Maths for the future

Domain of m

The set

Number of solutions

She chooses her moves

The numbers from 1 to 100 are written in order around a circle. On each move Ana chooses
an even number, y, on the circle, erases it along with its two neighbors, x and z, and replaces
the three numbers with the sum of the two neighbors, x + z. She continues to make moves
until either all the remaining numbers are odd or there are fewer than 3 numbers remaining.
Prove that no matter how she chooses her moves, she will end up with 2 even numbers.

Winning Strategy

ABR and BILLU play the following game.
They start out with n dimes on a table and take turns with ABR starting.
In each step a player can take at most n/2 + 1 dimes from the table,
but he has to take at least one. If somebody takes all the dimes on the table
then he wins. For which values of n will ABR have a winning strategy?

Probability that the sum is Even

We have an urn with 1000 balls numbered from 1 to 1000.
We choose 9 balls randomly from the urn (without replacement)
and add the shown numbers. Determine the probability that the sum is even.

Divisible by Five

Show that if the sum of the fifth powers of five integers is divisible by 25
then one of the original integers is divisible by five.

ID and AB are parallel

Let ABC be a triangle with AB = AC, circumcenter O and incenter I.
Let D be the point on AC such that line OD is perpendicular to line CI.
Prove that ID and AB are parallel.