AMC 10 Problems and Solutions

Problem 1 :

Four positive integers a, b, c and d have a product of 8!. And satisfy ab + a + b = 524, bc + b + c = 146 and cd  + c + d = 104, what is a – d?

A)  4

B)  6

C)  8

D)  10

E)  12

Solution :

Problem 2 :

Let A, M and C be nonnegative integers such that A + M + C = 12. What is the maximum value of A ⋅ M ⋅ C + A ⋅ M + M ⋅ C + C ⋅ A?

A)  62

B)  72

C)  92

D)  102

E)  112

Solution :

Problem 3 :

If a, b, c and d are nonzero real numbers such that c and d are the solutions of x² + ax + b = 0 and a and b are the solutions of x² + cx + d = 0, then a + b + c + d equals

A)  0

B)  –2

C)  2

D)  4

E)  (–1 + √5)/2 

Solution :

Problem 4 :

For some real numbers a and b, the equation

8x³ + 4ax² + 2bx + a = 0

has three distinct positive roots. If the sum of the base 2 logarithms of the roots is 5, what is the value of a?

A)  –256

B)  –64

C)  –8

D)  64

E)  256 

Solution :

Problem 5 :

The polynomial P(x) = x³ + ax² + bx + c has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the y-intercept of the graph of y = P(x) is 2, what is b?

A)  –11

B)  –10

C)  –9

D)  1

E)  5 

Solution :

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