OPM 1999

A number is said to be balanced if one of its digits is average of the others. How many balanced $3$ -digit numbers are there?

How many positive integers are there such that $\frac{2n^2+4n+18}{3n+3}$ is an integer?

If two parallel chords of a circumference, $10$ mm and $14$ mm long, with distance $6$ mm from each other, how long is the chord equidistant from these two?

Given a number, we calculate its square and add $1$ to the sum of the digits in this square, obtaining a new number. If we start with the number $7$ we will obtain, in the first step, the number $1+(4+9)=14$ , since $7^2 = 49$ . What number will we obtain in the $1999$ th step?

Each of the numbers $a_1,...,a_n$ is equal to $1$ or $-1$ . If $a_1a_2 + a_2a_3 + ··· + a_{n-1}a_n + a_na_1 = 0$ , proves that $n$ is divisible by $4$ .

In the triangle $[ABC], D$ is the midpoint of $[AB]$ and $E$ is the trisection point of $[BC]$ closer to $C$ . If $\angle ADC= \angle BAE$ , find the measue of $\angle BAC$ .