MathDB

4

Part of 2021 China Team Selection Test

Problems(4)

Polynomial up to a Translation

Source: 2021 China TST, Test 1, Day 2 P4

3/17/2021
Let f(x),g(x)f(x),g(x) be two polynomials with integer coefficients. It is known that for infinitely many prime pp, there exist integer mpm_p such that f(a)≡g(a+mp)(modp)f(a) \equiv g(a+m_p) \pmod p holds for all a∈Z.a \in \mathbb{Z}. Prove that there exists a rational number rr such that f(x)=g(x+r).f(x)=g(x+r).
algebrapolynomialnumber theorymodular arithmetic
Totient function inside function

Source: China TST 2021, Test 2, Day 2 P4

3/22/2021
Find all functions f:Z+→Z+f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+ such that for all positive integers m,nm,n with m≥nm\ge n, f(mφ(n3))=f(m)⋅φ(n3).f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3). Here φ(n)\varphi(n) denotes the number of positive integers coprime to nn and not exceeding nn.
number theorytotient functionfunction
NT function innequality

Source: 2021ChinaTST test3 day2 P1

4/13/2021
Proof that ∑m=1n5ω(m)≤∑k=1n⌊nk⌋τ(k)2≤∑m=1n5Ω(m). \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .
number theory functionnumber theoryInnequalityfloor functioninequalities
maximum of sum of xi^2(x(i+1)-x(i-1))

Source: 2021ChinaTST test4 day2 P1

4/14/2021
Suppose x1,x2,...,x60∈[−1,1]x_1,x_2,...,x_{60}\in [-1,1] , find the maximum of ∑i=160xi2(xi+1−xi−1), \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}), where xi+60=xix_{i+60}=x_i.
InequalityalgebraTST