MathDB

3

Part of 2013 Middle European Mathematical Olympiad

Problems(2)

Isosceles triangle

Source: Middle European Mathematical Olympiad 2013 I-3

5/17/2014
Let ABCABC be an isosceles triangle with AC=BCAC=BC. Let NN be a point inside the triangle such that 2∠ANB=180∘+∠ACB2 \angle ANB = 180 ^\circ + \angle ACB . Let D D be the intersection of the line BNBN and the line parallel to ANAN that passes through CC. Let PP be the intersection of the angle bisectors of the angles CANCAN and ABNABN. Show that the lines DPDP and ANAN are perpendicular.
geometryperpendicular bisectorgeometry proposedanglechasing
different sequences of number plates

Source: Middle European Mathematical Olympiad 2013 T-3

5/17/2014
There are n≥2n \ge 2 houses on the northern side of a street. Going from the west to the east, the houses are numbered from 1 to nn. The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the currnet number plates are swapped exactly once during the day. How many different sequences of number plates are possible at the end of the day?
combinatorics proposedcombinatorics