Yeet1
Yeet2
Yeet3
Yeet4
100
Points 
and 
are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?
Drawing segments 
and 
, the number of triangles outside square 
is the same as the number of triangles inside the square. Thus areas must be equal so the area of is half the area of the larger square which is 
.
100
Suppose m and n are positive odd integers. Which of the following must also be an odd integer?
Assume WLOG that 
and 
are both 
. Plugging into each of the choices, we get 
and 
. The only odd integer is 
.
100
A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove from the drawer to be certain there will be 5 socks of the same color?
The Martian can pull out 
socks, 
of each color, without having 
of the same kind yet. However, the next one he pulls out must be the fifth of one of the colors so he must remove 
socks.
100
What is the largest power of 
that is a divisor of 
?
is 
200
Annie and Bonnie are running laps around a 
-meter oval track. They started together, but Annie has pulled ahead, because she runs 
faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?
Call 
the distance Annie runs. If Annie is faster than Bonnie, then Bonnie will run a distance of 
. For Annie to meet Bonnie, she must run an extra meters, the length of the track. So 
, which is 
laps.
200
The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, and are all equal. What is the value of ?
Notice that the mean of this set of numbers, in terms of , is:
Because we know that the mode must be 
(it can't be any of the numbers already listed, as shown above, and no matter what is, either or a new number, it will not affect being the mode), and we know that the mode must equal the mean, we can set the expression for the mean and equal:
200
Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
The probability that both show a green bean is 
. The probability that both show a red bean is 
. Therefore the probability is 
200
A 
rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?
A semicircle has symmetry, so the center is exactly at the midpoint of the 2 side on the rectangle, making the radius, by the Pythagorean Theorem, 
. The area is 
.
300
The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers?
Since the remainder is the same for all numbers, then we will only need to find the lowest common multiple of the three given numbers, and add the given remainder. The 
is 
. Since 
, and that is in the range of 
300
A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of 
-graders to 
-graders is 
, and the the ratio of -graders to 
-graders is 
. What is the smallest number of students that could be participating in the project?
We multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:
Therefore, the ratio of 8th graders to 7th graders to 6th graders is 
. Since the ratio is in lowest terms, the smallest number of students participating in the project is 
.
300
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?
For notation purposes, let Alice be A, Bob be B, Carla be C, Derek be D, and Eric be E. We can split this problem up into two cases:
A sits on an edge seat.
Then, since B and C can't sit next to A, that must mean either D or E sits next to A. After we pick either D or E, then either B or C must sit next to D/E. Then, we can arrange the two remaining people in two ways. Since there are two different edge seats that A can sit in, there are a total of 
.
A does not sit in an edge seat.
In this case, then only two people that can sit next to A are D and E, and there are two ways to permute them, and this also handles the restriction that D can't sit next to E. Then, there are two ways to arrange B and C, the remaining people. However, there are three initial seats that A can sit in, so there are 
seatings in this case.
Adding up all the cases, we have 
.
300
At a gathering of 
people, there are 
people who all know each other and 
people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur within the group?
Each one of the ten people has to shake hands with all the other people they don’t know. So 
. From there, we calculate how many handshakes occurred between the people who don’t know each other. This is simply counting how many ways to choose two people to shake hands, or 
. Thus the answer is 
.
400
In the right triangle 
, 
, 
, and angle 
is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
We can reflect triangle over line 
This forms the triangle 
and a circle out of the semicircle. Let us call the center of the circle 
We can see that Circle 
is the incircle of 
We can use the formula for finding the radius of the incircle to solve this problem: Area of a triangle = Semi-perimeter 
inradius . The area of is 
The semiperimeter is 
Simplifying 
Our answer is therefore 
400
A semicircle is inscribed in an isosceles triangle with base 
and height 
so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?
First, we draw a line perpendicular to the base of the triangle and cut the triangle in half. The base of the resulting right triangle would be 8, and the height would be 15. Using the Pythagorean theorem, we can find the length of the hypotenuse, which would be 17. Using the two legs of the right angle, we can find the area of the right triangle, . 
times results in the radius, which is the height of the right triangle when using the hypotenuse as the base. The answer is 
.
400
After Euclid High School's last basketball game, it was determined that 
of the team's points were scored by Alexa and 
were scored by Brittany. Chelsea scored points. None of the other 
team members scored more than points. What was the total number of points scored by the other team members?
Since 
and 
are integers, we have 
. We see that the number of points scored by the other team members is less than or equal to 
and greater than or equal to 
. We let the total number of points be 
and the total number of points scored by the other team members, which means that 
, which means 
. The only value of that satisfies all conditions listed is 
, so 
.
400
The area of rectangle is 72. If point 
and the midpoints of 
and 
are joined to form a triangle, the area of that triangle is
B