Tottenham10

The problem tells us that every member of the math club is a member of the science club. Also, we are given that there are 6 members in the science club. Since the 3 members of the math club are part of the 6 members of the science club, the answer is 6 members.

Feel free to tell me if I did anything wrong! :D

This solution assumes diagonals don't count as being next to each other.

There are three possible circles to choose from in the first row, there are two possible circles to choose from in the second row (because we chose one from the first row), and there is one possible circle to choose from in the third row. Therefore, there are 3*2=6 different ways.

Feel free to tell me if I did anything wrong! :D

Let's use the simple d=rt formula. Because Will has a 45-minute head start, he would have rowed 45*50=2250 meters by the time Gracie started. Now, because the two are heading towards each other, the distance between them decreases by 80 meters per minute. 550/80 is equal to 6.875, meaning they will meet 6.875 minutes after 2:45. 0.875 is equal to 7/8, so 0.875 minutes is equal to 52.5 seconds. Therefore, they will meet at 2:51:52.50.

Feel free to tell me if I did anything wrong! :D

Using the arithmetic series formula, we find the sum is \(\frac{(140+1)(70)}{2}\), which is 141*35. Now that we have a factorization, we can easily find the greatest prime divisor. The prime factorization is 3*5*7*47, so the greatest prime divisor is \(47\).

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First, let's assume the psychology textbooks are one bundle. Then, there are obviously 5!=120 ways to organize the four textbooks and the bundle. Next, there are 3!=6 ways to arrange the different psychology books inside the bundle. Therefore, there are 120*6= \(720\) ways to organize the textbooks.

Feel free to tell me if I made a mistake! :D

Lets assume d as the distance from the origin to the beach. Then, by the d=rt formula, it would have taken d/40 hours to get to the beach. If the speed was increased to 55mph, it would have taken d/55 hours. We are given this value is 25 minutes (or 5/12 hours) smaller than d/40. We can now set up the equation \(\frac{d}{55}+\frac{5}{12}=\frac{d}{40}\) and multiply the gcd of 55, 12, and 40 to both sides. The gcd happens to be 1320, so the equation turns into \(24d+110=33d\). Therefore, d=110/9 (notice how I got too lazy to use latex).

I may have made a mistake here, so feel free to tell me if I did. :D

Notice how the two perpendicular bisectors of a triangle meet at a point. This means point O is the circumcenter of the triangle (I do comp math not proofs plz forgive). By the inscribed angle theorem, BCA is 45 degrees (because point O is the center and BOA is 90 degrees). Because angles OMC and ONC are both equal to 90 degrees, quadrilateral OMCN is cyclic. Therefore, angles NOM and NCM must add up to 180 degrees, meaning angle MON is equal to 135 degrees.

Feel free to tell my if I made a mistake! :D

A dilation of -1 means the points X, Y, and Z are just reflected over point G to create points X', Y', and Z'. Assume D is where line YG meets XZ. G is the centroid of the triangle, so the ratio of YG to GD is 2:1. Now, you just do some length chasing (I'm sorry I'm dumb to find a way to prove it) to figure out that triangle XYZ can be expressed as 9 congruent equilateral triangles and 6 of those triangles form the regular hexagon known as region A. Therefore, the ratio of A/[XYZ] would be 6/9=2/3

P.S. Tell me if I did anything wrong! :D

P.P.S. CPhill or the other people would be much better at explaining this

P.P.P.S. Plz tell me how to use latex in this website thx