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1. Consider the set {1,11,111, …, ((10^2007) – 1)/9}. At least one of these numbers is divisible by 2007. Is the same true for 2008 (replacing 10^2007 with 10^2008, of course)?
2. You have 7 generals and a safe with many locks. You assign the generals keys in such a way that EVERY set of four generals has enough keys between them to open ALL the locks; however, NO set of three generals is able to open ALL the locks. How many locks do you need, and list how many keys does the first general get, the second, … Is there more than one way that works?
3. Yes, the machine only requires that one of its candies be returned it need not be one purchased with the 75 cents.
4. There is a machine with 20 pieces of candy. Five of those candies are butterscotch. If you put in a 25 cents, one candy is provided at random. If you put in 75 cents, two candies are dropped at random but you may give the machine back one candy in exchange for a 25 cents. And if you put in \$1.50 you receive 5 pieces of candy at random but are guaranteed at least one butterscotch. How much should I expect to spend to get all of the butterscotch?
5. multiples of your six seem to work too
6. This contradicts the solution below. As your a > b > c > d > e > f > g solution does not follow the condition that a + b must equal the sum of the rest
7. A balance and a set of metal weights are given, with no two the same. If any pair of these weights is placed in the left pan of the balance, then it is always possible to counterbalance them with one or several of the remaining weights placed in the right pan. What is the smallest possible number of weights in the set?
8. let a = ln(ln(x))/(ln(x) let b= xa let y = eb find dy/dx
9. Find all continuous positive functions that possess the following property: integral from 0 to 1 of f(x) dx = 1 and integral from 0 to 1 of x*f(x) dx = a and integral from 0 to 1 of x2*f(x) dx = a2
10. I am not disputing either argument, I am curious though on your thoughts that something like the pistol would be better than the shotgun because 2/3's of the time it is equal or better than the shotgun and only 1/3 of the time is it not.
11. I was playing a mobile video game based on the fall out series. In this game, you need to arm your citizens with weapons to protect themselves. When the citizens use the weapons they will hit their target for an amount of damaged indicated within the range given. Currently, I have a surplus of weapons and would like to allocate the best weapons to my people. Place the weapons in order of best to worst: Knife: 0 - 9 damage Shotgun: 6 damage Pistol: 5 - 7 damage Rail gun: 4 - 7 damage Rifle: 3 - 7 damage Grenade: 0, 11 (only 2 possible outcomes)
12. Working with lengths that are whole units from 1 - 100, how many obtuse triangles can be formed?
13. At a circus there was a stunt completed by six bears, 3 brown, and 3 black. Initially, the 3 brown bears were each standing in a square to the left of a center square and the 3 black bears were located right of center. The bears either scooted left/right one square or one of the bears jumped over another until each group of bears were on the other group's set of squares. If the bears needed 15 moves (scoots, jumps) to completely switch sides, how many scoots and how many jumps were used? Bonus points: what is the largest total amount of bears (assuming equal amounts of brown and black bears) that could complete this task in fifteen moves? Start: br br br ___ bl bl bl End: bl bl bl ___ br br br
14. Suppose the post office only sold 5 cent and 7 cent stamps. Now, if the amount of postage on your mail had to be a precise amount equivalent to double of the distance (e.g. sending a letter five miles costs ten cents), what whole number locations are you not able to mail to?
15. Say 5939 is a "right" prime because it remains prime after dropping any number of digits from the right: 5939, 593, 59, and 5 are all prime. How many right primes are there less than 1000?
16. Four dogs are positioned at the corners of a square (d=1m), chase each other in clockwise direction with the same constant speed . As their target is moving, they will follow a curved path, eventually colliding in the center of the square. Why is the total length of the path just 1m?
17. Suppose g(x) = 3 + 1/x and f(g(x)) = 1/(x^2) what does f(x) equal?
18. The only day where you cannot have a partial car built is the 7th day. The other days must have a whole car built to meet your quota but you can have part of a car as long as you don't make two in a given day. You must build something each shift.
19. You are in charge of building cars, you are tasked to build exactly one car a day (no more, no less) and to be clear a partial car is as good as not building a car. Your shift is separated into two parts in which you could either build a whole, half, third, fourth, or fifth of a car in a given shift. By the end of the week you are to have built 7 cars with no partial cars left over. How many ways can this be done assuming a 7-day work week?
20. If the antiderivative of u^-1 = ln |u| + c then why does this not follow: integrate 1/(2x) dx set u =2x, then du = 2 dx, then dx = (1/2)du Then we could integrate (1/2)(1/u)du By the definition above we get 1/2 ln|u| + c which means that the integration of 1/(2x) = 1/2 ln|2x|| + c However, this is a false statement.
21. there is a faster time
22. 5 college students need to cross a river in a small boat. Two of these students are on the rowing team and can cross it in one minute if either one or both are rowing. The other three are not and will take four minutes to cross even if they are with the rowing team members. The boat can only hold up to three people at a time. What is the shortest amount of time it will take to cross the river?
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