MathDB

3

Part of 2010 China Team Selection Test

Problems(8)

China 2010 quiz1 Problem 3

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9/5/2010
Fine all positive integers m,n≥2m,n\geq 2, such that (1) m+1m+1 is a prime number of type 4k−14k-1; (2) there is a (positive) prime number pp and nonnegative integer aa, such that m2n−1−1m−1=mn+pa.\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.
modular arithmeticnumber theory unsolvednumber theory
China 2010 quiz2 problem 3

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9/8/2010
Let AA be a finite set, and A1,A2,⋯ ,AnA_1,A_2,\cdots, A_n are subsets of AA with the following conditions: (1) ∣A1∣=∣A2∣=⋯=∣An∣=k|A_1|=|A_2|=\cdots=|A_n|=k, and k>∣A∣2k>\frac{|A|}{2}; (2) for any a,b∈Aa,b\in A, there exist Ar,As,At (1≤r<s<t≤n)A_r,A_s,A_t\,(1\leq r<s<t\leq n) such that a,b∈Ar∩As∩Ata,b\in A_r\cap A_s\cap A_t; (3) for any integer i,j (1≤i<j≤n)i,j\, (1\leq i<j\leq n), ∣Ai∩Aj∣≤3|A_i\cap A_j|\leq 3. Find all possible value(s) of nn when kk attains maximum among all possible systems (A1,A2,⋯ ,An,A)(A_1,A_2,\cdots, A_n,A).
combinatorics unsolvedcombinatorics
China 2010 quiz3 problem 3

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9/11/2010
Let k>1k>1 be an integer, set n=2k+1n=2^{k+1}. Prove that for any positive integers a1<a2<⋯<ana_1<a_2<\cdots<a_n, the number ∏1≤i<j≤n(ai+aj)\prod_{1\leq i<j\leq n}(a_i+a_j) has at least k+1k+1 different prime divisors.
pigeonhole principleceiling functionnumber theory unsolvednumber theory
China 2010 quiz4 problem 3

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9/12/2010
For integers n>1n>1, define f(n)f(n) to be the sum of all postive divisors of nn that are less than nn. Prove that for any positive integer kk, there exists a positive integer n>1n>1 such that n<f(n)<f2(n)<⋯<fk(n)n<f(n)<f^2(n)<\cdots<f^k(n), where fi(n)=f(fi−1(n))f^i(n)=f(f^{i-1}(n)) for i>1i>1 and f1(n)=f(n)f^1(n)=f(n).
number theory unsolvednumber theory
China 2010 quiz5 Problem 3

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9/13/2010
Given positive integer kk, prove that there exists a positive integer NN depending only on kk such that for any integer n≥Nn\geq N, (nk)\binom{n}{k} has at least kk different prime divisors.
number theoryp-adic
China 2010 quiz6 Problem 3

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9/14/2010
An (unordered) partition PP of a positive integer nn is an nn-tuple of nonnegative integers P=(x1,x2,⋯ ,xn)P=(x_1,x_2,\cdots,x_n) such that ∑k=1nkxk=n\sum_{k=1}^n kx_k=n. For positive integer m≤nm\leq n, and a partition Q=(y1,y2,⋯ ,ym)Q=(y_1,y_2,\cdots,y_m) of mm, QQ is called compatible to PP if yi≤xiy_i\leq x_i for i=1,2,⋯ ,mi=1,2,\cdots,m. Let S(n)S(n) be the number of partitions PP of nn such that for each odd m<nm<n, mm has exactly one partition compatible to PP and for each even m<nm<n, mm has exactly two partitions compatible to PP. Find S(2010)S(2010).
combinatorics unsolvedcombinatorics
China TST 2010, Problem 3

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8/28/2010
Let n1,n2,⋯ ,n26n_1,n_2, \cdots, n_{26} be pairwise distinct positive integers satisfying (1) for each nin_i, its digits belong to the set {1,2}\{1,2\}; (2) for each i,ji,j, nin_i can't be obtained from njn_j by adding some digits on the right. Find the smallest possible value of ∑i=126S(ni)\sum_{i=1}^{26} S(n_i), where S(m)S(m) denotes the sum of all digits of a positive integer mm.
floor functioninequalitiesalgorithmcombinatorics unsolvedcombinatorics
China TST 2010, Problem 6

Source:

8/28/2010
Given integer n≥2n\geq 2 and real numbers x1,x2,⋯ ,xnx_1,x_2,\cdots, x_n in the interval [0,1][0,1]. Prove that there exist real numbers a0,a1,⋯ ,ana_0,a_1,\cdots,a_n satisfying the following conditions: (1) a0+an=0a_0+a_n=0; (2) ∣ai∣≤1|a_i|\leq 1, for i=0,1,⋯ ,ni=0,1,\cdots,n; (3) ∣ai−ai−1∣=xi|a_i-a_{i-1}|=x_i, for i=1,2,⋯ ,ni=1,2,\cdots,n.
inequalitiesinductionalgebra unsolvedalgebra