MathDB

1

Part of 2002 Vietnam Team Selection Test

Problems(2)

interior angle bisector of BC intersects the trisectors

Source: Vietnam TST 2002 for the 43th IMO, problem 1

6/26/2005
Find all triangles ABCABC for which ∠ACB\angle ACB is acute and the interior angle bisector of BCBC intersects the trisectors (AX,(AY(AX, (AY of the angle ∠BAC\angle BAC in the points N,PN,P respectively, such that AB=NP=2DMAB=NP=2DM, where DD is the foot of the altitude from AA on BCBC and MM is the midpoint of the side BCBC.
geometryangle bisectorperpendicular bisectorgeometry unsolved
Half of the elements in the array are colored in red

Source: Vietnam TST 2002 for the 43th IMO, problem 4

6/26/2005
Let n≥2n\geq 2 be an integer and consider an array composed of nn rows and 2n2n columns. Half of the elements in the array are colored in red. Prove that for each integer kk, 1kk rows such that the array of size k×2nk\times 2n formed with these kk rows has at least k!(n−2k+2)(n−k+1)(n−k+2)⋯(n−1) \frac { k! (n-2k+2) } {(n-k+1)(n-k+2)\cdots (n-1)} columns which contain only red cells.
floor functioncombinatorics unsolvedcombinatorics