CMO 2000

The sides $a,b,c$ of triangle $ABC$ satisfy $a\le b\le c$ . The circumradius and inradius of triangle $ABC$ are $R$ and $r$ respectively. Let $f=a+b-2R-2r$ . Determine the sign of $f$ by the measure of angle $C$ .

A sequence $(a_n)$ is defined recursively by $a_1=0, a_2=1$ and for $n\ge 3$ , $$a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right).$$ Find a closed-form expression for $f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1$ .

A table tennis club hosts a series of doubles matches following several rules: (i) each player belongs to two pairs at most; (ii) every two distinct pairs play one game against each other at most; (iii) players in the same pair do not play against each other when they pair with others respectively. Every player plays a certain number of games in this series. All these distinct numbers make up a set called the "set of games". Consider a set $A=\{a_1,a_2,\ldots ,a_k\}$ of positive integers such that every element in $A$ is divisible by $6$ . Determine the minimum number of players needed to participate in this series so that a schedule for which the corresponding set of games is equal to set $A$ exists.

Given an ordered $n$ -tuple $A=(a_1,a_2,\cdots ,a_n)$ of real numbers, where $n\ge 2$ , we define $b_k=\max{a_1,\ldots a_k}$ for each k. We define $B=(b_1,b_2,\cdots ,b_n)$ to be the "innovated tuple" of $A$ . The number of distinct elements in $B$ is called the "innovated degree" of $A$ . Consider all permutations of $1,2,\ldots ,n$ as an ordered $n$ -tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to $2$

Find all positive integers $n$ such that there exists integers $n_1,\ldots,n_k\ge 3$ , for some integer $k$ , satisfying $$n=n_1n_2\cdots n_k=2^{\frac{1}{2^k}(n_1-1)\cdots (n_k-1)}-1.$$

A test contains $5$ multiple choice questions which have $4$ options in each. Suppose each examinee chose one option for each question. There exists a number $n$ , such that for any $n$ sheets among $2000$ sheets of answer papers, there are $4$ sheets of answer papers such that any two of them have at most $3$ questions with the same answers. Find the minimum value of $n$ .