Κυριακή, 11 Νοεμβρίου 2018

Tournament of Towns: Problem Archives Toronto

Tournament of Towns: Problem Archives: Tournament of Towns is a mathematical competition of international standards. While preserving the best Olympiad traditions, it has aquired a character of its own. Its held four times a year in cities around the world. And Toronto is one of them.

Πέμπτη, 8 Νοεμβρίου 2018

POL YA PROBLEM-SOL VING SEMINAR WEEK 5: INVARIANTS AND MONOV ARIANT

07putnam5.pdf



Tournament of Towns - Διαγωνισμός των Πόλεων Καναδάς

Tournament of Towns: Problem Archives: Tournament of Towns is a mathematical competition of international standards. While preserving the best Olympiad traditions, it has aquired a character of its own. Its held four times a year in cities around the world. And Toronto is one of them.

Ε1997Β1

(86) ΕΥΚΛΕΙΔΗΣ 1997 - Β ΛΥΚΕΙΟΥ - mathematica.gr



Re: ΕΥΚΛΕΙΔΗΣ 1997 - Β ΛΥΚΕΙΟΥ





  • #3

    Δημοσίευση

    από socrates » 07 Νοέμ 2012, 19:34
    1. Έστω \displaystyle{\alpha,\beta\in\mathbb{N}^*} και \displaystyle{A=\frac{\alpha^3+1}{\beta+1}+\frac{\beta^3+1}{\alpha+1} \in\mathbb{N}^*}. Να δειχτεί ότι οι αριθμοί \displaystyle{\frac{\alpha^3+1}{\beta+1},\frac{\beta^3+1}{\alpha+1}} είναι φυσικοί..
    Αρκεί να παρατηρήσουμε ότι για τους αριθμούς \displaystyle{x:=\frac{\alpha^3+1}{\beta+1}} και \displaystyle{y:=\frac{\beta^3+1}{\alpha+1}}



    ισχύει x+y, xy \in \Bbb{Z}....