MathDB

3

Part of 2021 China Team Selection Test

Problems(4)

On estimation of the number of solutions

Source: 2021 China TST, Test 2, Day 1 P3

3/21/2021
Given positive integers a,b,ca,b,c which are pairwise coprime. Let f(n)f(n) denotes the number of the non-negative integer solution (x,y,z)(x,y,z) to the equation ax+by+cz=n.ax+by+cz=n. Prove that there exists constants α,β,γ∈R\alpha, \beta, \gamma \in \mathbb{R} such that for any non-negative integer nn, ∣f(n)−(αn2+βn+γ)∣<112(a+b+c).|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).
algebranumber theoryDiophantine equation
Represent as Differences

Source: 2021 China TST, Test 1, Day 1 P3

3/17/2021
Given positive integer nn. Prove that for any integers a1,a2,⋯ ,an,a_1,a_2,\cdots,a_n, at least ⌈n(n−6)19⌉\lceil \tfrac{n(n-6)}{19} \rceil numbers from the set {1,2,⋯ ,n(n−1)2}\{ 1,2, \cdots, \tfrac{n(n-1)}{2} \} cannot be represented as ai−aj(1≤i,j≤n)a_i-a_j (1 \le i, j \le n).
number theoryAdditive Number Theory
an inequality about geometric average

Source: 2021ChinaTST test3 day1 P3

4/13/2021
Determine the greatest real number C C , such that for every positive integer n≥2 n\ge 2 , there exists x1,x2,...,xn∈[−1,1] x_1, x_2,..., x_n \in [-1,1], so that ∏1≤i<j≤n(xi−xj)≥Cn(n−1)2\prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}}.
inequalitiesalgebraTSTChina TST
sum of digits changes regularly during addition

Source: 2021ChinaTST test4 day1 P3

4/14/2021
Find all positive integer n(≥2)n(\ge 2) and rational β∈(0,1)\beta \in (0,1) satisfying the following: There exist positive integers a1,a2,...,ana_1,a_2,...,a_n, such that for any set I⊆{1,2,...,n}I \subseteq \{1,2,...,n\} which contains at least two elements, S(∑i∈Iai)=β∑i∈IS(ai). S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). where S(n)S(n) denotes sum of digits of decimal representation of nn.
number theorysum of digitsdecimal representationChina TSTChinanumber theory proposed