Pure maths, physics, logic (braingames.ru): non-trade-related brain games - page 45
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Yes, corrected - will hit with 2/3 and split in half. A third ship will come towards it.
Z.Y. Not 2/3, but 5/6 :)
The criterion for scoring is the amount of fuel consumed all together. I got 5 tanks - only then did it count. Who has less or the same amount?
(3)
Once upon a time there was a farmer who had two joys in life: raising sheep and having children. He had 24 children and many more sheep. When he had no money left, the farmer went to a fair and sold all his sheep. When he returned home, he wanted to divide the proceeds equally among his children, but no such luck. No sharing, that's all. He goes to his neighbour, Megamozg, and complains about his unfair life. Megamozg asks: "How much money did you make?" The farmer can't remember," he says, "I only remember that the number of sheep was a large prime number, and I sold each sheep for as many dollars as there were originally in the flock. The megabrain replied: "No big deal, give me 1 dollar for the advice, and divide the rest of the money for your own health". The farmer is delighted, goes home and thinks, "Has Megamogg made a mistake? How could he have known that the rest of the money could be divided equally?"
(3)
The flask contains22010 bacteria, and a virus enters the flask. In the 1st second the virus destroys one bacteria and immediately after that both the virus and the bacteria split in half. In the 2nd second, 2 viruses destroy 2 bacteria, and immediately after that both viruses and bacteria divide in half, and so on. Will there come a point in time when no bacteria remain?
(3)
Among the 101 coins there are exactly 50 fake ones. The weights of the real coins are equal, the weight of each counterfeit differs from the real one by 1 gram (the counterfeit coins may have different weights). How can you determine in one weighing on a two-cup scale with an arrow and a scale (without weights) whether a coin is false or not?
(3) - was published on "quaternary", but no solution has been received from anyone:
Megamozg bought 53 packs of apple moth repellent. The remedy starts working three days after it is applied to the tree. However, he discovers that one of the packs is defective and also repels the fruit pollinator, without which no apples will grow. Megamogg can't risk his orchard, but he has six wild apple trees on which he can test the repellents. But after six days, the apple moth starts to go on a feeding frenzy, and there won't be much left of his orchard if he doesn't treat it. How can Megamosk find the defective product and save his crop?
Note: Megamosk has only one day to apply products to the test trees - Day 0. On Day 3, he should already know everything and completely treat the garden.
(5)
A ray of light, directed from the top of a cube, reflected 2008 times from its interior mirror walls (by the law "angle of incidence equals angle of reflection") and not hitting any of its edges, ends its path at another vertex of the cube. What is the shortest distance a ray of light can travel if the cube's edge length is equal to 1?
(3)
Once upon a time there was a farmer who had two joys in life: raising sheep and having children. He had 24 children and many more sheep. When he had no money left, the farmer went to a fair and sold all his sheep. When he returned home, he wanted to divide the proceeds equally among his children, but no such luck. No sharing, that's all. He goes to his neighbour, Megamozg, and complains about his unfair life. Megamozg asks: "How much money did you make?" The farmer can't remember," he says, "I only remember that the number of sheep was a large prime number, and I sold each sheep for as many dollars as there were originally in the flock. The megabrain replied: "No big deal, give me 1 dollar for the advice, and divide the rest of the money for your own health". The farmer is delighted, goes home and thinks, "Has Megamogg made a mistake? How could he have known that the rest of the money could be divided equally?"
All numbers of the form (d^2-1, where d is prime) are divisible by 24. Exception = 2^2-1 and 3^2 -1
Proof: Let us decompose the difference d^2-1 into (d-1) and (d+1)
Obviously, if c is prime, then it is not divisible by 3(the remainder of the division by 3 is either 1 or 2), so one of the numbers (d-1) or (d+1) is divisible by 3. The exception is the square of the triplet itself minus one.
Next, both numbers (d-1) and (d+1) are obviously even, and since one of them is greater than the other by 2, one of them is divisible by four, which means that their product is divisible by 8. The exception is the square of two minus one.
By convention, d >> 24 >> 3
We get (d-1) * (d+1) = 2*4*3 * x = 24 * x
Ъ
(3)
There are22010 bacteria in the flask and a virus enters the flask. In the 1st second the virus destroys one bacteria and immediately afterwards both the virus and the bacteria split in half. In the 2nd second, 2 viruses destroy 2 bacteria, and immediately after that both viruses and bacteria divide in half, and so on. Will there come a point in time where no bacteria remain?
It will occur after exactly 2010 cycles (seconds).
// Probably much earlier - the bacteria will run out of food. But there are other possibilities - for example, they will mutate and become resistant to this virus, etc.
The answer is obvious if you put the bacteria in a table measuring22010 x22010.
At the first moment, the bacteria occupy one column, with one row affected by the virus. Every second that follows, the number of columns occupied by the bacteria doubles, but the virus multiplies synchronously and eats away one row per second.
After22010 seconds all the rows will have been eaten.
Ъ
Mathemat:
MetaDriver: Not that. We've already established that. Now, we are looking for a situation where passing is only possible in one direction.
???
Generally only periodic with several detours around the ring. Period = 100 km.
The question is quite different. In your solution, when you diversions to the right and left of the ring, will the minimum required tank volumes match?
--
Since the minimum tank volume is equal to the sum of the minimum and maximum distances between the tanks, it obviously doesn't matter which way you put the ruler.
(5)
A ray of light, directed from the vertex of a cube, reflected 2008 times from its inner mirror walls (according to the law "angle of incidence equals angle of reflection") and not hitting any edge, ends its path at another vertex of the cube. What is the shortest distance a ray of light could travel if the cube edge length is equal to 1?
S = sqrt(1 + 2 / 2009^2) * 2009
// What is the question, what is the answer - I won't explain, the question doesn't ask for proofs.
Reproduction is only invented in the task to intimidate and confuse. It is superfluous. Then the problem becomes simply trivial.
And the cycles should not be 2010, but 2^2010.
S = sqrt(1 + 2 / 2009^2) * 2009
// What is the question, is the answer - I will not explain, the question does not ask for proof.
OK, the answer is wrong. Despite its "threescore", the Zadacha requires finding an optimum.
My answer is approximately 1161.05943 (I do not give the exact value on purpose). Yours is much larger.
P.S. According to braingames.ru rules, justification is not required only if it is explicitly stated.
(4)
Five mega-brains and a parrot are stranded on a desert island. During the day they picked a huge pile of pineapples, put them all together and went to bed. At night one of the megabrains, fearing the 'unfair' division of the pineapples, woke up to claim his share. He divided the pineapples into five equal piles and gave one leftover pineapple to the parrot. Then he took his share and hid it, and put the rest in one pile. An hour later, for the same reason, a second megamogeon woke up and did the same thing. All five megamogs did the same (i.e. each took a fifth part and one gave it to the parrot). In the morning, the megamrains woke up and divided the remaining pineapples equally among them and gave the remaining pineapple to the parrot. What minimum number of pineapples could the megamrains collect?
(4)
Two pipes lead to the pool: a drain at the bottom and a filler at the top. An empty pool fills with water in 1 hour. A full pool takes 20 minutes to drain completely. How long will it take for a full pool to come down completely if the two pipes are open at the same time.
(The problem is not a mathematical problem, but a physical one!)
(4)
Every day a car picks up Megamind at school. It arrives exactly at the end of school. One day the school ended an hour early and Megamozg decided to go to meet the car. He met it and got home 10 minutes earlier than usual. How many times the speed of Megamozg is less than the speed of the car?
(Better with a graphical justification - otherwise you'll get confused like I did.)
(4)
Megamozg worked as a telephonist, and one day he got a call from an office dispatcher asking him to find a buried cable. The cable was buried at a shallow depth in a straight line running exactly 5 km from the point where Megamogg was located. Unfortunately, communication broke down and the dispatcher did not have time to clarify in which direction the cable ran. Megamogg has a metal detector that rings exactly above the cable. Can he plan his way so that he is guaranteed to find the cable, while walking no more than 32 km?
(Drawing is compulsory.)
There are two pipes leading into the pool: a drain at the bottom and a filler at the top. An empty pool fills with water in 1 hour. A full pool takes 20 minutes to drain completely. How long does it take for a full pool to come completely down if the two pipes are open at the same time?