5

Part of 2008 JBMO Shortlist

Problems(3)

1/x+4/y+9/z=3, x+y+z<=12, x,y,z >0

Source: JBMO 2008 Shortlist A5

Find all triples

(x, y, z)

of real positive numbers, which satisfy the system

\begin{cases} \frac{1}{x}+\frac{4}{y}+\frac{9}{z}=3 \\ x + y + z \le 12 \end{cases}

JBMOalgebrasystem of equations

2008 JBMO Shortlist G5

Source: 2008 JBMO Shortlist G5

Is it possible to cover a given square with a few congruent right-angled triangles with acute angle equal to

{{30}^{o}}

? (The triangles may not overlap and may not exceed the margins of the square.)

1^1, 2^2,..., 2008^{2008} rearrangement for perfect square

Source: JBMO 2008 Shortlist N5

Is it possible to arrange the numbers

1^1, 2^2,..., 2008^{2008}

one after the other, in such a way that the obtained number is a perfect square? (Explain your answer.)

JBMOnumber theoryPerfect Squares