MathDB

2

Part of 2007 China Team Selection Test

Problems(8)

Another 'good' number problem

Source: China TST 2007, P2

12/29/2008
A rational number x x is called good if it satisfies: x\equal{}\frac{p}{q}>1 with p p, q q being positive integers, \gcd (p,q)\equal{}1 and there exists constant numbers α \alpha, N N such that for any integer n≥N n\geq N, |\{x^n\}\minus{}\alpha|\leq\dfrac{1}{2(p\plus{}q)} Find all the good numbers.
floor functionnumber theory unsolvednumber theory
A Geometric Inequality?

Source: China TST 2007, Problem 5

12/29/2008
Let x1,…,xn x_1, \ldots, x_n be n>1 n>1 real numbers satisfying A\equal{}\left |\sum^n_{i\equal{}1}x_i\right |\not \equal{}0 and B\equal{}\max_{1\leq in n vectors αi⃗ \vec{\alpha_i} in the plane, there exists a permutation (k1,…,kn) (k_1, \ldots, k_n) of the numbers (1,…,n) (1, \ldots, n) such that \left |\sum_{i\equal{}1}^nx_{k_i}\vec{\alpha_i}\right | \geq \dfrac{AB}{2A\plus{}B}\max_{1\leq i\leq n}|\alpha_i|.
Geometry inequality
Perpendicular

Source: Chinese TST 2007 1st quiz P2

1/4/2009
Let I I be the incenter of triangle ABC. ABC. Let M,N M,N be the midpoints of AB,AC, AB,AC, respectively. Points D,E D,E lie on AB,AC AB,AC respectively such that BD\equal{}CE\equal{}BC. The line perpendicular to IM IM through D D intersects the line perpendicular to IN IN through E E at P. P. Prove that AP⊥BC. AP\perp BC.
geometryincenterPythagorean Theoremgeometry proposed
Sequence

Source: Chinese TST 2007 2nd quiz P2

1/3/2009
Find all positive integers n n such that there exists sequence consisting of 1 1 and −1:a1,a2,⋯ ,an - 1: a_{1},a_{2},\cdots,a_{n} satisfying a1⋅12+a2⋅22+⋯+an⋅n2=0. a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.
modular arithmeticalgebra proposedalgebra
(1-x)(1-x^2)... [particular case of pentagonal number thm.]

Source: Chinese TST 2007 4th quiz P2

3/26/2007
After multiplying out and simplifying polynomial (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1), getting rid of all terms whose powers are greater than 2007, 2007, we acquire a new polynomial f(x). f(x). Find its degree and the coefficient of the term having the highest power. Find the degree of f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007}) (mod (mod x2008). x^{2008}).
algebrapolynomialEulernumber theory unsolvednumber theory
K-digits

Source: Chinese TST 2007 3rd quiz P2

1/3/2009
Given an integer k>1. k > 1. We call a k \minus{}digits decimal integer a1a2⋯ak a_{1}a_{2}\cdots a_{k} is p \minus{}monotonic, if for each of integers i i satisfying 1\le i\le k \minus{} 1, when ai a_{i} is an odd number, a_{i} > a_{i \plus{} 1}; when ai a_{i} is an even number, a_{i}
combinatorics proposedcombinatorics
Color

Source: Chinese TST 2007 5th quiz P2

1/4/2009
Given n n points arbitrarily in the plane P1,P2,…,Pn, P_{1},P_{2},\ldots,P_{n}, among them no three points are collinear. Each of Pi P_{i} (1≤i≤n1\le i\le n) is colored red or blue arbitrarily. Let S S be the set of triangles having {P1,P2,…,Pn} \{P_{1},P_{2},\ldots,P_{n}\} as vertices, and having the following property: for any two segments PiPj P_{i}P_{j} and PuPv, P_{u}P_{v}, the number of triangles having PiPj P_{i}P_{j} as side and the number of triangles having PuPv P_{u}P_{v} as side are the same in S. S. Find the least n n such that in S S there exist two triangles, the vertices of each triangle having the same color.
combinatorics proposedcombinatorics
Nice geometry

Source: Chinese TST 2007 6th quiz P2

10/12/2007
Let ABCD ABCD be the inscribed quadrilateral with the circumcircle ω \omega.Let ζ \zeta be another circle that internally tangent to ω \omega and to the lines BC BC and AD AD at points M,N M,N respectively.Let I1,I2 I_1,I_2 be the incenters of the △ABC \triangle ABC and △ABD \triangle ABD.Prove that M,I1,I2,N M,I_1,I_2,N are collinear.
geometrycircumcircleincentersymmetryEulerprojective geometrycyclic quadrilateral