MathDB

4

Part of 2015 Dutch IMO TST

Problems(2)

each of numbers 1-2014 colored red and blue, half each, max sum 2 balls

Source: Dutch IMO TST 2015 day 2 p4

8/30/2019
Each of the numbers 11 up to and including 20142014 has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number kk of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of kk that can be obtained.
maximum valuemaximumcombinatoricsSumColoring
perpendicular wanted, intersecting circles and parallel given

Source: Dutch IMO TST1 2015 P4

8/5/2019
Let Γ1\Gamma_1 and Γ2\Gamma_2 be circles - with respective centres O1O_1 and O2O_2 - that intersect each other in AA and BB. The line O1AO_1A intersects Γ2\Gamma_2 in AA and CC and the line O2AO_2A intersects Γ1\Gamma_1 in AA and DD. The line through BB parallel to ADAD intersects Γ1\Gamma_1 in BB and EE. Suppose that O1AO_1A is parallel to DEDE. Show that CDCD is perpendicular to O2CO_2C.
geometryperpendicularparallelcircles