MathDB

1

Part of 2012 Vietnam National Olympiad

Problems(2)

Sequence of reals x_1=3 and x_n={n+2}/{3n}(x_{n-1}+2).

Source: Vietnamese National Mathematical Olympiad 2012-P1

2/8/2012
Define a sequence {xn}\{x_n\} as: {x1=3xn=n+23n(xn−1+2)  for n≥2.\left\{\begin{aligned}& x_1=3 \\ & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right. Prove that this sequence has a finite limit as n→+∞.n\to+\infty. Also determine the limit.
limitalgebra proposedalgebra
Group of 5 girls and 12 boys sitting in 17 seats

Source: Vietnamese National Mathematical Olympiad 2012-P5

2/8/2012
For a group of 5 girls, denoted as G1,G2,G3,G4,G5G_1,G_2,G_3,G_4,G_5 and 1212 boys. There are 1717 chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met: (a) Each chair has a proper seat. (b) The order, from left to right, of the girls seating is G1;G2;G3;G4;G5.G_1; G_2; G_3; G_4; G_5. (c) Between G1G_1 and G2G_2 there are at least three boys. (d) Between G4G_4 and G5G_5 there are at least one boy and most four boys. How many such arrangements are possible?
combinatorics proposedcombinatorics