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Find the largest integer N [Spain 2000]

Source:

February 3, 2011
modular arithmeticfloor functionnumber theory proposednumber theory

Problem Statement

Find the largest integer NN satisfying the following two conditions:
(i) [N3]\left[ \frac N3 \right] consists of three equal digits;
(ii) [N3]=1+2+3+⋯+n\left[ \frac N3 \right] = 1 + 2 + 3 +\cdots + n for some positive integer n.n.
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