Categories

## Nilpotent matrix eigenvalues (TIFR 2014 problem 11)

Question:

Let $A$ be an $nxn$ matrix with real entries such that $A^k=0$ (0-matrix) for some $k\in\mathbb{N}$.

Then

A. A has to be the 0-matrix.

B. trace(A) could be non-zero.

C. A is diagonalizable.

D. 0 is the only eigenvalue of A

Discussion:

Let $v$ be an eigenvector of $A$ with eigenvalue $\lambda$.

Then $v \neq 0$ and $Av=\lambda v$.

Again, $A^2 v=A(Av)=A(\lambda v)=\lambda Av= (\lambda)^2v$.

We continue to apply A, applying it k times gives: $A^k v=(\lambda)^k v$.

By given information, the left hand side of the above equality is 0.

So $\lambda^k v=0$ and remember $v \neq 0$.

So $\lambda =0$.

Therefore $0$ is the only eigenvalue for $A$.

So D is true.

We analyse the question a little bit further, to check it satisfies no other options above.

We know $trace(A)=$ sum of eigenvalues of A= $\sum 0 =0$

So option B is false.

Take $A=\begin{bmatrix} 0 & 1 // 0 & 0 \end{bmatrix}$.

Then $A^2 =0$. But $A$ is not the zero matrix.

Also, if $A$ were diagonalizable then the corresponding diagonal matrix would be the zero matrix. Which would then imply that $A$ is the zero matrix, which in this case it is not. (See diagonalizable-nilpotent-matrix-tifr-2013-problem-8 ) So this disproves options A and C.

Categories

## Theory of Equation (TIFR 2014 problem 10)

Question:

Let $C \subset \mathbb{ZxZ}$ be the set of integer pairs $(a,b)$ for which the three complex roots $r_1,r_2,r_3$ of the polynomial $p(x)=x^3-2x^2+ax-b$ satisfy $r_1^3+r_2^3+r_3^3=0$. Then the cardinality of $C$ is

A. $\infty$

B. 0

C. 1

D. $1<|C|<\infty$

Discussion:

We have $r_1+r_2+r_3=-(-2)=2$

$r_1r_2+r_2r_3+r_3r_1=a$ and

$r_1r_2r_3=-(-b)=b$

Also, of the top of our head, we can think of one identity involving the quantities $r_1^3+r_2^3+r_3^3=0$ and the three mentioned just above.

Let’s apply that:

$r_1^3+r_2^3+r_3^3-3r_1r_2r_3=(r_1+r_2+r_3)(r_1^2+r_2^2+r_3^2-(r_1r_2+r_2r_3+r_3r_1))$ …$…(1)$

Also, note that $r_1^2+r_2^2+r_3^2=(r_1+r_2+r_3)^2-2(r_1r_2+r_2r_3+r_3r_1)$

So $r_1^2+r_2^2+r_3^2=(2)^2-2(a)=4-2a$.

So by equation $(1)$,

$0-3b=2(4-2a-a)$

or, $6a-3b=8$.

Now, we notice the strangest thing about the above equation: $3|(6a-3b)$ but $3$ does not divide $8$.

Why did this contradiction occur? We didn’t start off by saying “assume that … something …”. Well, even though we pretend to not assume anything, we did assume that this equation has a solution for some $a,b$. So, the

ANSWER: $|C|=0$.

Categories

## Light through Prisms (KVPY ’10)

White light is split into a spectrum by a prism and it is seen on a screen. If we put another identical inverted prism behind in contact, what will be seen on the screen?

(A) Violet will appear where red was

(B) The spectrum will remain the same

(C) There will be no spectrum but only the original light with no deviation

(D) There will be no spectrum, but the original will be laterally displaced

Discussions:

The system will behave as a slab since an inverted prism is put behind in contact with the first prism. Hence, there will be no spectrum, but only original light with no deviation.

Categories

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

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.

Categories

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Categories

## AMC 10A 2016

1. What is the value of $\dfrac{11!-10!}{9!}$?
(A) 99
(B) 100
(C) 110
(D) 121
(E) 132
2. For what value of $x$ does $10^x \cdot 100^{2x} = 1000^5$?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
3. For every dollar Ben spent on bagels, David spent 25 cents less. Ben paid $12.50 more than David. How much did they spend in the bagel store together? (A)$37.50
(B) $50.00 (C)$87.50
(D) $90.00 (E)$ 92.50
4. The remainder function can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by $$rem (x,y) – x – y \left \lfloor \frac{x}{y} \right \rfloor$$
where $\left \lfloor \frac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\frac{x}{y}$. What is the value of $rem(\frac{3}{8}, – \frac{2}{5})$
(A) $-\frac{3}{8}$
(B) $-\frac{1}{40}$
(C) 0
(D) $\frac{3}{8}$
(E) $\frac{31}{40}$
5. A rectangular box has integer side lengths in the ratio 1:3:4. What is the volume of the box?
(A) 48
(B) 56
(C) 64
(D) 96
(E) 144
6. Ximena lists the whole numbers 1 through 30 once. Emilio copies Ximena’s numbers, replacing each occurrence of the digit 2 by digit 1. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena’s sum than Emilio’s?
(A) 13
(B) 26
(C) 102
(D) 103
(E) 110
7. The mean, median, and mode of the 7 data values 60, 100, x, 40, 50, 200, 90 are all equal to $x$. What is the value of $x$ ?
(A) 50
(B) 60
(C) 75
(D) 90
(E) 100
8. Trickster Rabbit agrees with Foolish Fox to double Fox’s money every time Fox crosses the bridge by Rabbit’s house, as long as Fox pays 40 coins in toll to Rabbit after each crossing. The payment is made after the doubling. Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?
(A) 20
(B) 30
(C) 35
(D) 40
(E) 45
9. A triangular array of 2016 coins has 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on upto $N$ coins in the $N$th row. What is the sum of the digits of $N$ ?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
10. A rug is made with three different colors as shown. the areas of the three differently colored regions from an arithmetic progression. The inner rectangle is one foot wide, and each 0f two shaded region is1 foot wide on all four sides. What is the length in feet of the inner rectangle?

(A) 1
(B) 2
(C) 4
(D) 6
(E) 8
11. What is the area of the shaded region of the given 8 X 5 rectangle?

(A) $4 \dfrac{3}{4}$
(B) 5
(C) $5 \dfrac{1}{4}$
(D) $6 \dfrac{1}{4}$
(E) 8
12. Three distinct integers are selected at random between 1 and 2016, inclusive. What should be the correct statement about the probability $p$ that the product of the three integers is odd?
(A)  $p > \frac{1}{8}$
(B)  $p = \frac{1}{8}$
(C) $\frac{1}{8} < p < \frac{1}{3}$
(D) $p = \frac{1}{3}$
(E) $p < \frac{1}{3}$
13. Five friends sat in a movie theatre in a row containing 5 seats, numbered 1 to 5 from left to right. (The direction “left” and “right” are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned. she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
14. How many ways are there to write 2016 as the some of twos and threes, ignoring order? (For example,$1008 \cdot 2 + 0 \cdot 3$ and $402 \cdot 2 + 404 \cdot 3$ are two such ways.)
(A) 236
(B) 336
(C) 337
(D) 403
(E) 672
15. Seven cookies of radius 1 inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the centre cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?

(A) $\sqrt{2}$
(B) 1.5
(C) $\sqrt{\pi}$
(D) $\sqrt{2\pi}$
(E) $\pi$
16. A triangle with vertices $A(0,2)$, $B(-3,2)$, and $C(-3,0)$ is reflected about the x axis; then the image $\triangle A’B’C’$ is rotated counterclockwise around the origin by $90^{\circ}$ to produce $\triangle A”B”C”$. What is the transformation will return $\triangle A”B”C”$ to $\triangle ABC$ ?
(A) counterclockwise rotation around the origin by $90^{\circ}$
(B) clockwise rotation around the origin by $90^{\circ}$
(D) reflection about the line y-x
17. Let $N$ be a positive multiple of 5. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)$=1 and that  $P(N)$ approaches  $\frac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \frac{321}{400}$ ?
(A) 12
(B) 14
(C) 16
(D) 18
(E) 20
18. Each vertex of a cube is to be labeled with an integer from 1 through 8, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
(A) 1
(B) 3
(C) 6
(D) 12
(E) 24
19. In rectangle ABCD, AB=6 and BC=3. Point E between B and C, and point F between E and C are such that BE=EF=FC. segment $\bar{AE}$ and $\bar{AF}$ intersect $\bar{BD}$ at P and Q respectively. The ratio BP:PQ:QD can be written as r:s:t, where the greatest common factor of r,s, and t is 1. what is $r+s+t$ ?
(A) 7
(B) 9
(C) 12
(D) 15
(E) 20
20. For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables, a,b,c and d, each to some positive power. What is $N$ ?
(A) 9
(B) 14
(C) 16
(D) 17
(E) 19
21. Circles with centres P,Q, and R, having radii 1,2, and 3, respectively, lie on the same side of line l and are tangent to l at P’,Q’, and R’, respectively, with Q’ between P’ and R’. The circle with center Q is ex tangent to each of the othe other two circles. What is the area of $\triangle PQR$ ?
(A) 0
(B) $\sqrt{\frac{2}{3}}$
(C) 1
(D) $\sqrt{6} – \sqrt{2}$
(E) $\sqrt{\frac{3}{2}}$
22. For some positive integer $n$, the number $110x^2$ has 110 positive integer divisors, including 1 and the number $110x^2$. How many positive integer divisors does the $81x^2$ have?
(A) 110
(B) 191
(C) 261
(D) 325
(E) 425
23. A binary operation $\diamondsuit$ has the properties that $a\diamondsuit(b\diamondsuit c)-(a\diamondsuit b) \cdot c$ and that $a\diamondsuit a=1$ for all nonzero real numbers a,b, and c. (Here the dot. represents the usual multiplication operation.) the solution to the equation $2016 \diamondsuit (6 \diamondsuit x) – 100$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$?
(A) 109
(B) 201
(C) 301
(D) 3049
(E) 33601
24. A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length 200. What is the length of the fourth side?
(A)  200
(B)  $200\sqrt{2}$
(C)   $200\sqrt{3}$
(D) $300\sqrt{2}$
(E) 500
25. How many ordered triples $(x,y,z)$ of positive integers satisfy lcm(x,y)=72, lcm(x,z)=600, and lcm(y,z)=900?
(A) 15
(B) 16
(C) 24
(D)  27
(E) 64
Categories

## Beautiful Books

This is an (ever-growing and ever-changing) list of books, useful for school and college mathematics students. If you are working toward Math Olympiad programs or intense college mathematics, these books may prove to be your best friend.

If you are taking a Cheenta Advanced Math Program, chances are that you will referred to use this post.

Categories

## Beautiful Books for Mathematics

This is an (ever-growing and ever-changing) list of books, useful for school and college mathematics students. If you are working toward Math Olympiad, I.S.I., C.M.I. entrance programs or intense college mathematics, these books may prove to be your best friend.

If you are taking a Cheenta Advanced Math Program, chances are that you will referred to use this post.

Categories

## Geometry Problems in AIME; problems and discussions.

Let’s have a problem discussion of Geometry problems in AIME. (American Invitational Mathematics Competitions). Give them a try.

1. In $\Delta ABC, AB = 3, BC = 4$, and CA = 5. Circle $\omega$ intersects$\overline{AB} at E and B, \overline{BC}$ at B and D, and $\overline{AC}$ at F and G. Given that EF=DF and $\displaystyle \dfrac{DG}{EG} = \frac{3}{4}$ , length $\displaystyle DE=\dfrac{a\sqrt{b}}{c}$, where a and c are relatively prime positive integers, and b is a positive integer not divisible by the square of any prime. Find a+b+c. (2014 AIME I Problems/Problem 15)
2. Circle C with radius 2 has diameter $\overline{AB}$. Circle D is internally tangent to circle C at A. Circle E is internally tangent to circle C, externally tangent to circle D, and tangent to $\overline{AB}$. The radius of circle D is three times the radius of circle E, and can be written in the form $\sqrt{m}-n$, where m and n are positive integers. Find m+n. (2014 AIME II Problems/Problem 8)
3. A rectangle has sides of length a and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length a can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length a parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find $a^2$. (2014 AIME II Problems/Problem 3)
4. In $\triangle{ABC}, AB=10, \angle{A}=30^\circ$ , and $\angle{C=45^\circ}$. Let H, D, and M be points on the line BC such that $AH\perp{BC}, \angle{BAD}=\angle{CAD}$, and BM=CM. Point N is the midpoint of the segment HM, and point P is on ray AD such that $PN\perp{BC}$.  Then $AP^2=\dfrac{m}{n}$, where m and n are relatively prime positive integers. Find m+n. (2014 AIME II Problems/Problem 14)
5. In triangle RED, measured $\angle DRE=75^{\circ}$ and measured $\angle RED=45^{\circ}. abs{RD}=1$. Let M be the midpoint of segment $\overline{RD}$. Point C lies on side $\overline{ED}$ such that $\overline{RC} \perp \overline{EM}$. Extend segment $\overline{DE}$ through E to point A such that CA=AR. Then $AE=\frac{a-\sqrt{b}}{c}$, where a and c are relatively prime positive integers, and b is a positive integer. Find a+b+c. (2014 AIME II Problems/Problem 11)
6. In triangle ABC, AB= $\frac{20}{11}$ AC. The angle bisector of angle A intersects BC at point D, and point M is the midpoint of AD. Let P be the point of the intersection of AC and BM. The ratio of CP to PA can be expressed in the form $\dfrac{m}{n}$, where m and n are relatively prime positive integers. Find m+n. (2011 AIME II Problems/Problem 4)
7. The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle. (2011 AIME II Problems/Problem 3)
8. On square ABCD, point E lies on side AD and point F lies on side BC, so that BE=EF=FD=30. Find the area of the square ABCD. (2011 AIME II Problems/Problem 2)
9. Point P lies on the diagonal AC of square ABCD with AP > CP. Let $O_{1} and O_{2}$ be the circumcenters of triangles ABP and CDP respectively. Given that AB = 12 and ${\angle O_{1}PO_{2}}$ = $120^{\circ}$, then $AP = \sqrt{a} + \sqrt{b}$, where a and b are positive integers. Find a + b. (2011 AIME II Problems/Problem 13)
10. Gary purchased a large beverage, but only drank m/n of it, where m and n are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only 2/9 as much beverage. Find m+n. (2011 AIME II Problems/Problem 1)
11. Let ABCDEF be a regular hexagon. Let G, H, I, J, K, and L be the midpoints of sides AB, BC, CD, DE, EF, and AF, respectively. The segments $\overbar{AH}$, $\overbar{BI}, \overbar{CJ}, \overbar{DK}, \overbar{EL}$, and $\overbar{FG}$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of ABCDEF be expressed as a fraction $\frac {m}{n}$ where m and n are relatively prime positive integers. Find m + n. (2010 AIME II Problems/Problem 9)
12. Triangle ABC with right angle at C, $\angle BAC < 45^\circ$ and AB = 4. Point P on \overbar{AB} is chosen such that $\angle APC = 2\angle ACP$ and CP = 1. The ratio $\frac{AP}{BP}$ can be represented in the form $p + q\sqrt{r}$, where p, q, r are positive integers and r is not divisible by the square of any prime. Find p+q+r. (2010 AIME II Problems/Problem 14)
13. Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is 8: 7. Find the minimum possible value of their common perimeter. (2010 AIME II Problems/Problem 12)
14. In triangle{ABC} with AB = 12, BC = 13, and AC = 15, let M be a point on \overline{AC} such that the incircles of triangle{ABM} and triangle{BCM} have equal radii. Let p and q be positive relatively prime integers such that $\frac {AM}{CM}$ = $\frac {p}{q}$. Find p + q. (2010 AIME I Problems/Problem 15)
15. Rectangle ABCD and a semicircle with diameter AB are coplanar and have nonoverlapping interiors. Let $\mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line ell meets the semicircle, segment AB, and segment CD at distinct points N, U, and T, respectively. Line ell divides region $\mathcal{R}$ into two regions with areas in the ratio 1: 2. Suppose that AU = 84, AN = 126, and UB = 168. Then DA can be represented as $m\sqrt {n}$, where m and n are positive integers and n is not divisible by the square of any prime. Find m + n. (2010 AIME I Problems/Problem 13)
16. Let $\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \le 10$ and $3y - x \ge 15$. When $\mathcal{R}$ is revolved around the line whose equation is 3y – x = 15, the volume of the resulting solid is $\frac {m\pi}{n\sqrt {p}}$, where m, n, and p are positive integers, m and n are relatively prime, and p is not divisible by the square of any prime. Find m + n + p. (2010 AIME I Problems/Problem 11)