Hippopotamus Hexagon Hopping, Part 2July 7, 2011

Posted by Alex in : bonus , 1 comment so far

There exists a two-row hexagonal grid, shown below, that extends infinitely to the right, continuing the numbering pattern shown. Albert is on hexagon $1$. A hippopotamus is on hexagon $12345$. Every second, the hippopotamus jumps and lands on, with equal probability, any adjacent hexagon that is not further away* from Albert than the current hexagon (there is a chance the hippopotamus will not move at all). If the expected value of the number of seconds it will take for the hippopotamus to reach Albert is $t$, compute $1000 t$. *Note: the distance between two hexagons is defined as the length of the shortest sequence of adjacent hexagons that goes between them.

Billy’s PizzaMay 31, 2011

Posted by Saketh in : bonus , 2 comments

Billy obtained a triangular slice of pizza $ABC$ to go with his soda. If $m\angle A = m\angle B = 72 ^\circ$ and $AB = 1$, compute the exact value of $BC$.

Playing with Cyclic QuadrilateralsMay 27, 2011

Posted by Saketh in : bonus , add a comment

Alex has created a model of a cyclic quadrilateral using some modelling clay. He observes that it has the following properties:

• It has a side of length $3$ and a side of length $5$.
• Its diagonals are perpendicular to each other.
• The sum of the lengths of its diagonals is $10$.

SeungIn sees a spider crawling on the model and attempts to swat it. Unfortunately, he ends up squishing the quadrilateral instead. Both of its diagonals shrink by exactly $1$ unit, but they remain perpendicular to each other. Determine the area of the new shape.

Happy April First!April 1, 2011

Posted by Albert in : bonus , 3 comments

Compute the probability that three positive integers chosen at random are relatively prime. Your answer must be in terms of rational numbers and $\pi$.

A Two-Player Board GameMarch 20, 2011

Posted by Saketh in : bonus , add a comment

On a 5×5 board, two players, Albert and Billy, alternately mark numbers on empty cells. The first player, Albert, always marks 1′s, the second, Billy, 0′s. One number is marked per turn, until the board is filled. For each of the nine 3 x 3 squares the sum of the nine numbers on its cells is computed. How large can the Albert force the maximum of these sums to be regardless of Billy’s behavior?