MathDB

1

Part of 2017 Taiwan TST Round 3

Problems(3)

Many cats

Source: 2017 Taiwan TST, 3rd round, Quiz 1

4/23/2017
In an n×nn\times{n} grid, there are some cats living in each cell (the number of cats in a cell must be a non-negative integer). Every midnight, the manager chooses one cell: (a) The number of cats living in the chosen cell must be greater than or equal to the number of neighboring cells of the chosen cell. (b) For every neighboring cell of the chosen cell, the manager moves one cat from the chosen cell to the neighboring cell. (Two cells are called "neighboring" if they share a common side, e.g. there are only 22 neighboring cells for a cell in the corner of the grid) Find the minimum number of cats living in the whole grid, such that the manager is able to do infinitely many times of this process.
combinatorics
Easy inequality from Taiwan TST

Source: 2017 Taiwan TST Round 3

4/13/2018
There are mm real numbers xi≥0x_i \geq 0 (i=1,2,...,mi=1,2,...,m), n≥2n \geq 2, ∑i=1mxi=S\sum_{i=1}^{m} x_i=S. Prove that\\ ∑i=1mxiS−xin≥2, \sum_{i=1}^{m} \sqrt[n]{\frac{x_i}{S-x_i}} \geq 2, The equation holds if and only if there are exactly two of xix_i are equal(not equal to 00), and the rest are equal to 00.
inequalities
Interesting recurring sequence with floor function

Source: 2017 Taiwan TST Round 3

4/13/2018
Let {an}n≥0\{a_n\}_{n\geq 0} be an arithmetic sequence with difference dd and 1≤a0≤d1\leq a_0\leq d. Denote the sequence as S0S_0, and define SnS_n recursively by two operations below: Step 11: Denote the first number of SnS_n as bnb_n, and remove bnb_n. Step 22: Add 11 to the first bnb_n numbers to get Sn+1S_{n+1}. Prove that there exists a constant cc such that bn=[can]b_n=[ca_n] for all n≥0n\geq 0, where [][] is the floor function.
arithmetic sequencealgebrafloor functionfunction