MathDB

2

Part of 2003 China Team Selection Test

Problems(10)

find A with largest possible cardinality

Source: China Team Selection Test 2003, Day 1, Problem 2

10/13/2005
Suppose A⊆{0,1,…,29}A\subseteq \{0,1,\dots,29\}. It satisfies that for any integer kk and any two members a,b∈Aa,b\in A(a,ba,b is allowed to be same), a+b+30ka+b+30k is always not the product of two consecutive integers. Please find AA with largest possible cardinality.
quadraticsmodular arithmeticcombinatorics unsolvedcombinatorics
Find the number of M-partitions of A

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

10/13/2005
Suppose A={1,2,…,2002}A=\{1,2,\dots,2002\} and M={1001,2003,3005}M=\{1001,2003,3005\}. BB is an non-empty subset of AA. BB is called a MM-free set if the sum of any two numbers in BB does not belong to MM. If A=A1∪A2A=A_1\cup A_2, A1∩A2=∅A_1\cap A_2=\emptyset and A1,A2A_1,A_2 are MM-free sets, we call the ordered pair (A1,A2)(A_1,A_2) a MM-partition of AA. Find the number of MM-partitions of AA.
combinatorics unsolvedcombinatorics
Area ratio inequality

Source: China TST 2003

6/29/2006
In triangle ABCABC, the medians and bisectors corresponding to sides BCBC, CACA, ABAB are mam_a, mbm_b, mcm_c and waw_a, wbw_b, wcw_c respectively. P=wa∩mbP=w_a \cap m_b, Q=wb∩mcQ=w_b \cap m_c, R=wc∩maR=w_c \cap m_a. Denote the areas of triangle ABCABC and PQRPQR by F1F_1 and F2F_2 respectively. Find the least positive constant mm such that F1F2<m\frac{F_1}{F_2}<m holds for any △ABC\triangle{ABC}.
geometryratioinequalitiesanalytic geometrylinear algebramatrixgeometry unsolved
Find integer value of P

Source: China TST 2003

6/29/2006
Let x<yx<y be positive integers and P=x3−y1+xyP=\frac{x^3-y}{1+xy}. Find all integer values that PP can take.
number theory unsolvednumber theory
Find real numbers

Source: China TST 2003

6/29/2006
Can we find positive reals a1,a2,…,a2002a_1, a_2, \dots, a_{2002} such that for any positive integer kk, with 1≤k≤20021 \leq k \leq 2002, every complex root zz of the following polynomial f(x)f(x) satisfies the condition ∣Im z∣≤∣Re z∣|\text{Im } z| \leq |\text{Re } z|, f(x)=ak+2001x2001+ak+2000x2000+⋯+ak+1x+ak,f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k, where a2002+i=aia_{2002+i}=a_i, for i=1,2,…,2001i=1,2, \dots, 2001.
algebrapolynomialalgebra unsolved
Find minimum n

Source: China TST 2003

6/29/2006
Positive integer nn cannot be divided by 22 and 33, there are no nonnegative integers aa and bb such that ∣2a−3b∣=n|2^a-3^b|=n. Find the minimum value of nn.
modular arithmeticquadraticsnumber theory unsolvednumber theory
Functional equation

Source: China TST 2003

6/29/2006
Find all functions f,gf,g:R→RR \to R such that f(x+yg(x))=g(x)+xf(y)f(x+yg(x))=g(x)+xf(y) for x,y∈Rx,y \in R.
functionalgebrafunctional equationalgebra unsolved
Find AP [in an isosceles right triangle ABC]

Source: China tst 2003

6/8/2005
Denote by (ABC)\left(ABC\right) the circumcircle of a triangle ABCABC. Let ABCABC be an isosceles right-angled triangle with AB=AC=1AB=AC=1 and ∡CAB=90∘\measuredangle CAB=90^{\circ}. Let DD be the midpoint of the side BCBC, and let EE and FF be two points on the side BCBC. Let MM be the point of intersection of the circles (ADE)\left(ADE\right) and (ABF)\left(ABF\right) (apart from AA). Let NN be the point of intersection of the line AFAF and the circle (ACE)\left(ACE\right) (apart from AA). Let PP be the point of intersection of the line ADAD and the circle (AMN)\left(AMN\right). Find the length of APAP.
geometrycircumcirclesymmetryprojective geometrypower of a pointradical axisgeometry proposed
Function

Source: China TST 2003

6/29/2006
Let SS be a finite set. ff is a function defined on the subset-group 2S2^S of set SS. ff is called \textsl{monotonic decreasing} if when X⊆Y⊆SX \subseteq Y\subseteq S, then f(X)≥f(Y)f(X) \geq f(Y) holds. Prove that: f(X∪Y)+f(X∩Y)≤f(X)+f(Y)f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y) for X,Y⊆SX, Y \subseteq S if and only if g(X)=f(X∪{a})−f(X)g(X)=f(X \cup \{ a \}) - f(X) is a \textsl{monotonic decreasing} funnction on the subset-group 2S∖{a}2^{S \setminus \{a\}} of set S∖{a}S \setminus \{a\} for any a∈Sa \in S.
functioninductionalgebra unsolvedalgebra
Find a sequence

Source: China TST 2003

6/29/2006
Given an integer a1a_1(a1≠−1a_1 \neq -1), find a real number sequence {an}\{ a_n \}(ai≠0,i=1,2,⋯ ,5a_i \neq 0, i=1,2,\cdots,5) such that x1,x2,⋯ ,x5x_1,x_2,\cdots,x_5 and y1,y2,⋯ ,y5y_1,y_2,\cdots,y_5 satisfy bi1x1+bi2x2+⋯+bi5x5=2yib_{i1}x_1+b_{i2}x_2+\cdots +b_{i5}x_5=2y_i, i=1,2,3,4,5i=1,2,3,4,5, then x1y1+x2y2+⋯+x5y5=0x_1y_1+x_2y_2+\cdots+x_5y_5=0, where bij=∏1≤k≤i(1+jak)b_{ij}=\prod_{1 \leq k \leq i} (1+ja_k).
algebra unsolvedalgebra