MathDB

3

Part of 2004 China Team Selection Test

Problems(8)

China TST 2004 product inequality

Source: China Team Selection Test 2004, Day 1, Problem 3

10/14/2005
Let k≥2,1<n1<n2<…<nkk \geq 2, 1 < n_1 < n_2 < \ldots < n_k are positive integers, a,b∈Z+a,b \in \mathbb{Z}^+ satisfy ∏i=1k(1−1ni)≤ab<∏i=1k−1(1−1ni) \prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right) Prove that: ∏i=1kni≥(4⋅a)2k−1. \prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}.
inequalitiesLaTeXinequalities unsolved
Perimeter does not exceed 2 * Pi

Source: China Team Selection Test 2004, Day 2, Problem 3

10/14/2005
Let a,b,ca, b, c be sides of a triangle whose perimeter does not exceed 2⋅π.2 \cdot \pi., Prove that sin⁡a,sin⁡b,sin⁡c\sin a, \sin b, \sin c are sides of a triangle.
geometryperimetertrigonometrygeometry solved
p doesn't divide n^m-m

Source: China TST 2004 Quiz

2/1/2009
Find all positive integer m m if there exists prime number p p such that n^m\minus{}m can not be divided by p p for any integer n n.
number theoryrelatively primenumber theory unsolved
Polynomial

Source: China TST 2004 Quiz

2/1/2009
Given arbitrary positive integer a a larger than 1 1, show that for any positive integer n n, there always exists a n-degree integral coefficient polynomial p(x) p(x), such that p(0) p(0), p(1) p(1), ⋯ \cdots, p(n) p(n) are pairwise distinct positive integers, and all have the form of 2a^k\plus{}3, where k k is also an integer.
algebrapolynomialcalculusintegrationnumber theory
Find n

Source: China TST 2004 Quiz

2/1/2009
Find all positive integer n n satisfying the following condition: There exist positive integers m m, a1 a_1, a2 a_2, ⋯ \cdots, a_{m\minus{}1}, such that \displaystyle n \equal{} \sum_{i\equal{}1}^{m\minus{}1} a_i(m\minus{}a_i), where a1 a_1, a2 a_2, ⋯ \cdots, a_{m\minus{}1} may not distinct and 1 \leq a_i \leq m\minus{}1.
algebra unsolvedalgebra
Max of abcd

Source: China TST 2004 Quiz

2/1/2009
In convex quadrilateral ABCD ABCD, AB\equal{}a, BC\equal{}b, CD\equal{}c, DA\equal{}d, AC\equal{}e, BD\equal{}f. If \max \{a,b,c,d,e,f \}\equal{}1, then find the maximum value of abcd abcd.
geometry unsolvedgeometry
Subset of Some Integers

Source: China TST 2004 Quiz

2/1/2009
S S is a non-empty subset of the set {1,2,⋯ ,108} \{ 1, 2, \cdots, 108 \}, satisfying: (1) For any two numbers a,b∈S a,b \in S ( may not distinct), there exists c∈S c \in S, such that \gcd(a,c)\equal{}\gcd(b,c)\equal{}1. (2) For any two numbers a,b∈S a,b \in S ( may not distinct), there exists c′∈S c' \in S, c′≠a c' \neq a, c′≠b c' \neq b, such that gcd⁡(a,c′)>1 \gcd(a, c') > 1, gcd⁡(b,c′)>1 \gcd(b,c') >1. Find the largest possible value of ∣S∣ |S|.
number theory unsolvednumber theory
Primes and Arithmetic Progressions

Source: China TST 2004 Quiz

2/1/2009
The largest one of numbers p1α1,p2α2,⋯ ,ptαt p_1^{\alpha_1}, p_2^{\alpha_2}, \cdots, p_t^{\alpha_t} is called a <spanclass=′latex−bold′>GoodNumber</span> <span class='latex-bold'>Good Number</span> of positive integer n n, if \displaystyle n\equal{} p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdots p_t^{\alpha_t}, where p1 p_1, p2 p_2, ⋯ \cdots, pt p_t are pairwisely different primes and α1,α2,⋯ ,αt \alpha_1, \alpha_2, \cdots, \alpha_t are positive integers. Let n1,n2,⋯ ,n10000 n_1, n_2, \cdots, n_{10000} be 10000 10000 distinct positive integers such that the <spanclass=′latex−bold′>GoodNumbers</span> <span class='latex-bold'>Good Numbers</span> of n1,n2,⋯ ,n10000 n_1, n_2, \cdots, n_{10000} are all equal. Prove that there exist integers a1,a2,⋯ ,a10000 a_1, a_2, \cdots, a_{10000} such that any two of the following 10000 10000 arithmetical progressions \{ a_i, a_i \plus{} n_i, a_i \plus{} 2n_i, a_i \plus{} 3n_i, \cdots \}( i\equal{}1,2, \cdots 10000) have no common terms.
number theory unsolvednumber theory