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AMC8 2018 #22 © MAA

Point (E) is on side (AB) of regular pentagon (ABCDE). If (BE = frac{1}{3} cdot AB), what fraction of the area of (ABCDE) is triangle (BEF), where (F) is the center of the pentagon?

AMC8 2017 #4 © MAA

When 0.000315 is multiplied by 7,928,564 the product is closest to which of the following?

AMC8 2014 #23 © MAA

Three members of the Euclid Middle School girls’ softball team had the following conversation.

Ashley: I just realized that our uniform numbers are all $2$ -digit primes.

Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month.

Caitlin: That’s funny. The sum of your two uniform numbers is the date of my birthday later this month.

Ashley: And the sum of your two uniform numbers is today’s date.

What number does Caitlin wear?

Solution

The maximum amount of days any given month can have is

31

, and the smallest two-digit primes are

11, 13,

and

17

. There are a few different sums that can be deduced from the following numbers, which are

24, 30,

and

28

, all of which represent the three days. Therefore, since Bethany says that the other two people's uniform numbers are earlier, so that means Caitlin and Ashley's numbers must add up to

24

. Similarly, Caitlin says that the other two people's uniform numbers are later, so the sum must add up to

30

. This leaves

28

as today's date. From this, Caitlin was referring to the uniform wearers

13

and

17

, telling us that her number is

11

, giving our solution as

boxed{(A) 11}

. == Video Solution by OmegaLearn == https://youtu.be/6xNkyDgIhEE?t=735 ~ pi_is_3.14 ==Video Solution== https://youtu.be/yESMPOzgdug ~savannahsolver

AMC8 2014 #1 © MAA

Harry and Terry are each told to calculate $8-(2+5)$ . Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$ . If Harry’s answer is $H$ and Terry’s answer is $T$ , what is $H-T$ ?

Solution

We have

H=8-7=1

and

T=8-2+5=11

. Clearly

1-11=-10

, so our answer is

boxed{textbf{(A)}-10}

Four numbers are written in a row. The average of the first two is $21,$ the average of the middle two is $26,$ and the average of the last two is $30.$ What is the average of the first and last of the numbers?

Solution

Note that the sum of the first two numbers is

21cdot2=42,

the sum of the middle two numbers is

26cdot2=52,

and the sum of the last two numbers is

30cdot2=60.

It follows that the sum of the four numbers is

42+60=102,

so the sum of the first and last numbers is

102-52=50.

Therefore, the average of the first and last numbers is

50div2=boxed{textbf{(B) } 25}.

~MRENTHUSIASM

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