MathDB

1024 marbles, 2 players game

Source: 2017 Saudi Arabia JBMO TST 1.3

5/28/2020

On the table, there are

1024

marbles and two students,

A

and

B

, alternatively take a positive number of marble(s). The student

A

goes first,

B

goes after that and so on. On the first move,

A

takes

k

marbles with

1 < k < 1024

. On the moves after that,

A

and

B

are not allowed to take more than

k

marbles or

0

marbles. The student that takes the last marble(s) from the table wins. Find all values of

k

the student

A

should choose to make sure that there is a strategy for him to win the game.

combinatorics

tangents' intersection passes through fixed point and perpendicularity wanted,

Source: 2017 Saudi Arabia JBMO TST 2.3

5/28/2020

Let

(O)

be a circle, and

BC

be a chord of

(O)

such that

BC

is not a diameter. Let

A

be a point on the larger arc

BC

of

(O)

, and let

E, F

be the feet of the perpendiculars from

B

and

C

to

AC

and

AB

, respectively. 1. Prove that the tangents to

(AEF)

at

E

and

F

intersect at a fixed point

M

when

A

moves on the larger arc

BC

of

(O)

. 2. Let

T

be the intersection of

EF

and

BC

, and let

H

be the orthocenter of

ABC

. Prove that

TH

is perpendicular to

AM

.

geometryFixed pointperpendicularTangentsorthocenter

perpendicular from D to ST passes through midpoint of MN, max triangle area

Source: 2017 Saudi Arabia JBMO TST 3.3

5/28/2020

Let

BC

be a chord of a circle

(O)

such that

BC

is not a diameter. Let

AE

be the diameter perpendicular to

BC

such that

A

belongs to the larger arc

BC

of

(O)

. Let

D

be a point on the larger arc

BC

of

(O)

which is different from

A

. Suppose that

AD

intersects

BC

at

S

and

DE

intersects

BC

at

T

. Let

F

be the midpoint of

ST

and

I

be the second intersection point of the circle

(ODF)

with the line

BC

. 1. Let the line passing through

I

and parallel to

OD

intersect

AD

and

DE

at

M

and

N

, respectively. Find the maximum value of the area of the triangle

MDN

when

D

moves on the larger arc

BC

of

(O)

(such that

D \ne A

). 2. Prove that the perpendicular from

D

to

ST

passes through the midpoint of

MN

geometryFixed pointperpendicularmidpointmaxarea

p^3 - q^5 = (p + q)^2

Source: 2017 Saudi Arabia JBMO Training Tests 3

5/28/2020

Find all pairs of primes

(p, q)

such that

p^3 - q^5 = (p + q)^2

.

number theoryDiophantine equationdiophantine

3

Problems(4)