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Problem of the Day #103: Triples with Log June 30, 2011

Posted by Arjun in : potd , add a comment

Find the number of ordered triples $(a,b,c)$ of positive integers such that $ab + c \le 87$ and $\log_a b$ is a positive integer.

Problem of the Day #102: Billy’s Cities June 29, 2011

Posted by Sreenath in : potd , add a comment

Billy is in charge of creating plans for roads between a number of cities. To create a plan for $n$ cities, he flips a coin for each of the $\binom{n}{2}$ possible roads. If it lands heads, he builds the road, and if it lands tails, he doesn’t. A plan is considered acceptable if for every choice of two cities, there exists a path between them (can go through other cities). If Billy creates $4620$ plans for $3$ cities using this method, compute the expected value of the number of acceptable plans.

Problem of the Day #101: Square Roots are Hard June 28, 2011

Posted by Seungln in : potd , add a comment

If $a = 18.36$, what is the approximate value of $$\sqrt{a^2 \cdot 123456789}$$ Round to the nearest integer. No calculators!

Problem of the Day #100: Tangent Circles June 27, 2011

Posted by Billy in : potd , 2 comments

Circle $A$ has radius $3$ and circle $B$ has radius $2$. Their centers are $13$ units from one another. A common internal tangent $\overline{PQ}$ is drawn such that $P$ lies on $A$ and $Q$ lies on $B$. Next, circles $A’$ and $B’$ are constructed outside of circles $A$ and $B$ such that circle $A’$ is tangent to $\overline{AB}$, $\overline{PQ}$, and circle $A$, and circle $B’$ is tangent to $\overline{AB}$, $\overline{PQ}$, and circle $B$. The distance between the centers of $A’$ and $B’$ can be expressed in the form $\frac{a\sqrt{b}-c\sqrt{d}}{e}$, where $a$, $c$, and $e$ are relatively prime and $b$ and $d$ are not divisible by the square of any prime. Find $a+b+c+d+e$.

Problem of the Day #99: Transaction Fees June 26, 2011

Posted by Saketh in : potd , add a comment

Mitchell is purchasing cookies at the local bakery. He can buy cookies one at a time: the first will cost $\$1$, the second $\$2$, and so on. That is, the $n^{\mathrm{th}}$ cookie will cost $\$n$. However, his debit card company deducts $\frac{1}{7}$ of the funds on his card before charging for any purchase.

Suppose he arrives at the bakery with an integer number of dollars on his card. If he buys some cookies and ends up with exactly $0$ dollars, what is the greatest amount of money he could have started with?

Problem of the Day #98: A Serpent with Regenerating Heads June 25, 2011

Posted by Alex in : potd , 5 comments

There exists a magical serpent with a single head of size $234$. Saketh the Valiant and Sreenath the Troll wish to slay this magical serpent. They know that when they pulverize a head of size $n$, there is a $\frac{2}{3}$ chance that the head will regenerate into two new heads, each of size $m$, where $m$ is a random integer between $0$ and $n$, inclusive (the two new heads may have different sizes). To slay the magical serpent, they must pulverize all of its heads. What is the expected value of the total size of all the heads that Saketh the Valiant and Sreenath the Troll will have to pulverize in order to slay the serpent?

Summer Contest 2011 Prizes June 25, 2011

Posted by Saketh in : announcement , add a comment

The MPotD Summer 2011 Contest is now underway, and we are pleased to announce that competitors will be able to earn significant prizes throughout the summer.

Short-term rankings will be maintained for each of the first four two-week periods of the contest, and the top solver from each period will receive a $10 gift card of their choice. However, each competitor can only win this prize once.

Additionally, overall rankings will be maintained for the duration of the competition. At the end of the summer, top solvers will receive prizes in the form of gift cards of their choice. The top two finishers will also receive Bitcable gift cards, good for 6 months of web hosting.

$1^{\mathrm{st}}$ $\$100$ Gift Card+ $\$15$ Bitcable
$2^{\mathrm{nd}}$ $\$75$ Gift Card+ $\$15$ Bitcable
$3^{\mathrm{rd}}$ $\$50$ Gift Card
$4^{\mathrm{th}}$ $\$25$ Gift Card
Top $10$ Math Problem of the Day T-Shirt
Special Prize $\$10$ Gift Card or MPotD T-Shirt

Problem of the Day #97: Logs and Products and Roots June 24, 2011

Posted by Albert in : potd , add a comment

There are $n$ numbers $a_i$ such that $a_i \in (1,\infty)$ for all $a_i$. Given:

$$\prod\limits_{i=1}^{n} \sqrt[n]{a_i^5} = 1000$$

Find the maximum possible value of:

$$25\sqrt[n]{\log_{10}(a_1) \times \log_{10}(a_2) \times \cdots \times \log_{10}(a_n)}$$

Problem of the Day #96: Sum of Five Nonzero Squares June 23, 2011

Posted by Saketh in : potd , add a comment

Determine the greatest integer that cannot be expressed as the sum of five nonzero squares.

Problem of the Day #95: A cube in a sphere in a cube June 22, 2011

Posted by Sreenath in : potd , add a comment

A sphere is inscribed within a cube. A cube is then inscribed within the sphere such that the faces of the inner cube are all parallel to the corresponding faces on the outer cube. Let $S$ be the set of the distances from the center of one face of the larger cube to the vertices of the smaller cube. If the side length of the smaller cube is $1$, compute the sum of the squares of the elements in $S$.