OPM 2015

A number of three digits is said to be firmwhen it is equal to the product of its unit digit by a number formed by the remaining digits. For example, $153$ is firm because $153 = 3 \times 51$ . How many firm numbers are there?

The numbers from $1$ to $2015$ are written on sheets so that if if $n-m$ is a prime, then $n$ and $m$ are on different sheets. What is the minimum number of sheets required?

Let $[ABCD]$ be a parallelogram and $P$ a point between $C$ and $D$ . The line parallel to $AD$ that passes through $P$ intersects the diagonal $AC$ in $Q$ . Knowing that the area of $[PBQ]$ is $2$ and the area of $[ABP]$ is $6$ , determine the area of $[PBC]$ .

A sequence of integers $(a_0,...,a_k)$ is said to be medaled if, for each $i = 0,...,k$ , there are exactly $a_i$ elements of the sequence equal to $i$ . For example, $(1,2,1,0)$ is a medaled seqence. Indicates all medaled sequences $(a_0,...,a_{2015})$ .