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1. Suppose superman can survive the vacuum of space (the comics are inconsistent with this ability). Let's say he is in the front of a spaceship that just crossed, what I believe in English is called, the event horizon for a black hole. That point where matter gets sucked into hole. Is it possible for superman to move to the rear of the ship and escape the black hole, assuming that the back of the ship hasn't crossed the event horizon?
2. Find a continuous function where the following identity is true: f(2x) = 3f(x)
3. How many points would you need to have to uniquely determine an ellipse given that you know a foci is located at (0,0).
4. I intended your interpretation but came up with a much different answer than the ones reported forgive my lack of parenthesis what I meant for the problem is 1/(2x)
5. His initial bet is \$1250. Every time he wins, he calculates his total and takes 1/4 of it to bet. So initially he has 5000, 5000 x .25 = 1250. If he were to win, he would earn double his bet so he would have \$6250. Recalculating he would bet \$1562.50. If he loses his bet he would maintain that amount in his next bet. He keeps at it until he is unable to make his desired bet.
6. Think more logically
7. Abraham is tasked with reviewing damaged planes coming back from sorties over Germany in the Second World War. He has to review the damage of the planes to see which areas must be protected even more. Abraham finds that the fuselage and fuel system of returned planes are much more likely to be damaged by bullets or flak than the engines. What should he recommend to his superiors?
8. A gambler has \$5,000 and is playing a game of chance with a win probability of .95. Every time he wins, he raises his stake to 1/4, of his bankroll. The gambler doesn't reduce his stake when he loses If he keeps at it, what are his expected winnings?
9. A town's population of size x doubled after 30 years (2x). How long ago was this population 1/2x?
10. You are to color a 5x5 square grid using green, red, blue, and yellow. You must color it in a way where a square cannot share a side or a vertex with another square of the same color. What is the fewest amount of yellow squares needed to color this appropriately? Now instead lets divide each square diagonally from the top left corner to the bottom right corner. What is the fewest amount of yellow colorings needed?
11. ## Probability of selecting two blue discs back to back

If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken at random, it can be seen that the probability of taking two blue discs, P(BB) = (15/21)×(14/20) = 1/2. The next such arrangement, for which there is exactly 50% chance of taking two blue discs at random, is a box containing eighty-five blue discs and thirty-five red discs. By finding the first arrangement to contain over 1012 = 1,000,000,000,000 discs in total, determine the number of blue discs that the box would contain.
12. ? arccos(-1/3) = 109.5 degrees.
13. Your solution does simplify to something really nice. I will see if anyone sees it before I give it away. Nice work.
14. Yes it follows this pattern: 1+5x7x11xN Where N=0,1,2,3,4,5,... This will ensure the remainder is always one.
15. In retrospect, when I thought up the question, I realized that we shouldn't assume that we have a ninety degree angle. In geometric terms, if you've measured so all the sides are the correct length (opposite sides are the same length), then you have a parallelogram, and what I want to know is how far the angles are from right angles, based on the difference in the diagonals.
16. Here is an algorithmic approach to solving this problem that may help with finding the equation. Assume that between L and W is a right angle, use a string to find the length of the hypotenuse. Use that string to arc out the possible endpoints of the other diagonal using the length from the string. Use the same rope to find the length of L or W. Go to the end point of either L or W away from the given ninety degree angle. Arc the line again and where it crosses with the other arc is where you need to put your fourth corner. Done!
17. I don't see how that's possible if we don't know whether the fake is heavier or lighter in advance.
18. Let's say that there are three congruent circles that are tangent to each other. Draw a fourth congruent circle with its center at the center of the gap formed between the three circles. If all four circles are unit circles, what is the area of the overlap?
19. You understood it correctly. You have one of the numbers but there are actually more that meet the requirements. I just want the lowest answers.
20. I do see that, which is why we must keep L and W arbitrary
21. Everyday I make the same number of cookies. On Monday, five people came over for cookies. I gave each of them the same amount and kept one for me. The next day, I baked the same amount but this time eleven people came over. I gave them all an equal share and kept one for me. On the third day, I baked the same amount of cookies as the other days, gave thirty-five people the same amount, and of course kept one for me. What are the three possible lowest amounts of batches I made each day?
22. Not anywhere otherwise there is no way someone would confuse the shape for a rectangle as it would lose its fourth side and by symmetry much of that circle is immaterial.
23. If the bag contains coins that equal 2^m then it would take m+1 try to find the fake coin. The spoiler option seems to have disappeared from the mobile app.
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