MathDB

2017 Taiwan TST Round 3

Part of TST Round 3

Subcontests

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Strange number theory from 2017 Taiwan TST

Choose a rational point P0(xp,yp)P_0(x_p,y_p) arbitrary on ellipse C:x2+2y2=2098C:x^2+2y^2=2098. Define P1,P2,⋯P_1,P_2,\cdots recursively by the following rules:
(1)(1) Choose a lattice point Qi=(xi,yi)∉CQ_i=(x_i,y_i)\notin C such that ∣xi∣<50|x_i|<50 and ∣yi∣<50|y_i|<50. (2)(2) Line PiQiP_iQ_i intersects CC at another point Pi+1P_{i+1}.
Prove that for any point P0P_0 we can choose suitable points Q0,Q1,⋯Q_0,Q_1,\cdots such that ∃k∈N∪{0}\exists k\in\mathbb{N}\cup\{0\}, OPk‾2=2017\overline{OP_k}^2=2017.

Normal Geometry Problem

△ABC\triangle ABC satisfies ∠A=60∘\angle A=60^{\circ}. Call its circumcenter and orthocenter O,HO, H, respectively. Let MM be a point on the segment BHBH, then choose a point NN on the line CHCH such that HH lies between C,NC, N, and BM‾=CN‾\overline{BM}=\overline{CN}. Find all possible value of MH‾+NH‾OH‾\frac{\overline{MH}+\overline{NH}}{\overline{OH}}

Interesting Polynomial

Prove that there exists a polynomial with integer coefficients satisfying the following conditions: (a)f(x)=0f(x)=0 has no rational root. (b) For any positive integer nn, there always exists an integer mm such that n∣f(m)n\mid f(m).
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Many cats

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.

Easy inequality from Taiwan TST

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.

Interesting recurring sequence with floor function

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.