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Mathematical Challenge of Christmas Cheer (MCCC)

Merry Christmas ladies and gentlemen, today I have procured a set of Jolly Problems for you to solve, good luck!

Problem 1

Santa has brought 5 gifts for five people $A, B, C, D$ and $E$ and has placed them around the Christmas tree in a circular arrangement. If each of the gifts contains a surprise of one of the three types: toy, gadget and sweet, then the number of ways of distributing the surprises such that the gifts placed in adjacent positions get different surprise is ............

Problem 2

Santa's elves have prepared a nutcracker festival and have arranged them as a triangle $\triangle ABC$ . They want to know whether there is a line $\textit{\textrm{l}}$ in the plane of $\triangle ABC$ such that the intersection of the interior of $\triangle ABC$ and the interior of its reflection $\triangle A'B'C'$ in $\textit{\textrm{l}}$ has an area more than $\frac{2}{3}$ the area of $\triangle ABC$ . Show the elves why such a line exists.

Problem 3

Little Timmy has been good this year and wishes for a mecharobot suit for Christmas. As his parents are associated with the Mafia, Santa has no choice but to comply with Timmy's wishes. The elves make a puzzle for Timmy to solve, and only if he solves it will he get his desired present.

The puzzle talks of a number $N$ , which is defined as the smallest positive integer such that this number multiplied by the peak gaming year, i.e., $2008 \cdot N$ , is a perfect square and $2007 \cdot N$ is a perfect cube. Timmy needs to find the remainder when $N$ is divided by $25$ .

Of course, you don't want to die either, so you need to help Timmy find the solution.

Problem 4

Santa has been busy in the workshop making little Timmy's gift, so he puts the responsibility of the nice and naughty lists on his elves.

According to the elves, each child can be represented as a lattice point $(x, y)$ . A child is added to the nice list if $x, y$ are natural numbers and $x,y \le 100$ . Children who do not satisfy these conditions are added to the naughty list.

Find the minimum number of lines with gradient $\frac{3}{7}$ we should draw in the way that each child belonging to the nice list lies on at least one of these lines.

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