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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?

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.

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 $)$.

Peter bought an automatic machine at the store, which for 5 pence multiplies any number entered into it by 3, and for 2 pence adds 4 to any number. Peter wants, starting with a unit that can be entered free of charge to get the number 1981 on the machine number whilst spending the smallest amount of money. How much will the calculations cost him? What happens if he wants to get the number 1982?

In a set there are 100 weights, each two of which differ in mass by no more than 20 g. Prove that these weights can be put on two cups of weighing scales, 50 pieces on each one, so that one cup of weights is lighter than the other by no more than 20 g.

The White Rook pursues a black horse on a board of $3 \times 1969$ cells $($ they walk in turn according to the usual rules $)$. How should the rook play in order to take the horse? White makes the first move.

a$)$ Could an additional 6 digits be added to any 6-digit number starting with a 5, so that the 12-digit number obtained is a complete square?

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b$)$ The same question but for a number starting with a 1.

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c$)$ Find for each n the smallest k = k $(n)$ such that to each n-digit number you can assign k more digits so that the resulting $(n + k)$ – digit number is a complete square.

The case of Brown, Jones and Smith is being considered. One of them committed a crime. During the investigation, each of them made two statements. Brown: “I did not do it. Jones did not do it. ” Smith: “I did not do it. Brown did it. “Jones:” Brown did not do it. This was done by Smith. “Then it turned out that one of them had told the truth in both statements, another had lied both times, and the third had told the truth once, and he had lied once. Who committed the crime?

Multiplication of numbers. Restore the following example of the multiplication of natural numbers if it is known that the sum of the digits of both factors is the same.

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There are 6 locked suitcases and 6 keys to them. At the same time, it is not known to which suitcase each key fits. What is the smallest number of attempts you need to make in order to open all the suitcases for sure? And how many attempts will it take there are not 6 but 10 keys and suitcases?

Decipher the following rebus $($ see the figure $)$. Despite the fact that only two figures are known here, and all others are replaced by asterisks, the example can be restored.

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Three people A, B, C counted a bunch of balls of four colors $($ see table $)$.

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Each of them correctly distinguished some two colors, and confused the numbers of the other two colours: one mixed up the red and orange, another – orange and yellow, and the third – yellow and green. The results of their calculations are given in the table.

How many balls of each colour actually were there?

A family went to the bridge at night. The dad can cross over it in 1 minute, the mom can cross it in 2, the child takes 5 minutes, and grandmother in 10 minutes. They have one flashlight. The bridge can only withstands two people at a time. How can they all cross the bridge in 17 minutes? $($ If two people pass, then they go at the lower of their speeds. $)$ You can not move along a bridge without a flashlight. You can not shine it from a distance.

On the island of Contrast, both knights and liars live. Knights always tell the truth, liars always lie. Some residents said that the island has an even number of knights, and the rest said that the island has an odd number of liars. Can the number of inhabitants of the island be odd?

There is a rectangular table. Two players start in turn to place on it one pound coin each, so that these coins do not overlap one another. The player who cannot make a move loses. Who will win with the correct strategy?

a$)$ Two players play in the following game: on the table there are 7 two pound coins and 7 one pound coins. In a turn it is allowed to take coins worth no more than three pounds. The one who takes the last coin wins. Who will win with the correct strategy?

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b$)$ The same question, if there are 12 one pound and 12 two pound coins.

On Easter Island, people ask each other questions, to which only “yes” or “no” can be answered. In this case, each of them belongs exactly to one of the tribes either A or B. People from tribe A ask only those questions to which the correct answer is “yes”, and from tribe B – those questions to which the correct answer is “no.” In one house lived a couple Ethan and Violet Russell. When Inspector Krugg approached the house, the owner met him on the doorstep with the words: “Tell me, do Violet and I belong to tribe B?”. The inspector thought and gave the right answer. What was the right answer?

Gary drew an empty table of $50 \times 50$ and wrote on top of each column and to the left of each row a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down a product of numbers written at the top of its column and to the left of the row $($ the “multiplication table” $)$. What is the largest number of products in this table which could be rational numbers?

The equations $ax^2$ + bx + c = 0 $(1)$ and – $ax^2$ + bx + c $(2)$ are given. Prove that if $x_1$ and $x_2$ are, respectively, any roots of the equations $(1)$ and $(2)$, then there is a root $x_3$ of the equation ½ $ax^2$ + bx + c such that either $x_1$ $≤$ $x_3$ $≤$ $x_2$ or $x_1$ $≥$ $x_3$ $≥$ $x_2$.