jump to navigation

Problem of the Day #308: Squares January 21, 2012

Posted by Saketh in : potd , add a comment

Given a $10$ by $10$ grid, what is the least number of cells you must remove so that no four of the remaining cells form the corners of a square parallel to the sides of the grid?

Problem of the Day #307: Counting Binary Strings January 20, 2012

Posted by Saketh in : potd , 1 comment so far

Determine, in terms of $n$, the number of binary strings of length $2n-1$ that do not contain any substrings of length $n$ that are all $0$ or all $1$.

Problem of the Day #306: The Clock January 19, 2012

Posted by Saketh in : potd , add a comment

Alex is stuck in a room with a very strange clock. Every time the minute hand of the clock reaches the $12$, the clock picks a real number $n$ between $1$ and $N$. The minute hand will then complete its next full rotation in $1/n$ real minutes.

Although Alex cannot keep track of the time exactly using this clock, he can estimate the current real time based on what the clock shows. Devise a formula he can use to do this.

Bonus: Devise a function $f(k)$ that gives the probability that Alex’s estimate of the real time is correct at $k$ clock hours.

Problem of the Day #305: Triangle Containing the Origin January 18, 2012

Posted by Alex in : potd , add a comment

Point $A$ has coordinates $(1, 1)$ and point $b$ has coordinates $(-3, -4)$. If isosceles $\triangle ABC$ contains the origin, find the least possible area of $\triangle ABC$.

Problem of the Day #304: Towers of Hanoi January 17, 2012

Posted by Alex in : potd , 1 comment so far

This problem features a variation on the traditional Towers of Hanoi puzzle. Suppose we have two pegs and $7$ discs of size $1$, $2$, $3$, $4$, $5$, $6$, and $7$. Disc $7$ is under disc $6$, which is under disc $5$, and so on, all on the same peg. A disc of size $x$ can only be placed on top of a disc with size $x + k$, where $k$ is a positive odd integer. One step consists of moving a disc from one peg to another peg. The puzzle is solved when all the discs are moved from one peg to another. How many pegs must be added before the puzzle can be solved, and what is the length of the shortest sequence of steps needed to solve the puzzle?

Problem of the Day #303: Triangle with Tangent Base January 16, 2012

Posted by Alex in : potd , 1 comment so far

A circle with radius $2$ has a center at point $C$. $\triangle ABC$ is constructed such that $\overline{AB}$ is tangent to circle $C$ at point $D$ and $D$ is the midpoint of $\overline{AB}$. Let $W$ be the area of the region inside $\triangle ABC$ but outside circle $C$ and $Q$ be the area of the region inside both $\triangle ABC$ and circle $C$. Find $\frac{W}{Q}$.

Problem of the Day #302: Optimal Path January 15, 2012

Posted by Albert in : potd , 1 comment so far

Alex is trying to run to Saketh as fast as he can in Coordinatesville. Alex starts off at $(0 \text{m}, 0 \text{m})$ and runs to Saketh at $(10 \text{m} , 10 \text{m})$. Given that Coordinatesville gets increasingly muddy for increasing $x$, such that Alex’s speed $v$ (in meters per second) is given by $v(x) = e^{x^2} \frac{\text{m}}{\text{s}}$, find the minimum amount of time it takes for Alex to reach Saketh.

Problem of the Day #301: Triangle Boundaries January 14, 2012

Posted by Alex in : potd , add a comment

$\triangle ABC$ has side lengths $13$, $14$, $15$. Let $R$ be the region containing all the points within $3$ of at least one of the sides of $\triangle ABC$. Find the area of $R$.

Problem of the Day #300: Lucky Days (a.k.a. Friday the 13th) January 13, 2012

Posted by Alex in : potd , add a comment

Suppose the calendar days were numbered starting from $1$, and day $1$ is a Friday. A day is lucky if it is a Friday and the sum of the digits of its number is a multiple of $13$. How many days from $1$ to $1,000,000$ are lucky?

Problem of the Day #299: Rolling an Arithmetic Sequence January 12, 2012

Posted by Alex in : potd , add a comment

Saketh rolls a fair six-sided dice until the last four rolls form an arithmetic sequence. Find the expected number of rolls needed before Saketh stops.