( JBMO 2018 )

$n$ numbers are written on a board, satisfying the following properties:

- all the numbers have 3 digits
- none of them contain a 0
- two different numbers never have the same hundreds digit, or the same tens digit, or the same units digit
- the sum of the digits of each number is 9

What is the maximum possible value for $n$?
We already observe that $n \leqslant 7$, because we cannot use digits larger than 7 and we must use $n$ different unit digits. The sum of the $3 n$ digits used must be equal to $9 n$. The sum of the unit digits, for example, is at least: $1+\ldots+n=\frac{n(n+1)}{2}$. Hence the inequality satisfied by $n$:

$$
9 n \geqslant \frac{3 n(n+1)}{2}
$$

We have: $9 \times 7=63<\frac{3 \times 7 \times 8}{2}=84$ and $9 \times 6=54<\frac{3 \times 6 \times 7}{2}=63$, but we observe that 5 satisfies the inequality, and it is even an equality in this case: $9 \times 5=45=\frac{3 \times 5 \times 6}{2}$

If it is possible with 5 numbers, we then know that only the digits 1, 2, 3, 4, 5 are used. We arrive, for example, at the following 5 numbers, which allows us to verify that 5 is indeed the maximum (attained): $135,252,324,441,513$.

## 4 Geometry

## 1 Angle Hunting

The course partly reviewed the angle hunting course given to group B during the Montpellier olympic stage in 2016 (available here). The concepts actually covered are:

- Alternate interior angles, corresponding angles
- Sum of the angles of a triangle, of a quadrilateral (with proof)
- Isosceles triangle
- Central angle theorem (with proof)
- Inscribed angle theorem (with proof, without limit cases)
- Right triangle, English Thales' theorem (with proof)
- Cyclic points, Fifi's theorem (proof of the direct sense)

## - Exercises covered -