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Problem of the Day #314: $\pi$ January 27, 2012

Posted by Albert in : potd , add a comment

Saketh graphs $f(x) = \frac{1}{1+x^2}$ for $x \in [0,1]$. He cuts out the square $0 \leq x \leq 1$, $0 \leq y \leq 1$, then cuts along the line $f(x)$, producing piece 1 (below $f(x)$) and piece 2 (above $f(x)$). He finds that the ratio of the weights of piece 1 to piece 2 is $R$. Given this, help him find an expression for $\pi$.

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 #178: Twins! September 13, 2011

Posted by Seungln in : potd , add a comment

Andrew Tao has finally found his long-lost twin, Tandrew Ao.

Tandrew Ao, who came from the parallel universe, gives Andrew Tao an unusual gift – a cylinder that constantly changes its radius and height but has the same volume ($400 \pi$ cubic inches). Andrew Tao can carry it only in a rectangular box container made of Aonium; otherwise the cylinder will explode into a burst of $\gamma$-rays. Aonium is expensive and therefore Andrew Tao wants to use as little Aonium as possible. If a cubic inch of Aonium is $\$1$, the Aonium container has to be at least half an inch thick, and the Aonium is sold only in cubic inches, what is the least amount of money that Andrew Tao has to spend to bring his gift back home?

Moving Across the Coordinate Plane May 13, 2011

Posted by Billy in : calculus , add a comment

Albert is standing at the origin of the Cartesian plane, desperately in need of cake. Looking around, he spots some delicious chocolate cake at the point $(100,100)$. Albert immediately departs for the cake. He knows that if he goes outside the square with corners $(0,0)$, $(100,0)$, $(100,100)$, and $(0,100)$ the cake will disappear and he will starve. When Albert is at the point $(x,y)$, the maximum speed he can move is given by $v(x,y) = 5+\frac{y}{20}$. What is the minimum time required for Albert to reach the cake?

A Tricky Exponent March 25, 2011

Posted by Saketh in : calculus , 1 comment so far

Determine the exact value of  $$\int^{\frac{\pi}{2}}_0 \frac{1}{1+(\tan{x})^{\sqrt{2}}} \,dx$$

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An Integral March 25, 2011

Posted by Arjun in : calculus , 3 comments

Find $$\int^{\frac{\pi}{2}}_0 \frac{1}{\sqrt{\tan{x}}} \,dx$$