Free Sample AMC 10

Expressing 2001 as a product of its primes you get . Since there are only three numbers being multiplied and they are prime the only combination for three positive integers with a product of 2001 is .

Therefore, the largest possible value of the sum I + M + O is .

Since , then

Applying the product rule for exponents:

Since M and N are the midpoints of sides PA and PB, segment MN is the midsegment of ΔPAB. By definition, the length of the midsegment MN is always half the length of AB, regardless of where P is located as long as it stays parallel to AB, the length of segment MN will not change.
As point P moves along the line parallel to side AB in triangle ΔPAB, the length of sides PA and PB will change and also the perpendicular height of ΔPAB will change. So the perimeter and area of ΔPAB changes as P moves.
The lengths of AB and MN will remain unchanged but the perpendicular height of the trapezoid will change. Therefore the area of trapezoid ABMN changes.
Therefore, 3 out of the 4 quantities change.

At the end of the second day, 20% of that day's jellybeans were eaten so the amount that remained would be 80% of the start of the second day. Let x represent the amount at the start of the second day.

80% of  x = 32

Therefore, , x = 40

The amount at the beginning of day 2 would be the same amount that ended day 1. This amount would represent 80% of the total jellybeans at the beginning of the day since Jenny ate 20%.

Let n represent the amount at the start of the first day.

80% of  n = 40

Therefore, , n = 50

Therefore 50 jellybeans were in the jar originally.

Let ‘x’ be the product of the hourly charge for connect time and the hours used in December and ‘f’ be the fixed monthly fee.

Writing two equations in two unknowns:

December: x + f = $12.48          …(equation 1)

For January, since twice as much connect time was used we can represent this by 2x.

January: 2x + f = $17.54             ...(equation 2)

Solving for x and f, using method of elimination by subtracting equation 1 from 2 : 

x + f = 12.48

2x + f = 17.54 

Substituting 5.06 for x, in equation 1 to find f.

,

So, the fixed monthly fee is $7.42 .

Based on the elements listed, the digits in the units position are 1,  2, 3, 5, and 8. Let's list the next 20 terms in this sequence to see which of the digits 0, 4, 6, 7, and 9 would come last.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, ... 

In the units position for the next 20 elements, the digits appear for the first time in the order 4, 9, 7 then 0. Thus 6 is the last digit to appear in the units position.

If one can of soda holds 12 ounces, then the number of cans needed to provide a gallon would be

cans would be in between 10 cans and 11 cans. 10 cans would provide less than the gallon so the minimum cans of soda needed to provide a gallon would be 11 cans.

To determine which of the given values cannot be the total value of four coins picked from the piggy bank we need to check for at least one combination of these coins for each option using only 4 coins.

15 :  3 nickels or 1 dime and 1 nickel . These combinations uses 3 coins. Switching a nickel to pennies gives 5 pennies. Which is 5 coins.

25: 3 nickels and 1 dime

35: 1 nickel and 3 dimes

45: 2 nickels, 1 dime and 1 quarter

55: 3 dimes and 1 quarter

Therefore the value from exactly 4 coins cannot sum to 15 cents.

Starting with the fraction in the denominator.

If Eric averages 2 miles per hour in the -mile swim, then it will take hour to swim mile.

If Eric averages 6 miles per hour in the 3-mile run, then it will take hour to run 3 miles.

For the first two parts, he would take hour.

Therefore Eric will need to take hours to complete the 15-mile bicycle ride.

Average Speed = distance time

Therefore Eric's average speed in the 15-mile bicycle ride needs to be to complete the race in 2 hours.