OMA 2014

$201$ positive integers are written on a line, such that both the first one and the last one are equal to $19999$ . Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line.

Given several numbers, one of them, $a$ , is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called good if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.

Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$ , are placed inside an angle whose vertex is $O$ . $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$ , and the other one intersects the rays of the angle at points $A$ and $B$ , with $AO=BO$ . Find the distance of point $A$ to the line $OB$ .

Consider the following $50$ -term sums: $S=\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+...+\frac{1}{99\cdot 100}$ , $T=\frac{1}{51\cdot 100}+\frac{1}{52\cdot 99}+...+\frac{1}{99\cdot 52}+\frac{1}{100\cdot 51}$ .

Express $\frac{S}{T}$ as an irreducible fraction.

An integer $n \geq 3$ is called special if it does not divide $\left ( n-1 \right )!\left ( 1+\frac{1}{2}+\cdot \cdot \cdot +\frac{1}{n-1} \right )$ . Find all special numbers $n$ such that $10 \leq n \leq 100$ .

Determine whether there exists positive integers $a_{1}<a_{2}< \cdot \cdot \cdot <a_{k}$ such that all sums $a_{i}+a_{j}$ , where 1 $\leq i < j \leq k$ , are unique, and among those sums, there are $1000$ consecutive integers.