5

Part of 2021 Alibaba Global Math Competition

Problems(2)

AGMC 2021 Prelim Q5

Source:

For the complex-valued function

f(x)

which is continuous and absolutely integrable on

\mathbb{R}

, define the function

(Sf)(x)

\mathbb{R}

(Sf)(x)=\int_{-\infty}^{+\infty}e^{2\pi iux}f(u)du

. (a) Find the expression for

S(\frac{1}{1+x^2})

S(\frac{1}{(1+x^2)^2})

. (b) For any integer

k

f_k(x)=(1+x^2)^{-1-k}

k\geq 1

, find constant

c_1

c_2

such that the function

y=(Sf_k)(x)

satisfies the ODE with second order:

xy''+c_1y'+c_2xy=0

calculuscomplex analysisOdedifferential equation

Points with unit distance almost on a sphere

Source: Alibaba Global Math Competition 2021, Problem 5

A

is a finite subset of

\mathbb{R}^d

such that
(a) every three distinct points in

A

contain two points that are exactly at unit distance apart, and
(b) the Euclidean norm of every point

v

A

\sqrt{\frac{1}{2}-\frac{1}{2\vert A\vert}} \le \|v\| \le \sqrt{\frac{1}{2}+\frac{1}{2\vert A\vert}}.

Prove that the cardinality of

A

2d+4

Euclidean spacegeometrylengthcollege contests3D geometrysphere