MathDB

8

Part of 2008 JBMO Shortlist

Problems(3)

Σx_i \cdot Σ1/x_i >= 4 (Σχ_i/(x_ix_j+1))^3 , i=1,2,3

Source: JBMO 2008 Shortlist A8

10/14/2017
Show that (x+y+z)(1x+1y+1z)≥4(xxy+1+yyz+1+zzx+1)2(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2 , for all real positive numbers x,yx, y and zz.
JBMOinequalitiesalgebra
2008 JBMO Shortlist G8

Source: 2008 JBMO Shortlist G8

10/10/2017
The side lengths of a parallelogram are a,ba, b and diagonals have lengths xx and yy. Knowing that ab=xy2ab = \frac{xy}{2}, show that (a,b)=(x2,y2)\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right) or (a,b)=(y2,x2)\left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right).
geometryJBMO
1 square is sum of 3 squares in 2 different ways

Source: JBMO 2008 Shortlist N8

10/14/2017
Let a,b,c,d,e,fa, b, c, d, e, f are nonzero digits such that the natural numbers abc‾,def‾\overline{abc}, \overline{def} and abcdef‾\overline{abcdef } are squares. a) Prove that abcdef‾\overline{abcdef} can be represented in two different ways as a sum of three squares of natural numbers. b) Give an example of such a number.
JBMOnumber theory