MathDB

21

Part of 2013 Online Math Open Problems

Problems(2)

2012-2013 Winter OMO #21

Source:

1/16/2013
Dirock has a very neat rectangular backyard that can be represented as a 32×3232\times 32 grid of unit squares. The rows and columns are each numbered 1,2,…,321,2,\ldots, 32. Dirock is very fond of rocks, and places a rock in every grid square whose row and column number are both divisible by 33. Dirock would like to build a rectangular fence with vertices at the centers of grid squares and sides parallel to the sides of the yard such that [*] The fence does not pass through any grid squares containing rocks; [*] The interior of the fence contains exactly 5 rocks. In how many ways can this be done?
Ray Li
Online Math OpengeometryrectangleLaTeX
2013-2014 Fall OMO #21

Source:

10/30/2013
Let ABCABC be a triangle with AB=5AB = 5, AC=8AC = 8, and BC=7BC = 7. Let DD be on side ACAC such that AD=5AD = 5 and CD=3CD = 3. Let II be the incenter of triangle ABCABC and EE be the intersection of the perpendicular bisectors of ID‾\overline{ID} and BC‾\overline{BC}. Suppose DE=abcDE = \frac{a\sqrt{b}}{c} where aa and cc are relatively prime positive integers, and bb is a positive integer not divisible by the square of any prime. Find a+b+ca+b+c.
Proposed by Ray Li
Online Math Opengeometryincentertrigonometryinradiusanalytic geometrycircumcircle