MathDB

5

Part of 2017 Thailand TSTST

Problems(3)

P(x)-x|P^n(x)-x

Source: 2016 Thailand October Camp 2.5

2/26/2022
Prove that for all polynomials P∈R[x]P \in \mathbb{R}[x] and positive integers nn, P(x)−xP(x)-x divides Pn(x)−xP^n(x)-x as polynomials.
polynomialalgebra
collinear wanted, 2 circles and exsimilicenter center

Source: 2016 Thailand October Camp 1.5

10/15/2020
Let ω1,ω2\omega_1, \omega_2 be two circles with different radii, and let HH be the exsimilicenter of the two circles. A point X outside both circles is given. The tangents from XX to ω1\omega_1 touch ω1\omega_1 at P,QP, Q, and the tangents from XX to ω2\omega_2 touch ω2\omega_2 at R,SR, S. If PRPR passes through HH and is not a common tangent line of ω1,ω2\omega_1, \omega_2, prove that QSQS also passes through HH.
geometrycollinearcircles
Another Weird Inequality

Source: 2016 Thailand October Camp 3.5

2/28/2022
Let a,b,c∈R+a, b, c \in \mathbb{R}^+ such that a+b+c=3a + b + c = 3. Prove that ∑cyc(a3+1a2+1)≥127(ab+bc+ca)4.\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)\geq\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4.
inequalities