KMO 2002

For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$ , let $$\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p, p\nmid 4a^3+27b^2\}$$ For $(a,b), (a',b') \in \mathbb{E}_p$ we say that $(a,b)$ and $(a',b')$ are equivalent if there is a non zero element $c\in \mathbb{Z}_p$ such that $p\mid (a' -ac^4)$ and $p\mid (b'-bc^6)$ . Find the maximal number of inequivalent elements in $\mathbb{E}_p$ .

Find all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(x-y)=f(x)+xy+f(y)$ for every $x \in \mathbb{R}$ and every $y \in \{f(x) \mid x\in \mathbb{R}\}$ , where $\mathbb{R}$ is the set of real numbers.

The following facts are known in a mathematical contest:

[list] (a) The number of problems tested was $n\ge 4$ (b) Each problem was solved by exactly four contestants. (c) For each pair of problems, there is exactly one contestant who solved both problems [/list]

Assuming the number of contestants is greater than or equal to $4n$ , find the minimum value of $n$ for which there always exists a contestant who solved all the problems.

For $n \ge 3$ , let $S=a_1+a_2+\cdots+a_n$ and $T=b_1b_2\cdots b_n$ for positive real numbers $a_1,a_2,\ldots,a_n, b_1,b_2 ,\ldots,b_n$ , where the numbers $b_i$ are pairwise distinct.

(a) Find the number of distinct real zeroes of the polynomial $$f(x)=(x-b_1)(x-b_2)\cdots(x-b_n)\sum_{j=1}^n \frac{a_j}{x-b_j}$$ (b) Prove the inequality $$\frac1{n-1}\sum_{j=1}^n \left(1-\frac{a_j}{S}\right)b_j > \left(\frac{T}{S}\sum_{j=1}^{n} \frac{a_j}{b_j}\right)^{\frac1{n-1}}$$

Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the perpendicular line from $A$ to $BC$ meet $\omega$ at $D$ . Let $P$ be a point on $\omega$ , and let $Q$ be the foot of the perpendicular line from $P$ to the line $AB$ . Prove that if $Q$ is on the outside of $\omega$ and $2\angle QPB = \angle PBC$ , then $D,P,Q$ are collinear.

Let $p_n$ be the $n^{\text{th}}$ prime counting from the smallest prime $2$ in increasing order. For example, $p_1=2, p_2=3, p_3 =5, \cdots$ (a) For a given $n \ge 10$ , let $r$ be the smallest integer satisfying $$2\le r \le n-2, n-r+1 < p_r$$ and define $N_s=(sp_1p_2\cdots p_{r-1})-1$ for $s=1,2,\ldots, p_r$ . Prove that there exists $j, 1\le j \le p_r$ , such that none of $p_1,p_2,\cdots, p_n$ divides $N_j$ .

(b) Using the result of (a), find all positive integers $m$ for which $$p_{m+1}^2 < p_1p_2\cdots p_m$$