CN-SMO 2005

Let $a \in \mathbb{R}$ be a parameter.

(1) Prove that the curves of $y = x^2 + (a + 2)x - 2a + 1$ pass through a fixed point; also, the vertices of these parabolas all lie on the curve of a certain parabola.

(2) If the function $x^2 + (a + 2)x - 2a + 1 = 0$ has two distinct real roots, find the value range of the larger root.

Circle $C$ (with center $O$ ) does not have common point with line $l$ . Draw $OP$ perpendicular to $l$ , $P \in l$ . Let $Q$ be a point on $l$ ( $Q$ is different from $P$ ), $QA$ and $QB$ are tangent to circle $C$ , and intersect the circle at $A$ and $B$ respectively. $AB$ intersects $OP$ at $K$ . $PM$ , $PN$ are perpendicular to $QB$ , $QA$ , respectively, $M \in QB$ , $N \in QA$ . Prove that segment $KP$ is bisected by line $MN$ .

Let $n$ be positive integer, set $M = \{ 1, 2, \ldots, 2n \}$ . Find the minimum positive integer $k$ such that for any subset $A$ (with $k$ elements) of set $M$ , there exist four pairwise distinct elements in $A$ whose sum is $4n + 1$ .

Find all positive integer solutions $(a, b, c)$ to the function $a^{2} + b^{2} + c^{2} = 2005$ , where $a \leq b \leq c$ .

Line $l$ tangents unit circle $S$ in point $P$ . Point $A$ and circle $S$ are on the same side of $l$ , and the distance from $A$ to $l$ is $h$ ( $h > 2$ ). Two tangents of circle $S$ are drawn from $A$ , and intersect line $l$ at points $B$ and $C$ respectively.

Find the value of $PB \cdot PC$ .

Let $P(A)$ be the arithmetic-means of all elements of set $A = \{ a_1, a_2, \ldots, a_n \}$ , namely $P(A) = \frac{1}{n} \sum^{n}_{i=1}a_i$ . We denote $B$ "balanced subset" of $A$ , if $B$ is a non-empty subset of $A$ and $P(B) = P(A)$ . Let set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$ .

Find the number of all "balanced subset" of $M$ .

(1) Find the possible number of roots for the equation $|x + 1| + |x + 2| + |x + 3| = a$ , where $x \in R$ and $a$ is parameter.

(2) Let $\{ a_1, a_2, \ldots, a_n \}$ be an arithmetic progression, $n \in \mathbb{N}$ , and satisfy the condition $$\sum^{n}_{i=1}|a_i| = \sum^{n}_{i=1}|a_{i} + 1| = \sum^{n}_{i=1}|a_{i} - 2| = 507.$$ Find the maximum value of $n$ .

Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$ . Prove that $$\tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} .$$