8 : Ralph : A reminder of the fact that a Klein bottle can be decomposed
into a sum of two Moebius bands. This was followed by a show and tell
of a glass Klein bottle.
Second half of the session was why we do proofs. The students
computed the number of lines and regions in a circle with
n points on the perimeter.
9 : Zoltan : 1984 Int AMC paper with some solutions, advice and tricks.
10 : Peter : How many ways can you tie a tie?
11 & 12 : Srinivasa : Induction -- size of powerset of a set.
Boolean Algebra -- De Morgan's Laws and relevance to computer programming.
Towers of Hanoi -- variants.
8 : Srinivasa : Solve a^b=b^a for a,b in Z.
Linear search versus binary search (complexity best vs worst case)
Show 2^2+2b^2+c^2 >= 2(ab+bc) for all reals a,b,c.
9 : Elizabeth : Discussed polite numbers, i.e. integers that can be written as a
sum of two or more consecutive integers.
10 : Zoltan : Horrible game -- flipping breakdown.
2013 AIC -- Part A Intermediate and Senior.
11&12 : Ralph : Shared the results of the search for an infinite number of
golden tetrahedra. We discussed what could happen if
k equations intersect in n-dimensional space where 0<=k<=n+1.
This was followed by a glimpse of rational points on curves and surfaces.
8 : Tamiru : Basic facts about circles.
9 : Ralph : Iterated surds and the rational root theorem.
10 : Peter : What is r^s (for r,s in the real field)?
What is R^2 as a 2-dim real vector space, Cartesian plane?
Field properties of C. (H is a skew-field).
Discussed definition e^x = sum_0^inf x^k/k!.
Claim e^x e^y = e^(x+y) (Pascal's triangle / binomial theorem)
11&12 : Zoltan : AIC 2012 Q1.
8 : Elizabeth : Show that the corner of squares in a triangle lies on a straight line. (Polya)
Parabolic Problem Q9. If p and q are twin primes, 3<p<q show that pq+1 is
divisible by 36.
9 : Ralph : Show that if squares have bases on the base of a given triangle
and top left corners on the left edge of a triangle then the top right
corners of all the squares lie on a straight line. (Polya)
10&11&12 : Zoltan & Melinda : Horrible game y = 1+greatest prime factor(x+21) (3.5 stars?)
AIC 2012 Q5 (last 15 min)
8 : Srinivasa : Number of divisors of an integer n=p_1^r_1 * ... * p_q^r_q = ?
GCD and LCM in terms of prime factorisations.
ab = GCD(a,b)LCM(a,b)
9 : Ralph : Clark's square-circle problem. A square pinned to a circle at two vertices of one edge
and is tangent to the opposite side. Which has greater perimeter?
Compute primes as a sum of two squares. Look for patterns.
One student found the modulo 4 pattern in the odd primes.
10 : Peter : Functions, domain --> range.
e^x = sum_{k=0}^infinity x^k/k! ==> e^0 = 1, e^1 = e.
Claim e^x . e^y = e^(x+y)
cos(x) = even part (e^(ix)) = (e^ix + e^-ix) / 2
sin(x) = (e^ix - e^-ix) / 2
( f_even)^2 - (f_odd)^2 = f(x)f(-x)
cos^2 - i^2 sin^2 = 1.
11&12 : Elizabeth & Zoltan : Parabolic Problem Q3.
8 : Srinivasa : Induction. Sum of first n integers and first n squares.
9 : Elizabeth : ???
10 : Zoltan & Paut : Horrible game y = 2x^2-3x+2. Chocolate game, topological chocolate.
1984 AMC Int.
11&12 : Peter : Modular arithmetic Z/pqZ.
Groups, rings, fields, Q,R,C.
8 : Srinivasa : Towers of Hanoi. How many moves?
9 : Elizabeth : ???
10 : Zoltan & Paul : Horrible game y = x+x^2+Floor(sqrt(x)). Chocolate game, topological chocolate.
1984 AMC Int.
11&12 : Peter : Rings Z, Z/NZ, (Z/NZ)*, and calculating the size of (Z/NZ)*
Factoring methods Fermat, Kraitchick, NFS, Shor
8 : Srinivasa : ???
9 : Elizabeth & Paul : If p and q are twin primes, 3<p<q, show pq+1 is divisible by 36.
Q1 Parabolic Problems.
10 : Zoltan : ???
11&12 : Peter : Theorems about (Z/NZ)*,
RSA Encryption