## Problem 4.

A box contains 900 cards, numbered from 100 to 999. Cards are drawn at random (without replacement) from the box, and the sum of the digits of each card drawn is noted. What is the minimum number of cards that must be drawn to ensure that at least three of these sums are the same?
## Solution:

There are 27 possible outcomes for the sum of the digits (from 1 to 27). The sums 1 and 27 can only be obtained in one way (100 and 999). In the worst-case scenario, after drawing $52 (27 + 25)$ cards, all sums will repeat twice, and on the next draw (53rd draw), one of them will appear for the third time. Therefore, the required number is $27 + 25 + 1 = 53$.

The centroid of triangle $\triangle \mathrm{ABC}$ is $\mathrm{G}$. We denote by $g_{a}, \mathrm{~g}_{\mathrm{b}}, \mathrm{g}_{\mathrm{c}}$ the distances from $\mathrm{G}$ to the sides a, b, and c, respectively.

Let $r$ be the radius of the inscribed circle. Prove that:

i) $g_{a} \geq \frac{2 r}{3}, \mathrm{~g}_{\mathrm{b}} \geq \frac{2 r}{3}, \mathrm{~g}_{\mathrm{c}} \geq \frac{2 r}{3}$

ii) $\frac{g_{a}+g_{b}+g_{c}}{r} \geq 3$