MathDB

min of max {xy, (x- 1)(y - 1), x + y - 2xy} for 0 <= x, y <= 1

Source: 2022 Dutch IMO TST 1.3

12/3/2022

For real numbers

x

and

y

we define

M(x, y)

to be the maximum of the three numbers

xy

,

(x- 1)(y - 1)

, and

x + y - 2xy

. Determine the smallest possible value of

M(x, y)

where

x

and

y

range over all real numbers satisfying

0 \le x, y \le 1

.

algebrainequalities

divisors on the exponent

Source: All-Russian 2021/9.2

4/19/2021

Let

n

be a natural number. An integer

a>2

is called

n

-decomposable, if

a^n-2^n

is divisible by all the numbers of the form

a^d+2^d

, where

d\neq n

is a natural divisor of

n

. Find all composite

n\in \mathbb{N}

, for which there's an

n

-decomposable number.

number theoryRussiaAll Russian Olympiad

15 numbered lights on the ceiling of a room, 15 switches each for 2 lights

Source: 2022 Dutch IMO TST 2.3

12/3/2022

There are

15

lights on the ceiling of a room, numbered from

1

to

15

. All lights are turned off. In another room, there are

15

switches: a switch for lights

1

and

2

, a switch for lights

2

and

3

, a switch for lights

3

en

4

, etcetera, including a sqitch for lights

15

and

1

. When the switch for such a pair of lights is turned, both of the lights change their state (from on to off, or vice versa). The switches are put in a random order and all look identical. Raymond wants to find out which switch belongs which pair of lights. From the room with the switches, he cannot see the lights. He can, however, flip a number of switches, and then go to the other room to see which lights are turned on. He can do this multiple times. What is the minimum number of visits to the other room that he has to take to determine for each switch with certainty which pair of lights it corresponds to?

combinatorics

3

Problems(3)