## Math Club Geometry Training Session

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:

https://pgadey.ca/teaching/2019-math-club/geometry-training-talk/geometry-training-talk.pdf (tex)

**1. Advice and Suggestions **

- 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 1Congurence conditions for triangles: (SAS, ASA, SSS, AAS)

Side-angle-side, angle-side-angle, side-side-side, angle-angle-side.

Question 2Why is angle-side-sidenota congruence condition?

Question 4Show that using the Fact~3.

Fact 5 (Transversals)and are parallel iff .

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

Question 7For any triangle , the exterior angle at is the sum of the interior angles at and .

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 ).
- The diagonals of a parallelogram bisect each other.

(Idea: Introduce triangles.)

Question 9 (Isoceles triangles)A triangle is isoceles if . Show that: If is isoceles then .

Question 10Let be a triangle. Let , , and be the midpoints of , , and respectively. Show that the lines: , , and , dissect in to congruent triangles.

(Big Idea: Introduce parallels to and at . These will create new “phantom points” on and . 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 13Suppose that is outside a circle. If and are tangent to the circle, then .

Question 14 (Inscribed Angle Theorem)Consider a circle centered at with points on the perimeter of the circle. Show that . (Idea: Chase angles.)

Question 15Consider a semi-circle with base .

If is on the perimeter of the semi-circle, show that .

Question 16 (Circumcircles)The circumcircle of a triangle is a circle passing through , , and .

- 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 is a circle tangent to , , and .

- 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 be a circle and a point in the plain. Each line through that interesects determines a chord of . Show that the midpoints of these chords form a circle.

Question 19 (Rochester 2012)Let be an isoceles triangle with and . Consider and interior point of this triangle such that and . Find with proof, the measure .

Question 20 (Rochester 2013)Let be a triangle with . Suppose the bisectors of , , and meet , , and at points , , , respectively. Prove that is a right angled triangle.

## Math Club Number Theory Training Session

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:

https://pgadey.ca/teaching/2019-math-club/number-theory-training-talk/number-theory-training-talk.pdf (tex)

**1. Advice and Suggestions **

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

**2. Facts and Questions **

Fact 1If we write for the statement “ divides .”

Formally, means for .

Question 2What is the largest such that is divisible by ? Idea: Find a factorization where is a small constant.

Fact 3The “divisors” of are all such that . We say is “prime” if its divisors are . We say that is “composite” if it is not prime.

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

Question 5Show that is irrational. Generalize!

Question 6Show that there are infinitely many primes. Euclid’s idea: Suppose there are finitely many and consider .

Question 7Show that there are arbitrarily large gaps between primes. That is, show that for any there are consecutive numbers which are all composite.

Question 8 (Germany 1995)Consider the sequence and . Show that this sequence contains infinitely many composite numbers.

**3. Congruence **

Fact 9 (The Division Algorithm)For any there is a unique pair such that and .

Fact 10We write if . For any there is \mbox{} such that . We say that “ is congruent to modulo ”. Congruence preserves the usual rules of arithmetic regarding addition and multiplication.

Question 11Suppose that has digits in decimal notation.

- Show that .
- Show that .
- Show that .
- Show that .

Question 12What are the last two digits of ?

Question 13Show that any perfect square is congruent to or . Conclude that no element of is a perfect square.

Question 14Show that 3 never divides .

**4. The Euclidean Algorithm **

Fact 15The “greatest common divisor” of and is:

Question 16Show that where and is the unique pair of numbers given by the division algorithm.

Question 17The Fibonacci numbers are defined so that , and for . Show that .

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 then the Euclidean algorithm takes at most steps to calculate . Check out this great write-up at Cut the Knot.

** 4.1. Parity **

Question 18Suppose that is odd and is a permutation. Show that the number

must be even.

Question 19A room starts empty. Every minute, either one person enters or two people leave. Can the room contain people after minutes?

Idea: Consider the “mod-3 parity” of room population.

**5. Contest Problems **

Question 20Show that is not an integer for any .

Idea: Consider the largest power . 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. . Prove that less than a tenth of them can be expressed in the form where , , and are positive integers.

Idea: None of , , or can be very big. For example, .

Question 22 (Rochester 2003)An -digit number is “-transposable” if and . For example, is -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 .

Find a number with a repeating decimal of length six.

Question 23Suppose that you write the numbers on the blackboard. You now proceed as follows: pick two numbers and , erase them from the board, and replace them with . Continue until there is a single number left. Does this number depend on the choices you made?

## MAT B41 — Exam Multiple Choice Statistics

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.

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

The handout for the talk is available here:

https://pgadey.ca/teaching/talks/science-unlimited-2018-storer-calculus.pdf

## Canada Math Camp — Storer Calculus

The handout for the talk is available here:

https://pgadey.ca/teaching/talks/cmc-2018-storer-calculus.pdf

## MAT B41 — Week 12

**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!

**Additional resources:**

- Khan Academy on triple integrals (Pt. 1).
- Khan Academy on triple integrals (Pt. 2).
- Kristal King on triple integrals.
- PatrickJMT on triple integrals.

**Suggested Exercises:**

- 6.1 Geometry of Maps from to : 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.

## Mock Final Exam!

**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.

## MAT B41 — Week 11

**Homework 5 is due! Homework 6 (tex) is now available!**

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

**Additional resources:**

- Khan Academy on triple integrals (Pt. 1).
- Khan Academy on triple integrals (Pt. 2).
- Kristal King on triple integrals.
- PatrickJMT on triple integrals.

**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:**

## Homework #5 Question 4

Consider a solid ball of radius . Cut a cylindrical hole, through the center of the ball, such that the remaining body has height . Call this the donut . Use Cavalieri’s principle to calculate the volume of . Calculate the volumes of and .

Several students have asked what looks like. Here are some pictures that I found to illustrate the concept. The donut 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 and , so that it has total height .

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