MathDB

2

Part of 2008 Saint Petersburg Mathematical Olympiad

Problems(2)

St Petersburgh 2008 #2

Source:

7/23/2011
In a kingdom, there are roads open between some cities with lanes both ways, in such a way, that you can come from one city to another using those roads. The roads are toll, and the price for taking each road is distinct. A minister made a list of all routes that go through each city exactly once. The king marked the most expensive road in each of the routes and said to close all the roads that he marked at least once. After that, it became impossible to go from city AA to city BB, from city BB to city CC, and from city CC to city AA. Prove that the kings order was followed incorrectly.
combinatorics proposedcombinatorics
Inscribed quadrilateral

Source: St Petersburg 2008 10th grade #2

7/28/2011
Point OO is the center of the circle into which quadrilateral ABCDABCD is inscribed. If angles AOCAOC and BADBAD are both equal to 110110 degrees and angle ABCABC is greater than angle ADCADC, prove that AB+AD>CDAB+AD>CD.
Fresh translation.
trigonometrygeometry proposedgeometry