MathDB

Strongly majorizing pairs of sequences

Source: St Petersburg 2022 10.2~11.4

9/28/2022

We will say that a set of real numbers

A = (a_1,... , a_{17})

is stronger than the set of real numbers

B = (b_1, . . . , b_{17})

, and write

A >B

if among all inequalities

a_i > b_j

the number of true inequalities is at least

3

times greater than the number of false. Prove that there is no chain of sets

A_1, A_2, . . . , A_N

such that

A_1>A_2> \cdots A_N>A_1

.
Remark: For 11.4, the constant

3

is changed to

2

and

N=3

and

17

is changed to

m

and

n

in the definition (the number of elements don't have to be equal).

combinatorics

Game with piles of stones

Source: St Petersburg 2022 9.5~10.4

9/28/2022

There are two piles of stones:

1703

stones in one pile and

2022

in the other. Sasha and Olya play the game, making moves in turn, Sasha starts. Let before the player's move the heaps contain

a

and

b

stones, with

a \geq b

. Then, on his own move, the player is allowed take from the pile with

a

stones any number of stones from

1

to

b

. A player loses if he can't make a move. Who wins?
Remark: For 10.4, the initial numbers are

(444,999)

combinatorics

Points between parabolas

Source: St Petersburg 2022 9.4

9/28/2022

We will say that a point of the plane

(u, v)

lies between the parabolas

y = f(x)

and

y = g(x)

if

f(u) \leq v \leq g(u)

. Find the smallest real

p

for which the following statement is true: for any segment, the ends and the midpoint of which lie between the parabolas

y = x^2

and

y=x^2+1

, then they lie entirely between the parabolas

y=x^2

and

y=x^2+p

.

parabolaalgebra

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