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 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?
Question 4 Show that
using the Fact~3.
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Fact 5 (Transversals)
and
are parallel iff
.
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Question 6 Show that the sum of the interior angles of a triangle is
. (Idea: Add a new parallel line.)
Question 7 For 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 10 Let
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 13 Suppose 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 15 Consider a semi-circle with base
.
Ifis 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 1 If
we write
for the statement “
divides
.”
Formally,means
for
.
Question 2 What is the largest
such that
is divisible by
? Idea: Find a factorization
where
is a small constant.
Fact 3 The “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 5 Show that
is irrational. Generalize!
Question 6 Show that there are infinitely many primes. Euclid’s idea: Suppose there are finitely many
and consider
.
Question 7 Show 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 10 We 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 11 Suppose that
has digits
in decimal notation.
- Show that
.
- Show that
.
- Show that
.
- Show that
.
Question 12 What are the last two digits of
?
Question 13 Show that any perfect square
is congruent to
or
. Conclude that no element of
is a perfect square.
Question 14 Show that 3 never divides
.
4. The Euclidean Algorithm
Fact 15 The “greatest common divisor” of
and
is:
Question 16 Show that
where
and
is the unique pair of numbers given by the division algorithm.
Question 17 The 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 18 Suppose that
is odd and
is a permutation. Show that the number
must be even.
Question 19 A 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 20 Show 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 23 Suppose 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!
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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