MathDB

3

Part of 2009 China Team Selection Test

Problems(7)

The number of integers

Source: ChInese TST 2009 P3

4/4/2009
Prove that for any odd prime number p, p, the number of positive integer n n satisfying p|n! \plus{} 1 is less than or equal to cp23. cp^\frac{2}{3}. where c c is a constant independent of p. p.
modular arithmeticinequalitiesnumber theory proposednumber theory
Chinese TST 2009,quiz 1

Source: Chinese TST 2009 1st quiz P3

3/19/2009
Let x1,x2,⋯ ,xm,y1,y2,⋯ ,yn x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n} be positive real numbers. Denote by X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y. Prove that 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|
inductioninequalitiesinequalities unsolved
Linear function

Source: China TST 2009, Quiz 2, Problem 3

3/21/2009
Consider function f:R→R f: R\to R which satisfies the conditions for any mutually distinct real numbers a,b,c,d a,b,c,d satisfying \frac {a \minus{} b}{b \minus{} c} \plus{} \frac {a \minus{} d}{d \minus{} c} \equal{} 0, f(a),f(b),f(c),f(d) f(a),f(b),f(c),f(d) are mutully different and \frac {f(a) \minus{} f(b)}{f(b) \minus{} f(c)} \plus{} \frac {f(a) \minus{} f(d)}{f(d) \minus{} f(c)} \equal{} 0. Prove that function f f is linear
functionalgebra proposedalgebra
On polynomial

Source: Chinese TST 2009 3rd quiz P3

3/22/2009
Let f(x) f(x) be a n \minus{}degree polynomial all of whose coefficients are equal to ±1 \pm 1, and having x \equal{} 1 as its m m multiple root. If m≥2k(k≥2,k∈N) m\ge 2^k (k\ge 2,k\in N), then n\ge 2^{k \plus{} 1} \minus{} 1.
algebrapolynomialinductionmodular arithmeticbinomial coefficientsalgebra proposed
Set

Source: Chinese TST 2009 4th P3

4/5/2009
Let X X be a set containing 2k 2k elements, F F is a set of subsets of X X consisting of certain k k elements such that any one subset of X X consisting of k \minus{} 1 elements is exactly contained in an element of F. F. Show that k \plus{} 1 is a prime number.
combinatorics proposedcombinatorics
Sequence of positive integers

Source: Chinese TST 2009 6th P3

4/4/2009
Let (an)n≥1 (a_{n})_{n\ge 1} be a sequence of positive integers satisfying (am,an)=a(m,n) (a_{m},a_{n}) = a_{(m,n)} (for all m,n∈N+ m,n\in N^ +). Prove that for any n∈N+,∏d∣nadμ(nd) n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}} is an integer. where d∣n d|n denotes d d take all positive divisors of n. n. Function μ(n) \mu (n) is defined as follows: if n n can be divided by square of certain prime number, then μ(1)=1;μ(n)=0 \mu (1) = 1;\mu (n) = 0; if n n can be expressed as product of k k different prime numbers, then μ(n)=(−1)k. \mu (n) = ( - 1)^k.
inductionnumber theoryprime numbersrelatively primeprime factorizationgreatest common divisorMobius function
Max

Source: Chinese TST 2009 5th P3

4/5/2009
Let nonnegative real numbers a1,a2,a3,a4 a_{1},a_{2},a_{3},a_{4} satisfy a_{1} \plus{} a_{2} \plus{} a_{3} \plus{} a_{4} \equal{} 1. Prove that max\{\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \minus{} 1} \plus{} a_{i \minus{} 1}^2 \plus{} a_{i \minus{} 1}a_{i \minus{} 2}}},\sum_{1}^4{\sqrt {a_{i}^2 \plus{} a_{i}a_{i \plus{} 1} \plus{} a_{i \plus{} 1}^2 \plus{} a_{i \plus{} 1}a_{i \plus{} 2}}}\}\ge 2. Where for all integers i, a_{i \plus{} 4} \equal{} a_{i} holds.
inequalitiesinequalities proposed