. What is the sum of the digits of $10^{2008}-2008$?

## 2 The solutions

 untranslated text has been directly output as requested.
. The number $10^{2008}$ - 2008 consists of 2008 digits. We have

$$
10^{2008}-2008=\left(10^{2008}-1\right)-2007=99 \cdots 9-2007=99 \cdots 97992
$$

and the sum of its digits is $2004 \times 9+7+9+9+2=18063$.

## vI. Various

## 1 Slides from Antoine's Presentation

Will this conference ever end?

Antoine Taveneaux

Olympic Training 2008
Computer programs are like recipes

Elementary functions to compose:

- If ... then ... else

If you have eggs, then put them in, else put in ham.

- For $i$ from 1 to $n$ do

For $i$ from 1 to 10, crack the $j^{\text{th}}$ egg from the box and put it in a bowl.

- While ... do

While the cake is not finished, keep eating.

An example

Let's calculate $a_{100}$ when $\left(a_{n}\right)$ is the Fibonacci sequence.

- I prepare 3 slots to store numbers.
- In the first two slots, I put the number 1.
- For $i$ from 2 to 100 do

In the third slot, I copy the number that was in the first slot.

In the first slot, I put the sum of the numbers that were in the first and second slots.

In the second slot, I put the number from the third slot.

I write down the number in the first slot.
Presentation Outline

What is a computer program? What does a computer do? A program is a number A program may not stop

What can a computer not do? Predict whether a program will stop Proof idea A function that grows very, very fast

A definable but not computable number

Computability is not useless

Summary

It's a bit more complicated

- The computer must also manage memory
- Just like a cook must manage their pots.
Calculating the GCD of two numbers

How to calculate the large number $i$ such that $a_{i}A(1000,1000)
$$

What is a computable number

For the numbers you know (like $\pi$ or e), there are programs that can give their decimals.

Do you know a number that no program could approximate as closely as desired?

The length of proofs

Question: What is the maximum length of a proof of a theorem of length $n$?

A computer can verify a proof (if given).

Let $T$ be a theorem and $|T|$ its length (the number of letters and spaces needed to write the theorem).

By contradiction.

If all theorems $T$ could be proven with a proof of length $2^{|T|}$, then to determine if a theorem is provable or not, it would suffice to write all texts of length $2^{|T|}$ and check if any of them are proofs of $T$.

We would thus find a proof if and only if the theorem is provable. But such a program does not exist.

Therefore, there exist theorems whose shortest proof is longer than $2^{|T|}$ (and even $A(|T|,|T|$ ).
Do you think all this will finally come to an end one day?

- Computer science is a science.
- We can do fun things.
- But also serious things.

## For Further Reading:

Gilles Dowek

The Metamorphoses of Calculus: An Astonishing History of Mathematics. Editions Le Pommier

## 2 IMO 2008 Problems