## IMC 2019 – Solutions to some problems from Day 1

**Problem 1.** Evaluate the product

*Solution: *When dealing with this kind of questions there aren’t that many options. Either use a known product, or find a telescopic argument, i.e. show that all but finitely many terms simplify.

The telescopic argument is the most likely to work. Indeed, we have the following

which is a product of two telescopic expressions:

and

I leave to you the pleasure of finding which of the terms remain after all simplifications.

**Problem 2.** A four-digit is called *very good* if the system

of linear equations in the variables has at least two solutions. Find all very good s in the st century (between and ). *Solution: *Well, this is not too complicated in the first part. The fact that the system has at least two solutions implies that the square matrix of the system is singular, i.e. the determinant of

is zero. Now, it is not difficult to note that the determinant can be factorized as follows:

The first two factors may be obtained by adding all lines to the first one, for example, factoring out and obtaining a line of ones. Then we do the classical trick of creating zeros in all positions except one on the first line by subtracting the first column from everything else. This reduces the dimension of the matrix. The second factor may be obtained with a similar trick, while the third comes directly by computing the determinant of a matrix. The rest of the exercise (the most difficult part, for me 🙂 ) is just casework.

**Problem 3.** Let be a twice differentiable function such that

Prove that

*Solution:* The first question here is what is the connection between the hypothesis and the conclusion? We have some derivatives in the hypothesis. It would be more useful if we could write them as a single derivative. In order to do this note that and there is a missing to have the complete expression. Therefore

Now, surprisingly, we find that the function is connected to in a natural way: . This means that the hypothesis simply says that the second derivative of is greater or equal than . Integrating this a few times is enough to obtain the desired inequality. A nice observation is the following: note that the hypothesis contains only derivatives of . Therefore, if we replace by where is a constant, the conclusion should not change. And indeed, it doesn’t, since . You may need this information at some point in order to simplify the computations.

**Problem 5.** Determine whether there exist an odd positive integer and matrices and with integer entries that satisfy the following conditions:

- .
- .

As usual denotes the identity matrix.

*Solution:* This problem is an example of an overkill. Having that many hypotheses makes you think that the solution is complicated. In fact, the last relation suffices, when looking at remainders modulo .

Passing to determinants we find that

which means that should be a quadratic residue modulo . However, since is odd, we find that , which is not a quadratic residue. Therefore no such matrices exist for odd .

## IMC 2019 – Problems from Day 2

**Problem 6.** Let be contiouous function such that is differentiable. Assume that . Show that there exists a point such that .

**Problem 7.** Let be the set of composite positive integers. For each let be the smallest positive integer such that is divisible by . Determine whether the following series converges:

**Problem 8.** Let be real numbers. For any set let . Assume that the function takes on at least values where runs over all subset of . Prove that the number of sets for which does not exceed .

**Problem 9.** Determine all positive integers for which there exist real invertible matrices and that satisfy .

**Problem 10.** points are chosen at random, independently, and distributed uniformly in the unit disk . Let be the convex hull of the chosen points. Which probability is larger: that is a polygon with three vertices or a polygon with four vertices?

Source: http://www.imc-math.org.uk

## IMC 2019 – Problems from Day 1

**Problem 1.** Evaluate the product

**Problem 2.** A four-digit is called *very good* if the system

of linear equations in the variables has at least two solutions. Find all very good s in the st century (between and ).

**Problem 3.** Let be a twice differentiable function such that

Prove that

**Problem 4.** Define the sequence of numbers by the following recurrence:

Prove that all terms of this sequence are integers.

**Problem 5.** Determine whether there exist an odd positive integer and matrices and with integer entries that satisfy the following conditions:

- .
- .

As usual denotes the identity matrix.

Source: imc-math.org.uk

## IMO 2019 Problem 1

**Problem 1.** Let be the set of integers. Determine all functions such that, for all integers and ,

*Solution:* As usual with this kind of functional equations the first thing that comes into mind is to pick simple cases of and .

## IMO 2019 – Problems

**Problem 1.** Let be the set of integers. Determine all functions such that, for all integers and ,

**Problem 2.** In triangle , point lies on the side and point lies on side . Let and be points on segments and , respectively, such that is parallel to . Let be a point on line such that lies strictly between and and . Similarly, let be a point on line such that lies strictly between and and .

Prove that points and are concyclic.

**Problem 3.** A social network has users, some pairs of whom are friends. Whenever user is friends with user , user is also friends with user . Events of the following kind may happen repeatedly, one at a time:

Three users such that is friends with both and but and are not friends change their friendship statuses such that and are now friends, but is no longer friends with and no longer friends with . All other friendship statuses are unchanged.

Initially, users have friends each and users haf friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

**Problem 4.** Find all pairs of positive integers such that

**Problem 5.** The Bank of Bath issues coins with an on a side and a on the other. Harry has of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly coins showing , then he turns over the th coin from the left; otherwise, all coins show and he stops. For example, if the process starting with the configuration would be

which stops after three operations.

- (a) Show that, for each initial configuration, Harry stops after a finite number of operations.
- (b) For each initial configuration , let be the number of operation before Harry stops. For example and . Determine the average value of over all possible initial configurations .

**Problem 6.** Let be the incenter of the acute triangle with . The incircle of is tangent to sides and at , respectively. The line through perpendicular to meets again at . The line meets again at . THe circumcircles of triangles and meet again at .

Prove that lines and meet on a line through , perpendicular to .

Source: http://www.imo-official.org

## Computing the Shape Derivative of the Wentzell Problem

Computing shape derivatives is a nice thing to know if you do numerical shape optimization. When you want to minimize a quantity depending on a shape in a more or less direct way, you should be able to differentiate it so that gradient algorithms could be applied. Computing shape derivatives is not that easy, but below there is a method which is not that hard once you practice some simple examples. For some bibliography on the subject I present you the following:

- Jean CEA,
*Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût*: this is in French, but it presents the method nicely. - Chicco-Ruiz, Morin, Pauletti,
*The shape derivative of the Gauss curvature*. This is not an original source of the classical results, but it contains a very nice summary of the basic notions of shape derivatives. - Delfour, Zolesio,
*Shapes and Geometries* - Henrot, Pierre,
*Shape Variation and Optimization* - Gregoire Allaire, see his the shape optimization course on his homepage

We start directly with a complicated example, containing some complex boundary terms. The Wentzell eigenvalue problem is given by the following:

It is not hard to see that the associated variational formulation is given by

## Putnam 2018 – Problem B4

**B4.** Given a real number , we define a sequence by , and for . Prove that if for some , then the sequence is periodic.

*Putnam 2018, Problem B4*

*Solution*: The path to the solution for this problem is really interesting. The first question that pops in mind when seeing the problem is: *How do I prove that a sequence is periodic?*. It’s not obvious from the given recurrence relation that this should happen, and what’s with the condition ?