MathDB

1

Part of 1991 Turkey Team Selection Test

Problems(2)

Turkey TST 1991 - P1

Source:

3/13/2011
Let C′,B′,A′C',B',A' be points respectively on sides AB,AC,BCAB,AC,BC of △ABC\triangle ABC satisfying AB′B′C=BC′C′A=CA′A′B=k \tfrac{AB'}{B'C}= \tfrac{BC'}{C'A}=\tfrac{CA'}{A'B}=k. Prove that the ratio of the area of the triangle formed by the lines AA′,BB′,CC′AA',BB',CC' over the area of △ABC\triangle ABC is (k−1)2(k2+k+1)\tfrac{(k-1)^2}{(k^2+k+1)}.
ratiogeometrygeometry proposed
Turkey TST 1991 - P4

Source:

3/13/2011
A frog is jumping on NN stones which are numbered from 11 to NN from left to right. The frog is jumping to the previous stone (to the left) with probability pp and is jumping to the next stone (to the right) with probability 1−p1-p. If the frog has jumped to the left from the leftmost stone or to the right from the rightmost stone, it will fall into the water. The frog is initially on the leftmost stone. If p<13p< \tfrac 13, show that the frog will fall into the water from the rightmost stone with a probability higher than 12\tfrac 12.
probabilityalgebra proposedalgebra