MathDB

3

Part of 2003 China Team Selection Test

Problems(10)

x_0 is divisible by any prime factor of x

Source: China TST 2003

6/29/2006
Let x0+2003y0x_0+\sqrt{2003}y_0 be the minimum positive integer root of Pell function x2−2003y2=1x^2-2003y^2=1. Find all the positive integer solutions (x,y)(x,y) of the equation, such that x0x_0 is divisible by any prime factor of xx.
functionnumber theory unsolvednumber theory
China TST 2003 inequality

Source: China Team Selection Test 2003, Day 1, Problem 3

10/13/2005
Suppose A⊂{(a1,a2,…,an)∣ai∈R,i=1,2…,n}A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}. For any α=(a1,a2,…,an)∈A\alpha=(a_1,a_2,\dots,a_n)\in A and β=(b1,b2,…,bn)∈A\beta=(b_1,b_2,\dots,b_n)\in A, we define γ(α,β)=(∣a1−b1∣,∣a2−b2∣,…,∣an−bn∣), \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), D(A)={γ(α,β)∣α,β∈A}. D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. Please show that ∣D(A)∣≥∣A∣|D(A)|\geq |A|.
inequalitiesvectorinductionfunctionceiling functioninequalities unsolved
How many integers at least belong to this sequence?

Source: China Team Selection Test 2003, Day 2, Problem 3

10/13/2005
Let (xn) \left(x_{n}\right) be a real sequence satisfying x0=0 x_{0}=0, x2=23x1 x_{2}=\sqrt[3]{2}x_{1}, and xn+1=143xn+43xn−1+12xn−2 x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2} for every integer n≥2 n\geq 2, and such that x3 x_{3} is a positive integer. Find the minimal number of integers belonging to this sequence.
inductionmodular arithmeticalgebrabinomial theoremalgebra unsolved
Jumping frogs

Source: China TST 2003

6/29/2006
There is a frog in every vertex of a regular 2n-gon with circumcircle(n≥2n \geq 2). At certain time, all frogs jump to the neighborhood vertices simultaneously (There can be more than one frog in one vertex). We call it as \textsl{a way of jump}. It turns out that there is \textsl{a way of jump} with respect to 2n-gon, such that the line connecting any two distinct vertice having frogs on it after the jump, does not pass through the circumcentre of the 2n-gon. Find all possible values of nn.
modular arithmeticcombinatorics unsolvedcombinatorics
Complex coefficient polynomial

Source: China TST 2003

6/29/2006
The n n roots of a complex coefficient polynomial f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n are z1,z2,⋯ ,zn z_1, z_2, \cdots, z_n. If \sum_{k \equal{} 1}^n |a_k|^2 \leq 1, then prove that \sum_{k \equal{} 1}^n |z_k|^2 \leq n.
algebrapolynomialalgebra unsolvedinequalities
Set and Function

Source: China TST 2003

6/29/2006
Let A={a1,a2,⋯ ,an}A= \{a_1,a_2, \cdots, a_n \} and B={b1,b2⋯ ,bn}B=\{b_1,b_2 \cdots, b_n \} be two positive integer sets and ∣A∩B∣=1|A \cap B|=1. C={all the 2-element subsets of A}∪{all the 2-element subsets of B}C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}. Function f:A∪B→{0,1,2,⋯2Cn2}f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \} is injective. For any {x,y}∈C\{x,y\} \in C, denote ∣f(x)−f(y)∣|f(x)-f(y)| as the \textsl{mark} of {x,y}\{x,y\}. If n≥6n \geq 6, prove that at least two elements in CC have the same \textsl{mark}.
functioncombinatorics unsolvedcombinatorics
A sequence

Source: China TST 2003

6/29/2006
Sequence {an}\{ a_n \} satisfies: a1=3a_1=3, a2=7a_2=7, an2+5=an−1an+1a_n^2+5=a_{n-1}a_{n+1}, n≥2n \geq 2. If an+(−1)na_n+(-1)^n is prime, prove that there exists a nonnegative integer mm such that n=3mn=3^m.
number theory unsolvednumber theory
Geometrical inequality

Source: China TST 2003

6/29/2006
(1) DD is an arbitary point in △ABC\triangle{ABC}. Prove that: BCmin⁡AD,BD,CD≥{2sin⁡A, ∠A<90o2, ∠A≥90o \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array} (2)EE is an arbitary point in convex quadrilateral ABCDABCD. Denote kk the ratio of the largest and least distances of any two points among AA, BB, CC, DD, EE. Prove that k≥2sin⁡70ok \geq 2\sin{70^o}. Can equality be achieved?
inequalitiestrigonometryratiogeometry unsolvedgeometry
Lattice points

Source: China TST 2003

6/29/2006
Given SS be the finite lattice (with integer coordinate) set in the xyxy-plane. AA is the subset of SS with most elements such that the line connecting any two points in AA is not parallel to xx-axis or yy-axis. BB is the subset of integer with least elements such that for any (x,y)∈S(x,y)\in S, x∈Bx \in B or y∈By \in B holds. Prove that ∣A∣≥∣B∣|A| \geq |B|.
analytic geometryinequalitiesgraph theorynumber theory unsolvednumber theory
Hard inequality

Source: China TST 2003 Quizzes

4/1/2006
Let a1,a2,...,ana_{1},a_{2},...,a_{n} be positive real number (n≥2)(n \geq 2),not all equal,such that ∑k=1nak−2n=1\sum_{k=1}^n a_{k}^{-2n}=1,prove that: ∑k=1nak2n−n2.∑1≤i<j≤n(aiaj−ajai)2>n2\sum_{k=1}^n a_{k}^{2n}-n^2.\sum_{1 \leq i<j \leq n}(\frac{a_{i}}{a_{j}}-\frac{a_{j}}{a_{i}})^2 >n^2
inequalitiesinequalities proposed