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Difference between revisions of "User:Shalomkeshet"
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Let the <math>n</math>th iteration of his pattern be <math>x_{n} = n^2 + (n-1)^2 - (n-3)^2 - (n-4)^2 + (n-5)^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2</math>, where the additions and subtractions alternate in twos. Frosty wants to know the remainder when <math>x_{100}</math> is divided by <math>1000</math>.
 
Let the <math>n</math>th iteration of his pattern be <math>x_{n} = n^2 + (n-1)^2 - (n-3)^2 - (n-4)^2 + (n-5)^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2</math>, where the additions and subtractions alternate in twos. Frosty wants to know the remainder when <math>x_{100}</math> is divided by <math>1000</math>.
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==Problem 10==
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On Christmas Eve, Santa’s elves are arranging gifts in different orders. There are <math>n</math> gifts, where <math>n</math> is an odd integer <math>>1</math>, and each gift has a value <math>v_1, v_2, \dots, v_n</math>.
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For each permutation <math>a = (a_1, a_2, \dots, a_n)</math> of <math>\{1, 2, \dots, n\}</math>, the total joy from that arrangement is given by <math>J(a) = \sum_{i=1}^n v_i a_i</math>.
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Santa wonders if there exist two distinct ways to arrange the gifts, <math>a,b</math> of <math>\{1, 2, \dots, n\}</math> where <math>a \neq b</math> such that the difference in the total joy, <math>J(a) - J(b)</math>, is divisible by <math>n!</math>. Prove that such distinct arrangements do indeed exist.

Revision as of 16:40, 24 December 2024

Welcome to Shalom Keshet's

Mathematical Challenge of Christmas Cheer (MCCC) [2024]

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.


Problem 5

The Grinch has cursed Rudolph the Red-Nosed Reindeer by trapping him in a Free-2-Play Mobile Game! Rudolph has to finish the game to escape this curse.

Each level he passes in the game starts an ad with a price of $$n.99$ to skip the ad for each nonnegative integer $n$. A "sales tax" of $7.5\%$ is applied on all levels.

If the total "cost" of completing the game, after tax, is an integer number of cents, find the minimum possible number of steps in the level. (Rudolph wants to spend as little time as possible so he pays to skip every ad)


Problem 6

Ginger1 and Ginger2 are two gingerbreadmen. They are getting bored because all their friends keep getting eaten. They decide to pass the time by playing a game.

The Gingers have two fair coins and a third coin that comes up heads with probability $\frac{4}{7}$. Ginger1 flings the three coins up, and then Ginger2 flings the three coins up. Let $\frac{m}{n}$ be the probability that Ginger1 gets the same number of heads as Ginger2, where $m$ and $n$ are coprime, positive integers. Find $m+n$.


Problem 7

Santa's sleigh appears as a right-angled triangle $\triangle OAB$ with $\angle A = 90^\circ$. It lies inside a right angle with vertex $O$. The altitude of $\triangle OAB$ from $A$ is extended beyond $A$ until it intersects the side of angle $\angle O$ at $M$. The distances from $M$ and $B$ to the second side of angle $\angle O$ are 2 and 1 respectively.

Find the value of $\overline{OA}$.


Problem 8

For some reason, all the different Ghosts of Christmas - past, present, and yet to come - have decided to meet up at their meeting place $x + iy$. If this meeting place is represented by $z$, then $|z|-2=0$ and $|z-i|-|z+5i|=0|$ hold true.

What is the gradient $m$ of the line of form $y=mx+c$?


Problem 9

Frosty the Snowman has been dormant all year, and now is finally his time to awaken from his deep slumber. He likes counting snowflakes and their intricate patterns. He observes the snowflakes and thinks of a certain pattern.

Let the $n$th iteration of his pattern be $x_{n} = n^2 + (n-1)^2 - (n-3)^2 - (n-4)^2 + (n-5)^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$, where the additions and subtractions alternate in twos. Frosty wants to know the remainder when $x_{100}$ is divided by $1000$.


Problem 10

On Christmas Eve, Santa’s elves are arranging gifts in different orders. There are $n$ gifts, where $n$ is an odd integer $>1$, and each gift has a value $v_1, v_2, \dots, v_n$.

For each permutation $a = (a_1, a_2, \dots, a_n)$ of $\{1, 2, \dots, n\}$, the total joy from that arrangement is given by $J(a) = \sum_{i=1}^n v_i a_i$.

Santa wonders if there exist two distinct ways to arrange the gifts, $a,b$ of $\{1, 2, \dots, n\}$ where $a \neq b$ such that the difference in the total joy, $J(a) - J(b)$, is divisible by $n!$. Prove that such distinct arrangements do indeed exist.

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