MathDB

5

Part of 2009 Puerto Rico Team Selection Test

Problems(2)

computational with a cyclic ABCD

Source: 2009 Puerto Rico Team Selection Test p5

10/23/2021
Let ABCD ABCD be a quadrilateral inscribed in a circle. The diagonal BD BD bisects AC AC. If AB=10 AB = 10, AD=12 AD = 12 and DC=11 DC = 11, find BC BC.
geometrycyclic quadrilateral
2009 integers around a circle, weird mean Puerto Rico Ibero IMO TST 2009.5

Source:

9/16/2021
The weird mean of two numbers a a and b b is defined as 2a2+3b25 \sqrt {\frac {2a^2 + 3b^2}{5}}. 2009 2009 positive integers are placed around a circle such that each number is equal to the the weird mean of the two numbers beside it. Show that these 2009 2009 numbers must be equal.
algebranumber theory