Problems(4)
9x9 board
Source: IZO 1 Junior Problem 1
12/16/2008
The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table?
combinatorics proposedcombinatorics
easy ineq
Source: IZO 1 Junior Problem 4
12/16/2008
For the positive real numbers prove the inequality
\frac {c}{a \plus{} 2b} \plus{} \frac {a}{b \plus{} 2c} \plus{} \frac {b}{c \plus{} 2a}\ge1.
inequalitiesinequalities proposed
x^5 + 31 = y^2
Source: First Zhautykov Olympiad 2005, Problem 1
1/19/2008
Prove that the equation x^{5} \plus{} 31 \equal{} y^{2} has no integer solution.
algebrapolynomialquadraticsmodular arithmeticnumber theory unsolvednumber theory
Familiar inequality expression greater than 4/3.
Source: First Zhautykov Olympiad 2005, Problem 4
12/22/2008
For the positive real numbers prove that \frac c{a \plus{} 2b} \plus{} \frac d{b \plus{} 2c} \plus{} \frac a{c \plus{} 2d} \plus{} \frac b{d \plus{} 2a} \geq \frac 43.
inequalities