2020 JBMO Shortlist

Part of JBMO ShortLists

Subcontests

(7)

1

JBMO Shortlist 2020 N7

Prove that there doesn’t exist any prime

p

such that every power of

p

is a palindrome (a palindrome is a number that is read the same from the left as it is from the right; in particular, a number that ends in one or more zeros cannot be a palindrome).

1

JBMO Shortlist 2020 N6

Are there any positive integers

m

n

satisfying the equation

m^3 = 9n^4 + 170n^2 + 289

1

JBMO Shortlist 2020 N5

The positive integer

k

A

of distinct integers from

1

3k

inclusively are such that there are no distinct

a

b

c

A

2b = a + c

. The numbers from

A

in the interval

[1, k]

will be called small; those in

[k + 1, 2k]

- medium and those in

[2k + 1, 3k]

- large. It is always true that there are no positive integers

x

d

x

x + d

x + 2d

3k

then the remainders belong to

A

x

x + d

are different and are: a) small?

\hspace{1.5px}

\hspace{1.5px}

c) large? (In this problem we assume that if a multiple of

3k

3k

then the remainder is

3k

0

1

JBMO Shortlist 2020 N4

Find all prime numbers

p

(x + y)^{19} - x^{19} - y^{19}

is a multiple of

p

for any positive integers

x

y

2

JBMO Shortlist 2020 A3

Find all triples of positive real numbers

(a, b, c)

so that the expression

M = \frac{(a + b)(b + c)(a + b + c)}{abc}

gets its least value.

JBMO Shortlist 2020 N3

Find the largest integer

k

k \ge 2

), for which there exists an integer

n

n \ge k

) such that from any collection of

n

consecutive positive integers one can always choose

k

numbers, which verify the following conditions:
1. each chosen number is not divisible by

6

7

8

; 2. the positive difference of any two distinct chosen numbers is not divisible by at least one of the numbers

6

7

8

4

3

JBMO Shortlist 2020 C1

Alice and Bob play the following game: starting with the number

2

written on a blackboard, each player in turn changes the current number

n

n + p

p

is a prime divisor of

n

. Alice goes first and the players alternate in turn. The game is lost by the one who is forced to write a number greater than

\underbrace{22...2}_{2020}

. Assuming perfect play, who will win the game.

JBMO Shortlist 2020 G1

\triangle ABC

be an acute triangle. The line through

A

perpendicular to

BC

BC

D

E

be the midpoint of

AD

\omega

the the circle with center

E

and radius equal to

AE

BE

\omega

X

X

B

are not on the same side of

AD

CE

\omega

Y

C

Y

are not on the same side of

AD

. If both of the intersection points of the circumcircles of

\triangle BDX

\triangle CDY

lie on the line

AD

AB = AC

JBMO Shortlist 2020 N1

Determine whether there is a natural number

n

8^n + 47