# Can you explain how to solve several AMC 10 problems from the year 2000?

• k1point618
In summary, the conversation is about several problems from the 2000 AMC 10 exam. The first problem involves finding the number of people in a family based on the amounts of coffee and milk they drink. The second problem involves finding the sum of all possible values of x in a non-constant arithmetic progression. The final problem asks for the sum of all values of z for which a given function equals 7. The answers to the problems are (c), (e), and (b) respectively. The solution to the first problem involves setting up a system of equations and using restrictions on the variables to solve for the number of people. The solution to the third problem involves finding the form of a given function and using it to solve for
k1point618
AMC 10 year 2000

here are several problems that i found while practicing for AMC 10, really wish someone can give a thorough explanation of how they are to be solved. The answers are at the end.

22) One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Anegla drank a qarter of the total amount of milk and a sixth of the totalamount of coffee. How many people are in the family?
a)3 b)4 c)5 d)6 e)7

23) When the mean, median and moode of the list:
10, 2, 5, 2, 4, 2, x
are arranged in increasing order, they form a "non-constant arithmetic progressions". What is the sum of all possible real value of x?
a)3 b)6 c)9 d)17 e)20
*also would you please explain non-constant arithmetic progression? I think i have some confusion on that THANK YOU

24) Lef f be a function ofr which f(x/3) = x^2 + x + 1. Find the sum of all values of z for which f(3z) = 7
a) -1/3 b) -1/9 c)0 d) 5/9 e) 5/3
this problem i tried to solve. since 3z is to be plugged into f(x/3) then it can be written as f(3z/3) = f(z) = 7 = x^2 + x + 1; i solved that and got -1/3 (a), but i don't think it is correct.

The answers to the problems: 22:(c) 23:(e) 24:(b)

I'll indulge you with the first one:
Let x be the total amount of ounces milk, y the total amount of ounces coffee, and z the number of persons in Anfgela's family.

What restrictions MUST lie on x, y and z:
Clearly, all three numbers must be positive!
Furthermore, z must be a whole number!

Now, the total amount of milk+coffee clearly satisfies the equation:

x+y=z*8 (1)

For Angie, we have the following equation that also holds for x and y:

x/4+y/6=8 (2)

Let us multiply all terms in (2) with 12 and all terms in (1) with 2.

This yields the system of equations:

2x+2y=16z (1*)

3x+2y=96 (2*)

Subtracting (1*) from (2*) yields:

3x-2x+2y-2y=96-16z

or:

x=16*(6-z) (**)

Now, deduce your result by (**) and that (1) must hold, with the above-mentioned restrictions on x,y and z.

THANK YOU so much for that one!

24)

If f( x/3 ) = x^2 + x + 1 then f(x) must have the form a x^2 + b x + c.
So let f(x) = a x^2 + b x + c which gives f( x/3 ) = a x^2 / 9 + b x / 3 + c.
Identifying coefficients gives a = 9, b = 3 and c = 1.
Solving f( 3z ) = 7 yields z1 = -1/3 and z2 = 2/9 which sums to -1/9.

## 1. What is the AMC 10?

The AMC 10 is an annual mathematics competition open to students in grades 10 and below. It is one of the first steps in the selection process for the United States International Mathematical Olympiad team.

## 2. How many problems are on the AMC 10?

The AMC 10 consists of 25 multiple-choice questions that must be completed in 75 minutes.

## 3. What types of problems are on the AMC 10?

The problems on the AMC 10 cover a wide range of topics in algebra, geometry, counting, probability, and number theory. They require problem-solving skills and creative thinking.

## 4. Are calculators allowed on the AMC 10?

No, calculators are not allowed on the AMC 10. This is to test students' ability to solve problems without the aid of technology.

## 5. How are AMC 10 problems scored?

Each correct answer on the AMC 10 is worth 6 points, each incorrect answer is worth 0 points, and each unanswered question is worth 1.5 points. The total raw score is then converted to a scaled score out of 150.

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