MathDB

1

if 2 of 6 quadrilaterals are concyclic, then all 6 are concyclic

P

lies in the interior of the triangle

ABC

. The lines

AP, BP

, and

CP

intersect

BC, CA

, and

AB

at points

D, E

, and

F

, respectively. Prove that if two of the quadrilaterals

ABDE, BCEF, CAFD, AEPF, BFPD

, and

CDPE

are concyclic, then all six are concyclic.

G3

1

many angleswith point on the arc of circumcircles

Consider triangle

ABC

such that

AB \le AC

. Point

D

on the arc

BC

of thecircumcirle of

ABC

not containing point

A

and point

E

on side

BC

are such that

\angle BAD = \angle CAE < \frac12 \angle BAC

. Let

S

be the midpoint of segment

AD

. If

\angle ADE = \angle ABC - \angle ACB

prove that

\angle BSC = 2 \angle BAC

.

G2

1

prove that D is circumcenter of MNK

Let

ABC

be an acute triangle such that

AB

is the shortest side of the triangle. Let

D

be the midpoint of the side

AB

and

P

be an interior point of the triangle such that

\angle CAP = \angle CBP = \angle ACB

. Denote by M and

N

the feet of the perpendiculars from

P

to

BC

and

AC

, respectively. Let

p

be the line through

M

parallel to

AC

and

q

be the line through

N

parallel to

BC

. If

p

and

q

intersect at

K

prove that

D

is the circumcenter of triangle

MNK

.

G1

1

parallelogram, circle and collinear points,

Given a parallelogram

ABCD

. The line perpendicular to

AC

passing through

C

and the line perpendicular to

BD

passing through

A

intersect at point

P

. The circle centered at point

P

and radius

PC

intersects the line

BC

at point

X

, (

X \ne C

) and the line

DC

at point

Y

, (

Y \ne C

). Prove that the line

AX

passes through the point

Y

.

A2

1

minimum of \sqrt{\frac{a}{b(3a+2)}}+\sqrt{\frac{b}{a(3b+2)}}

Let

a

and

b

be positive real numbers such that

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

. Find the minimum value of

A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}}

A1

1

\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2 }+\sqrt{c^2+a^2+2}\ge 6

Let

a, b, c

be positive real numbers such that

a + b + c + ab + bc + ca + abc = 7

. Prove that

\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6

.

C1

1

coloring with 3 colors sides of a regular 2n + 1-gon

Consider a regular

2n + 1

-gon

P

in the plane, where n is a positive integer. We say that a point

S

on one of the sides of

P

can be seen from a point

E

that is external to

P

, if the line segment

SE

contains no other points that lie on the sides of

P

except

S

. We want to color the sides of

P

in

3

colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to

P

, at most

2

different colors on

P

can be seen (ignore the vertices of

P

, we consider them colorless). Find the largest positive integer for which such a coloring is possible.

C3

1

2 piles with 2000 and 2017 coins winning strategy

We have two piles with

2000

and

2017

coins respectively. Ann and Bob take alternate turns making the following moves: The player whose turn is to move picks a pile with at least two coins, removes from that pile

t

coins for some

2\le t \le 4

, and adds to the other pile

1

coin. The players can choose a different

t

at each turn, and the player who cannot make a move loses. If Ann plays first determine which player has a winning strategy.

NT5

1

find positive n so that exists prime p with p^n-(p-1)^n$ a power of 3

Find all positive integers

n

such that there exists a prime number

p

, such that

p^n-(p-1)^n

is a power of

3

.
Note. A power of

3

is a number of the form

3^a

where

a

is a positive integer.

NT4

1

diopantine 5^t + 3^x4^y = z^2 , solve in nonnegative

Solve in nonnegative integers the equation

5^t + 3^x4^y = z^2

.

NT3

1

2^x + 3^y a perfect square, find positive integers x,y

Find all pairs of positive integers

(x,y)

such that

2^x + 3^y

is a perfect square.

NT2

1

fin n so that n^2 / (n - 1)!.

Determine all positive integers n such that

n^2/ (n - 1)!

A3

1

Inequality

let

a\le b\le c \le d

show that:

ab^3+bc^3+cd^3+da^3\ge a^2b^2+b^2c^2+c^2d^2+d^2a^2

2017 JBMO Shortlist

Subcontests

if 2 of 6 quadrilaterals are concyclic, then all 6 are concyclic

many angleswith point on the arc of circumcircles

prove that D is circumcenter of MNK

parallelogram, circle and collinear points,

minimum of \sqrt{\frac{a}{b(3a+2)}}+\sqrt{\frac{b}{a(3b+2)}}

\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2 }+\sqrt{c^2+a^2+2}\ge 6

coloring with 3 colors sides of a regular 2n + 1-gon

2 piles with 2000 and 2017 coins winning strategy

find positive n so that exists prime p with p^n-(p-1)^n$ a power of 3

diopantine 5^t + 3^x4^y = z^2 , solve in nonnegative

2^x + 3^y a perfect square, find positive integers x,y

fin n so that n^2 / (n - 1)!.

Inequality