MathDB

3

Part of 2012 China Team Selection Test

Problems(6)

Infinit many finite sets of positive integers

Source: 2012 China TST Quiz 1 Day 1 P3

3/14/2012
Let xn=(2nn)x_n=\binom{2n}{n} for all n∈Z+n\in\mathbb{Z}^+. Prove there exist infinitely many finite sets A,BA,B of positive integers, satisfying A∩B=∅A \cap B = \emptyset , and ∏i∈Axi∏j∈Bxj=2012.\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.
number theory proposednumber theory
Find all functions

Source: 2012 China TST- Quiz 1 - Day 2 -P6

3/15/2012
nn being a given integer, find all functions f ⁣:Z→Zf\colon \mathbb{Z} \to \mathbb{Z}, such that for all integers x,yx,y we have f(x+y+f(y))=f(x)+nyf\left( {x + y + f(y)} \right) = f(x) + ny.
functiongroup theoryabstract algebrafloor functioninductionalgebra proposedalgebra
periodic sequence

Source: 2012 China TST Test 2 p3

3/19/2012
Let a1<a2a_1<a_2 be two given integers. For any integer n≥3n\ge 3, let ana_n be the smallest integer which is larger than an−1a_{n-1} and can be uniquely represented as ai+aja_i+a_j, where 1≤i<j≤n−11\le i<j\le n-1. Given that there are only a finite number of even numbers in {an}\{a_n\}, prove that the sequence {an+1−an}\{a_{n+1}-a_{n}\} is eventually periodic, i.e. that there exist positive integers T,NT,N such that for all integers n>Nn>N, we have aT+n+1−aT+n=an+1−an.a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.
pigeonhole principlecombinatorics proposedcombinatorics
the least real c

Source: 2012 China TST,Test 3,Problem 3

3/25/2012
Find the smallest possible value of a real number cc such that for any 20122012-degree monic polynomial P(x)=x2012+a2011x2011+…+a1x+a0P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0 with real coefficients, we can obtain a new polynomial Q(x)Q(x) by multiplying some of its coefficients by −1-1 such that every root zz of Q(x)Q(x) satisfies the inequality ∣Im⁡z∣≤c∣Re⁡z∣. \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert.
algebrapolynomialinequalitiesgeometrygeometric transformationalgebra unsolved
good functions

Source: 2012 China TST Test 2 p6

3/20/2012
Given an integer n≥2n\ge 2, a function f:Z→{1,2,…,n}f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\} is called good, if for any integer k,1≤k≤n−1k,1\le k\le n-1 there exists an integer j(k)j(k) such that for every integer mm we have f(m+j(k))≡f(m+k)−f(m)(modn+1).f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. Find the number of good functions.
functionmodular arithmeticalgebrafunctional equationnumber theory
beetles in a grid

Source: 2012 China TST Test 3 p6

3/26/2012
In some squares of a 2012×20122012\times 2012 grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle BB flies to the centre of the square on which it lands is called the translation vector of beetle BB. For all possible starting and ending configurations, find the maximum length of the sum of the translation vectors of all beetles.
vectorgeometrygeometric transformationanalytic geometrycombinatorics proposedcombinatorics