MathDB

3

Part of 2005 ITAMO

Problems(2)

1 or 2 is written in a 4X4 grid

Source: ItaMO 2005, P3

3/9/2012
In each cell of a 4×44 \times 4 table a digit 11 or 22 is written. Suppose that the sum of the digits in each of the four 3×33 \times 3 sub-tables is divisible by 44, but the sum of the digits in the entire table is not divisible by 44. Find the greatest and the smallest possible value of the sum of the 1616 digits.
combinatorics proposedcombinatorics
CD bisects arc EF

Source: ItaMO 2005, P6

3/9/2012
Two circles γ1,γ2\gamma_1, \gamma_2 in a plane, with centers AA and BB respectively, intersect at CC and DD. Suppose that the circumcircle of ABCABC intersects γ1\gamma_1 in EE and γ2\gamma_2 in FF, where the arc EFEF not containing CC lies outside γ1\gamma_1 and γ2\gamma_2. Prove that this arc EFEF is bisected by the line CDCD.
geometrycircumcircleincentergreatest common divisorangle bisectorgeometry proposed