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Problem of the Day #42: Subtracting divisors April 30, 2011

Posted by Mitchell in : potd , add a comment

A number $n$ is written on a blackboard. In one turn, Albert may choose a divisor $d$ of the number $k$ written on the board and replace $k$ by $k – d$. He wishes to change the number to $1$ in the least possible number of moves; call this number $f(n)$.

For how many integers $n < 1000$ is $2^{f(n) – 1} \leq n$?

Problem of the Day #41: Last Digits of an Exponentiated Factorial April 29, 2011

Posted by Albert in : potd , 1 comment so far

Albert is mad at everyone for stipulating the so-called “Albert week” on MPOTD. To get his revenge, he makes everyone else work on the following problem:

  1. Prove that there exists a function $g(k)$ such that the last $k$ digits of $f(n) = 2^{n!}$ becomes constant for all $n \geq g(k)$.
  2. Find the smallest integer $m$ such that the last $32$ digits of $f(n)$ are constant for all $n \geq m$.

Problem of the Day #40: Painting a Face April 28, 2011

Posted by Alex in : potd , add a comment

Albert wishes to paint his face a very special color, RGB #FF6EC7. He does not want to waste a single drop of paint, but he also does not want to leave a spot of his face unpainted. Luckily, Albert can find the volume of paint (of thickness 2) he needs to get because he has found a way to represent his face geometrically.

Let a circle with center $O$ and radius $8$ represent the face. Let octagon $ABCDEFGH$ be a regular octagon inscribed in circle $O$. Let $I$, $J$, $K$, and $L$ be the midpoints of $\stackrel{\frown{}}{AB}$, $\stackrel{\frown{}}{BC}$, $\stackrel{\frown{}}{CD}$, $\stackrel{\frown{}}{DE}$, respectively. Albert’s hair consists of four circles, each marked by two radii:

  1. $IA$ and $IB$
  2. $JB$ and $JC$
  3. $KC$ and $KD$
  4. $LD$ and $LE$

The two eyes are the largest circles inside $\triangle{}AOB$ and $\triangle{}DOE$, respectively, that do overlap with any hair. Albert’s mouth is quadrilateral $AZEG$, where Z is the midpoint of $\overline{OG}$. This geometric representation of Albert’s face does not contain any other facial features.

Given that Albert does not want to paint his eyes, his hair, or his mouth, calculate the volume of paint needed.

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Problem of the Day #39: Albert’s cakes April 27, 2011

Posted by Sreenath in : potd , 1 comment so far

Albert now has two orange cakes, four green cakes, and five pink cakes. He randomly arranges them in a line. Find the probability that no two green cakes are next to each other.

Problem of the Day #38: You will be baked (and then there will be) cake April 26, 2011

Posted by Saketh in : potd , add a comment

Mitchell is decorating a cube-shaped cake using his collection of icing pens. The design he has in mind requires three colors on each face, as shown in the diagram below:

Mitchell would like to add some color to his cake, and has $6$ different icing pens available for this purpose. Although the diagram shows a cube with the same coloring on all of its faces, he is going to color each face of his cake independently. He thus has a total of $18$ different color choices to make while icing the cake.

How many distinct colorings of Mitchell’s cake are possble? Two colorings are not considered distinct if one can be rotated to obtain the other. The orientation of the hearts on the faces of the cube is negligible.

Problem of the Day #37: Winner April 25, 2011

Posted by Aziz in : potd , add a comment

Albert and Arjun are competing in a running tournament with $128$ total participants. They are the best of buddies, but also very competitive, so they’d like to race against each other in the tourney.

The tourney follows a double elimination format, where the winners of the first round move into the Winner Bracket while the losers go into the Loser Bracket.

Every match result thereafter follows the Best of Three format, where a runner needs to win two races against his opponent before moving on in the Bracket.

Compute the number of ways that when Arjun competes against Albert, he has less losses than Albert.

Problem of the Day #36: Fingers April 24, 2011

Posted by Arjun in : potd , 2 comments

Albert has been watching a particularly scary horror anime. In this anime, fingernails are ripped off of fingers. However, there are a few limitations on what can show up on screen (due to censorship laws). Firstly, there is a limited screentime of $23$ minutes, and different fingers have different times taken for the nail to be ripped off, as well as a different amount of horror points associated with them. Thumbs take $50$ seconds, and are worth $10$ horror points. Index fingers take $40$ seconds, and are worth $8$ horror points. Middle fingers take $50$ seconds, and are worth $10$ horror points. The remaining $2$ fingers have already been ripped off, so viewing them on screen takes $10$ seconds and provides a measly $1$ horror point. Due to weird censorship laws, at least $3$ fingers from each hand must be shown on screen. What is the maximum amount of horror points possible?

Problem of the Day #35: No Calculators Please April 23, 2011

Posted by Seungln in : potd , add a comment

Albert is trying to solve a problem using only his head, a pencil or a pen, and some paper. Help him find the last 3 digits of $$\large 3^{999999} + 3^{999998} + \cdots + 3^1 + 1$$.

Problem of the Day #34: GIRP April 22, 2011

Posted by Sreenath in : potd , add a comment

Credit for this problem goes to Zeming Lin

GIRP1

Saketh is climbing a mountain, starting at point (0,0) going to point (4,4). His maximum reach is $\sqrt{2}$. There are no holds on the mountain, but Sreenath can put type A holds and type B holds on any lattice point as he sees fit, though he only has 2 type A holds and 3 type B holds. Saketh cannot climb from a type A hold to another type A hold, or a type B hold to another type B hold. How many ways can Albert place the holds such that Saketh can get to (4, 4)?

Problem of the Day #33: Stacking coins April 21, 2011

Posted by Mitchell in : potd , add a comment

In a certain restaurant, there are $100$ distinguishable tables. Each table can support one stack of zero or more coins, but no more than $100$ coins can be placed on a single table. How many ways are there to stack $250$ distinguishable coins on the tables? (Two arrangements are considered the same if each table has the same coins in the same order in both arrangements.)