# 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. 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 $|A|\leq 19\sqrt{n}+5$.

## 5 thoughts on “China Team Selection Test 2007”

1. Ngô Quốc Anh says:

Hóa ra đây là nhà của QuanVu hả?

2. trungtuan says:

Moi nguoi van con nham anh la QUANVU o DDTH a? 😀

3. tychuot057 says:

giup em voi ; co ai giai giup em bai nay ko
Σ(can(a+b-c))/(can(a)+can(b)-can(c)) ≤ 3

4. vdmedragon says:

Lam sao de hien thi cong thuc Toan duoc ha T.Tuan? Co can cai plugin gi ko?

5. trungtuan says:

Go binh thuong thoi, sau do them chu latex vao. Vi du \$latex…