## Problem of the Day #124: Circle Inscribed Within an Isosceles TrapezoidJuly 21, 2011

Posted by Saketh in : potd , add a comment

A circle is inscribed inside isosceles trapezoid $ABCD$. The circle intersects diagonal $\overline{AC}$ twice, once at point $E$ and again at point $F$. Suppose $E$ lies between $A$ and $F$. The value of

$$\frac{AF \cdot EC}{AE \cdot FC}$$

can be expressed in the form $a + \sqrt{b}$, where $a$ and $b$ are positive integers. Compute the value of $ab$.

## Problem of the Day #123: The Alberts’ Triangle GameJuly 20, 2011

Posted by Billy in : potd , add a comment

Albert Sr. (Senior) and Albert Jr. (Junior) are playing a game. Senior picks a random point inside a triangle with sides $5$, $7$, and $9$. He then draws three lines, one through each of the vertices of the triangle, that also go through the point he picked. This divides the triangle into six smaller triangles. Junior then picks the three smaller triangles with the largest areas. If the sum of the areas of the three triangles is more than $\frac{2}{3}$ of the area of the original triangle, Junior wins the game. If Senior and Junior play $10000$ games, what is the closest integer to the number of games Junior can expect to win?

## End of Second Two-Week PeriodJuly 19, 2011

Posted by Sreenath in : announcement , add a comment

Please join us in congratulating Lewis Chen, the winner of the second two-week period. This period spanned problems $108$ through $121$.