problem of the week

05/19

What is the number of integer solutions to 1 < x^2 < 25 ?

05/01/08

A bag contains 10 red balls, 10 green balls and 6 white balls. Two balls are drawn at random without replacement. What is the probability that they are of different color? 

Solution: P(different color) = 1 - P(same color) = 1 - [P(RR) + P(GG) + P(WW)]=

1- (10/26*9/25 + 10/26*9/25 + 6/26*5/25) = 1 -210/650 = 440/650=44/65 

4/17/08

Twenty people are to greet each other by shaking hands. How many hand shakes are there?

Solution: 190 handshakes 

4/01/08 (Problem of the week will be changed after the spring break. So if you solve this problem now, bring it to me when we get back to school, 4/14)

For what values of k, the graph of y = x^2 -3x + k will never meet the x-axis?

Solution: k>9/4

3/21/08

Find the coefficient of x^3 in the expansion of (1 - x)^5*(1 + x)^7

Note: * means multiplication sign

Solution: - 10

3/5/08

Find the smallest number divisible by every number from 1 to 10.

Solution: 2520 

2/20/08

What are the last 2 digits of 7^1000 ? 

Solution: Doing 7^0 = 1, 7^1=7, 7^2=49, 7^3=343, 7^4=2401, 7^5=16807, you should notice the pattern. (Similar to powers of the imaginary number i) The last 2 digits will depend on the remainder when the exponent is divided by 4.

There is no remainder when 1000 is divided by 4, so that will be the first case in the pattern. Therefore the last 2 digits are 01.

1/18/08

If  a^b = a + b - 5,  and  a~b = 2a + 2b, evaluate ( - 3^10) ~7.

-3^10= -3+10-5 = 2

Now 2 ~ 7 = twice2 + twice7= 18

1/6/08
 Log Value

What is the value of log₃16 • log₂27?
  Solution:

Use the change of base formula and the power propery for logs, to change the expression into:  (4log2/log3)*(3log3/log2) = 4*3 = 12

12/10/07

Real Value Sum

Find the sum of the real values of x satisfying the equality |x + 2| = 2|x - 2|.

Solution
6 2/3. We have
x + 2 = 2(x - 2)
x + 2 = 2x - 4
6 = x

or

x + 2 = -2(x - 2)
x + 2 = -2x + 4
x = 2/3.

The sum is 6 2/3.

 12/03/07

Opposite Sides
Two opposite sides of a square are increased by 10 cm. The other two opposite sides of the square are decreased by 10 cm. Does either the area or the perimeter stay the same?  

Solution:  The perimeter stays the same, and the area decreases by 100 square cm.

Monday, 11/12/07

Triangle XYZ

Suppose that the lengths of the sides of ΔXYZ are the smallest possible sequence of positive integers with an even sum. Find the area of ΔXYZ.

Solution:. Examine the smallest sequences of positive numbers: 1, 2, 3. This sequence would not form a triangle, since the sum of the two smallest sides is not greater than the length of the third side. The sequence 2, 3, 4 does not yield an even sum. The sequence 3, 4, 5 does yield an even sum and satisfies the triangle inequality theorem. A triangle with side lengths of 3, 4, and 5 is a right triangle with a hypotenuse of 5, so the area = (1/2) • (3) • (4) = 6.

Monday, 11/05/07

Simplify the expression

Solution: The answer follows from