## Math Club Geometry Training Session

Posted in Uncategorized by pgadey on 2019/01/31

These are some questions that I prepared for Math Club. The problems follow Paul Zeitz’s excellent book The Art and Craft of Problem Solving. You can find this hand-out is here:

• Draw big diagrams. Lots of them.
• Use multiple colours to keep track on information.
• Draw several examples with different lengths/angles.
• Avoid coordinates.

2. Facts and Questions

Fact 1 Congurence conditions for triangles: (SAS, ASA, SSS, AAS)
Side-angle-side, angle-side-angle, side-side-side, angle-angle-side.

Question 2 Why is angle-side-side not a congruence condition?

Fact 3 If an angle ${\angle A}$ measures a straight line, then ${\angle A = \pi = 180^\circ}$.

Question 4 Show that ${\angle ECD = \angle ACB}$ using the Fact~3.

Fact 5 (Transversals) ${AB}$ and ${CD}$ are parallel iff ${\angle B = \angle D}$.

Question 6 Show that the sum of the interior angles of a triangle is ${\pi}$. (Idea: Add a new parallel line.)

Question 7 For any triangle ${\triangle ABC}$, the exterior angle at ${A}$ is the sum of the interior angles at ${B}$ and ${C}$.

Question 8 (Parallelograms) Use congruence and parallels to show:

• The opposite sides of a parallelogram have the same length.
• The opposite angles of a parallelogram are equal, and adjacent angles are “supplementary” (sum to ${\pi}$).
• The diagonals of a parallelogram bisect each other.
(Idea: Introduce triangles.)

Question 9 (Isoceles triangles) A triangle ${\triangle ABC}$ is isoceles if ${|AB| = |AC|}$. Show that: If ${\triangle ABC}$ is isoceles then ${\angle B = \angle C = 90^\circ - (\angle A)/2}$.

Question 10 Let ${\triangle ABC}$ be a triangle. Let ${D}$, ${E}$, and ${F}$ be the midpoints of ${BC}$, ${AC}$, and ${AB}$ respectively. Show that the lines: ${DE}$, ${EF}$, and ${FD}$, dissect ${\triangle ABC}$ in to congruent triangles.

(Big Idea: Introduce parallels to ${BC}$ and ${AC}$ at ${D}$. These will create new “phantom points” on ${BC}$ and ${AC}$. These new points will be very helpful because of our theory of parallelograms.)

Fact 11 (Circles and Chords) If any two are true, then all three are true.

• The line passes through the center of the circle.
• The line passes through the midpoint of the chord.
• The line line is perpendicular to the chord.

Fact 12 (Circles and Tangents) A tangent to a circle is perpendicular to the radius at the point of tangency. Also, a perpendicular to a tangent line placed at the point of tangency, will pass through the center of the center of the circle.

Question 13 Suppose that ${A}$ is outside a circle. If ${AX}$ and ${AY}$ are tangent to the circle, then ${|AX|=|AY|}$.

Question 14 (Inscribed Angle Theorem) Consider a circle centered at ${O}$ with points ${A,B,C}$ on the perimeter of the circle. Show that ${\angle ABC = 2\angle AOC}$. (Idea: Chase angles.)

Question 15 Consider a semi-circle with base ${AC}$.
If ${B}$ is on the perimeter of the semi-circle, show that ${\angle ABC = 90^\circ}$.

Question 16 (Circumcircles) The circumcircle of a triangle ${\triangle ABC}$ is a circle passing through ${A}$, ${B}$, and ${C}$.

• Suppose that a circumcircle exists. Show that its center is the intersection point of the perpendicular bisectors of the sides of the triangle. (If a circumcircle exists, then it has a unique center.)
• Given a triangle, any two perpendicular bisectors will intersect in a point equidistant from all three vertices. (Any triangle has a circumcenter.)
• All three perpendicular bisectors intersect in a unique point, the “circumcenter”.

Question 17 (Incircles) The incircle of a triangle ${\triangle ABC}$ is a circle tangent to ${AB}$, ${BC}$, and ${AC}$.

• Suppose that an incircle exists. Show that its center is the intersection of any two angle bisectors.
• Show that the intersection of any two angle bisectors is the center of an incircle and this point is unique.
• All three angle bisectors intersect in a unique point, the “incenter”.

3. Contest Problems

Question 18 (Canada 1991) Let ${C}$ be a circle and ${P}$ a point in the plain. Each line through ${P}$ that interesects ${C}$ determines a chord of ${C}$. Show that the midpoints of these chords form a circle.

Question 19 (Rochester 2012) Let ${\triangle ABC}$ be an isoceles triangle with ${AC = BC}$ and ${\angle ACB = 80^\circ}$. Consider and interior point of this triangle such that ${\angle MBA = 30^\circ}$ and ${\angle MAB = 10^\circ}$. Find with proof, the measure ${\angle AMC}$.

Question 20 (Rochester 2013) Let ${\triangle ABC}$ be a triangle with ${\angle BAC = 120^\circ}$. Suppose the bisectors of ${\angle BAC}$, ${\angle ABC}$, and ${\angle ACB}$ meet ${BC}$, ${AC}$, and ${AB}$ at points ${D}$, ${E}$, ${F}$, respectively. Prove that ${\triangle DEF}$ is a right angled triangle.

## Math Club Number Theory Training Session

Posted in Lecture Notes, Math by pgadey on 2019/01/31

These are some questions that I prepared for Math Club. The problems follow Paul Zeitz’s excellent book The Art and Craft of Problem Solving. You can find this hand-out is here:

• Try out lots of examples.
• The small numbers are your friends.

2. Facts and Questions

Fact 1 If ${a, b \in \mathbb{Z}}$ we write ${a | b}$ for the statement “${a}$ divides ${b}$.”
Formally, ${a|b}$ means ${b = ka}$ for ${k \in \mathbb{Z}}$.

Question 2 What is the largest ${n}$ such that ${n^3 + 100}$ is divisible by ${n+10}$? Idea: Find a factorization ${n^3+100 = (n+10)( ... ) \pm C}$ where ${C}$ is a small constant.

Fact 3 The “divisors” of ${k}$ are all ${d}$ such that ${d | k}$. We say ${p}$ is “prime” if its divisors are ${\{1, p\}}$. We say that ${k}$ is “composite” if it is not prime.

Fact 4 (Fundamental Theorem of Arithmetic) Any natural number ${n}$ is a product of a unique list of primes.

Question 5 Show that ${\sqrt{2}}$ is irrational. Generalize!

Question 6 Show that there are infinitely many primes. Euclid’s idea: Suppose there are finitely many ${\{ p_1, p_2, \dots, p_n\}}$ and consider ${N = p_1 p_2 \dots p_k + 1}$.

Question 7 Show that there are arbitrarily large gaps between primes. That is, show that for any ${k}$ there are ${k}$ consecutive numbers ${n, n+1, \dots, n+k}$ which are all composite.

Question 8 (Germany 1995) Consider the sequence ${x_0 = 1}$ and ${x_{n+1} = ax_n + b}$. Show that this sequence contains infinitely many composite numbers.

3. Congruence

Fact 9 (The Division Algorithm) For any ${a, b \in \mathbb{N}}$ there is a unique pair ${(k,r)}$ such that ${b = ka + r}$ and ${0 \leq r < a}$.

Fact 10 We write ${a \equiv b \mod n}$ if ${n | (a-b)}$. For any ${a \in \mathbb{Z}}$ there is \mbox{${r \in \{0, 1, \dots, n-1\}}$} such that ${a \equiv r \mod n}$. We say that “${a}$ is congruent to ${r}$ modulo ${n}$”. Congruence preserves the usual rules of arithmetic regarding addition and multiplication.

Question 11 Suppose that ${n}$ has digits ${n = [d_1 \dots d_k]}$ in decimal notation.

1. Show that ${n \equiv d_1 + d_2 + \dots + d_k \mod 9}$.
2. Show that ${n \equiv d_k \mod 10}$.
3. Show that ${n \equiv \sum_{k=0}^n (-1)^k d_k \mod 11}$.
4. Show that ${n \equiv [d_{k-1}d_k] \mod 100}$.

Question 12 What are the last two digits of ${7^{40001}}$?

Question 13 Show that any perfect square ${n^2}$ is congruent to ${0}$ or ${1 \mod 4}$. Conclude that no element of ${\{11, 111, 1111, \dots\}}$ is a perfect square.

Question 14 Show that 3 never divides ${n^2 + 1}$.

4. The Euclidean Algorithm

Fact 15 The “greatest common divisor” of ${a}$ and ${b}$ is:

$\displaystyle \gcd(a,b) = \max\{ d : d|a \textrm{ and } d|b \}$

Question 16 Show that ${\gcd(a,b) = \gcd(a,r)}$ where ${b = ak + r}$ and ${(k,r)}$ is the unique pair of numbers given by the division algorithm.

Question 17 The Fibonacci numbers are defined so that ${F(1) = 1, F(2) = 1}$, and ${F(n) = F(n-1) + F(n-2)}$ for ${n>2}$. Show that ${\gcd(F_n, F_{n-1}) = 1}$.

The Fibonacci numbers have the following curious property: Consecutive Fibonacci numbers are the worst-case scenario for the Euclidean Algorithm. In 1844, Gabriel Lamé showed: If ${a \leq b \leq F_n}$ then the Euclidean algorithm takes at most ${n}$ steps to calculate ${\gcd(a,b)}$. Check out this great write-up at Cut the Knot.

4.1. Parity

Question 18 Suppose that ${n = 2k + 1}$ is odd and ${f : \{1, 2, \dots, n\} \rightarrow \{1, 2, \dots, n\}}$ is a permutation. Show that the number

$\displaystyle (1 - f(1))(2 - f(2)) \dots (n - f(n))$

must be even.

Question 19 A room starts empty. Every minute, either one person enters or two people leave. Can the room contain ${2401}$ people after ${3000}$ minutes?
Idea: Consider the “mod-3 parity” of room population.

5. Contest Problems

Question 20 Show that ${\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}}$ is not an integer for any ${n > 1}$.

Idea: Consider the largest power ${2^k < n}$. Divide out by this largest power. This will make all of the denominators odd. (In fancy number theory terms, you’re using a 2-adic valuation.)

Question 21 (Rochester 2012) Consider the positive integers less than or equal to one trillion, i.e. ${1 \leq n \leq 10^{12}}$. Prove that less than a tenth of them can be expressed in the form ${x^3 + y^3 + z^4}$ where ${x}$ , ${y}$ , and ${z}$ are positive integers.

Idea: None of ${x}$, ${y}$, or ${z}$ can be very big. For example, ${x < \sqrt[3]{10^{12}} = 10^4}$.

Question 22 (Rochester 2003) An ${n}$-digit number is “${k}$-transposable” if ${N = [d_1 d_2 \dots d_n]}$ and ${kN = [d_2 d_3 \dots d_n d_1]}$. For example, ${3 \times 142857 = 428571}$ is ${3}$-transposable. Show that there are two 6-digit numbers which are 3-transposable and find them.

\noindent Big Idea: Consider repeating decimal expansions.
Observe that ${10 \times 0.[d_1 d_2 d_3 \dots] = d_1 . [d_2 d_3 d_4 \dots]}$.
Find a number with a repeating decimal of length six.

Question 23 Suppose that you write the numbers ${\{1, 2, \dots, 100\}}$ on the blackboard. You now proceed as follows: pick two numbers ${x}$ and ${y}$, erase them from the board, and replace them with ${xy + x + y}$. Continue until there is a single number left. Does this number depend on the choices you made?

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## MAT B41 — Exam Multiple Choice Statistics

Posted in Teaching and Learning by pgadey on 2018/08/17

MAT B41 wrote their exam today!

The multiple choice have been graded and we got the following information.
It looks like the class did alright, the average is quite good.

Average: 70
Median: 73
Standard deviation: 16.7

These are just the statistics for the multiple choice questions.
The full grades should be available early next week.

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## Science Unlimited — Knot Theory and Cat’s Cradle: A Brief Introduction to Storer Calculus

Posted in Lecture Notes by pgadey on 2018/08/14

This slideshow requires JavaScript.

The handout for the talk is available here:

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## MAT B41 — Final Exam Details

Posted in 2018 -- MAT B41 by pgadey on 2018/08/01

## Canada Math Camp — Storer Calculus

Posted in Math by pgadey on 2018/07/31

This slideshow requires JavaScript.

The handout for the talk is available here:

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## MAT B41 — Week 12

Posted in 2018 -- MAT B41, Lecture Notes by pgadey on 2018/07/31

You made it to the last week! You’re done!

On Homework 5, you solved the Napkin Ring Problem. Check it out! That is super cool!

Suggested Exercises:

• 6.1 Geometry of Maps from $\mathbb{R}^2$ to $\mathbb{R}^2$: 1, 3, 6, 11
• 6.2 The Change of Variables Theorem: 3, 4, 7, 10, 11, 21, 23, 26, 28
• 6.3 Applications : 1, 3, 4, 5, 6, 11, 13, 16

Notes:

The notes are available here.

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## Mock Final Exam!

Posted in 2018 -- MAT B41 by pgadey on 2018/07/28
The Mock Final is now available!

Thanks everyone, who came out and wrote today! We had about thirty people in total. The last writer finished at approximated 14:50pm. It seems like the final will take approximately three hours. Please attempt the mock final, it is the best preparation for the real final.

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## MAT B41 — Week 11

Posted in 2018 -- MAT B41, Lecture Notes by pgadey on 2018/07/24

(Archimedes Thoughtful by Domenico Fetti 1620 from Wikimedia)
Homework 5 is due! Homework 6 (tex) is now available!

The Mock Midterm will be Friday July 27 in SY110 from 12–3pm.

Suggested Exercises:

• 5.4 Changing the Order of Integration: 2,3,7,9,14
• 5.5 The Triple Integral: 1,3,4,9,10,11,12,16,18,20,21

Notes:

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## Homework #5 Question 4

Posted in Math by pgadey on 2018/07/20

Consider a solid ball of radius $R$. Cut a cylindrical hole, through the center of the ball, such that the remaining body has height $h$. Call this the donut $D(R,h)$. Use Cavalieri’s principle to calculate the volume of $D(R,h)$. Calculate the volumes of $D(25,6)$ and $D(50,6)$.

Several students have asked what $D(R,h)$ looks like. Here are some pictures that I found to illustrate the concept. The donut $D(R,h)$ is the region between the red sphere and blue cylinder. The golden balls below show various views of the donut. The donut should fit between the two planes $z=h/2$ and $z=-h/2$, so that it has total height $h$.

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