## Romanian Masters in Mathematics contest – 2018

**Problem 1.** Let be a cyclic quadrilateral an let be a point on the side The diagonals meets the segments at The line through parallel to mmets the extension of the side beyond at The line through parallel to meets the extension of the side beyond at Prove that the circumcircles of the triangles and are tangent .

**Problem 2.** Determine whether there exist non-constant polynomials and with real coefficients satisfying

**Problem 3.** Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?

**Problem 4.** Let be positive integers such that and . Let be the set of values attained by as runs through the positive integers. Show that is the set of all positive divisors of some positive integer.

**Problem 5.** Let be positive integer and fix distinct points on a circle. Determine the number of ways to connect the points with arrows (oriented line segments) such that all of the following conditions hold:

- each of the points is a startpoint or endpoint of an arrow;
- no two arrows intersect;
- there are no two arrows and such that , , and appear in clockwise order around the circle (not necessarily consecutively).

**Problem 6.** Fix a circle , a line to tangent , and another circle disjoint from such that and lie on opposite sides of . The tangents to from a variable point on meet at and . Prove that, as varies over , the circumcircle of is tangent to two fixed circles.

Source: Art of Problem Solving forums

**Some quick ideas:** For Problem 1 just consider the intersection of the circle with the circle . You’ll notice immediately that this point belongs to the circle . Furthermore, there is a common tangent to the two circles at this point.

For Problem 2 we have . Eliminate the highest order term from both sides and look at the next one to get a contradiction.

Problem 4 becomes easy after noticing that if divides and then divides .

In Problem 5 try to prove that the choice of start points determines that of the endpoints. Then you have a simple combinatorial proof.

Problem 6 is interesting and official solutions use inversions. Those are quite nice, but it may be worthwhile to understand what happens in the non-inverted configuration.

I will come back to some of these problems in some future posts.

## 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.

## IMC 2017 – Day 2 – Problems

**Problem 6.** Let be a continuous function such that exists (finite or infinite).

Prove that

**Problem 7.** Let be a nonconstant polynomial with real coefficients. For every positive integer let

Prove that there are only finitely many numbers such that all roots of are real.

**Problem 8.** Define the sequence of matrices by the following recurrence

where is the identity matrix.

Prove that has distinct integer eigenvalues with multiplicities , respectively.

**Problem 9.** Define the sequence of continuously differentiable functions by the following recurrence

Show that exists for every and determine the limit function.

**Problem 10.** Let be an equilateral triangle in the plane. Prove that for every there exists an with the following property: If is a positive integer and are non-overlapping triangles inside such that each of them is homothetic to with a negative ratio and

then

## 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 Problems – Day 2

**Problem 6.** Let be a sequence of positive real numbers satisfying . Prove that

**Problem 7.** Today, Ivan the Confessor prefers continuous functions satisfying for all . Fin the minimum of over all preferred functions.

**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 .

**Problem 9.** Let be a positive integer. For each nonnegative integer let be the number of solutions of the inequality . Prove that for every we have .

**Problem 10.** Let be a complex matrix whose eigenvalues have absolute value at most . Prove that

(Here for every matrix and for every complex vector .)

Official source and more infos here.

## Balkan Mathematical Olympiad – 2016 Problems

**Problem 1.** Find all injective functions such that for every real number and every positive integer ,

**Problem 2.** Let be a cyclic quadrilateral with . The diagonals intersect at the point and lines and intersect at the point . Let and be the orthogonal projections of onto lines and respectively, and let , and be the midpoints of , and respectively. Prove that the second intersection point of the circumcircles of triangles and lies on the segment .

**Problem 3.** Find all monic polynomials with integer coefficients satisfying the following condition: there exists a positive integer such that divides for every prime for which is a positive integer.

**Problem 4.** The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of colours so that no rectangle with perimeter contains two squares of the same colour. Show that no rectangle of size or contains two squares of the same colour.