MathDB

3

Part of 1992 Turkey Team Selection Test

Problems(2)

Turkey TST 1992 - P3: x_1.x_2...x_(n+1) >= n^(n+1)

Source:

3/13/2011
x1,x2,⋯ ,xn+1x_1, x_2,\cdots,x_{n+1} are posive real numbers satisfying the equation 11+x1+11+x2+⋯+11+xn+1=1\frac{1}{1+x_1} + \frac{1}{1+x_2} + \cdots + \frac{1}{1+x_{n+1}} =1
Prove that x1x2⋯xn+1≥nn+1x_1x_2 \cdots x_{n+1} \geq n^{n+1}.
inequalitiesinequalities proposed
Turkey TST 1992 - P6: Find circle with 11 pts out of 251 pts

Source:

3/13/2011
A circle with radius 44 and 251251 distinct points inside the circle are given. Show that it is possible to draw a circle with radius 11 and containing at least 1111 of these points.
geometrycombinatorics proposedcombinatorics