10

Part of 1989 IMO Longlists

Problems(2)

4x^3 + 4x^2y - 15xy^2 - 18y^3 - 12x^2 + 6xy + 36y^2 + 5x...

Source: IMO Longlist 1989, Problem 10

Given the equation 4x^3 \plus{} 4x^2y \minus{} 15xy^2 \minus{} 18y^3 \minus{} 12x^2 \plus{} 6xy \plus{} 36y^2 \plus{} 5x \minus{} 10y \equal{} 0, find all positive integer solutions.

algebra unsolvedalgebra

Find the maximum number c

Source: IMO Longlist 1989, Problem 109

Find the maximum number

c

such that for all

n \in \mathbb{N}

\{n \cdot \sqrt{2}\} \geq \frac{c}{n}

where \{n \cdot \sqrt{2}\} \equal{} n \cdot \sqrt{2} \minus{} [n \cdot \sqrt{2}] and

[x]

is the integer part of

x.

Determine for this number

c,

n \in \mathbb{N}

for which \{n \cdot \sqrt{2}\} \equal{} \frac{c}{n}.

algebra unsolvedalgebra