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I.S.I. and C.M.I. Entrance Corner USA Math Olympiad

American Mathematical Contest

Overview of Math Olympiads in United States

The American Mathematics Competitions (AMC) are the first of a series of competitions in middle school and high school mathematics that lead to the United States team for the International Mathematical Olympiad (IMO).

AMC has three levels:

  • AMC 8 – grade 8 and below
  • AMC 10 – grades 10 and below
  • AMC 12 – grades 12 and below

The AMCs lead to International Math Olympiad (IMO), world’s most prestigious math contest at school level. The basic route is as follows

  1. AMC 10 or AMC 12 ===>American Invitational Mathematics Examination (AIME)
  2. AIME ===> United States of America Mathematical Olympiad (USAMO).
  3. USAMO (typically around 30 students) ===> Mathematical Olympiad Summer Program (MOSP or more commonly, MOP)
  4. Six students are selected from the top twelve scorers on the USAMO (through the Team Selection Test (TST)) to form the United States Math Team that goes to International Math Olympiad (held in June-July).
INFOAMC 8AMC 10AMC 12
When?A Tuesday in NovemberA Tuesday in February or 15 days laterA Tuesday in February or 15 days later
Problems252525
Time (min)407575
ScoreCorrect +1
Incorrect 0
Not Attempted 0
Correct +6
Incorrect 0
Not Attempted +1.5
Correct +6
Incorrect 0
Not Attempted +1.5
Next LevelOnly awards, no next levelAIME if scores 120+AIME if scores 100+

How to Prepare for AMC

The American Math Contest is designed to encourage independent and critical thinking in young mind. Hence the problems are non-standard in nature. Hence key factors of AMC training are

  1. Strong foundation of mathematical concepts.
  2. Plenty of problem practice

All problems in AMC (or other Math olympiads) can be (and should be) solved without calculus. Though most problems are interdisciplinary in nature (that is use several concepts from secondary mathematics) 33% problems are from Geometry making it the single most important topic in AMC. The trend actually continues at all higher level olympiads, For example in IMO, 2 out of 6 problems are from Geometry.

Books

We suggest two types of books/resources for AMC preparation. The first kind is for concept building. The second is for problem solving. Nothing beats the old Russian books as far as Math Olympiads are concerned. In fact American Math Olympiads are largely ‘copies’ of Russian Math Olympiads and Math Circles.

Starting Point

  • Mathematical Circles; A Russian Experience by Fomin
  • Lines and Curves by Vasiliyev
  • AMC 8 Problems

Both of these are outstanding books for concept building as well as problem solving. Lines and Curves present a novel method to introduce geometry to young people. Mathematical Circles covers most of the topics asked at AMC (or even higher levels). They are two light weight books for a great beginning.

Next Level

  • Challenges and Thrills of Pre College Mathematics by Venkatchala
  • Elementary Algebra and Higher Algebra by Hall and Knight
  • AMC 10 and AMC 12 Problems

Challenges and Thrills of Pre College Mathematics covers all the necessary topics of AMC. Hall and Knight’s work are classics. They are useful for conceptual strengthening of skills in Algebra

Our Courses use several other resources in conjunction with these basic books. For example the outstanding articles of ‘Quant’, Mathematical Gems (Dolciani Series), Excursions into Mathematics or books by Martin Gardener are regularly used in a selective manner. Nowadays, thanks to internet, resource for math olympiad is unlimited. Hence the critical task is to point at selective few resources and fully utilize them.

There can be no quick-fix book for Math Olympiads. It is better to avoid undue sales gimmick and focus on classics. The sole purpose of math olympiads is to facilitate deep mathematical learning in young minds. This depth cannot be achieved in one day.

Am I ready for AMC?

The only way to understand whether you are ready for AMC (8, 10 or 12), or figure out your strengths and weaknesses, is to give a diagnosis test (AMC like test). You may either take an original AMC or take our diagnosis test. We will send you a report on the basis of this test with a review of your strengths and weaknesses.

You may request for a diagnosis test here (fill in the form).

Categories
I.S.I. and C.M.I. Entrance USA Math Olympiad

American Mathematical Competitions

Overview of Math Olympiads in United States

The American Mathematics Competitions (AMC) are the first of a series of competitions in middle school and high school mathematics that lead to the United States team for the International Mathematical Olympiad (IMO).

AMC has three levels:

  • AMC 8 – grade 8 and below
  • AMC 10 – grades 10 and below
  • AMC 12 – grades 12 and below

The AMCs lead to International Math Olympiad (IMO), world’s most prestigious math contest at school level. The basic route is as follows

  1. AMC 10 or AMC 12 ===>American Invitational Mathematics Examination (AIME)
  2. AIME ===> United States of America Mathematical Olympiad (USAMO).
  3. USAMO (typically around 30 students) ===> Mathematical Olympiad Summer Program (MOSP or more commonly, MOP)
  4. Six students are selected from the top twelve scorers on the USAMO (through the Team Selection Test (TST)) to form the United States Math Team that goes to International Math Olympiad (held in June-July).
INFOAMC 8AMC 10AMC 12
When?A Tuesday in NovemberA Tuesday in February or 15 days laterA Tuesday in February or 15 days later
Problems252525
Time (min)407575
ScoreCorrect +1
Incorrect 0
Not Attempted 0
Correct +6
Incorrect 0
Not Attempted +1.5
Correct +6
Incorrect 0
Not Attempted +1.5
Next LevelOnly awards, no next levelAIME if scores 120+AIME if scores 100+

How to Prepare for AMC

The American Math Contest is designed to encourage independent and critical thinking in young mind. Hence the problems are non-standard in nature. Hence key factors of AMC training are

  1. Strong foundation of mathematical concepts.
  2. Plenty of problem practice

All problems in AMC (or other Math olympiads) can be (and should be) solved without calculus. Though most problems are interdisciplinary in nature (that is use several concepts from secondary mathematics) 33% problems are from Geometry making it the single most important topic in AMC. The trend actually continues at all higher level olympiads, For example in IMO, 2 out of 6 problems are from Geometry.

Books

We suggest two types of books/resources for AMC preparation. The first kind is for concept building. The second is for problem solving. Nothing beats the old Russian books as far as Math Olympiads are concerned. In fact American Math Olympiads are largely ‘copies’ of Russian Math Olympiads and Math Circles.

Starting Point

  • Mathematical Circles; A Russian Experience by Fomin
  • Lines and Curves by Vasiliyev
  • AMC 8 Problems

Both of these are outstanding books for concept building as well as problem solving. Lines and Curves present a novel method to introduce geometry to young people. Mathematical Circles covers most of the topics asked at AMC (or even higher levels). They are two light weight books for a great beginning.

Next Level

  • Challenges and Thrills of Pre College Mathematics by Venkatchala
  • Elementary Algebra and Higher Algebra by Hall and Knight
  • AMC 10 and AMC 12 Problems

Challenges and Thrills of Pre College Mathematics covers all the necessary topics of AMC. Hall and Knight’s work are classics. They are useful for conceptual strengthening of skills in Algebra

Our Courses use several other resources in conjunction with these basic books. For example the outstanding articles of ‘Quant’, Mathematical Gems (Dolciani Series), Excursions into Mathematics or books by Martin Gardener are regularly used in a selective manner. Nowadays, thanks to internet, resource for math olympiad is unlimited. Hence the critical task is to point at selective few resources and fully utilize them.

There can be no quick-fix book for Math Olympiads. It is better to avoid undue sales gimmick and focus on classics. The sole purpose of math olympiads is to facilitate deep mathematical learning in young minds. This depth cannot be achieved in one day.

Am I ready for AMC?

The only way to understand whether you are ready for AMC (8, 10 or 12), or figure out your strengths and weaknesses, is to give a diagnosis test (AMC like test). You may either take an original AMC or take our diagnosis test. We will send you a report on the basis of this test with a review of your strengths and weaknesses.

You may request for a diagnosis test here (fill in the form).

Some Important Links:

Math Olympiad Program

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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.

Categories
I.S.I. and C.M.I. Entrance USA Math Olympiad

AMC 10 (2013) Solutions

12. In (\triangle ABC, AB=AC=28) and BC=20. Points D,E, and F are on sides (\overline{AB}, \overline{BC}), and (\overline{AC}), respectively, such that (\overline{DE}) and (\overline{EF}) are parallel to (\overline{AC}) and (\overline{AB}), respectively. What is the perimeter of parallelogram ADEF?

(\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }56\qquad\textbf{(D) }60\qquad\textbf{(E) }72\qquad )

http://cache.artofproblemsolving.com/asyforum/c/e/e/cee0c4038bfb9b14cfc95050e9b3d20bf808167d.png

Solution: Perimeter = 2(AD + AF). But AD = EF (since ABCD is a parallelogram).
Hence perimeter = 2(AF + EF).
Now ABC is isosceles (AB = AC = 28). Thus angle B = angle C. But EF is parallel to AB. Thus angle FEC = angle B which in turn is equal to angle C.
Hence triangle CEF is isosceles. Thus EF = CF.
Perimeter = 2(AF + EF) = 2(AF + EF) =2AC = (2 \times 28) = 56.

Ans. (C) 56

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I.S.I. and C.M.I. Entrance USA Math Olympiad

USAJMO 2012 questions

  1. Given a triangle ABC, let P and Q be the points on the segments AB and AC, respectively such that AP = AQ. Let S and R be distinct points on segment BC such that S lies between B and R, ∠BPS = ∠PRS, and ∠CQR = ∠QSR. Prove that P, Q, R and S are concyclic (in other words these four points lie on a circle).
  2. Find all integers (n \ge 3 ) such that among any n positive real numbers ( a_1 , a_2 , ... , a_n ) with \displaystyle {\text(\max)(a_1 , a_2 , ... , a_n) \le n) (\min)(a_1 , a_2 , ... , a_n)}  there exist three that are the side lengths of an acute triangle.
  3. Let a, b, c be positive real numbers. Prove that \displaystyle {(\frac{a^3 + 3 b^3}{5a + b} + \frac{b^3 + 3c^3}{5b +c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2))} .
  4. Let (\alpha) be an irrational number with (0 < \alpha < 1), and draw a circle in the plane whose circumference has length 1. Given any integer (n \ge 3 ), define a sequence of points (P_1 , P_2 , ... , P_n ) as follows. First select any point (P_1) on the circle, and for ( 2 \le k \le n ) define (P_k) as the point on the circle for which the length of the arc (P_{k-1} P_k) is (\alpha), when travelling counterclockwise around the circle from (P_{k-1} ) to (P_k). Suppose that (P_a) and (P_b) are the nearest adjacent points on either side of (P_n). Prove that (a+b \le n).
  5. For distinct positive integers a, b < 2012, define f(a, b) to be the number of integers k with (1le k < 2012) such that the remainder when ak divided by 2012 is greater than that of bk divided by 2012. Let S be the minimum value of f(a, b), where a and b range over all pairs of distinct positive integers less than 2012. Determine S.
  6. Let P be a point in the plane of triangle ABC, and (\gamma) be a line passing through P. Let A’, B’, C’  be the points where reflections of the lines PA, PB, PC with respect to (\gamma) intersect lines BC, AC, AB, respectively. Prove that A’, B’ and C’ are collinear.