MathDB

Set with strange consitions

Source: Russian TST 2022, Day 3 P3

3/21/2023

The set

A{}

of positive integers satisfies the following conditions:
[*]If a positive integer

n{}

belongs to

A{}

, then

2n

also belongs to

A{}

; [*]For any positive integer

n{}

there exists an element of

A{}

divisible by

n{}

; [*]There exist finite subsets of

A{}

with arbitrarily large sums of reciprocals of elements. Prove that for any positive rational number

r{}

there exists a finite subset

B\subset A

such that

\sum_{x\in B}\frac{1}{x}=r.

algebranumber theory

Changing the series of natural numbers

Source: Russian TST 2022, Day 6 P3

3/21/2023

Write the natural numbers from left to right in ascending order. Every minute, we perform an operation. After

m

minutes, we divide the entire available series into consecutive blocks of

m

numbers. We leave the first block unchanged and in each of the other blocks we move all the numbers except the first one one place to the left, and move the first one to the end of the block. Prove that throughout the process, each natural number will only move a finite number of times.

combinatorics

NT with congruences

Source: Russian TST 2022, Day 7 P3

3/21/2023

Let

n = 2k + 1

be an odd positive integer, and

m

be an integer realtively prime to

n{}

. For each

j =1,2,\ldots,k

we define

p_j

as the unique integer from the interval

[-k, k]

congruent to

m\cdot j

modulo

n{}

. Prove that there are equally many pairs

(i,j)

for which

1\leqslant i<j\leqslant k

which satisfy

|p_i|>|p_j|

as those which satisfy

p_ip_j<0

.

number theorycongruence

Hard inequality

Source: Russian TST 2022, Day 8 P3

3/21/2023

Let

n\geqslant 3

be an integer and

x_1>x_2>\cdots>x_n

be real numbers. Suppose that

x_k>0\geqslant x_{k+1}

for an index

k{}

. Prove that

\sum_{i=1}^k\left(x_i^{n-2}\prod_{j\neq i}\frac{1}{x_i-x_j}\right)\geqslant 0.

algebraInequality

P3

Problems(4)