MathDB

Romania TST 2021 Day 1 P2

Source:

5/15/2021

For any positive integer

n>1

, let

p(n)

be the greatest prime factor of

n

. Find all the triplets of distinct positive integers

(x,y,z)

which satisfy the following properties:

x,y

and

z

form an arithmetic progression, and

p(xyz)\leq 3.

number theoryRomanian TSTTSTBritishMathematicalOlympiad

Romania TST 2021 Day 2 P2

Source:

6/13/2021

Consider the set

M=\{1,2,3,...,2020\}.

Find the smallest positive integer

k

such that for any subset

A

of

M

with

k

elements, there exist

3

distinct numbers

a,b,c

from

M

such that

a+b, b+c

and

c+a

are all in

A.

combinatoricsSetsromaniaRomanian TSTTST

Romania TST 2021 Day 3 P2

Source:

6/14/2021

Let

N\geq 4

be a fixed positive integer. Two players,

A

and

B

are forming an ordered set

\{x_1,x_2,...\},

adding elements alternatively.

A

chooses

x_1

to be

1

or

-1,

then

B

chooses

x_2

to be

2

or

-2,

then

A

chooses

x_3

to be

3

or

-3,

and so on. (at the

k^{th}

step, the chosen number must always be

k

or

-k

)
The winner is the first player to make the sequence sum up to a multiple of

N.

Depending on

N,

find out, with proof, which player has a winning strategy.

combinatoricsgameromaniaRomanian TSTTST

2

Problems(3)