Mathcounts State Questions: Algebra

 Problems: (Solutions are provided below; however, refrain yourself from reading the solutions too soon.)

1. 2008 state #19 -- Positive integers x and y have a product of 56 and x < y. Seven times the reciprocal of the smaller integer plus 14 times the reciprocal of the larger integer equals 4. What is the value of x?

2.2008 state  #24 -- The areas of two squares differ by 100 square units, and the perimeters of the two squares differ by 10 units. What is the perimeter, in units, of the smaller square?

3. 2009 state team # 5. If x + 1/y = 1 and y + 1/z = 1, what is the value of the product xyz?

Solutions: 

1. xy = 56 and x < y (given)
7/x + 14/y = 4 ; Consolidate and you have (7y + 14x) / xy = 4 or (7y + 14x)/ 56 = 4
Times 56 -- 7y + 14 x = 224
Divided by 7-- y + 2x = 32
x = 2 and y = 28   The answer is 2.

2. Let x and y be the side length of the two squares and x > y

x² - y² = 100, ( x + y ) ( x - y) = 100 -- (1) difference of squares.

4x - 4y = 10 so x - y = 2.5

Plug in equation (1) and you have ( x + y ) times 2.5 = 100

x + y = 40-- (2)

x - y = 2.5-- (3) 

We are trying to find the perimeter of the smaller square so we should use (2) - (3) to get rid of x. 

(2) - (3) 2y = 37.5, 4y (the perimeter of the smaller square) = 75 The answer is 75.

3. (1) x + 1/y = 1 (given) times y and you have xy + 1 = y (3)

    (2) y + 1/z = 1 (given) times z and  you have yz + 1 = z (4)  and yz = z -1

    (3) times  z -- xyz + z = yz = z-1; xyz = z - 1 - z = -1  The answer is -1

Mathcounts Chapter Prep Daily 5 Problems

E-mail me your answers through comments, My students and I will check if they are right.
Have fun problem solving and good luck on your chapter competition.

#1: When x² - 3x + 2C is divided by x + 1, the remainder is -6, what is C?

#2:  13_b + 23_b = 40_b, what is base b?

#3:  If you increase 5 inches to the side of the original square, the area would increase 245 square inches, what is the original area of the square?







#4:  The first and the sixth number in a sequence are both 10 and starting with the third number, each number is the sum of the previous two numbers. What is the fourth number?

#5: If you spend 1/3 of the money you have on the first day, 2/5 of what you have left on the second day and 1/4 of what you have left on the third day. Now you have 12 dollars left. How much do you originally have?