問題1.Armenia Team Selection Test 2011
證明對任意正實數$a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$有

$\displaystyle\sum\limits_{i=1}^{n}{\frac{a_{i}(a_{1}+a_{2}+\cdots+a_{i})}{b_{1}+b_{2}+\cdots+b_{i}}}\le 2\sum\limits_{i=1}^{n}{\frac{a_{i}^{2}}{b_{i}}},$

並且等號不成立.

問題2.Morocco National Olympiad 2011 Day 1
正數$a,b$滿足$a+b=ab$,求證:

$\displaystyle\frac{a}{b^2+4}+\frac{b}{a^2+4}\ge \frac{1}{2}.$

問題3.USA Mathematical Olympiad 2011
如果正數$a,b,c$滿足$a^2+b^2+c^2+(a+b+c)^2\le 4$,求證:

$\displaystyle\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\ge 3.$

問題4.Balkan Mathematical Olympiad 2011
對任意滿足$x+y+z=0$的實數$x,y,z$. 求證:

$\displaystyle\frac{x(x+2)}{2x^{2}+1}+\frac{y(y+2)}{2y^{2}+1}+\frac{z(z+2)}{2z^{2}+1}\ge 0.$