MathDB

6

Part of 2021 Korea Winter Program Practice Test

Problems(2)

Imaginary sequence

Source: 2021 Korea Winter Program Test1 Day2 #6

2/14/2021
Is there exist a sequence a0,a1,a2,⋯a_0,a_1,a_2,\cdots consisting of non-zero integers that satisfies the following condition?
Condition: For all integers nn (≥2020\ge 2020), equation anxn+an−1xn−1+⋯+a0=0a_n x^n+a_{n-1}x^{n-1}+\cdots +a_0=0 has a real root with its absolute value larger than 2.0012.001.
algebra
Another cute geo

Source: 2021 Korea Winter Program Test2 Day2 #6

2/17/2021
The acute triangle ABCABC satisfies AB‾<BC‾<CA‾\overline {AB}<\overline {BC}<\overline {CA}. Denote the foot of perpendicular from A,B,CA,B,C to opposing sides as D,E,FD,E,F. Let PP a foot of perpendicular from FF to DEDE, and Q(≠F)Q(\neq F) a intersection point of line FPFP and circumcircle of BDFBDF. Prove that ∠PBQ=∠PAD\angle PBQ=\angle PAD.
geometry