## IMO 2018 Problems – Day 2

**Problem 4.** A *site* is any point in the plane such that and are both positive integers less than or equal to 20.

Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to . On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone.

Find the greatest such that Amy can ensure that she places at least red stones, no matter how Ben places his blue stones.

**Problem 5.** Let be an infinite sequence of positive integers. Suppose that there is an integer such that, for each , the number

is an integer. Prove that there is a positive integer such that for all .

**Problem 6.** A convex quadrilateral satisfies . Point lies inside so that and . Prove that .

Source: AoPS

## Balkan Mathematical Olympiad 2018

**Problem 1.** A quadrilateral is inscribed in a circle , where and is not parallel to . Point is the intersection of the diagonals and and the perpendicular from to intersects the segment at the point . If bisects the angle , prove that is a diameter of the circle .

**Problem 2.** Let be a positive rational number. Two ants are initially at the same point in the plane. In the -th minute each of them chooses whether to walk due north, east, south or west and then walks the distance of meters. After a whole number of minutes, they are at the same point in the plane (non necessarily ), but have not taken exactly the same route within that time. Determine all the possible values of .

**Problem 3.** Alice and Bob play the following game: They start with two non-empty piles of coins. Taking turns, with Alice playing first, each player chooses a pile with an even number of coins and moves half of the coins of this pile to the other pile. The came ends if a player cannot move, in which case the other player wins.

Determine all pairs of positive integers such that if initially the two piles have and coins, respectively, then Bob has a winning strategy.

**Problem 4.** Find all primes and such that divides .

Source: https://bmo2018.dms.rs/wp-content/uploads/2018/05/BMOproblems2018_English.pdf

## When is arccos a rational multiple of pi?

Recently I had to test if the of some algebraic number is a rational multiple of or not. I found this solution online and since it is quite nice, I write it here in a more generalized form.

Suppose that is an algebraic number. Under which circumstances we have

Suppose that with . Then and applying we see that

It is well known that the function is polynomial for . If we denote then is a polynomial of degree with leading coefficient . Since we have , the minimal polynomial associated to must divide . This gives us the following necessary condition:

*If then there exist , such that (the minimal polynomial of ) divides (where is the Chebyshev polynomial of degree ).*

It is not difficult to see that this condition is also sufficient. Now, using this we have the following application:

**Consequence.** If is an algebraic number whose minimal polynomial has a leading coefficient which is not a power of , then

Since the minimal polynomial has a leading coefficient which is not a power of , it cannot divide for some positive integer . Therefore, following the arguments stated above, .

**Applications.**

- if .
- for if and is not a power of .

## SEEMOUS 2018 – Problems

**Problem 1.** Let be a Riemann integrable function. Show that

**Problem 2.** Let and let the matrices , , , be such that

Prove that .

**Problem 3.** Let such that and , where is the identity matrix. Prove that if then .

**Problem 4.** (a) Let be a polynomial function. Prove that

(b) Let be a function which has a Taylor series expansion at with radius of convergence . Prove that if converges absolutely then converges and

Source: official site of SEEMOUS 2018

**Hints: **1. Just use and . The strict inequality comes from the fact that the Riemann integral of strictly positive function cannot be equal to zero. This problem was too simple…

2. Use the fact that , therefore is symmetric and positive definite. Next, notice that . Notice that is diagonalizable and has eigenvalues among . Since it is also positive definite, cannot be an eigenvalue. This allows to conclude.

3. First note that the commutativity allows us to diagonalize using the same basis. Next, note that both have eigenvalues of modulus one. Then the trace of is simply the sum where are eigenvalues of and , respectively. The fact that the trace equals and the triangle inequality shows that eigenvalues of are a multiple of eigenvalues of . Finish by observing that they have the same eigenvalues.

4. (a) Integrate by parts and use a recurrence. (b) Use (a) and the approximation of a continuous function by polynomials on compacts to conclude.

I’m not sure about what others think, but the problems of this year seemed a bit too straightforward.

## Putnam 2017 A2 – Solution

**Problem A2.** We have the following recurrence relation

for , given and . In order to prove that is always a polynomial with integer coefficients we should prove that divides somehow. Recurrence does not seem to work very well. Also, root based arguments might work, but you need to take good care in the computation.

A simpler idea, which is classic in this context, is to try and linearize the recurrence relation. In order to do this, let’s write two consecutive recurrence relations

We add them and we obtain the following relation

which leads straightforward to a telescoping argument. Finally, we are left with a simple linear recurrence with integer coefficient polynomials, and the result follows immediately.

## Balkan Mathematical Olympiad 2017 – Problems

**Problem 1.** Find all ordered pairs of positive integers such that:

**Problem 2.** Consider an acute-angled triangle with and let be its circumscribed circle. Let and be the tangents to the circle at points and , respectively, and let be their intersection. The straight line passing through the point and parallel to intersects in point . The straight line passing through the point and parallel to intersects in point . The circumcircle of the triangle intersects in , where is located between and . The circumcircle of the triangle intersects the line (or its extension) in , where is located between and .

Prove that , , and are concurrent.

**Problem 3.** Let denote the set of positive integers. Find all functions such that

for all

**Problem 4.** On a circular table sit students. First, each student has just one candy. At each step, each student chooses one of the following actions:

- (A) Gives a candy to the student sitting on his left or to the student sitting on his right.
- (B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right.

At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps. Find the number of legitimate distributions.

(Two distributions are different if there is a student who has a different number of candy in each of these distributions.)

Source: AoPS

## IMC 2016 – Day 2 – Problem 8

**Problem 8.** Let be a positive integer and denote by the ring of integers modulo . Suppose that there exists a function satisfying the following three properties:

- (i) ,
- (ii) ,
- (iii) for all .

Prove that modulo .