8

. Prove that

a^2+b^2+c^2+d^2+x^2+y^2+z^2+t^2\le 28

A7

1

JBMO 2018. Shortlist Algebra

Let

A

be a set of positive integers satisfying the following :

a.)

If

n \in A

, then

n \le 2018

.

b.)

If

S \subset A

such that

|S|=3

, then there exists

m,n \in S

such that

|n-m| \ge \sqrt{n}+\sqrt{m}

What is the maximum cardinality of

A

?

A1

1

JBMO 2018. Shortlist Algebra

Let

x,y,z

be positive real numbers . Prove:

\frac{x}{\sqrt{\sqrt[4]{y}+\sqrt[4]{z}}}+\frac{y}{\sqrt{\sqrt[4]{z}+\sqrt[4]{x}}}+\frac{z}{\sqrt{\sqrt[4]{x}+\sqrt[4]{y}}}\geq \frac{\sqrt[4]{(\sqrt{x}+\sqrt{y}+\sqrt{z})^7}}{\sqrt{2\sqrt{27}}}

NT3

1

Number theory shortlist jbmo 2018

Find all positive integers

abcd=a^{a+b+c+d} - a^{-a+b-c+d} + a

, where

abcd

is a four-digit number

NT2

1

JBMO 2018. Shortlist NT

Find all ordered pairs of positive integers

(m,n)

such that :

125*2^n-3^m=271

NT4

1

JBMO 2018. Shortlist NT

Prove that there exist infinitely many positive integers

n

such that

\frac{4^n+2^n+1}{n^2+n+1}

is a positive integer.

A2

1

m^3+n^3>(m+n)^2

Find the maximum positive integer

k

such that for any positive integers

m,n

such that

m^3+n^3>(m+n)^2

, we have

m^3+n^3\geq (m+n)^2+k

Proposed by Dorlir Ahmeti, Albania

A6

1

JBMO shortlist 2018 inequality

For

a,b,c

positive real numbers such that

ab+bc+ca=3

, prove:

\frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}

Proposed by Dorlir Ahmeti, Albania

A3

1

2019 Junior Pre-selection question

Let

a,b,c

be positive real numbers . Prove that

\frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.

2018 JBMO Shortlist

Subcontests

JBMO 2018 P3, the shortlist version, min k for the nxn system

MP \cdot XY \ge 2 \cdot QX \cdot PY, related to a chord

strange area condition given (GCE)=1/2 (a^3/b+ab) inside a rectangle

sum R^4/P_i^2 >= 16, geometric inequality with areas P_i, by incenter I

< NA'T = < ADT wanted, starting with a right triangle, symmetric, projections

MN bisects segment CH

Alice and Bob play, 8x8 table, white red black, minimum n for victory

if x \in S => (x - 1) or (x+1) \in S, S has at least 4 elements

0<= a,b,c,d<=1<=x,y,z,t and a+b+c+d +x+y+z+t=8 show sum of squares<=28

JBMO 2018. Shortlist Algebra

JBMO 2018. Shortlist Algebra

Number theory shortlist jbmo 2018

JBMO 2018. Shortlist NT

JBMO 2018. Shortlist NT

m^3+n^3>(m+n)^2

JBMO shortlist 2018 inequality

2019 Junior Pre-selection question

2018 JBMO Shortlist

Subcontests

JBMO 2018 P3, the shortlist version, min k for the nxn system

MP \cdot XY \ge 2 \cdot QX \cdot PY, related to a chord

strange area condition given (GCE)=1/2 (a^3/b+ab) inside a rectangle

sum R^4/P_i^2 &gt;= 16, geometric inequality with areas P_i, by incenter I

&lt; NA'T = &lt; ADT wanted, starting with a right triangle, symmetric, projections

MN bisects segment CH

Alice and Bob play, 8x8 table, white red black, minimum n for victory

if x \in S =&gt; (x - 1) or (x+1) \in S, S has at least 4 elements

0&lt;= a,b,c,d&lt;=1&lt;=x,y,z,t and a+b+c+d +x+y+z+t=8 show sum of squares&lt;=28

JBMO 2018. Shortlist Algebra

JBMO 2018. Shortlist Algebra

Number theory shortlist jbmo 2018

JBMO 2018. Shortlist NT

JBMO 2018. Shortlist NT

m^3+n^3&gt;(m+n)^2

JBMO shortlist 2018 inequality

2019 Junior Pre-selection question

sum R^4/P_i^2 >= 16, geometric inequality with areas P_i, by incenter I

< NA'T = < ADT wanted, starting with a right triangle, symmetric, projections

if x \in S => (x - 1) or (x+1) \in S, S has at least 4 elements

0<= a,b,c,d<=1<=x,y,z,t and a+b+c+d +x+y+z+t=8 show sum of squares<=28

m^3+n^3>(m+n)^2