Revision as of 14:58, 24 December 2024

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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 puzzles 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 have to help Timmy find the solution.

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 ==Problem 2==
-Santa's elves have prepared a nutcracker festival and have arranged them as a triangle <math>\triangle ABC</math>. They want to know whether there is a line <math>\textit{\textrm{l}}</math> in the plane of <math>\triangle ABC</math> such that the intersection of the interior of <math>\triangle ABC</math> and the interior of its reflection <math>\triangle A'B'C'</math> in <math>\textit{\textrm{l}}</math> has an area more than <math>\frac{2}{3}</math> the area of <math>\triangle ABC</math>.
+Santa's elves have prepared a nutcracker festival and have arranged them as a triangle <math>\triangle ABC</math>. They want to know whether there is a line <math>\textit{\textrm{l}}</math> in the plane of <math>\triangle ABC</math> such that the intersection of the interior of <math>\triangle ABC</math> and the interior of its reflection <math>\triangle A'B'C'</math> in <math>\textit{\textrm{l}}</math> has an area more than <math>\frac{2}{3}</math> the area of <math>\triangle ABC</math>. 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 puzzles talks of a number <math>N</math>, which is defined as the smallest positive integer such that this number multiplied by the peak gaming year, i.e., <math>2008 \cdot N</math>, is a perfect square and <math>2007 \cdot N</math> is a perfect cube. Timmy needs to find the remainder when <math>N</math> is divided by <math>25</math>. Of course, you don't want to die either, so you have to help Timmy find the solution.

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