Three members of the Euclid Middle School girls’ softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all -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 , and the smallest two-digit primes are and . There are a few different sums that can be deduced from the following numbers, which are and , 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 . Similarly, Caitlin says that the other two people's uniform numbers are later, so the sum must add up to . This leaves as today's date. From this, Caitlin was referring to the uniform wearers and , telling us that her number is , giving our solution as .
== Video Solution by OmegaLearn ==
https://youtu.be/6xNkyDgIhEE?t=735
~ pi_is_3.14
==Video Solution==
https://youtu.be/yESMPOzgdug ~savannahsolver
Four numbers are written in a row. The average of the first two is the average of the middle two is and the average of the last two is What is the average of the first and last of the numbers?
Solution
Note that the sum of the first two numbers is the sum of the middle two numbers is and the sum of the last two numbers is
It follows that the sum of the four numbers is so the sum of the first and last numbers is Therefore, the average of the first and last numbers is
~MRENTHUSIASM
A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?
Rectangle ABCD has sides CD=3 and DA=5. A circle of radius 1 is centered at A, a circle of radius 2 is centered at B, and a circle of radius 3 is centered at C. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?
The area in the rectangle but outside the circles is the area of the rectangle minus the area of all three of the quarter circles in the rectangle.
The area of the rectangle is 3cdot5 =15. The area of all 3 quarter circles is frac{pi}{4}+frac{pi(2)^2}{4}+frac{pi(3)^2}{4} = frac{14pi}{4} = frac{7pi}{2}. Therefore the area in the rectangle but outside the circles is 15-frac{7pi}{2}. pi is approximately dfrac{22}{7}, and substituting that in will give 15-11=boxed{text{(B) }4.0}
Which of the following numbers is not a perfect square?
Solution
Our answer must have an odd exponent in order for it to not be a square. Because 4 is a perfect square, 4^{2019} is also a perfect square, so our answer is boxed{textbf{(B) }2^{2017}}.