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College Mathematics I.S.I. and C.M.I. Entrance Math Olympiad Problem Solving Marathon Themes in Mathematics USA Math Olympiad

The Dhaba Problem | ISI and CMI Entrance

Suppose on a highway, there is a Dhaba. Name it by Dhaba A.

You are also planning to set up a new Dhaba. Where will you set up your Dhaba?

Model this as a Mathematical Problem. This is an interesting and creative part of the BusinessoMath-man in you.

You have to assume something for Mathematical simplicity to model a real life phenomenon via math.

Assumptions:

  • The length of the highway is 1 unit. Assume the highway is denoted by the interval [0,1].
  • The number of Persons in a road is proportional to the road length. P = c. R, where P is the Persons and R is the road length, c is an arbitrary constant.
  • The profit is modeled as the Total Number of Persons arriving in the Dhaba.
  • A person will visit a dhaba which is nearer over another which is further away.
  • When a customer is at the same distance from two shops then the customer choose randomly.
  • Your aim is to maximize your profits.

Observe that the assumptions are valid and are actually followed in real life. Just sit and think for some time placing yourself in that position.

The Explicit Mathematical Problem

distance of the dhaba
distance of the dhaba
dhaba problem distance

Profit Calculation of your Dhaba B

Now based on the lengths of the roads above and the assumptions made, we have to calculate the profit of Dhaba B.

solution of the dhaba problem
solution

So the profit made by B = c.R , where R is the length of the road on which B has monopoly. Here, as shown R = \c.( x + \frac{|d-x|}{2}\).

We have assumed in this case that \(0 \leq x \leq d\) by the diagram.

So, the profit for the Dhaba B is c.\(\frac{d+x}{2}\) .

Hence as \(0 \leq x \leq d\), the profit is maximized if x = d.

Exercise: Show that the profit of Dhaba B as a function of x is c.(\( 1 – \frac{d+x}{2}\)) if \(d \leq x \leq 1\) .

Also, observe that in this case the profit is maximized at x = d.

Exercise: Calculate the maximum profit in both cases. Do you observe something fishy? What steps and what arguments will you give to understand the fishiness?

Please share your views in the comments section.

Eager to listen to your beautiful ideas.

Other useful links:-

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Math Olympiad Corner Themes in Mathematics USA Math Olympiad

Thousand Flowers Program : a paradigm shift in Olympiad Training

The central theme of the thousand flowers program is: connected ideas and connected problems. We will illustrate the idea using some examples.

But before we do so, let’s point out the theoretical motivation behind such a program. It is greatly borrowed from the pedagogical experiments of Rabindranath Thakur. (Reference: https://bn.m.wikisource.org/wiki/বিশ্বভারতী). One of his major criticisms of existing pedagogical methods is this (I won’t try to translate this):

এই শিক্ষাপ্রণালীর সকলের চেয়ে সাংঘাতিক দোষ এই যে, এতে গোড়া থেকে ধরে নেওয়া হয়েছে যে আমরা নিঃস্ব। যা-কিছু সমস্তই আমাদের বাইরে থেকে নিতে হবে—আমাদের নিজের ঘরে শিক্ষার পৈতৃক মূলধন যেন কানাকড়ি নেই। এতে কেবল যে শিক্ষা অসম্পূর্ণ থাকে তা নয়, আমাদের মনে একটা নিঃস্ব-ভাব জন্মায়। আত্মাভিমানের তাড়নায় যদি-বা মাঝে মাঝে সেই ভাবটাকে ঝেড়ে ফেলতে চেষ্টা করি তা হলেও সেটাও কেমনতরো বেসুরো রকম আস্ফালনে আত্মপ্রকাশ করে। আজকালকার দিনে এই আস্ফালনে আমাদের আন্তরিক দীনতা ঘোচে নি, কেবল সেই দীনতাটাকে হাস্যকর ও বিরক্তিকর করে তুলেছি।

Opposing the piggy bank method of education (where there is a teacher who ‘knows’ and a student who ‘does not know’) we want to seek the students’ input to solve problems with their own creativity. We are assuming that the student ‘knows’ and is ‘creative’; that he/she can do stuff. While he/she explores that inner strength, we catalyze the process with some inputs (skills, interesting problems) from time to time.

(Note that this program is run by Cheenta for young students, usually of age 7/8 to 10/11. It is designed as a launching pad for advanced Olympiad programs. Our central goal is to expose the students to rigorous creative problem-solving. This cannot be achieved by simple formula-learning. We must allow the young minds to be creative. It takes years of hard work).

Examples

(each of the themes presented below may span over 6 to 8 classes (of 90 minutes). They should be punctuated by exercises and software simulations):

Primes and Algorithm of the Sieve of Eratosthenes (Connecting Mathematics and Computer Science)

  • We begin with a description of prime numbers and how they are useful to ‘build’ other numbers. Next, we try to find methods of checking if a number is prime. An elementary number theoretic investigation reveals that to check n is prime, it is sufficient to perform \( \sqrt n \) divisions.
  • A natural follow up question would be: how many primes are there between 1 and n? This leads to Sieve of Eratosthenes. It is unnatural to compute the sieve by hand. Here comes the introduction to algorithms. A simple implementation using Python does the trick.This theme is usually spread over 4 to 5 sessions. It is a fantastic introduction to elementary number theory and computer programming cum algorithms.
  • Exercises: Divisibility 1 from Mathematical Circles by Fomin // Simple algorithms

Area, Irrationals and Algebraic Identities (Connecting Algebra and Geometry)

  • We begin with simple algebraic identities such as \( (a+b)^2 = a^2 + 2ab + b^2 \). We show the geometric implementation of these identities. This immediately brings us to the discussion of ‘area’. We define the area of a unit square as 1 and follow up with a development of area formula as a product of length and width (students realize that we are actually counting the number of unit squares.
  • We immediately define rational numbers (as ratios of integers) and show that geometrically we can chop off the unit square into smaller pieces. Next, we use the area to draw pictures of more intricate identities. Finally, we show that some numbers (like \( \sqrt 2 \) cannot be expressed as ratios of integers. We prove that using parity argument and also present a geometric construction of such numbers (compass-straight edge construction)
  • Exercises: Compass -straight edge construction of rational and irrational numbers starting with integers, Geometric proof of intricate algebraic identities, parity -argument proof of irrationality.

Vectors, Angle and Motion (Connecting Mathematics and Physics)

  • We begin with a simple description of points on the plane (Cartesian).  We clearly describe that points can be visualized as static objects or as a representation of motion. For example (1, 2) can be regarded as just a point or some physical phenomena with a magnitude of \( \sqrt {1^2 + 2^2} \) and direction \( \tan^{-1} 2 \).This is a great point to introduce the notion of ‘angle’. We describe it as a measure of rotation (an isometry of the plane). We specify that angle is just a ratio of arc over radius (and geometrically why that ratio is important).Next, we go over some physical phenomena that can be described using points.
  • We draw pictures of position -time graphs. We define velocity and acceleration draw graphs of several combinations of those quantities. We solve problems of kinematics and describe the physical and mathematical aspects of it. Simulations in Geogebra or other software may aid the process.
  • Exercises: Prove that arc over radius is invariant in the rotation. Plot points and vectors and differentiate them. Plot position -time, velocity-time graphs, Kinematics problems from Irodov.

As students, we mostly want to be inspired. A closely followed (connected) second would be to get intellectually challenged. A holistic problem-oriented approach to mathematical science (mathematics + computer science + physics and part of natural sciences) may serve both purposes.

Note for teachers: It is useless to lecture for a large span of time. In fact, it is extremely important to throw clever problems every now and then, that puts all the beads of ideas together.

Note for students/parents: The world is not separated by subjects and classrooms. It is important to approach a problem in a holistic manner. This approach, in fact, provides room for creativity and experiments.

We want to create a futuristic program for our children who will grow in a world of artificial intelligence and advanced technologies. If children of today are not allowed to be creative, then they won’t be able to respond to a world where most mundane tasks will be done by machines anyways.

I will add more ideas and themes here. Let me know your opinion in the comments section or at helpdesk@cheenta.com. We are still a development stage for this program.

Categories
Math Olympiad Themes in Mathematics USA Math Olympiad

Thousand Flowers Program: Paradigm shift in Olympiad Training

The central theme of the thousand flowers program is: connected ideas and connected problems. We will illustrate the idea using some examples.

But before we do so, let’s point out the theoretical motivation behind such a program. It is greatly borrowed from the pedagogical experiments of Rabindranath Thakur. (Reference: https://bn.m.wikisource.org/wiki/বিশ্বভারতী). One of his major criticisms of existing pedagogical methods is this (I won’t try to translate this):

এই শিক্ষাপ্রণালীর সকলের চেয়ে সাংঘাতিক দোষ এই যে, এতে গোড়া থেকে ধরে নেওয়া হয়েছে যে আমরা নিঃস্ব। যা-কিছু সমস্তই আমাদের বাইরে থেকে নিতে হবে—আমাদের নিজের ঘরে শিক্ষার পৈতৃক মূলধন যেন কানাকড়ি নেই। এতে কেবল যে শিক্ষা অসম্পূর্ণ থাকে তা নয়, আমাদের মনে একটা নিঃস্ব-ভাব জন্মায়। আত্মাভিমানের তাড়নায় যদি-বা মাঝে মাঝে সেই ভাবটাকে ঝেড়ে ফেলতে চেষ্টা করি তা হলেও সেটাও কেমনতরো বেসুরো রকম আস্ফালনে আত্মপ্রকাশ করে। আজকালকার দিনে এই আস্ফালনে আমাদের আন্তরিক দীনতা ঘোচে নি, কেবল সেই দীনতাটাকে হাস্যকর ও বিরক্তিকর করে তুলেছি।

Opposing the piggy bank method of education (where there is a teacher who ‘knows’ and a student who ‘does not know’) we want to seek the students’ input to solve problems with their own creativity. We are assuming that the student ‘knows’ and is ‘creative’; that he/she can do stuff. While he/she explores that inner strength, we catalyze the process with some inputs (skills, interesting problems) from time to time.

(Note that this program is run by Cheenta for young students, usually of age 7/8 to 10/11. It is designed as a launching pad for advanced Olympiad programs. Our central goal is to expose the students to rigorous creative problem-solving. This cannot be achieved by simple formula-learning. We must allow young minds to be creative. It takes years of hard work).

Examples

(each of the themes presented below may span over 6 to 8 classes (of 90 minutes). They should be punctuated by exercises and software simulations):

Primes and Algorithm of the Sieve of Eratosthenes (Connecting Mathematics and Computer Science)

  • We begin with a description of prime numbers and how they are useful to ‘build’ other numbers. Next, we try to find methods of checking if a number is prime. An elementary number theoretic investigation reveals that to check n is prime, it is sufficient to perform \( \sqrt n \) divisions.
  • A natural follow up question would be: how many primes are there between 1 and n? This leads to Sieve of Eratosthenes. It is unnatural to compute the sieve by hand. Here comes the introduction to algorithms. A simple implementation using Python does the trick.This theme is usually spread over 4 to 5 sessions. It is a fantastic introduction to elementary number theory and computer programming cum algorithms.
  • Exercises: Divisibility 1 from Mathematical Circles by Fomin // Simple algorithms

Area, Irrationals and Algebraic Identities (Connecting Algebra and Geometry)

  • We begin with simple algebraic identities such as \( (a+b)^2 = a^2 + 2ab + b^2 \). We show the geometric implementation of these identities. This immediately brings us to the discussion of ‘area’. We define the area of a unit square as 1 and follow up with a development of area formula as a product of length and width (students realize that we are actually counting the number of unit squares.
  • We immediately define rational numbers (as ratios of integers) and show that geometrically we can chop off the unit square into smaller pieces. Next, we use the area to draw pictures of more intricate identities. Finally, we show that some numbers (like \( \sqrt 2 \) cannot be expressed as ratios of integers. We prove that using parity argument and also present a geometric construction of such numbers (compass-straight edge construction)
  • Exercises: Compass -straight edge construction of rational and irrational numbers starting with integers, Geometric proof of intricate algebraic identities, parity -argument proof of irrationality.

Vectors, Angle and Motion (Connecting Mathematics and Physics)

  • We begin with a simple description of points on the plane (Cartesian).  We clearly describe that points can be visualized as static objects or as a representation of motion. For example (1, 2) can be regarded as just a point or some physical phenomena with a magnitude of \( \sqrt {1^2 + 2^2} \) and direction \( \tan^{-1} 2 \).This is a great point to introduce the notion of ‘angle’. We describe it as a measure of rotation (an isometry of the plane). We specify that angle is just a ratio of arc over radius (and geometrically why that ratio is important). Next, we go over some physical phenomena that can be described using points.
  • We draw pictures of position-time graphs. We define velocity and acceleration draw graphs of several combinations of those quantities. We solve problems of kinematics and describe the physical and mathematical aspects of it. Simulations in Geogebra or other software may aid the process.
  • Exercises: Prove that arc over radius is invariant in the rotation. Plot points and vectors and differentiate them. Plot position -time, velocity-time graphs, Kinematics problems from Irodov.

As students, we mostly want to be inspired. A closely followed (connected) second would be to get intellectually challenged. A holistic problem-oriented approach to mathematical science (mathematics + computer science + physics and part of natural sciences) may serve both purposes.

Note for teachers: It is useless to lecture for a large span of time. In fact, it is extremely important to throw clever problems every now and then, that puts all the beads of ideas together.

Note for students/parents: The world is not separated by subjects and classrooms. It is important to approach a problem in a holistic manner. This approach, in fact, provides room for creativity and experiments.

We want to create a futuristic program for our children who will grow in a world of artificial intelligence and advanced technologies. If children of today are not allowed to be creative, then they won’t be able to respond to a world where most mundane tasks will be done by machines anyways.

I will add more ideas and themes here. Let me know your opinion in the comments section or at helpdesk@cheenta.com. We are still a development stage for this program.

Link to our YouTube Channel: https://www.youtube.com/channel/UCK2CP6HbbQ2V6gn8Xp_vM-Q

Categories
Math Olympiad Themes in Mathematics USA Math Olympiad

Number Theory in Math Olympiad – Beginner’s Toolbox

This article is aimed at entry level Math Olympiad (AMC and AIME in U.S. , SMO Junior in Singapore, RMO in India). We have complied some of the most useful results and tricks in elementary number theory that helps in problem solving at this level. Note that only with a lot of practice and conceptual discussions, you may make practical use of these ideas.

Hunch

A proper guess or a hunch is sometimes instrumental in the solution of a problem.

  • Say x, y and n are positive integers such that xy = n^2 . It is sometimes useful to show that GCD (x, y) = 1 because in that case x and y are individually perfect squares.
  • GCD of two consecutive numbers is 1 i.e. GCD(n, n+1) =1
  • GCD (a, b) = GCD (a, a-b)
  • The possible candidates for GCD of two numbers a and b are always less than (at best equal to) a-b.
  • Only numbers that have odd number of divisors are perfect squares.
  • Sum of perfect squares is zero implies each one is 0.
  • Well Ordering Principle; Every set of natural numbers (positive integers) has a least element.
  • To check if a number is a perfect square (or to disprove it) show that the number is divisible by a particular prime but not by it’s square. Generally the easiest case is to show that the number is divisible by 3 (that is sum of the digits is divisible by 3) but not by 9.
  • Non Linear Diophantine Equations: Simplest situations are like this x^2 - y^2 = 31 . In cases like this factorize both sides and consider all possibilities (keeping in mind that x and y are integers).

Formula

  • Fermat’s Theorem: a^p \equiv a  mod p if a is any positive number and p is a prime.
  • Euler’s Totient function: \phi (n) = n \times (1 - \frac{1}{p_1})(1 - \frac{1}{p_2}) ... (1 - \frac{1}{p_k}) where n = \prod _{i=1}^k {p_i}^{r_i} is the prime factorization of n. This gives the number of numbers smaller than and co prime to n.
  • Pythagorean Triplet: If a^2 + b^2 = c^2 be a pythagorean equation (a, b and c positive integers). Then there exists positive integers u and  v such that a = u^2 - v^2 , b = 2 u v , c = u^2 + v^2 provided GCD(a, b, c) =1.
  • Number Theoretic Functions: If n = \prod _{i=1}^k {p_i}^{r_i} is the prime factorization of n then
    • Number of Divisors of n = (r_1 +1 ) (r_2 + 1) + ... + (r_k +1) = d(n)
    • Sum of the Divisors of n = \prod_{i=1}^{k} \frac{ {p_i}^{{r_i} + 1} -1 }{{p_i} -1 }
    • Product of Divisors of n = n^{d(n) / 2}
  • Highest power of a prime in n! = \sum _{k=1}^{\infty} [ {\frac{n}{p^k}} ] where [x] = greatest integer smaller than x.
  • Bezout’s Theorem: Suppose a and b be two positive integers and x, y be arbitrary integers (positive, negative or zero). Then the set ax + by is precisely the set of multiples of the gcd(a, b). More importantly there exists integers x, y (not unique) such that ax + by = d where d = gcd (a, b).
  • Congruence Notation:
    • a \equiv b mod m if m divides a – b.
    • You may raise both sides of a congruency to same power, multiply, add or subtract constants.
    • However note that ac \equiv bc mod m does not necessarily imply a \equiv b mod m. If m does not divides c then the above follows.
  • Wilson’s Theorem: (p-1)! \equiv -1 latex mod p if p is a prime. The converse also holds. That is if (n-1)! \equiv -1 mod n then n is a prime.
  • Sophie Germain Identity: a^4 + 4b^4 is factorizable
  • x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 +y^2 + z^2 - xy - yz - zx ) \frac {1}{2} (x + y + z) ( (x-y)^2 + ( y-z)^2  + (z-x)^2 ) $
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Themes in Mathematics USA Math Olympiad

Initiating a child into the world of Mathematical Science

“How do I involve my son in challenging mathematics? He gets good marks in school tests but I think he is smarter than school curriculum.”

“My daughter is in 4th grade. What competitions in mathematics and science can she participate? How do I help her to perform well in those competitions?”

“I have a 6 years old kid. He hates math. How do I change that?”

We often get queries and requests like these from parents around the world. Literally. In fact the first one came from Oregon, United States, second one from Cochin, India and last one from Singapore.

We have created this article to answer these kind of questions and help you to help your children.

Categories
Themes in Mathematics USA Math Olympiad

How do I involve my child in challenging mathematics?

“How do I involve my child in challenging mathematics? He gets good marks in school tests but I think he is smarter than school curriculum.”

“My daughter is in 4th grade. What competitions in mathematics and science can she participate in? How do I help her to perform well in those competitions?”

“I have a 6 years old kid. He hates math. How do I change that?”

We often get queries and requests like these from parents around the world. Literally. In fact, the first one came from Oregon, United States, the second one from Cochin, India, and the last one from Singapore.

We have created this article to answer these kinds of questions and help you to help your children.