China Team Selection Test 2007


Day 1
1. Let A,B be two distinct points on a given circle O and let M be the midpoint of arc AB,C be an arbitrary point outside the circle O.CS and CT are two lines passing through C and tangent to the circle O at S and T respectively.

Let E=MS\cap AB,F=MT\cap AB , two lines passing through E and F respectively and perpendicular to AB intersect OS and OT at X and Y respectively. A line through C cuts the circle O at P and Q ,let R=MP\cap AB , Z be the center of the circumcircle of \Delta PQR .Show that X,Y,Z are collinear.
2. A national number x is called “good” if it satisfies: x=p/q>1 with p,q being positive integers, \gcd (p,q)=1 and there exists constant numbers \alpha,N such that for any integer n\geq N,

|\{x^n\}-\alpha|\leq\dfrac{1}{2(p+q)}

Find all the “good” numbers.

3.There are 63 points arbitrarily on the circle C with its diameter being 20. Let S denote the number of triangles whose vertices are three of the 63 points and the length of its sides is no less than 9. Fine the maximum of S .

Day 2

4. Fine all functions f:\mathbb{Q}^+\to\mathbb{Q}^+ such that

f(x)+f(y)+2xyf(xy)=\dfrac{f(xy)}{f(x+y)}\, \forall x,y\in\mathbb{Q}^+.

.Here we denote by \mathbb{Q}^+ be the positive rational number set.

5. Let x_1,\cdots x_n(n>1) be n real numbers satisfing:

A=|x_1+\cdots x_n|\not =0 and B=\max_{1\leq i<j\leq n}|x_j-x_i|\not =0. Prove that for any n vectors \vec{\alpha_i} in the plane, there exists a permutations (k_1,\cdots,k_n) of the numbers (1,\cdots,n) such that |\sum_{i=1}^nx_{k_i}\vec{\alpha_i}|\geq\dfrac{AB}{2A+B}\max_{1\leq i\leq n}|\vec{\alpha_i}|.

6.Let n be a positive integer, let A be a subset of \{1,2,\cdots,n\} ,satisfing for any two numbers x,y\in A ,the least common multiple of x,y not more than n.

Show that |A|\leq 19\sqrt{n}+5.

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