MathDB

2

Part of 2005 China Team Selection Test

Problems(8)

ball weights

Source: China TST 2005-2

6/14/2005
Given positive integer n(n≥2)n (n \geq 2), find the largest positive integer λ\lambda satisfying : For nn bags, if every bag contains some balls whose weights are all integer powers of 22 (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least λ\lambda.
combinatorics unsolvedcombinatorics
Chinese TST 2005

Source: Chinese TST 2005

5/25/2005
Let n be a positive integer,and x be a positive real number. Prove that http://www.artofproblemsolving.com/Forum/latexrender/pictures/39ef05ce98cc12f269294d6d1d6854a7.gif [x] here means Gauss function.
Gaussfunctioninductionsearchinequalities unsolvedinequalities
sum 1/(a^2-bc+1) <= 3

Source: China 2005 TST1

3/21/2005
Let aa, bb, cc be nonnegative reals such that ab+bc+ca=13ab+bc+ca = \frac{1}{3}. Prove that 1a2−bc+1+1b2−ca+1+1c2−ab+1≤3\frac{1}{a^{2}-bc+1}+\frac{1}{b^{2}-ca+1}+\frac{1}{c^{2}-ab+1}\leq 3
inequalitiesinequalities unsolved
Max and min of perimeter

Source: China TST 2005

6/27/2006
Cyclic quadrilateral ABCDABCD has positive integer side lengths ABAB, BCBC, CACA, ADAD. It is known that AD=2005AD=2005, ∠ABC=∠ADC=90o\angle{ABC}=\angle{ADC} = 90^o, and max⁡{AB,BC,CD}<2005\max \{ AB,BC,CD \} < 2005. Determine the maximum and minimum possible values for the perimeter of ABCDABCD.
geometryperimetercyclic quadrilateralPythagorean Theoremgeometry unsolved
Rational or not

Source: China TST 2005

6/27/2006
Determine whether 10012+1+10022+1+⋯+20002+1\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1} be a rational number or not?
logarithmsalgebra unsolvedalgebra
Cyclic points and concurrency [1st Lemoine circle]

Source: China TST 2005

6/27/2006
Let ω\omega be the circumcircle of acute triangle ABCABC. Two tangents of ω\omega from BB and CC intersect at PP, APAP and BCBC intersect at DD. Point EE, FF are on ACAC and ABAB such that DE∥BADE \parallel BA and DF∥CADF \parallel CA. (1) Prove that F,B,C,EF,B,C,E are concyclic. (2) Denote A1A_{1} the centre of the circle passing through F,B,C,EF,B,C,E. B1B_{1}, C1C_{1} are difined similarly. Prove that AA1AA_{1}, BB1BB_{1}, CC1CC_{1} are concurrent.
geometrycircumcircleparallelogramgeometry unsolved
Residual

Source: China TST 2005

6/27/2006
Given prime number pp. a1,a2⋯aka_1,a_2 \cdots a_k (k≥3k \geq 3) are integers not divible by pp and have different residuals when divided by pp. Let Sn={n∣1≤n≤p−1,(na1)p<⋯<(nak)p} S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} Here (b)p(b)_p denotes the residual when integer bb is divided by pp. Prove that ∣S∣<2pk+1|S|< \frac{2p}{k+1}.
number theory unsolvednumber theory
Triangle centres

Source: China TST 2005

6/27/2006
In acute angled triangle ABCABC, BC=aBC=a,CA=bCA=b,AB=cAB=c, and a>b>ca>b>c. I,O,HI,O,H are the incentre, circumcentre and orthocentre of △ABC\triangle{ABC} respectively. Point D∈BCD \in BC, E∈CAE \in CA and AE=BDAE=BD, CD+CE=ABCD+CE=AB. Let the intersectionf of BEBE and ADAD be KK. Prove that KH∥IOKH \parallel IO and KH=2IOKH = 2IO.
geometryincenterratioEulergeometry unsolved