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?

September 23

Homework:

• Each next line is obtained from the previous one using a simple rule. Find it! Add 2-3 more lines using that rule:
  1
  1 1
  2 1
  1 2 1 1
  1 1 1 2 2 1
  3 1 2 2 1 1
  .....
  
• Out of 1000 numbers, starting with 0 and ending with 999, how many do not have digit 3 in them?
  
• Find midpoints of the intervals:
• [-7, 1000]
• [-1000, -7]

September 16, 2018

• Could chess knight jump over all squares of the $5\times 5$ board, visiting each just once?
  Answer - Yes!
• Could it do so and return to its starting point?
  Answer - No!
• A standard die is rolled to the right, towards you, to the left, away from you, returning to the original site; each time around an edge, without sliding. What will be the number on the top?
  How many times one will need to repeat this operation to return to the original orientation?
  
• How many different routes are there from A to F (moving always eastwards)?