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Problem of the Day #65: Trigonometric Expressions and Factorials May 23, 2011

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Evaluate $2b^4 + 2a^2b^2 + a^2 – b^2 + 567$ if $a = \cos(567!)$ and $b = \sin(567!)$.

Problem of the Day #64: Spider on a Triangle May 22, 2011

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Sreenath put a spider on triangle $ABC$ circumscribed by circle $O$, on which SeungIn is travelling.

Having seen the spider on vertex $A$, SeungIn, who is on vertex $B$, tries to maximize his distance from the spider. He is allowed to travel either on an arc of the circle or on an edge of the triangle.

Given that $AB$ = 13, $BC$ = 14, and $CA$ = 15, determine the maximum linear distance that SeungIn can attain from the spider.

Bonus: Now generalize this problem for any $ABC$.

Problem of the Day #63: BATMAN May 21, 2011

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The legendary Indian national hero Sid has many magical powers. For example, he can shoot lazer beams from his right eye. He can fly at 100 MPH, but that’s boring. He is also very skilled at writing poems. Sid uses his magical powers to create the Button thAt creaTes Many Arachnid moNsters (BATMAN). When the BATMAN is pressed,$$0.2x\cdot (\sin(2\pi \cdot x)+0.2x^2+1.2)$$ spiders are created, where x is the time since the last button press in minutes (or 0 if the button has never been pressed). Seungln is locked in a room with the BATMAN. Sreenath presses the button at time 0 and he will press it again one hour later. Seungln is allowed to press the button up to three times. Help Seungln minimize the number of spiders created.

Problem of the Day #62: An Infinite Structure of Cubes May 20, 2011

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Albert has an infinite number of cubes, with side lengths $1$, $\frac{1}{2}$, $\frac{1}{4}$, etc. He aligns the largest cube such that each of its $6$ faces is parallel to either the $xy$, $yz$, or $xz$ plane. Albert constructs a structure in the following way: he picks up the next largest cube that he hasn’t used in his structure yet, chooses any face of the structure so far, and places the cube somewhere on that face. The structure begins with the cube of side length $1$, and Albert continues building until he has used up all the cubes. Let $x_{min}$ be the minimum $x$-coordinate of any of the vertices in his structure, and let $x_{max}$ be the maximum $x$-coordinate of any of the vertices in his structure. Define $y_{min}$, $y_{max}$, $z_{min}$, and $z_{max}$ similarly. What is the maximum possible value of $(x_{max}-x_{min})(y_{max}-y_{min})(z_{max}-z_{min})$?

Problem of the Day #61: Evaluation of a certain two-variable polynomial May 19, 2011

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The polynomial $P(x, y)$ satisfies the equality \[P(x, y) = \begin{cases} 1 &  x=y \\0 &  \text{else} \end{cases}\] for $x, y \in \{0, 1, 2, \cdots, 60\}$. Additionally, the degree of $P$ in $x$ and the degree of $P$ in $y$ are both at most $60$. Find $P(61, 61)$.

Problem of the Day #60: HUM Field Trip Refreshments May 18, 2011

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Albert, Alex, Arjun, Saketh and SeungIn are travelling together on a HUM field trip. They each have a certain amount of money and wish to purchase food. (more…)

Problem of the Day #59: Infinite Semicircular Arcs May 17, 2011

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Let image $1$ be the figure formed by drawing four unique and non-overlapping circular arcs, each with a diameter that is a side of a square with side length $1$. Let image $i + 1$ be the figure formed when, for each semicircular arc in image $i$, two new unique semicircular arcs are drawn, each with one point at the end of the original arc and the second point at the midpoint of the arc. See the images below for examples. In each image, the “outside perimeter” is represented by solid lines.

Find the outside perimeter of figure $22$.
Alternatively, find the outside perimeter of figure $i$ as $i$ approaches $\infty$, or prove that the answer does not exist.


Problem of the Day #58: Pythagorean Triples May 16, 2011

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Let $ALBERT$ be a hexagon inscribed in the circle $O$ with radius $r$. $ALBERT$ has several interesting properties.

1. $AL = ER, LB = RT, BE = TA$

2. $\cos ALE = \cos LBR = \cos BET $

                   $= \cos ERA = \cos RTL = \cos TAB= 0$

3. $\cos AEL = \cos EAR = \frac{12}{13}$

4. $\cos LRB = \cos RLT = \frac{15}{17}$

5. $\cos BTE = \cos TBA = \frac{84}{85}$

6. $3 | r$

If $r$ is the hypotenuse of a Pythagorean triple, then there are four possible values for the length of the shorter leg, only one of which is not the length of one of the sides of $ALBERT$. What is the value?

Problem of the Day #57: Geological Expedition May 15, 2011

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Arjun wishes to obtain a sample of Guralite, a rare mineral that is found only on the cliffside of the cone-shaped Mount Rieger. He knows that there is a line running from the top of the mountain down to a point on its base along which deposits of Guralite can be accessed from the surface. Arjun wishes to travel to any point on this line.

Unfortunately, he sets up camp at the point on the mountain’s base that is diametrically opposite to the foot of this line. If Mount Rieger is $10$ km tall with a base of radius $4$ km, determine the shortest path Arjun can take along the mountain’s surface to reach a deposit of Guralite. Then, use a calculator or computer to approximate its length to the nearest thousandth of a kilometer.

Problem of the Day #56: Spring Cleaning May 14, 2011

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Five people, Sreenath, Mitchell, Alex, Albert and Sam, live in the same house and need to clean up before their parents come visit.

Each person gets his own room. A person will clean only their own room. Each person’s own room has a certain mess level, greater values requiring more cleaning. A cycle consists of everyone cleaning a certain amount of their own room and dumping the rest of their mess into another person’s room. Each person is required to clean between $5$ and $20$ of their mess each cycle. Whatever is left uncleaned is dumped into a randomly chosen other person’s room. Whenever someone dumps their remaining mess, the person receiving it will gain an additional 5 mess units as a cost for dumping.

Mess Factor of:
Sreenath — 150
Mitchell — 80
Alex — 50
Albert — 10
Sam — 70

The goal is to minimize the number of cycles it takes to clean the house.

Determine the expected number of cycles it would take if everyone acted independently and if they came up with and followed an optimal plan before beginning the cleanup.