MathDB

Inequality on the unit circle

Source: 2022 China TST, Test 1, P5

3/24/2022

Let

C=\{ z \in \mathbb{C} : |z|=1 \}

be the unit circle on the complex plane. Let

z_1, z_2, \ldots, z_{240} \in C

(not necessarily different) be

240

complex numbers, satisfying the following two conditions: (1) For any open arc

\Gamma

of length

\pi

on

C

, there are at most

200

of

j ~(1 \le j \le 240)

such that

z_j \in \Gamma

. (2) For any open arc

\gamma

of length

\pi/3

on

C

, there are at most

120

of

j ~(1 \le j \le 240)

such that

z_j \in \gamma

.
Find the maximum of

|z_1+z_2+\ldots+z_{240}|

.

inequalitiescomplex numbers

A function on the set of positive divisors

Source: 2022 China TST, Test 2, P5

3/29/2022

Given a positive integer

n

, let

D

is the set of positive divisors of

n

, and let

f: D \to \mathbb{Z}

be a function. Prove that the following are equivalent:
(a) For any positive divisor

m

of

n

,

n ~\Big|~ \sum_{d|m} f(d) \binom{n/d}{m/d}.

(b) For any positive divisor

k

of

n

,

k ~\Big|~ \sum_{d|k} f(d).

number theoryDivisorsbinomial coefficients

No perfect squares in A-A

Source: 2022 China TST, Test 3 P5

4/30/2022

Show that there exist constants

c

and

\alpha > \frac{1}{2}

, such that for any positive integer

n

, there is a subset

A

of

\{1,2,\ldots,n\}

with cardinality

|A| \ge c \cdot n^\alpha

, and for any

x,y \in A

with

x \neq y

, the difference

x-y

is not a perfect square.

number theoryPerfect Squares

Two every other sequences with sum not exceeding 1

Source: 2022 China TST, Test 4 P5

4/30/2022

Let

n

be a positive integer,

x_1,x_2,\ldots,x_{2n}

be non-negative real numbers with sum

4

. Prove that there exist integer

p

and

q

, with

0 \le q \le n-1

, such that \sum_{i=1}^q x_{p+2i-1} \le 1 \mbox{ and } \sum_{i=q+1}^{n-1} x_{p+2i} \le 1, where the indices are take modulo

2n

.
Note: If

q=0

, then

\sum_{i=1}^q x_{p+2i-1}=0

; if

q=n-1

, then

\sum_{i=q+1}^{n-1} x_{p+2i}=0

.

algebraInequalitycombinatorics

5

Problems(4)