Feb 10

Warmup problems:

• In an analog clock, how often small and big hands overlap during 24 hours?
• …how often they are opposite?
• …at 90 degrees?

1/13=0.0769(53841)...  periodic decimal fraction.

3/17=0.1765... - takes a while, but still decimal expansion is periodic.

We (sort of) proved: a number is rational if and only if it has periodic decimal expansion.

Areas of parallelograms: determinant: for a pair of vectors (a,b) and (c,d), the area of the parallelogram spanned by them is ad-bc (this expression is called determinant).

Sometimes long vectors generate parallelograms of small area, for example (13,21) and (21,34)... (Hmm.... Fibonacci numbers here for some reason...). Sometimes the area of the paralellogram is just 0! (For example, for (2,2), (5,5))

Game: each  player starts with $6, contribute any amount to the general fund. Then the general fund is distributed equally among with exception of those who contributed least.

 Homework

• 0.12345678910111213141516... (we just write down all the natural numbers in their decimal form). Is this number rational?
• 22/23=0.95652... Can you find period here? 
• Find a lot of pairs of vectors with zero area.
  
  

February 4

How to turn periodic decimal fractions into usual fractions...
For example: 0.303030303... Call this number x. Then 100x=30.3030303....=30+x. So 99x=30, and x=30/99=10/33.

How to find areas of parallelograms...

And the game was to disentangle yourself.

Homework:

1. Write 0.270270270270... as a fraction.

2. Find the area of parallelogram in the picture:

3. Who is linked with whom:

Dec. 2

Recall that {x} is the fractional part of a number, that is how far it is from the larger integer less or equal to x.

Exercises:

• Try to compare in your head 192/133 and 79/53.
• If {x}=1/2, and {y}=1/2, what are possible options for {xy}?
• What are possible options for {x}+{-x}?

november 4

Homework:

• Is it possible to mark some cells in a 6x8 array with x's so that each row has exactly 3 x's and each column exactly 2 x's?
• Find all x such that |x|+|x-2|≤ 4.
• The side disks of a spool are twice the radius of the inner cylinder. If one pulls the line, which way the spool will roll? (There is no sliding.)

October 28

Homework

1. Buses stopping at Pete's home can take him either to math circle or to pool. Bus in each of these directions comes every ten minutes.
   Pete leaves home at random time (no one has a watch there!) and takes the first bus that appears. Yet, he goes to pool 4 times more frequently that to math circle. How is this possible?
2. Plot y=|x|-|x-2|.
3. Rope running along the equator (its length is 2πR, where R is the radius of the Earth - approximately 25,000 miles) is extended by 2 feet and lifted uniformly above the surface. Will a cat be able to squeeze beneath the rope?

October 7

Recall that |x| is the distance from x to 0, that is x if x is a positive number, and -x if it is negative.

Homework

• Rope was folded in 2 three times and cut through (resulting in 9 pieces). One of the pieces is 2 in, another 3in. What was the length of the rope? 
• Mark on the real line all the points a where 
1. |a-5|>3;
2. a+|a|=0;
3. |a-2|+|a-2|=3.

September 30

Homework:

• Mark on the number line all numbers x for which x/2+3 is an integer.
• Find the point that splits the interval [-6,8] in relation 2:5.
•  How many numbers from 000 to 999 have no even digits?
• Pete earns less than twice what John earns, and John earns less than half of what Dan earns. Who earns more, Dan or Pete?