Filter Problems

Showing 41 to 60 of 2116 entries

In the dense dark forest ten sources of dead water are erupting from the ground: named from N 1 to N 10. Of the first nine sources, dead water can be taken by everyone, but the source N 10 is in the cave of the dark wizard, from which no one, except for the dark wizard himself, can collect water. The taste and color of dead water is no different from ordinary water, however, if a person drinks from one of the sources, then he will die. Only one thing can save him: if he then drinks poison from a source whose number is greater. For example, if he drinks from the seventh source, then he must necessarily drink poison from the N 8, N 9 or N 10 sources. If he doesn’t drink poison from the seventh source, but does from the ninth, only the poison from the source N 10 will save him. And if he originally drinks the tenth poison, then nothing will help him now. Robin Hood summoned the dark wizard to a duel. The terms of the duel were as follows: each brings with him a mug of liquid and gives it to his opponent. The dark wizard was delighted: “Hurray, I will give him poison No. 10, and Robin Hood can not be saved!” And I’ll drink the poison, which Robin Hood brings to me, then ill drink the N10 poison and that will save me! ” On the appointed day, both opponents met at the agreed place. They honestly exchanged mugs and drank what was in them. However, afterwards erupted the joy and surprise of the inhabitants of the dark forest, when it turned out that the dark wizard had died, and Robin Hood remained alive! Only the Wise Owl was able to guess how Robin Hood had managed to defeat dark wizard. Try and guess as well.

What figure should I put in place of the “?” in the number 888 … 88? 99 … 999 $($ eights and nines are written 50 times each $)$ so that it is divisible by 7?

In a certain kingdom there were 32 knights. Some of them were vassals of others $($ a vassal can have only one suzerain, and the suzerain is always richer than his vassal $)$. A knight with at least four vassals is given the title of Baron. What is the largest number of barons that can exist under these conditions?

$($ In the kingdom the following law is enacted: ” the vassal of my vassal is not my vassal”$)$.

A numerical sequence is defined by the following conditions:

$\\$

Prove that among the terms of this sequence there are an infinite number of complete squares.

In the first pile there are 100 sweets and in the second there are 200. Consider the game with two players where: in one turn a player can take any amount of sweets from one of the piles. The winner is the one who takes the last sweet. Which player would win by using the correct strategy?

The function f $(x)$ on the interval [a, b] is equal to the maximum of several functions of the form $y = C \times 10^{- | x-d |}$ $($where d and C are different, and all C are positive$)$. It is given that f $(a)$ = f $(b)$. Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.

A numerical sequence is defined by the following conditions:

$\\$

$\\$

How many complete squares are found among the first members of this sequence, not exceeding 1,000,000?

A game with three piles of rocks. There are three piles of rocks: in the first pile there are 10 rocks, 15 in the second pile and 20 in the third pile. In this game (with two players), in one turn a player is allowed to divide one of the piles into two smaller piles. The loser is the one who cannot make a move. Which player would be the winner?

A game with 25 coins. In a row there are 25 coins. For a turn it is allowed to take one or two neighbouring coins. The player who has nothing to take loses.

The numbers a and b are such that the first equation of the system

$cos x = ax + b$

$sin x + a = 0$

has exactly two solutions. Prove that the system has at least one solution.

The numbers a and b are such that the first equation of the system

$sin x + a = bx$

$cos x = b$

has exactly two solutions. Prove that the system has at least one solution.

Sage stated the sum of some three natural numbers, and the Patricia named their product.

“If I knew,” said Sage, “that your number is greater than mine, then I would immediately name the three numbers that are needed.”

“My number is smaller than yours,” Patricia answered, “and the numbers you want are …, … and ….”

What numbers did Patricia name?

Initially, a natural number A is written on a board. You are allowed to add to it one of its divisors, distinct from itself and one. With the resulting number you are permitted to perform a similar operation, and so on.

Prove that from the number A = 4 one can, with the help of such operations, come to any given in advance composite number.

A student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number. The result was three times greater.

Find these numbers.

There is a chocolate bar with five longitudinal and eight transverse grooves, along which it can be broken $($ in total into 9 * 6 = 54 squares $)$. Two players take part, in turns. A player in his turn breaks off the chocolate bar a strip of width 1 and eats it. Another player who plays in his turn does the same with the part that is left, etc. The one who breaks a strip of width 2 into two strips of width 1 eats one of them, and the other is eaten by his partner. Prove that the first player can act in such a way that he will get at least 6 more chocolate squares than the second player.

During a ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced the previous dance with, or more intelligent, and one man danced with a girl who was at the same time both more beautiful and more intelligent. Could this be possible? $($ There was an equal number of both boys and girls $)$.

During the ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced with during the previous dance, or more intelligent, but most of the men $($ at least 80% $)$ – with a girl who was at the same time more beautiful and more intelligent. Could this happen? $($ There was an equal number of boys and girls at the ball.$)$

A 1 × 10 strip is divided into unit squares. The numbers 1, 2, …, 10 are written into squares. First, the number 1 is written in one square, then the number 2 is written into one of the neighboring squares, then the number 3 is written into one of the neighboring squares of those already occupied, and so on $($ the choice of the first square is made arbitrarily and the choice of the neighbor at each step $)$. In how many ways can this be done?

On a plane there is a square, and invisible ink is dotted at a point P. A person with special glasses can see the spot. If we draw a straight line, then the person will answer the question of on which side of the line does P lie $($ if P lies on the line, then he says that P lies on the line $)$.

What is the smallest number of such questions you need to ask to find out if the point P is inside the square?

n numbers are given as well as their product, p. The difference between p and each of these numbers is an odd number.

Prove that all n numbers are irrational.