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Problem of the Day #94: Fractional Trigonometry June 21, 2011

Posted by Saketh in : potd , add a comment

Given that $x = \pi^{\pi^{\pi}}$, evaluate $$ \frac{1}{\frac{1}{4}(\sin^4{x}+\cos^4{x})-\frac{1}{6}(\sin^6{x}+\cos^6{x})} $$

MPotD Summer Contest June 21, 2011

Posted by Saketh in : announcement , add a comment

We are pleased to announce that with today’s problem, our summer competition will officially commence. Readers are encouraged to submit their answers to the daily problems and earn points.

Simply click the title of any PotD posted on or after today to access the submission form. For example, you can submit your answers to the first problem here.

For each problem solved correctly, competitors will receive $1$ point. Additionally, solvers are given a small score bonus based on the total number of times a problem is solved as well as the number of people to solve it before them.

The bonus function is monotonically decreasing in both the number of solvers as well as the rank of the competitor. The intent here is that faster solvers and solvers of harder problems will receive a small score bonus.

In order to give everyone a fair shot at solving problems quickly, new problems will be posted every day at $4$ PM EDT. We wouldn’t want to force you to actually wake up in the morning.

Top solvers will receive prizes at the end of the summer. You can view real-time point totals on the Scoreboard to see where you stand.

Points are associated with your forum account, so you will have to register if you have not done so already.

Please note that unless otherwise indicated, the use of computational aids such as calculators or computer software is strictly prohibited. Additionally, guessing will result in ineligibility to receive points for the problem(s) involved.

Feel free to email us if you have any questions.

Happy problem solving!

Problem of the Day #93: Two Functions June 20, 2011

Posted by Seungln in : potd , add a comment

Let $f(x)$ be the sum of the digits of $x$. For example,

$$f(123) = 1+2+3 = 6$$

and

$$f(1337) = 1 + 3 + 3 + 7 = 14$$

Let $g(x)$ be the difference between the sum of every other digit from the units digit and the sum of every other digit from the tens digit of $x$. For example, $$g(1337) = (3 + 7) – (1 + 3) = 10 – 4 = 6$$ and $$g(7654321) = (1 + 3  + 5 + 7) – (2 + 4 + 6) = 16 – 12 = 4$$ Compute the number of integers $x$ in the range $[1,100000]$ such that $f(x) = g(x)$.

Problem of the Day #92: Binomial Coefficients Modulo 5 June 19, 2011

Posted by Billy in : potd , add a comment

Find the number of coefficients of $(1+x)^{493}$ that are congruent to $0$ mod $5$.

Problem of the Day #91: Circular Tubes June 18, 2011

Posted by Albert in : potd , add a comment

Circle $O$ has radius $r$ and diameter $d = \overline{AB}$. The circle is curved so that it forms part of the surface of a cylinder and point $A$ touches point $B$. The locus of all lines $\overline{PQ}$ such that points $P$ and $Q$ are on circle $O$ and $\overline{PQ}$ is parallel to $d$ is now drawn in, forming a closed shape. What is the ratio of this shape’s volume to $r^3$?

Problem of the Day #90: Transporting Cake through a Tunnel June 17, 2011

Posted by Alex in : potd , add a comment

Saketh wants to bring a cake, shaped like a rectangular prism, to Nebraska in order to celebrate the awesomeness of Mitchell. However, he must do this by taking the cake through a secret path, which consists of two cylindrical tunnels joined at an angle of $120$ degrees (in a V shape). Both cylindrical tunnels have a radius of $8$. Assume that the length of each tunnel is significantly greater than the radii and that Saketh takes up no space. Find the volume of the largest cake that Saketh can bring to Mitchell.

Problem of the Day #89: Darwin’s Diophantine June 16, 2011

Posted by Saketh in : potd , 1 comment so far

Darwin is trying to find positive integers $x$ and $y$ such that $161x + 391y = 1863$. Determine the number of ordered pairs of positive integers $(x,y)$ that satisfy this equation.

Problem of the Day #88: Cow Bracelets June 15, 2011

Posted by Saketh in : potd , add a comment

Albert has many cows. He wishes to identify the cows by giving each one an $11$-bead bracelet. Each bracelet will contain one bead marked with a positive integer. However, Albert can only use numbers up to (and including) $100$ or the cows will get confused.

Additionally, Albert is free to choose from $3$ colors for each of the $11$ beads. If Albert wishes to make each bracelet unique, regardless of any rotations or flips that might occur as the cows roam about, what is the maximum number of cows he can tag?

Problem of the Day #87: Sinusoidal Thoughts June 14, 2011

Posted by Sreenath in : potd , add a comment

The expression $(\sin{x}+\cos{x})^n$ can be written as $k(n)\cdot\sin^n\left(f(x)\right)$, where $k(n)$ is a function of $n$ and $f(x)$ is a polynomial in $x$. Compute $k(10)$.

Problem of the Day #86: AMT Adrenaline June 13, 2011

Posted by Seungln in : potd , add a comment

As the end of the school year approaches, Saketh decides to start on all of the AMT homeworks that were due during 4th quarter. Saketh figures that the workload would keep him up all night.. The thought of that pumps adrenaline through his body.

The amount of adrenaline pumped inside Saketh can be represented by this function:

$$f(t) = \frac{1}{(9-t)^4}$$

When $t$ is the number of hours before 7:00, the time Saketh has to leave the house in order to arrive exactly at 8:30, when school starts.

If he requires 200 units of work (which will now be referred to as ‘psi’) to finish an AMT homework and he gathers up $$8000 \cdot \Sigma_{p=x}^{7} f(p)$$ psi every hour at the hour, what is the largest integer $x$ such that Saketh can start on his AMT homework at $x$ a.m. without having to do any of it at school?