# Understand the problem

We have a rectangle whose sides are mirrors. A ray of light enters from one of vertices of the rectangle and after being reflected several times, exits via the vertex opposite to the initial one. Prove that at some point of time, the ray passed through the centre of rectangle (Intersection of the diagonals.)

##### Source of the problem
Iranian Mathematical Olympiad 2019 (second round)
##### Topic
Geometry/Combinatorics

Hard
##### Comments
It is interesting to note how Olympiads reflect the current state of mathematical research. This is not unexpected, because olympiad problems usually originate from elementary corollaries of advanced Mathematics. This particular problem has to do with the problem of billiards in the field of Dynamical Systems. The interested reader can learn more here.

# Start with hints

Do you really need a hint? Try it first!

Instead of reflecting the ray and keeping the rectangle fixed, reflect the rectangle and keep the ray fixed.  Following hint 1, you will get a grid of rectangles and a straight line representing the path of light. If this straight line passes through one of the reflections of the opposite vertex, then in our original representation it has to pass through the opposite vertex. Similarly, if it passes through one of the reflections of the centre then in the original representation, it has to pass through the centre. Show that the reflections of $(a,b)$ are of the form $((2m-1)a,(2n-1)b)$. Also, the reflections of the centre $(a/2,b/2)$ are of the form $((p+1/2)a,(q+1/2)b$.
Suppose that the line indeed passes through a reflection of the opposite vertex. Then it is of the form $x(t)=(2m+1)at, y(t)=(2n+1)bt$. Taking $t=\frac{1}{2}$, we see that it passes through $((m+1/2)a,(n+1/2)b)$, which is a reflection of the centre.

# Connected Program at Cheenta

#### Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

# Similar Problems

## Solving a congruence

Understand the problemProve that the number of ordered triples in the set of residues of $latex p$ such that , where and is prime is . Brazilian Olympiad Revenge 2010 Number Theory Medium Elementary Number Theory by David Burton Start with hintsDo you really need...

## Inequality involving sides of a triangle

Understand the problemLet be the lengths of sides of a (possibly degenerate) triangle. Prove the inequalityLet be the lengths of sides of a triangle. Prove the inequalityCaucasus Mathematical Olympiad Inequalities Easy An Excursion in Mathematics Start with hintsDo...

## Vectors of prime length

Understand the problemGiven a prime number and let be distinct vectors of length with integer coordinates in an Cartesian coordinate system. Suppose that for any , there exists an integer such that all three coordinates of is divisible by . Prove that .Kürschák...

## Missing digits of 34!

Understand the problem34!=295232799cd96041408476186096435ab000000 Find $latex a,b,c,d$ (all single digits).BMO 2002 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!Get prepared to find the residue of 34! modulo...

## An inequality involving unknown polynomials

Understand the problemFind all the polynomials of a degree with real non-negative coefficients such that , . Albanian BMO TST 2009 Algebra Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!This problem is all about...

## Hidden triangular inequality (PRMO Problem 23, 2019)

Problem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...

## PRMO – 2019 – Questions, Discussions, Hints, Solutions

This is a work in progress. Please post your answers in the comment. We will update them here. Point out any error that you see here. Thank you. 1. 42. 133. 134. 725. 106. 297. 518. 499. 1410. 5511. 612. 1813. 1014. 5315. 4516. 4017. 3018. 2019. 1320. Bonus21. 1722....

## Bangladesh MO 2019 Problem 1 – Number Theory

A basic and beautiful application of Numebr Theory and Modular Arithmetic to the Bangladesh MO 2019 Problem 1.

## Functional equation dependent on a constant

Understand the problemFind all real numbers for which there exists a non-constant function satisfying the following two equations for all i) andii) Baltic Way 2016 Functional Equations Easy Functional Equations by BJ Venkatachala Start with hintsDo you really need...

## Pigeonhole principle exercise

Problem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...