MathDB

2

Part of 2010 Thailand Mathematical Olympiad

Problems(2)

computational, statring with an isosceles and circle

Source: 2010 Thailand Mathematical Olympiad day 1 p2

3/12/2021
Let △ABC\vartriangle ABC be an isosceles triangle with AB=ACAB = AC. A circle passing through BB and CC intersects sides ABAB and ACAC at DD and EE respectively. A point FF on this circle is chosen so that EF⊥BCEF\perp BC. If BC=xBC = x, CF=yCF = y, and BF=zBF = z, find the length of DFDF in terms of x,y,zx, y, z.
geometryisosceles
2010 students from 5 regions in a debate tournament into 3 topics

Source: Thailand Mathematical Olympiad 2010 day 2 p2

8/17/2020
The Ministry of Education selects 20102010 students from 55 regions of Thailand to participate in a debate tournament, where each pair of students will debate in one of the three topics: politics, economics, and societal problems. Show that there are 33 students who were born in the same month, come from the same region, are of the same gender , and whose pairwise debates are on the same topic.
combinatorics