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I sent an email on the 4th April 2015 to the new AMO/TT students. Unfortunately, not many students saw it. It seems almost everyone these days has an email account with a very aggressive spam filter. This means you will probably either need to check in your spam or Junk mail folder, regularly, or you need to find a way to "white-list" my email address, so that it is not filtered off as spam.
Tuesday, 11 Feb & Wednesday 12 Feb: AMO (Australian Mathematics Olympiad)
Tuesday 25 – Thursday, 27 March: KSF (Kangourou Sans Frontières)
No masks needed now. We will continue with
with SMC 2007 and AIMO 2000.
Below, I record what we covered in each of the 2-hour sessions for new students, so that if you miss a session, you can still keep up to some extent by checking out the i-lectures and notes below.
Introduction. Covered Set Theory and Logic, in order to explain the three types of proof: Direct Proof, Proof by Contradiction, and Proof by Induction. These topics are covered in Chapters 1 to 3. For i-lectures one should view all the Chapter 1 i-lectures, the Chapter 2 i-lectures up to the first Proof by Contradiction and then Chapter 3 i-lectures up to the first Proof by Induction.
We discussed what a function is (in the notes this is Section 14.6). We gave the answer to the question:
When can you apply a function to both sides of an inequality and know that it still holds?We also discussed Equivalence Relations (relational operators that are like "="), Partial Orders (relational operators that are like "≤"), and Full Orders (relational operators that are like "<"). These ideas are needed when writing proofs, and will hopefully appear in an appendix of the Notes in the future. All three of these relational operators possess the property of transitivity, e.g. (for "≤")
x ≤ y and y ≤ z implies x ≤ z.Then we used these ideas in some proofs.
We did some more proofs. In particular we did a harder Direct Proof, a harder Proof by Contradiction, and a harder Proof by Induction. These topics are covered in Chapters 2 to 3. For i-lectures one needs to view the remaining proofs by Contradiction and by Induction.
We did all of Chapter 4 except Horner's Method and Synthetic Division which we don't plan to do in these introductory sessions. We did Exercise 1 and 3, and suggested students try 4(i), 5, 7, 8, 9 for next time.
We demonstrated Remainder and Rational Zero Theorems. We left Exercises 2, 4(ii) for students to do on their own. We did Exercises 4(i), 5, 6, 7, 8 and 9 from Chapter 4. We summed Arithmetic and Geometric Series as an introduction to Chapter 5 and some problems related to this topic, including a problem from the 2008 AIMO, namely Problem 3.
We explained binomial coefficients of Chapter 5. The notes could be amplified at this point to discuss Combinatorics. We did a number of problems from Exercise Set 5. The i-lectures need to be improved and extended for Chapter 5. We started Number Theory, doing some exercises from Exercise Set 6.
For some private school students, this is the first day of school. For the students who were able to come we will do some more Number Theory. We'll probably skip some bits where there are i-lectures that were recorded last year. We hope to do the Euclidean Algorithm (Chapter 7).
Some Notation: sets commonly used in mathematics.
More Notation: set notation leading into notation for real intervals.
Sets vs Logic: each of these works according to the rules of Boolean Algebra. We need logic to understand what a theorem is.
Logic to Maths Theorem: We define what the converse and contrapositive are, and link Logic to Theorem satements in mathematics.
Direct Proof: we give the structure of a direct proof, and demonstrate it in an example.
Proof by Contradiction: we set up the idea, and then use it on an easy example.
Proof by Contradiction: another example. We prove a more difficult example, and ask also whether the converse is true.
When can one apply a function to both sides of an inequality?
Answer: The function must be increasing on the domain of the
inequality (in which case, the directionality of the inequality is
preserved), or,
the function must be decreasing on the domain of the
inequality (in which case, the directionality of the inequality is
reversed).
Common error: squaring both sides of an inequality, without checking that
both sides are positive first.
You may like to check out the i-lectures on Functions
and Common increasing functions (Chapter 14).
Proof ... where increasing function property is used:
Problem: Show that if \(2^n - 1\) is prime,
then \(n\) is prime.
We need the factorisation:
\( x^n - 1 = (x - 1)(x^{n-1} + x^{n-2} +\cdots+1) \),
and the knowledge that the exponential function \( x \mapsto 2^x \) is
increasing.
Proof by Mathematical Induction: with a first easy example.
Other forms of induction, with a harder example: we describe secondary, ternary and complete induction, and then do an example needing a form of secondary induction. Algebra. We need logic to understand what a theorem is.
Now try try Exercise 2 of these exercises. You will probably find it quite difficult, but in fact, if you apply persistence, patience and self-belief, you should discover it's actually quite easy. You do need to be systematic and layout your working in a nice way. Once you've had a go, check out my i-lecture solving this problem below.
Polynomials : initial definitions, introducing all the terminology.
Polynomials : degree. Properties of the degree are discussed.
Division Algorithm for Polynomials.
For polynomials \(u(x)\) and \(p(x)\),
there exist polynomials \(q(x)\) and \(r(x)\) with \(\partial(r) \lt \partial(u)\) such that
\[ p(x) = u(x) q(x) + r(x). \]
Polynomials: Remainder Theorem, Factor Theorem, Rational Zero Theorem, Viète's Theorem.
Rational Zero Theorem.
If \(p(x)\) is a monic polynomial over \(\mathbb{Z}\) and
\(\alpha\) is a rational zero of \(p(x)\)
then \(\alpha \in \mathbb{Z}\).
Quadratic polynomials.
Theorem. Let \(p(x) = a x^2 + b x + c\) be a quadratic polynomial over \(\mathbb{R}\).
Let \(\Delta = \text{discriminant}(p) := b^2 - 4 a c.\)
Then \(\;p(x)\) has real zeros \(\;\iff\; \Delta \ge 0\).
Remark. If \(p(x) = a x^2 + b x + c\) with \(a, b, c \in \mathbb{R}\) and \(a \ne 0\),
and \(\Delta(p) \ge 0\)
then the zeros of \(p(x)\) are
\[ \frac{-b - \Delta}{2 a} \text{ and } \frac{-b + \Delta}{2 a}. \]
So \(p(x)\) has:
\begin{alignat*}{5}
&\bullet\text{ two distinct real zeros,}&\;&\text{if }\Delta \gt 0,\\
&\bullet\text{ two equal real zeros,} &\;&\text{if }\Delta = 0,\\
&\bullet\text{ no real zeros,} &\;&\text{if }\Delta \lt 0.
\end{alignat*}
Theorem. Let \(p(x) = x^2 + b x + c\) be a monic polynomial over \(\mathbb{Z}\).
Then \(\;p(x)\) has integer zeros
\(\;\iff\;\Delta := b^2 - 4 bc\) is a perfect square.
Horner's Method and Synthetic Division.
Horner's Method is a rewriting of a polynomial \(p(x)\)
to reduce the number of operations in evaluating at a point \(x_0\),
but by the Remainder Theorem, the result is the remainder \(p(x_0)\)
when \(p(x)\) is divided by \(x - x_0\).
Writing in a tableau, we get synthetic division and a by-product
is the quotient of \(p(x)\) on division by \(x - x_0\).
Example 4.4.1 : Given \(p(x) = x^3 + 2 x^2 - 13 x + 10\), find \(p(3)\) by Horner's Method/synthetic division.
Ex Set 4 Q1 : Given quadratic equation \(x^2 - 3 x - 5 = 0\) has roots \(\alpha, \beta\) find:
Ex Set 4 Q2 : Given quadratic polynomial \(x^2 + 4 x + 1\) has zeros \(\alpha, \beta\) find:
Ex Set 4 Q3 : Solve \(2(x + 1/x)^2 - (x + 1/x) = 10\).
Ex Set 4 Q4(i) : Factor \(x^3 - 2 x^2 - 5 x + 6\).
Ex Set 4 Q4(i) : Factor \(x^3 - 2 x^2 - 5 x + 6\) by synthetic division.
Ex Set 4 Q5 : The quadratic polynomial \(a x^2 + b x - 4\)
has remainder 12 on division by \(x - 1\), and
has \(x + 2\) as a factor,
Find \(a, b\) and the zeros of the polynomial.
Ex Set 4 Q6 : Find a quadratic equation with roots \(2 + \sqrt{3}\) and \(2 - \sqrt{3}\).
Ex Set 4 Q7 : June solved a quadratic equation
\[ a x^2 + b x + c = 0 \]
and got 2 as a root.
Kay switched \(b\) and \(c\) and got 3 as a root.
Find June's equation.
Ex Set 4 Q8 : Given equation \(x^2 + a x + (b + 2) = 0\) has real roots,
find minimum value of \(a^2 + b^2\).
Ex Set 4 Q9 : Prove, \(a, b\) odd \(\;\implies\; x^2 + 2 a x + 2 b = 0\) has no rational roots.
Ex Set 4 Q10 : Given \(p(x) = 2 x^4 - 3 x^3 - 2 x^2 + x - 4\) find \(p(1), p(-2)\).
Ex Set 4 Q11(i) : Find quotient and remainder when \( p(x) = 3 x^5 + 2 x^4- 3 x^2 + 2 x + 7\) is divided by \(x + 2\).
Ex Set 4 Q11(ii) : Find quotient and remainder when \( p(x) = x^3 + 3 x^2 - 5 x + 6\) is divided by \((x - 1)(x + 2)\).
Ex Set 4 Q11(iii) : Find quotient and remainder when \( p(x) = x^4 - 2 x^3 + x^2 - 5 x + 11\) is divided by \(x^2 + 3 x + 2\).
Geometric series giving some important factorisations, then the Binomial Theorem, with discussion of factorials and properties of binomial coefficients.
Ex Set 5 Q1 : Expand
Ex Set 5 Q3 : Factor \(a^6 - b^6\).
Ex Set 5 Q4 : Fully factor over \(\mathbb{Q}\):
Ex Set 5 Q5 : Find all \( (x, y, z) \in \mathbb{Z}^3\) satisfying \[ x^2 + y^2 + z^2 = 10(x + y + z). \]
Ex Set 5 Q6 : Express \[2(a - b)(a - c) + 2(b - c)(b - a) + 2(c - a)(c - b)\] as the sum of squares.
Ex Set 5 Q7 : Given \(x + 3\) divides \(3 x^2 + x + k\) without remainder, find \(k\).
Ex Set 5 Q8 : Factor \(n^4 + 4\) as the product of two quadratics.
Find \(n \in \mathbb{N}\) such that \(n^4 + 4\) is prime.
Ex Set 5 Q9(i) : Factor \(1 + y(1 + x)^2 (1 + x y)\).
Ex Set 5 Q9(ii) : Factor \(1 - b - a^2 + a^3 b + a^2 b^3 - a^3 b^3\).
Ex Set 5 Q11 : Prove \begin{align*} &\text{“sum of $r^{\text{th}}$ and $(r + 1)^{\text{st}}$ term of $(1 + x)^n$”}\\ &= \text{“$(r + 1)^{\text{st}}$ term of $(1 + x)^{n + 1}$”} \end{align*} where we say 1 is the zeroth term of each expansion.
Ex Set 5 Q12 : Prove some binomial coefficient identities.
Ex Set 5 Q13 : Prove \[ \binom{n}{r} = \frac{n - r + 1}{r} \cdot \binom{n}{r-1} \] and hence find \(r\) that maximises \(\binom{n}{r}\).
Ex Set 5 Q14 : Prove \[ \binom{n}{1} + \binom{n}{3}+\cdots = \binom{n}{0} + \binom{n}{2} + \cdots \]
Ex Set 5 Q15 : Prove
Ex Set 5 Q15 : Prove
Ex Set 5 Q16 : Find short expressions for
Ex Set 5 Q17 : Factor \( a^2 (b - c) + b^2 (c - a) + c^2 (a - b) \).
Ex Set 5 Q18 : Find all \((x,y) \in \mathbb{N}^2\) such that \(x^2 - 871 = y^6\).
Ex Set 5 Q19 : Given \((a - 1/a)^2 = 3 \) and \( a - 1/a \gt 0\), find
Ex Set 5 Q20 : Given \(a = x - 1/x\) and \(b = x^2 - 1/x^2\)
prove \( a^2 (a^2 + 4) = b^2\).
Ex Set 5 Q21 : Prove \(1991 \mid 3500^n - 728^n - 785^n + 4^n\) for all \(n \in \mathbb{N}\).
The Division Algorithm and Divisibility terms and notation, Linear combination property, Fundamental Theorem of Arithmetic and Euclid's Lemma.
Modulo arithmetic.
This is essentially Ex. Set 6 Question 1.
Ex. Set 6 Question 3 : Show \(x^2 -y^2 = 2\) has no integer solutions.
Ex. Set 6 Question 6 : Is 167 prime?
Ex. Set 6 Question 7 : The number of positive divisors of a natural number.
Ex. Set 6 Question 10 : Show a natural number is a perfect square if and only if it has an odd number of positive divisors.
Ex. Set 6 Question 8,9 : What natural numbers have 3 (resp. 4) positive divisors?
Ex. Set 6 Question 11 : Which prisoners will be freed?
How much evidence is needed for a proof?
Ex. Set 6 Question 12 : Is the following statement true or false?
\( n^2 + n + 41 \) is prime for all non-negative integers \(n\).
Ex. Set 6 Question 14 : Deduce \(p\,|\,a\) given some conditions.
Euclidean Algorithm ... an efficient way of finding \( \gcd(a, b) \).
Congruence mod \(m\) is an Equivalence Relation.
The definition for an equivalence relation was given back at the
end of Chapter 2. We show here that congruence mod \(m\) has
similar properties to ‘\(=\)’.
Properties of congruence mod \(m\):
further properties that congruence mod \(m\) shares
with ‘\(=\)’.
Ex. Set 7 Qn 13 (“Fermat's Little Theorem”):
We prove Fermat's Little Theorem for small primes 2, 3, 5, and 7.
Bézout's Lemma: If \(d = \gcd(a, b)\) the there exist integers \(x, y\) such that \( ax + by = d \).
Sum of digits theorems, mod 3, 9 and 11.
Let \(S(n) \) be the sum of the digits of \(n\).
Let \(A(n) \) be the alternating sum of the digits of \(n\).
Theorem. \(S(n)\equiv n \pmod{m}\), where \(m = 3\), or 9.
Theorem. \(A(n)\equiv n \pmod{11}\).
Ex. Set 7 Qn 21 (4444 problem, from IMO 1975)
Let \(N = 4444^{4444}\), \(A = S(n)\), and \(B = S(A)\).
What's \(S(B)\)?
Cancellation lemmas, multiplicative inverse mod \(m\):
Lemma
(i) If \( (c, m) = 1 \) and \(ac\equiv bc\pmod{m} \)
then \(a\equiv b\pmod{m} \).
(ii) If \( (c, m) = d \) and \(ac\equiv bc\pmod{m} \)
then \(a\equiv b\pmod{m/d} \).
Definition. We say that \(x\) is a multiplicative inverse
of \(c\)
mod \(m\), if \(cx \equiv 1\pmod{m} \).
We can discover such an inverse, when it exists (i.e. when \( (c,m) = 1 \)), via Bézout's Lemma.
Chinese Remainder Theorem.
If \( x\equiv a_i \pmod{m_i} \), for \(i =1,\ldots,k \),
where \((m_i, m_j) = 1 \), for \(i\ne j\),
then there exists a unique solution mod \( M = \prod_i m_i \).
Chinese Remainder Theorem Example: we show with an example how easy it is to construct a table including the \(a_i, m_i, M_i, b_i\) of the theorem and thus the solution \(x\) mod \(M\), and then check that it works.
Euler phi function: Let
\(\mathcal{S}_n = \bigl\{ a \in\{1,2,\ldots,n\} : (a,n) = 1 \bigr\} \),
i.e. the set of natural numbers less than \(n\), that are coprime to
\(n\).
Then the Euler phi (or totient) function,
\(\varphi(n) = |\mathcal{S}_n|\), i.e. the cardinality of \(\mathcal{S}_n \).
A little Group Theory.
Definition. An abelian group is a set \(G\) with a binary
operation \(*\) satisfying:
G1 (closure) \(x,y\in G\implies x * y \in G\).If \(*\) is \(+\), traditionally the identity \(e = 0\).
G2 (associativity) \((x * y)*z = x * (y*z)\).
G3 (existence of identity) \(\exists e \in G\text{ such that }e * x = x * e = x\).
G4 (existence of inverses) \(\forall x \in G, \exists y \in G\text{ such that } x * y = y * x = e\).
G5 (commutativity) \( x * y = y * x\).
Examples and non-examples of groups.
\( (\mathbb{N}, +) \) is not a group.
It first fails G3, since \(0 \notin \mathbb{N}\).
\( (\mathbb{N} \cup \{0\}, +) \) is not a group.
It first fails G4, since e.g. \(-3 \notin \mathbb{N}\).
\( (\mathbb{Z}, +) \) is an abelian group.
\( (\mathbb{Q}, \cdot) \), is not a group. It fails G4, since \(0\) does not have a multiplicative inverse.
\( (\mathbb{Q}\setminus\{0\}, \cdot) \), is an abelian group.
\( (\mathbb{R}\setminus\{0\}, \cdot) \), is an abelian group.
Definition. A field is a set \(F\) with two binary operations, generally denoted by \(+\) and \(\cdot\) satisfying:
F1 \( (F, +) \) is an abelian group, where \(0\) denotes the additive identity.Here, \( (F, \cdot) \) only fails to be a group, due to \(0\) not having a multiplicative inverse.
F2 \( (F\setminus\{0\}, \cdot) \) is an abelian group, where \(1\) denotes the multiplicative identity.
F3 (distribution rule) \( x\cdot(y + z) = x\cdot y + x\cdot z\).
Examples and non-examples of fields.
\( (\mathbb{Z}, +, \cdot) \) only fails G4 of F2 (which means it is
instead a commutative ring).
\( (\mathbb{Q}, +, \cdot) \), is a field.
\( (\mathbb{R}, +, \cdot) \), is a field.
If we write \( (\mathbb{Z}_m, +, \cdot) \) to represent the set
\(\{0,1,2,\ldots,m-1\} \) with operations \(+\) and \(\cdot\) interpreted mod \(m\),
then:
\( (\mathbb{Z}_m, +, \cdot) \) is a (finite) commutative
ring.
\( (\mathbb{Z}_p, +, \cdot) \), where \(p\) is prime, is a finite
field.
Theorem (Corollary of Lagrange's Theorem).
If \( (G, \cdot) \) is a finite group and \(x \in G\) then \(x^{|G|} = 1\).
For a group \(G\), \( |G| \) is the cardinality of \( G \), though it is
termed the order of \(G\).
Euler's Theorem and Fermat's Little Theorem:
The set \(\mathcal{S}_m = \bigl\{ a \in\{1,2,\ldots,m\} : (a,m) = 1 \bigr\} \)
forms a group of order \( \varphi(m) \) under
multiplication mod \(m\), so that
the next result follows immediately from the Corollary to Lagrange's
Theorem mentioned in the last i-lecture:
Euler's Theorem. If \( (a,m) = 1 \) then \( a^{\varphi(m)} \equiv 1 \pmod{m} \).
The next result is a special case of Euler's Theorem where \( m = p\),
prime.
Fermat's Little Theorem.
If \( p \not|\,a \) and \(p\) prime then \( a^{p-1} \equiv 1 \pmod{p} \).
Ex. Set 7 Q22: Prove \( 7\,|\, 2222^{5555} + 5555^{2222} \).
Two forms of Fermat's Little Theorem:
we prove the following two forms of Fermat's Little Theorem are
equivalent.
Fermat's Little Theorem.
(i) If \( p \not|\,a \) and \(p\) prime then \( a^{p-1} \equiv 1 \pmod{p} \).
(ii) If \(p\) prime then \( a^p \equiv a \pmod{p} \).
Theorem (Pigeon Hole Principle).
If there are \(N\) pigeonholes into which,
we place \(N+1\) pigeons
then there is a pigeonhole with (at least) 2 pigeons.
If instead there are \(k N + 1\) pigeons
then one of the pigeonholes has at least \(k+1\) pigeons.
Ex Set 9 Q1 : Prove, in a group of 8 people there are 2 who were born on the same day of the week.
Ex Set 9 Q2 : Prove, if the sum of 3 natural nos. is 19 then one of the numbers is at least 7.
Ex Set 9 Q3 : There are 10 F,
20 S,
8 G,
15 R,
25 I books.
How many books must we (blindly) choose to ensure
12 books of the same language?
Ex Set 9 Q4 : There are 30 students.
One made 14 mistakes.
The remaining students made fewer mistakes.
Prove (at least) 3 students made the same no. of mistakes.
Ex Set 9 Q6 : There are 95 tables, around which are 465 chairs.
Can we be sure there is a table with 6 chairs?
Ex Set 9 Q7 : Prove, if there are 5 points in an equilateral triangle of side length 1,
there are 2 points whose distance apart is at most \(\frac{1}{2}\).
Ex Set 9 Q8 : Given 27 distinct positive odd numbers less than 100,
prove there are 2 whose sum is 102.
The integers 1 to 10 are arranged around a circle.
Prove there are 3 consecutive nos. whose sum is at least 17.
Solution 1 (8 min 31s)
Solution 2 (5 min 52s)
Ex Set 9 Q10 : Six swimmers swim several races.
Each swimmer either swims or watches the other swim.
Determine the minimum number of races such that
each swimmer watches each of the others swim.
Ex Set 9 Q11 : There are 11 people at a party.
Some of them exchange handshakes.
Prove that (at least) 2 people have shaken the same no. of hands.
Ex Set 9 Q12 : A computer is used for 99 billable hours over 12 days.
Prove that on some pair of consecutive days
the computer was used for at least 17 billable hours.
Note. If actual usage is \(x\) hours,
then billable hours\({} = \lceil x\rceil \).
Inequalities : Introduction.
Properties of \(>\).
Inequalities : Introduction.
Properties of \(>\) and discussion of what increasing/decreasing function
is applied with each of the properties.
Inequalities : Overview of Techniques.
Chap. 11 Ex 2 : Prove AM-GM for \(n = 2\), i.e.
\(a, b \ge 0 \implies (a + b)/2 \ge \sqrt{a b} \).
Chap. 11 Ex 3 : Prove \(a^2 + b^2 + c^2 \ge a b + b c + c a\).
Chap. 11 Ex 4 : Prove \(t^2 - t + 1/2 \gt 0.\)
Chap. 11 Ex 5 : Prove \(a - b^2, b - c^2, c - d^2, d - a^2\) cannot all be larger than \(1/4\).
Chap. 11 Ex 7 : Prove \( p, q \ge 0 \implies (p + 2)(q + 2)(p + q) \ge 16 p q\).
Chap. 11 Ex 7 : Prove \( p, q \ge 0 \implies (p + 2)(q + 2)(p + q) \ge 16 p q\).
This second solution uses one application of AM-GM for \(n = 8\).
Chap 11 Ex 8 : Prove \(a^2 (1 + b^4) + b^2 (1 + a^4) \le (1 + a^4)(1 + b^4)\).
Chap 11 Ex 9 : Prove \( x + y + z = 1 \implies x^2 + y^2 + z^2 \ge 1/3\).
Chap 11 Ex 10 : Prove \(n \in \mathbb{N} \implies (n + 1)^n \ge 2^n n!\).
Chap 11 Ex 10 : Prove \(n \in \mathbb{N} \implies (n + 1)^n \ge 2^n n!\).
This second solution uses one application of AM-GM for ‘\(n\)’\({} = n\).
Chap 11 Ex 11 : Prove \(a, b, c \ge 0 \implies (a + b)(b + c)(c + a) \ge 8 a b c\).
Chap 11 Ex 12 : Prove \(a^2 + b^2 + c^2 \ge a b + b c + c a\).
Chap 11 Ex 13 : Prove \(x \gt 0 \implies x (a - x) \le a^2/4\).
Chap 11 Ex 14 : Prove \(a \gt 0 \implies a + 1/a \ge 2\).
Chap 11 Ex 15 : Prove \[ x_1, x_2,\ldots,x_n \gt 0 \implies \frac{x_1}{x_2} + \frac{x_2}{x_3} + \cdots + \frac{x_n}{x_1} \ge n. \]
Chap 11 Ex 16 : Prove \(a, b, c, d \gt 0\) such that \(a + b + c + d = 1 \implies \sqrt{4 a + 1} + \sqrt{4 b + 1} + \sqrt{4 c + 1} + \sqrt{4 d + 1} \lt 6\).
AM-GM-HM from AM-GM : We deduce GM-HM from AM-GM.
Chap 11 Ex 17 : Prove \[ a, b, c \gt 0 \implies \frac{9}{a + b + c} \le \frac{2}{a + b} + \frac{2}{b + c} + \frac{2}{c + a} \le \frac{1}{a} + \frac{1}{b} + \frac{1}{c}. \]
Chap 11 Ex 18 : Prove \[ a, b, c \gt 0 \implies \frac{a}{b + c} + \frac{b}{c + a} + \frac{c}{a + b} \ge \frac{3}{2}. \]
Chap 11 Ex 19 : Prove \[ x_1, x_2, x_3, x_4\gt 0 \implies \frac{x_1 + x_3}{x_1 + x_2} + \frac{x_2 + x_4}{x_2 + x_3} + \frac{x_3 + x_1}{x_3 + x_4} + \frac{x_4 + x_2}{x_4 +x_3} \ge 4. \]
It is recommended to read Chapter 13. (Vectors) and watch the mini i-lectures there,
first.
Theorem (Cauchy-Schwarz Ineq.).
\( (a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + b_2^2 + \cdots b_n^2)
\ge (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n)^2 \),
with equality \({}\iff a_1 : b_1 = a_2 : b_2 = \cdots = a_n : b_n \).
Chap 11 Ex 20 : Prove, \[ h_1, h_2,\ldots, h_n \gt 0 \implies \sum_{i=1}^n \frac{a_i^2}{h_i} \ge \frac{\Bigl(\sum_{i=1}^n a_i\Bigr)^2}{\sum_{i=1}^n h_i}. \]
Chap 11 Ex 21 : Prove, \[ a, b, c, d \gt 0 \implies \frac{1}{a} + \frac{1}{b} + \frac{4}{c} + \frac{16}{d} \ge \frac{64}{a + b + c + d}. \]
Chap 11 Ex 22 : Prove, \(a^2 + b^2 + c^2 \ge a b + b c + c a\).
Theorem (Rearrangement Ineq.).
If \( \begin{aligned}[t]
&x_1 \le x_2 \le \cdots \le x_n \\
&y_1 \le y_2 \le \cdots \le y_n
\end{aligned} \)
and \((z_1, z_2,\ldots,z_n)\) is any permutation of \((y_1, y_2,\ldots,y_n)\)
then
\[ x_1y_n + x_2y_{n-1} +\cdots+x_ny_1 \le x_1 z_1 + x_2 z_2 + \cdots + x_n z_n
\le x_1 y_1 + x_2 y_2 + \cdots + x_n y_n. \]
Also discussed are: use of w.l.o.g.,
symmetry/asymmetry, and permutations.
Chap 11 Ex 25 : Prove, \(a^2 + b^2 + c^2 \ge a b + b c + c a\).
Chap 11 Ex 26 : Prove, if \(\begin{aligned}[t]
&x_1, x_2,\ldots, x_n \gt0\mbox{ and }\\
&(y_1, y_2,\ldots, y_n)\mbox{ is a permutation of }
(x_1, x_2,\ldots, x_n)
\end{aligned}\)
then
\[ \frac{x_1}{y_1} + \frac{x_2}{y_2} +\cdots+ \frac{x_n}{y_n} \ge n. \]
Chap 11 Ex 27 : Prove \[ a, b, c \gt 0 \implies \frac{a}{b + c} + \frac{b}{c + a} + \frac{c}{a + b} \ge \frac{3}{2}. \]
Optimisation application of AM-GM.
Chap 11 Ex 29 : Find the minimum value of \(a + 2 b + 7 c\)
such that \(a, b, c \gt 0\) and \(a^2 b^5 c = 1\).
Power Mean \(\text{PM}_\ell\).
\( \text{PM}_\ell(x_1, x_2,\ldots, x_n)
= \begin{cases}
\sqrt[\ell]{\dfrac{x_1^\ell+ x_2^\ell+\cdots, x_n^\ell}{n}},
&\text{if }0\ne\ell\in\mathbb{Z}\\
\sqrt[n]{x_1\cdot x_2\cdots x_n}&\text{if }\ell = 0.
\end{cases}\)
Theorem. If \(x_1, x_2,\ldots, x_n \gt 0\) then
\( \text{PM}_{\ell+1}(x_1, x_2,\ldots, x_n) \ge
\text{PM}_{\ell}(x_1, x_2,\ldots, x_n) \) for all \(\ell\in\mathbb{Z}\).
Note that Quadratic Mean \(\text{QM} = \text{PM}_2\),
\(\text{AM} = \text{PM}_1\),
\(\text{GM} = \text{PM}_0\),
\(\text{HM} = \text{PM}_{-1}\).
So we have: \( \cdots\ge \text{QM} \ge \text{AM} \ge \text{GM} \ge \text{HM} \ge \cdots \)
Chap 11 Ex 30 : Prove QM-AM for \(n = 2\), i.e.
\[ a, b \ge 0 \implies \sqrt{\frac{a^2 + b^2}{2}} \ge \frac{a + b}{2}. \]
Chap 11 Ex 31 : Prove, \[ a, b \gt 0 \text{ such that } a + b = 1 \implies (a + 1/a)^2 + (b + 1/b)^2 \ge 25/2. \]
Chap 11 Ex 32 : Prove, \[ x_1,x_2,\ldots,x_n \gt 0 \text{ such that } x_1 + x_2 + \cdots + x_n = 1, k \in \mathbb{N} \cup \{0\} \implies \frac{1}{x_1^k} + \frac{1}{x_2^k} + \cdots + \frac{1}{x_n^k} \ge n^{k + 1}. \]
Level 0 and Level 1 knowledge of Plane Geometry,
i.e. elements of geometry which students are expected to know already,
including:
What Level 2 background knowledge of Plane Geometry constitutes.
We prove the Sine Rule with triangle circumradius, and
briefly introduce some Circle theorems, and some Essential Trig.
which will be developed more comprehensively in later i-lectures.
Cosine Rule with a 5-line proof.
Aside: an application of Essential Trig. to define \( \sin(\theta), \cos(\theta) \) and hence any trig. formula in terms of \( t = \tan(\theta/2) \). The usual application is in integration, not geometry.
Ex Set 12 Q3: Prove for \( \triangle ABC \), \( |ABC| = a b c/(4 R) \), where \( R = \mbox{circumradius}(ABC) \).
Ex Set 12 Q4: For \( \triangle ABC \),
let \( p, q \) be the radii of two circles
passing through \( A \) that touch \( BC \)
at \( B, C \), respectively.
Prove \( p q = R^2 \).
Definition. A cevian is a line segment from
a vertex of a triangle to the opposite side.
Ceva's Theorem. Let \( AX, BY, CZ \) be cevians of triangle \(ABC\).
Then
\[ AX, BY, CZ \mbox{ concur }
\;\Longleftrightarrow\;
\frac{BX}{XC} \cdot \frac{CY}{YA}
\cdot \frac{AZ}{ZB} = 1. \]
Definition. The distance from a point to a line is the shortest (= perpendicular) distance
from the point to the line.
Loci.
Triangle centres.
A triangle \(ABC\) has 4 centres.
These centres, with their usual notation, and properties are:
Ex Set 12 Q5: Prove the medians of a triangle concur (at the centroid \(G\)).
Ex Set 12 Q6: Prove the altitudes of a triangle concur (at the orthocentre \(H\)).
Ex Set 12 Q7 : Let \(\triangle ABC\) and \(\triangle A'B'C'\) be non-congruent triangles whose corresponding sides are parallel.
Prove \(AA'\), \(BB'\), \(CC'\) concur.
Ex Set 12 Q8: Stewart's Theorem. If \( AX = p \) is a cevian of \( \triangle ABC \) such that \(BX = m\), \(XC = n\) then \(a (p^2 + m n) = b^2 m + c^2 n\).
Ex Set 12 Q9 : Prove medians of \( \triangle ABC \) dissect \(ABC\) into 6 smaller triangles of equal area.
Ex Set 12 Q10 : Prove the medians of a triangle divide each other in the ratio \(2 : 1\),
i.e. the medians of a triangle trisect one another.
Ex Set 12 Q11 : Theorem. If cevian \(AX\) is an angle bisector of \(\triangle ABC\) then \(BX : XC = c : b\).
Ex Set 12 Q12 : Prove the internal angle bisectors of a triangle concur (at the incentre \(I\)).
Ex Set 12 Q13 : Prove the circumcentre and orthocentre of an obtuse triangle lie outside the triangle.
Ex Set 12 Q14 : Find ratio “area of \(\triangle ABC\)” : “area of triangle whose sides have the same lengths as the medians of \(\triangle ABC\)”.
Ex Set 12 Q15 : Prove a triangle with two equal medians is isosceles.
Ex Set 12 Q16 : Prove a triangle with two equal altitudes is isosceles.
Ex Set 12 Q17 : If cevian \(AX\) of \(\triangle ABC\) is a median, find the length of \(AX\) in terms of \(a, b, c\).
Ex Set 12 Q18 : If cevian \(AX\) of \(\triangle ABC\) is an angle bisector, prove \[ AX^2 = b c\biggl(1 - \Bigl(\frac{a}{b + c}\Bigr)^2\biggr). \]
Ex Set 12 Q20 : Prove the product of two sides of \(\triangle ABC\) equals the product of the circumdiameter and altitude on the third side.
Ex Set 12 Q21 : For \( \triangle ABC\),
\begin{align*}
R &= \mbox{circumradius}(ABC),\\
I &= \mbox{incentre}(ABC),\\
r &= \mbox{inradius}(ABC),\\
s &= \mbox{semiperimeter}(ABC) = (a + b + c)/2.
\end{align*}
Let \(X, Y, Z\) be the feet of the perpendiculars dropped from \(I\) to sides \(BC, CA, AB\), resp., and
let \(x = AZ = AY\), \(y = BX = BZ\), \(z = CX = CY\). Then
prove \(x = s - a\), \(y = s - b\), \(z = s - c\).
Ex Set 12 Q22 : Prove \(|ABC| = s r\).
Ex Set 12 Q23 : Prove the external bisectors of two angles concur with the internal bisector of the third angle.
Ex Set 12 Q24 : If three circles with centres \(A, B, C\) all touch one another, prove their radii are \(s - a\), \(s - b\), \(s - c\), respectively.
Ex Set 12 Q25 : Prove \(abc = 4srR\).
Ex Set 12 Q26 : For \( \triangle ABC\), let \(X, Y, Z\) be the feet of the perpendiculars dropped from \(I\) to sides \(BC, CA, AB\), resp.
Prove the cevians \(AX, BY, CZ\) concur (at the Gergonne point \(\textit{Ge}\)).
Ex Set 12 Q27 : Let \(I_a, I_b, I_c\) be the pairwise intersections of the external angle bisectors of \( \triangle ABC\), opposite vertices \(A, B, C\), resp.
Prove \( \triangle ABC\) is the orthic triangle of \( \triangle I_aI_bI_c\).
Ex Set 12 Q28 : Let \(I_a, I_b, I_c\) be the pairwise intersections of the external angle bisectors of \( \triangle ABC\), opposite vertices \(A, B, C\), resp.
Then each of \(I_a, I_b, I_c\) is equidistant from the three sides of \( \triangle ABC\), externally,
and hence are the centres of circles (the excircles of \( \triangle ABC\)),
with radii (called exradii) \(r_a, r_b, r_c\), resp.
Prove \( |ABC| = (s - a)r_a = (s - b)r_b = (s - c)r_c \).
Ex Set 12 Q29 : If \(r_a, r_b, r_c\) are the exradii and \( r\) is the inradius of \( \triangle ABC\),
prove
\[ \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c} = \frac{1}{r}. \]
Ex Set 12 Q31 : Prove the orthocentre of an acute-angled triangle is the incentre of its orthic triangle.
Ex Set 12 Q32 : Let \(AD, BE, CF\) be the altitudes of \( \triangle ABC\).
Prove \( \triangle AEF \sim \triangle DBF \sim \triangle DEC \sim \triangle ABC \).
Ex Set 12 Q33 : Let \(H\) and \(O\) be the orthocentre and circumcentre, resp., of \( \triangle ABC\).
Prove \( \angle HAO = |\angle B - \angle C|\).
Bowtie Theorem. If two lines through a point \(P\) meet a circle \(K\)
at points \(A, A'\) and \(B, B'\), resp.
then \( PA \cdot PA' = PB \cdot PB'\).
Definition. The power \(\mathcal{P}(K, P)\) of a point \(P\) relative to a circle \(K\), is:
\begin{align*}
\mathcal{P}(K, P) &= d^2 - R^2\\
&= PA \cdot PA'
\end{align*}
where
\begin{align*}
AA'&\mbox{ is a chord of \(K\) that passes through \(P\)},\\
R &= \mbox{radius}(K),\\
O &= \mbox{centre}(K),\\
d &= OP,\mbox{ the distance \(P\) is from \(O\)},
\end{align*}
and \(PA, PA'\) are directed line segments.
Ex Set 12 Q34 : What is the minimum value of \(\mathcal{P}(K, P)\) for a fixed circle \(K\), and for what point \(P\) is this minimum value achieved?
Ex Set 12 Q35 : What is the locus of points of constant power \(\mathcal{P}(K, P)\), for a fixed circle \(K\)?
Ex Set 12 Q36 : Interpret \(t\) geometrically, if \(\mathcal{P}(K, P) = t^2\).
Ex Set 12 Q37 : Given \(PT\) and \(PU\) are tangents from \(P\) to 2 concentric circles,
with \(T\) on the smaller circle, and
\(PT\) meets the larger circle at Q,
prove
\[ PT^2 - PU^2 = QT^2. \]
Euler's Theorem. Let \(O, R, I, r\) be the circumcentre, circumradius, incentre, inradius of \(\triangle ABC\),
and let \(d = OI\). Then
\[ d^2 = R^2 - 2 r R. \]
Ex Set 12 Q38 : Prove, for \(\triangle ABC\), \(R \ge 2r\).
Ex Set 12 Q39 : If \( K = \mbox{circumcircle}(ABC)\), find \(\mathcal{P}(K, I)\) in terms of \(R\) and \(r\).
Ex Set 12 Q40 (AIMO 2013 Q8) :
A circle \(K\) meets equilateral \(\triangle ABC\)
in points \(D, E, F, G, H, I\) (in that order)
such that \(AE = 4\), \(ED = 26\), \(DC = 2\), \(FG =14\)
and a circle with diameter \(HI\) has area \(\pi b\).
Find \(b\).
Ex Set 12 Q41 : Stewart's Theorem (directed segment, symmetric version).
If \(P, A, B, C\) are 4 points such that \(A, B, C\) are collinear
then
\[ PA^2\cdot BC + PB^2\cdot CA + PC^2\cdot AB + BC\cdot CA\cdot AB = 0. \]
Ex Set 12 Q42 : A line through the centroid \(G\) of \(\triangle ABC\)
intersects the sides of the triangle in points \(X, Y, Z\).
Prove that:
\[ \frac{1}{GX} + \frac{1}{GY} + \frac{1}{GZ} = 0,\]
using the directed segment convention.
Vectors : introduction.
Vectors vs scalars vs points.
We discuss notation that helps us distinguish these from each other.
Vectors : algebra.
We define vector addition and discuss two equivalent geometric ways
of representing the addition of vectors, namely the Triangle
Law and the Parallelogram Law.
We also define scalar multiplication of vectors.
Vectors : example.
In this example we demonstrate the notation and the algebra of vectors.
Without care we can fail to communicate our intended meaning.
Length of a vector.
\( \| \underset{\sim}{\boldsymbol{a}} \| = \sqrt{a_1^2 + a_2^2 + \cdots+a_n^2}
= {}\)“\(\mbox{length of }\underset{\sim}{\boldsymbol{a}}\)”
Scalar multiples of vectors.
Lemma. \( \| k \underset{\sim}{\boldsymbol{v}} \|
= |k|\, \| \underset{\sim}{\boldsymbol{v}} \| \).
Definition. A unit vector is a vector of length 1.
Lemma. If \( \underset{\sim}{\boldsymbol{v}} \ne \underset{\sim}{\boldsymbol{0}} \)
then \( (1/\| \underset{\sim}{\boldsymbol{v}} \|)
\underset{\sim}{\boldsymbol{v}} \) is a unit vector.
Notation. If \( \underset{\sim}{\boldsymbol{v}} \ne \underset{\sim}{\boldsymbol{0}} \)
then \( \underset{\sim}{\boldsymbol{v}}\hat{\vphantom{v}} \) is the unit vector formed from
\( \underset{\sim}{\boldsymbol{v}} \).
Lemma. If \( \underset{\sim}{\boldsymbol{v}} \ne \underset{\sim}{\boldsymbol{0}} \)
then \( \underset{\sim}{\boldsymbol{v}}\hat{\vphantom{v}}
= (c \underset{\sim}{\boldsymbol{v}})\hat{\vphantom{v}} \),
for any \(c \gt 0 \).
Dot product.
Definition. The dot product (or scalar product)
of two vectors \( \underset{\sim}{\boldsymbol{a}}, \underset{\sim}{\boldsymbol{b}}
\in \mathbb{R}^n\) is defined by:
\begin{align*}
\underset{\sim}{\boldsymbol{a}} \boldsymbol{\cdot}
\underset{\sim}{\boldsymbol{b}}
&= \begin{pmatrix} a_1\\ a_2\\ \vdots \\ a_n \end{pmatrix}
\boldsymbol{\cdot}
\begin{pmatrix} b_1\\ b_2\\ \vdots \\ b_n \end{pmatrix}\\
&= a_1 b_1 + a_2 b_2 +\cdots+ a_n b_n.
\end{align*}
Properties.
Dot product Theorem.
\( \underset{\sim}{\boldsymbol{a}} \boldsymbol{\cdot}
\underset{\sim}{\boldsymbol{b}}
= \| \underset{\sim}{\boldsymbol{a}} \|\,
\| \underset{\sim}{\boldsymbol{b}} \|\,\cos(\theta) \),
where \(\theta\) is the angle between
\( \underset{\sim}{\boldsymbol{a}} \)
and \( \underset{\sim}{\boldsymbol{b}} \).
Four views of a function:
black box, graph, kidney diagram, algebraic map representation.
A function is a “to-one” map, between two sets;
a function maps from a domain to a codomain.
The subset of the codomain actually mapped to by
a function f is its range.
Properties that a function can have are injectivity (being
one-to-one) and surjectivity (being onto).
If a continuous function is one-to-one then it is
either strictly increasing or strictly decreasing.
Common increasing functions: aside from straight lines with positive
slope, there are loga (logarithm) and expa
(exponential) functions for a > 1.
These are the ones of immediate interest for us.
Limits – introduction.
Somehow we need a definition that captures the idea of a
small quantity getting close to zero, without it ever becoming zero.
Definition.
We say: the limit of \(f(x)\), as \(x\) approaches \(x_0\), is \(L\)
and write: \( \begin{aligned}[t]
&\lim_{x \to x_0} f(x) = L\\
&\qquad\text{or less formally,}\\
&f(x) \to L\text{ as }x \to x_0
\end{aligned} \)
\(\iff\;\forall\;\varepsilon \gt 0,
\exists\;\delta \gt 0\;\text{such that}\;
0 \lt |x - x_0| \lt\delta\implies |f(x) - L|\lt\varepsilon\).
Example. Prove \[\lim_{x\to 2} \dfrac{x^2 + x - 6}{x - 2} = 5. \]
Theorem (Triangle Inequality). \(|a + b| \le |a| + |b|\).
Corollary (Triangle Inequality Lower Bound). \(\bigl||a| - |b|\bigr| \le |a + b|\).
Example. Prove \(\lim_{x\to 2} x^2 - 1 = 3\).
Theorem (Algebra of Limits, some standard limits).
If \(f(x)\to L\) and \(g(x)\to M\) as \(x\to a\), \(c\in\mathbb{R}\) (constant),
\(n\in\mathbb{N}\)
then
Limits at infinity.
Definition. \( \lim_{x\to\infty} f(x) = L
\iff
\forall\varepsilon\gt 0\;\exists K \gt 0\;\text{such that}\;
(x \gt K \implies |f(x) - L| \lt \varepsilon)\).
One-sided limits.
Definition. \( \lim_{x\to a^+} f(x) = L
\iff
\forall\varepsilon\gt 0\;\exists\,\delta \gt 0\;\text{such that}\;
(0\lt x - a\lt\delta \implies |f(x) - L| \lt \varepsilon)\).
Definition. \( \lim_{x\to a^-} f(x) = L
\iff
\forall\varepsilon\gt 0\;\exists\,\delta \gt 0\;\text{such that}\;
(0\lt a - x\lt\delta \implies |f(x) - L| \lt \varepsilon)\).
“Infinite” limit.
Definition. \( \lim_{x\to a^+} f(x) = \infty
\iff
\forall K\gt 0\;\exists \delta \gt 0\;\text{such that}\;
(0\lt x - a\lt\delta \implies |f(x) - L| \lt \varepsilon)\).
Theorem. If \( \lim_{x\to a^+} f(x) = \infty \)
then \( \lim_{x\to a^+} f(x) = 0\).
Squeeze Theorem.
If \(f(x)\le g(x)\le h(x)\) near \(x = a\), and
\( \lim_{x\to a} f(x) = L = \lim_{x\to a} h(x) \)
then \( \lim_{x\to a} g(x) = L\).
Example 1. Prove \( \lim_{x\to 0} x \sin(1/x) = 0 \).
Example 2. Find \( \lim_{x\to 0} f(x)\), where \[ f(x) = \begin{cases} x^2,&\text{if }x\in\mathbb{Q}\\ 0, &\text{if }x\notin\mathbb{Q}. \end{cases} \]
Example 3. Find \( \displaystyle{\lim_{x\to 7} \frac{\sqrt{x + 2} - 3}{x - 7}}\).
Example 4. Find \( \displaystyle{\lim_{x\to -2} \frac{t^2 - 9}{2 t^2 + 7 t + 3}}\).
Example 5. Find \( \displaystyle{\lim_{x\to -2} \frac{x + 2}{x^2 + 8}}\).
Example 6. Find \( \displaystyle{\lim_{x\to \infty} \frac{3 x^2 + 2x + 1}{2 x^3 + x + 1}}\).
Continuity.
Definition.
Standard set of continuous functions.
Definition. Formally, a sequence is a function with domain \(\mathbb{N}\), \[ a : \begin{aligned}[t] \mathbb{N} & \to \mathbb{R} \\ n & \mapsto a_n,\qquad\text{i.e. }a(n) = a_n \end{aligned} \] usually represented by numbers (in order) separated by commas: \[ a_1, a_2, a_3, \ldots \]
Definition. We say \(a_n\) converges to the limit \(L\)
and formally write: \(\lim_{n \to \infty} a_n = L\)
and less formally write: \(a_n \to L\) (as \(n \to \infty\))
\( \iff \forall\,\varepsilon \gt 0\;\exists\,N\in\mathbb{N}\)
such that \( (n \gt N \implies |a_n - L| \lt\varepsilon)\).
Example. Prove \(1/n \to 0\).
Sequences – Standard Limits.
Examples. Find limit if it exists.
Example. Find limit if it exists.
Definition. A sequence is monotone if one of the following.
A sequence \(a_n\) is:
Definition. Given a sequence \(\bigl(a[n]\bigr)_{n = 1}^\infty\),
the corresponding series is \(\sum_{n = 1}^\infty a_n\).
Definition. The sequence of partial sums for \(\sum_{n = 1}^\infty a_n\)
is the sequence: \(a_1, a_1 + a_2, \ldots, \sum_{k = 1}^n a_k, \ldots\).
Definition. A series is said to converge if and only if
its sequence of partial sums converges.
Theorem.
\(\sum_{n = 1}^\infty a_n\) converges \(\;\implies a_n \to 0\).
Key series.
Definition. An invariant is a property \(P\) that is left unchanged after
a sequence of “allowed” operations are performed.
Chap 15 Ex 1 : The first 1998 natural nos. are written on the board.
At each step, 2 nos. are randomly chosen and replaced by their difference.
Is the last remaining no., odd or even?
Chap 15 Ex 2 : The polynomials \(x^2 + x\) and \(x^2 + 2\) are written on the board.
At each stage, the sum, difference or product
of 2 polynomials on the board,
is written on the board.
Can \(x\) ever be written on the board?
Chap 15 Ex 3 : Given 3 types of particle.
When 2 particles of different types collide,
result is a particle of 3rd type.
2 particles of the same type never collide.
Prove, if one starts with equal nos. of the 3 particle types
then one cannot finish with just one particle.
Cow, sheep, goat problem: respective pairs of the animals take
45 days, 60 days, and 90 days to eat out a paddock.
How many days would it take all 3 animals to eat out the paddock?
Gauss Student Notes, Chapter 5: Pythagoras’ Theorem, Exercise 3.
Given the side lengths of a triangle are 7, 11, and 14, how long is the
median to the longest side?
Prove there exist infinitely many \(a \in \mathbb{N}\), such that
\(n^4 + a\) is not prime for all \(n \in \mathbb{N}\).
\(AX\) and \(BY\) are altitudes,Is it necessary that \(\triangle ABC\) is isosceles?
\(AY\) and \(BT\) are angle bisectors,
\(\angle XAY = \angle ZBT\).
\(AB \lt AC\),then \(B, X, Y, M, C\) are in order on \(BC\).
cevians \(AX\), \(AY\) are an altitude and angle bisector, respectively,
\(M = \text{midpt}(BC)\)
comes to pile of sweets,Boys ‘round up’ and girls ‘round down’.
rounds #sweets/#“children left”,
takes resulting number of sweets, and
leaves the room.
\(E\) on side \(AB\),Prove \(\angle KPC\) is a right angle.
\(F\) on side \(AC\),
\(K\) on side \(AB\) produced,
such that \(AE = CF = BK\),
\(P = \text{midpt}(EF)\).
comes to pile of sweets,Boys ‘round up’ and girls ‘round down’.
rounds #sweets/#“children left”,
takes resulting number of sweets, and
leaves the room.
\(D\) is on side \(AB\),Determine the angles of \(\triangle BFG\).
\(E\) is on \(AC\) with \(DE\parallel BC\),
\(F = \text{midpoint}(CD)\), and
\(G = \text{circumcentre}(ADE)\).
AMO 2006 Q2: Let \( f : \mathbb{N} \to \mathbb{N} \) be a function such that:
(i) \(f(ab) = f(a)f(b) \),
(ii) \( f(a) \lt f(b) \), if \(a \lt b\),
(iii) \( f(3) \ge 7\).
AMO 2006 Q5: Let \( ABCD \) be a square with \(E\) on \(BD\).
Suppose \( O_1 = \text{circumcentre}(ABE) \), and
\( O_2 = \text{circumcentre}(ADE) \).
Prove \( AO_1EO_2 \) is a square.
side \(BC\) at points \(U\) and \(V\),where \(U,V,W,X,Y,Z\) lie on \(K\) in that cyclic order.
side \(CA\) at points \(W\) and \(X\), and
side \(AB\) at points \(Y\) and \(Z\),
This solution uses the property that a monic quadratic equation over \(\mathbb{Z}\) has integer solutions if and only if its discriminant is a perfect square.
This solution uses representations of AM\((a,b)\) and GM\((a,b)\) in geometry.
Note: As pointed out by Keith Wong, there is a typo. on
the second line of the proof.
It should read '\(\Longrightarrow\;CE \parallel AH\)'
... i.e. replace \(B\) by \(E\).
Note: This second version requires some background on Pell's equation.
\(ABCD\) is a parallelogram.
\(M = \text{midpoint}(BD)\).
\(P\) is on \(AD\) such that \(AD = 3 PD\).
\(|ABMP| = 1995.\)
Find \(|PDM|\).
Solution (14 min 20s)
Determine the largest \(d\in\mathbb{N}\) such that \[ d \mid n^4 (n - 1)^3 (n - 2)^2 (n - 3)\;\text{for all}\; n\in\mathbb{N}.\] Solution (22 min 44s)
Let \(N\in\mathbb{N}\) have \(2 n\ge 4\) digits such that
\[ N = 11\ldots122\ldots24, \]
where there are \(n - 1\) ones and \(n\) twos.
Prove that \(N = a b\), where \(a = 33\ldots34\), \(b = 33\ldots36\),
for some numbers of leading 3s.
Solution 1 by Linda Zou (15 min 49s)
Solution 2 by Ethan Yap (13 min 49s)
\(ABCD\) is cyclic.
\(E = BA \cap CD\).
\(F = CB \cap DA\).
\(A = \text{incentre}(CEF)\).
Find angles of \(\triangle ABD\).
Solution (18 min 59s)
Let \(f : \mathbb{R}\to \mathbb{R}\setminus\{0\}\) such that
\[ f(x + 2) = f(x - 1) f(x + 5) \]
for all \(x\in \mathbb{R}\).
Prove \(f\) is periodic.
Solution (10 min 32s)
Given \(|ABC| = 1\),
\(x \in (0, 1)\),
\(A', B', C'\) are points on \(BC, CA, AB\) respectively such that
\[ BA' : A'C = CB' : B'A = AC' : C'B = (1 - x) : x.\]
Find \(|A'B'C'|\) in terms of \(x\).
Solution (10 min 54s)
Let \(\begin{aligned}[t]
N_1 &= 123\ldots91011\ldots99100101\ldots99910001001\ldots1995,
\;\text{(the numbers }1,2,\ldots,1995\text{ concatenated),}\\
N_2 &= \text{“}N_1\text{ with even-place digits removed”}\\
N_3 &= \text{“}N_2\text{ with odd-place digits removed”}\\
N_4 &= \text{“}N_3\text{ with even-place digits removed”}\\
\text{etc.}
\end{aligned}\)
with the process continuing until some \(N_k\) which had just one digit.
What is \(N_k\)?
Solution (26 min 41s)
Determine all quadruples \((p_1, p_2, p_3, p_4)\) of primes such that
Determine all polynomials \(p(x)\) over \(\mathbb{R}\) such that
\[ x p(x - 1) = (x - 2) p(x) \]
for all \(x \in \mathbb{R}\).
Solution (12 min 16s)
Let \(\triangle\) be a right triangle such that
Statistics for 89 participants
\( \begin{array}{|l|c|c|c|c|c|}
\hline
\text{Question}& 1 & 2 & 3 & 4 & 5 \\
\hline
\text{Average} & 3.5 & 4.4 & 3.9 & 2.1 & 3.2\\
\hline
\end{array} \)
Let \(K\) be a semicircle on diameter \(AB\).
\(D\) is such that \(AB = AD\).
\(E\) is point other than \(A\) where \(AD\) and \(K\) intersect.
\(F\) is on \(AE\) such that \(DE = EF\).
\(C\) is point other than \(B\) where \(BF\) and \(K\) intersect.
Prove \(\angle BAE = 2\angle EAC\).
Solution 1 (8 min 50s)
Solution 2 (17 min 24s)
Find all functions \(f : \mathbb{R} \to \mathbb{R}\) such that
\[f(u + v) f(u - v) = 2 u + f(u^2 - v^2),\]
for all \(u, v \in \mathbb{R}\).
Solution 1 (12 min 17s)
Solution 2 (7 min 17s)
Given \(x + 1/x \in \mathbb{Z}\), \(x \ne 0\),
prove \(x^n + 1/x^n \in \mathbb{Z}\) for all \(n \in \mathbb{N}\).
Solution (11 min 31s)
Sequence \(a_1, a_2,\ldots, a_{1997}\) has properties:
- \( 0 \le a_n \le 1\) for all \(n \ge 0\),
- \(a_n \ge (a_{n-1} + a_{n+1})/2\) for all \(n \ge 1\).
Given \(\triangle ABC\) is acute,
\(\angle ACB = 60^\circ\),
\(h_a, h_b\) are altitudes from \(A, B\), respectively.
Prove \(\text{circumcentre}(ABC)\) lies on the
bisector of one of the 4 angles formed by \(h_a \cap h_b\).
Solution (21 min 21s)
Statistics for 80 participants
\( \begin{array}{|l|c|c|c|c|c|}
\hline
\text{Question}& 1 & 2 & 3 & 4 & 5 \\
\hline
\text{Average} & 6.1 & 3.2 & 3.2 & 3.7 & 2.3\\
\hline
\end{array} \)
There are 19971997 Ms and the same number of Fs.
The same number of Ms and Fs have P noses;
the rest have B noses.
Each M is randomly partnered with an F.
Prove the number of mixed-nose-colour pairings is even.
Solution (4 min 58s)
Given \(c\) is a circle,
\(A\) is on \(c\),
\(B, C\) are on \(c\) such that
\(BC\) is parallel to the tangent to \(c\) at \(A\).
\(P\) is on chord \(BC\),
\(Q\) is the point other than \(A\)
where \(AP\) intersects \(c\),
\(k\) is the circle touching \(BC\) at \(P\),
that passes through \(Q\).
Prove \(k\) touches \(c\) at \(Q\).
Solution (4 min 56s)
Determine all \((x,y) \in \mathbb{Z}^2\) such that \[ (x + 1)^4 - (x - 1)^4 = y^3. \] Solution (13 min 34s)
Let \(c\) be a circle and let \(P \in \text{interior}(c)\).
Let \(f : \mathbb{Z} \to \mathbb{Z}\) such that
\[ f(n) = \begin{cases}
1, &\text{if }n = 0,\\
0, &\text{if }n \in\{-5,-4,\ldots,-1\},\\
f(n - 6) + n, &\text{for all }n\in\mathbb{Z}.
\end{cases} \]
Prove \(\dfrac{(n + 1)(n + 5)}{12} \le f(n)
\le \dfrac{n^2 + 6 n + 12}{12}\)
for all \(n\in\mathbb{Z}\).
Solution 1 (16 min 17s)
Solution 2 (14 min 3s)
Statistics for 52 participants
\( \begin{array}{|l|c|c|c|c|c|}
\hline
\text{Question}& 1 & 2 & 3 & 4 & 5 \\
\hline
\text{Average} & 5.7 & 4.0 & 3.9 & 1.7 & 2.8\\
\hline
\end{array} \)
\(\triangle\) has side lengths \(a, b, c\),
\(\triangle_a\) has side lengths \(a', b, c\),
\(\triangle_b\) has side lengths \(a, b', c\),
\(\triangle_c\) has side lengths \(a, b, c'\).
Also, \(a \ne a'\), \(b \ne b'\), \(c \ne c'\),
\(|\triangle|= |\triangle_a| = |\triangle_b| = |\triangle_c| = 1.\)
Determine all \(n \in \mathbb{N}\) such that \[\sqrt{\frac{1 + \dfrac1{2^{n-1}}}2} \lt 1 - \frac2n.\] Solution (16 min 57s)
Let \(f : \mathbb{R} \to \mathbb{R}\) satisfy
Prove
- \(f(999 + x) = f(999 - x)\) for all \(x\in \mathbb{R}\),
- \(f(1998 + x) = -f(1998 - x)\) for all \(x\in \mathbb{R}\),
Statistics for 97 participants
\( \begin{array}{|l|c|c|c|c|c|}
\hline
\text{Question}& 1 & 2 & 3 & 4 & 5 \\
\hline
\text{Average} & 2.1 & 3.0 & 2.8 & 2.9 & 1.9\\
\hline
\end{array} \)
Circle \(k_1\) has its centre on circle \(k_2\).
\(\{A, C\} = k_1 \cap k_2\).
\(B\) is on \(k_2\)
\(BC\) intersects \(k_1\) again at \(D\).
Prove \(AB = BD\).
Let \(x, y, z \in \mathbb{Z}\)
such that \(\gcd(x, y, z) = 1\) and \(x^2 + y^2 = z^2\).
Prove exactly one of \(x, y, z\) is divisible by 5.
Solution (10 min 29s)
Let \(a_0, a_1, a_2 \in \mathbb{R}\) such that
\[ -1 \le a_0 + a_1 x + a_2 x^2 \le 1
\text{ for all }x \in [-1, 1]. \]
Prove \(-2 \le a_2 \le 2\).
Solution (5 min 29s)
Let \(A, B, C, D, E \in \mathbb{Z}^2\).
Prove that the midpoint of two of \(A, B, C, D, E\)
is in \(\mathbb{Z}^2\).
Solution (13 min 18s)
Let \(ABCD\) be cyclic such that
\(AC\perp BD\),
\(E = AC\cap BD\).
Let \(U, V, W, Z\) be on \(AB, BC, CD, DA\) respectively,
such that \(EU \perp AB\),
\(EV \perp BC\),
\(EW \perp CD\),
\(EZ \perp DA\).
Prove \(UVWZ\) is cyclic.
Solution (18 min 2s)
Statistics for 66 participants
\( \begin{array}{|l|c|c|c|c|c|}
\hline
\text{Question}& 1 & 2 & 3 & 4 & 5 \\
\hline
\text{Average} & 4.2 & 3.8 & 3.5 & 6.0 & 3.5\\
\hline
\end{array} \)
Let \(n, a \in \mathbb{N}\). Prove
\[ \frac{(a + 1)^(2 n + 1) + a^(n + 2)}{a(a + 1) + 1}
\in \mathbb{N} \text{ for all }n \in \mathbb{N}. \]
Solution (17 min 53s)
Determine all functions \(f : \mathbb{R} \to \mathbb{R}\)
such that
\[ f(x) + x f(1 - x) = x^2 - 1 \]
for all \(x \in \mathbb{R}\).
Solution (15 min 40s)
Prove \(\not\exists x, m, n\in \mathbb{N}\) such that
Prove
\[ \frac{n}{2^1 (n - 1)} + \frac{n}{2^2 (n - 2)} + \cdots
+ \frac{n}{2^n \cdot 1} \lt 4, \]
for all \(n\in \mathbb{N}\) such that \(n \gt 1\).
Solution (12 min 54s)
\(ABCD\) is convex,
\(\text{incircle}(ABC)\) touches \(\text{incircle}(ACD)\)
Prove \(\text{incircle}(BDA)\) touches \(\text{incircle}(BCD)\)
Solution 1 (17 min 26s)
Solution 2 (12 min 4s)
Statistics for 89 participants
\( \begin{array}{|l|c|c|c|c|c|}
\hline
\text{Question}& 1 & 2 & 3 & 4 & 5 \\
\hline
\text{Average} & 2.6 & 2.5 & 1.8 & 1.9 & 0.4\\
\hline
\end{array} \)
Prove there doesn't exist a function \(f : \mathbb{R} \to \mathbb{R}\) such that for all \(r \in \mathbb{R}\),
There are \(m\) houses around a pond, \(m \gt 1\), \(m\) odd.
In each house live \(n\) people.
Prove there is a female with \(n\) female neighbours, or
a male with \(n\) male neighbours.
Solution (7 min 45s)
The sequence
\(\ldots, a_{-2}, a_{-1}, a_0, a_1, a_2, \ldots \in \mathbb{R}\)
and satisfy
For \(\triangle ABC\), \(P_A, P_B, P_C\) lie on \(BC, CA, AB\)
respectively.
Let \(\ell_A, \ell_B, \ell_C\) be lines through \(P_A, P_B, P_C\)
respectively such that
\(\qquad\ell_A \perp BC\), \(\ell_B \perp CA\),
\(\ell_C \perp AB\).
Prove \(\ell_A, \ell_B, \ell_C\) concur
\(\;\iff\;
P_AC^2 + P_BA^2 + P_CB^2 = P_AB^2 + P_BC^2 + P_CA^2.\)
Solution (30 min 25s)
Show there doesn't exist
\((x, y) \in \mathbb{Z}^2\setminus\{(0,0), (0,-1)\}\) such that
\[ y + y^2 = x + x^2 + x^3. \]
Solution (34 min 19s)
Statistics for 75 participants
\( \begin{array}{|l|c|c|c|c|c|c|c|c|c|}
\hline
\underset{..\hphantom{12}}{\text{Qn}}\quad\text{Score:}
& 0& 1& 2& 3& 4& 5& 6& 7&\text{Average}\\
\hline
1 &39& 5& 2& 2& 2& 1& 3&21&2.6\\
\hline
2 & 7& 6& 5& 4& 3& 1& 6&43&5.1\\
\hline
3 &12& 4&10& 5&12& 6&13&13&3.8\\
\hline
4 &41& 3& 2&11& 4& 3& 3& 8&1.9\\
\hline
5 &46&18& 9& 2& 0& 0& 0& 0&0.6\\
\hline
\end{array} \)
Find all solutions of \begin{align*} \frac{x_1}{x_1+1} = \frac{x_2}{x_2+3} = \cdots &= \frac{x_{1001}}{x_{1001}+2001}\\ x_1 + x_2 +\cdots+x_{1001} &= 2002. \end{align*}
Statistics for 71 participants
\( \begin{array}{|l|c|c|c|c|c|c|c|c|c|}
\hline
\underset{..\hphantom{12}}{\text{Qn}}\quad\text{Score:}
& 0& 1& 2& 3& 4& 5& 6& 7&\text{Average}\\
\hline
1 & 6& 2& 1& 1& 2& 7&11&41&5.7\\
\hline
2 & 4& 9&10& 2&10& 3& 5&28&4.5\\
\hline
3 &21& 6& 5& 7& 3& 5& 2&22&3.4\\
\hline
4 &32&19& 8& 3& 1& 4& 4& 0&1.3\\
\hline
5 &26&11& 4& 6& 3& 3& 5&13&2.6\\
\hline
\end{array} \)
Determine all \( (x,y)\in\mathbb{Z}^2 \) that satisfy \[ (x + y + 11)^2 = x^2 + y^2 + 11^2. \]
Statistics for 69 participants
\( \begin{array}{|l|c|c|c|c|c|c|c|c|c|}
\hline
\underset{..\hphantom{12}}{\text{Qn}}\quad\text{Score:}
& 0& 1& 2& 3& 4& 5& 6& 7&\text{Average}\\
\hline
1 & 1&16&12& 6& 0& 4& 6&24&4.1\\
\hline
2 &28& 9& 8& 1& 2& 2& 1&18&2.6\\
\hline
3 &48& 0& 1& 0& 0& 0& 1&19&2.0\\
\hline
4 &52& 7& 0& 0& 0& 0& 2& 8&1.1\\
\hline
5 &55& 4& 1& 1& 1& 1& 0& 6&0.9\\
\hline
\end{array} \)
For finite sets \(A, B\) define
\(f(A, B) = |A \cup B \setminus A \cap B|\).
Suppose \(X, Y, Z\) are such that \(f(X, Y) = f(Y, Z) = f(Z, X)\).
Let \(A, B, C, D, E, F, G\) be distinct points in the plane such that
\[ AB = BC = CD = DE = EF = FA = AG = CG = EG.\]
Prove \(AD, BE, CF\) concur.
Solution (12 min 45s)
Prove there does not exist \(m, n \in \mathbb{N}\) such that
\[4 m^2 + 17 n^2 \;\text{and}\; 17 m^2 + 4 n^2\]
are both perfect squares.
Solution (13 min 49s)
Let \(ABCD\) be convex with \(AB = AD, CB = CD\).
Let point \(E\) on \(CD\) be such that \(AE \parallel BC\).
Let \(G = BD \cap AE\).
Let point \(F\) on \(BC\) be such that \(FG \parallel CE\).
Prove \(BE\) passes through \(\text{midpoint}(AF)\).
Solution (18 min 30s)
The polynomials \(p_{0}(x), p_{1}(x), p_{2}(x), \ldots\) are defined by:
\begin{align*}
p_{0}(x) &= 0 \\
p_{1}(x) &= x - 2013\\
p_{n}(x) &= (x - 2013) p_{n-1}(x) + (2014 - x) p_{n-2}(x)
\;\text{for}\;n \ge 2.
\end{align*}
Determine all real zeros of \(p_{n}(x)\) for all \(n \in \mathbb{N}\).
Solution (21 min 14s)
Statistics for 74 participants
\( \begin{array}{|l|c|c|c|c|c|c|c|c|c|}
\hline
\underset{..\hphantom{12}}{\text{Qn}}\quad\text{Score:}
& 0& 1& 2& 3& 4& 5& 6& 7&\text{Average}\\
\hline
1 &20& 2& 2& 1& 0&27& 6&16&3.9\\
\hline
2 &28&13& 1& 7& 1& 1& 1&22&2.8\\
\hline
3 &49& 9& 1& 1& 0& 0& 1&13&1.5\\
\hline
4 &31& 6& 2& 6& 1& 0& 2&26&3.1\\
\hline
5 &31&32& 3& 3& 0& 2& 1& 2&1.0\\
\hline
\end{array} \)
Given a \(3 \times 3\) cross-number grid
with index numbers 1, 2, 3 respectively, in first row
\(\quad\)1, 4, 5 respectively, in first column
\(\quad\)and bottom right square shaded and:
1A, 1D are squares of different primes,
5A, 2D are squares,
3D is prime,
find 4A.
Solution (5 min 38s)
A polygonal prism has 2004 edges.
Find the number of faces.
Solution (4 min 53s)
The average mass of 9 small pumpkins is calculated.
When a large pumpkin is added, the average mass doubles.
Given the large pumpkin is \(P\%\) of the total mass
of the pumpkins,
find P.
Solution (3 min 19s)
David swims upstream from a jetty.
After David swims \(400\,\)m,
he meets Julie who is floating downstream.
Then David turns around and swims back to the jetty,
\(\quad\)meeting Julie who arrives at the same time.
David's speed is constant relative to the water.
The current has constant speed \(x\,\)m/min.
Find \(x\).
Solution (11 min 52s)
3 circles (shaded within a semicircle) are tangent
to each other externally
\(\quad\)and tangent to the semicircle.
The “shaded area”\({} = 120\).
Find the “unshaded area”
(\(={} \) remaining area of semicircle).
Solution (12 min 53s)
Binary operation \(*\) satisfies
\(a, b, c \in \{1, 2, \ldots, 10\}\) are written on 3 cards,
respectively.
Then
\(AB, CD\) are chords of a circle of radius
\(\quad\)on opposite sides of the centre of that circle.
\(CB\) meets \(DA\) at \(P\) outside the circle.
\(\overset{\frown}{CD} = 4 \cdot \overset\frown{AB} = 8\pi/5\).
Find \(\#\text{degrees}(\angle APB)\).
Solution (11 min 27s)
Side \(AB\) of \(\triangle ABC\) is a diameter of a circle of
radius \(R\)
and \(C\) lies on the circle.
The angle bisector of \(\angle BAC\) meets \(BC\) at \(E\) and
the circle (again) at \(D\).
\(AC\) meets \(\text{circumcircle}(CED)\) again at \(F\).
\(BC = a\).
Find \(CF\) in terms of \(R, a\).
Solution (10 min 20s)
Statistics for 631 participants
\( \begin{array}{|l|c|c|c|c|c|c|c|c|}
\hline
\text{Question} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
\text{No. correct} & 451 & 481 & 523 & 217 & 258& 31&342&134\\
\hline
\end{array} \)
Average score (for all 10 questions): 12.2 (out of 35).
Find the area of rhombus of side length 29
whose diagonals differ by 2.
Solution (5 min 1s)
How many 4-digits numbers are there whose digit product is 60?
Solution (4 min 28s)
A base 7 3-digit number has its digits reversed
when written base 9.
What is the decimal representation of the number?
Solution (6 min 22s)
Primes \(p, q, r\) satisfy:
\begin{align*}
p q + p r &= 80\\
p q + q r &= 425
\end{align*}
Find \(p + q + r\).
Solution (6 min 52s)
How many pairs of 3-digit palindromes are there
such that their sum is a 4-digit palindrome?
Solution (21 min 56s)
\(\triangle ABC\) is equilateral with side length
\(2013\sqrt{3}\).
Find the largest diameter for a circle in one of the regions
between \(\triangle ABC\) and \(\text{incircle}(ABC)\).
Solution (17 min 4s)
Given \(a, b, c, d \in \mathbb{N}\)
such that \(a + b + c + d = 63\).
Find the maximum value of \(a b + b c + c d\).
Solution (11 min 49s)
A circle \(K\) meets equilateral \(\triangle ABC\)
in points \(D, E, F, G, H, I\) (in that order)
such that \(AE = 4\), \(ED = 26\), \(DC = 2\), \(FG =14\)
and a circle with diameter \(HI\) has area \(\pi b\).
Find \(b\).
Solution (23 min 54s)
Statistics for 809 participants
\( \begin{array}{|l|c|c|c|c|c|c|c|c|}
\hline
\text{Question} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
\hline
\text{No. correct} & 439 & 401 & 217 & 652 & 205& 200&284&54\\
\hline
\end{array} \)
Average score (for all 10 questions): 11.4 (out of 35).
Some wording issues:
8. The note in the second last sentence should say:
An empty word is related to any word consisting of two identical adjacent letters.10. The intended meaning of “completely visible” is: “not partially obscured by an adjacent tower”.