What is the largest integer that cannot be expressed as the sum of 100 not necessarily distinct composite numbers? (A composite number is one that can be expressed as the product of two integers greater than 1.)
The smallest composite number is 4, so no number smaller than 400 can be expressed as the sum of a hundred composite numbers. For $n \geq 400$, the

$$
n=99 \cdot 4+(n-396)
$$

representation is valid, provided that $(n-396)$ is also a composite number. This is always the case if $n$ is even, but for odd $n$, $(n-396)$ can also be a prime number. This situation occurs, for example, with $n=401$ and $n=403$. We will show that 403 is the number we are looking for.

The number 403 cannot be expressed as the sum of a hundred composite numbers, since an odd number is also needed in any possible representation. The smallest odd composite number, however, is 9, and since the sum of the other 99 numbers is at least $99 \cdot 4$, the sum would be at least $9+99 \cdot 4=405$.

We still need to show that any integer greater than 403 can be expressed as the sum of a hundred composite numbers. If $n$ is even, (1) is a valid representation. If $n$ is odd, a valid representation is, for example,

$$
n=98 \cdot 4+9+(n-401)
$$

since here $(n-401)$ is an even number greater than two, hence composite.

Note. No matter how simple our task, the solution involves at least three steps:

- we need to find the number we are looking for,
- we need to prove that it cannot be expressed as the sum of a hundred composite numbers, and
- we need to show that any number larger than it can be expressed as such. If we state that the smallest composite number among the even numbers is 4, and among the odd numbers is 9, then the sum of a hundred composite numbers is at least 400 in general, and at least 405 if there is an odd number among them. In this, everything is already "in it," but it cannot be known why we said these things, and how our statement "comes out of it" in the end. We recommend that our contestants make a sketch for their solution before writing it down, organize their material. A mathematical proof becomes definitive only when it is written down, and both its author and reviewers can measure through this whether it is complete, and if not, what is still missing from it. Let us not get into the habit of confusing the explanation with the proof. The explanation ends with understanding, and we try to use everything that can help; the completeness of a proof, however, does not depend on whether the listener believes the statement. A very common mistake is over-proving, when we also include arguments in the explanation that are not needed to prove the specific statement. For example, in our solution, it was not necessary to show that 401 cannot be expressed as the sum of a hundred composite numbers. In fact, it does not belong to the proof that numbers less than 400 cannot be expressed, we only needed this to find 403.