MathDB

2

Part of 2017 Taiwan TST Round 3

Problems(3)

Strange number theory from 2017 Taiwan TST

Source: 2017 Taiwan TST Round 3

4/13/2018
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.
conicsellipsenumber theory
Normal Geometry Problem

Source: 2017 Taiwan TST Round 3

4/13/2018
△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}}
geometrycircumcircle
Interesting Polynomial

Source: 2017 Taiwan TST Round 3

4/13/2018
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).
algebrapolynomial