A number set \(M\) contains \(2003\) distinct positive numbers, such that for any three distinct elements \(a, b, c\) in \(M\), the number \(a^2 + bc\) is rational. Prove that we can choose a natural number \(n\) such that for any \(a\) in \(M\) the number \(a\sqrt{n}\) is rational.
One day all the truth tellers on the planet decided to carry a clearly visible mark of truth in order to be distinguished from liars. Two truth tellers and two liars met and looked at each other. Which of them could say the phrase:
“All of us are truth tellers.”
“Only one of you is a truth teller.”
“Exactly two of you are truth tellers.”
Let’s denote any two digits with the letters \(A\) and \(X\). Prove that the six-digit number \(XAXAXA\) is divisible by 7 without a remainder.
In Neverland, only elves and gnomes live. Gnomes lie about their gold, but in any other instances they tell the truth. Elves lie when talking about gnomes, but in other instances they tell the truth. One day two neverlandians said:
\(A\): All my gold I stole from the Dragon.
\(B\): You’re lying.
Determine whether each of them is an elf or a gnome.
In the king’s prison, there are five cells numbered from 1 to 5. In each cell, there is one prisoner. Kristen persuaded the king to conduct an experiment: on the wall of each cell she writes at one point a number and at midnight, each prisoner will go to the cell with the indicated number (if the number on the wall coincides with the cell number, the prisoner does not go anywhere). On the following night at midnight, the prisoners again must move from their cell to another cell according to the instructions on the wall, and they do this for five nights. If the location of prisoners in the cells for all six days (including the first) is never repeated, then Kristen will be given the title of Wisdom, and the prisoners will be released. Help Kristen write numbers in the cells.
One day, Claudia, Sofia and Freia noticed that they brought the same toy cars to kindergarten. Claudia has a car with a trailer, a small car and a green car without a trailer. Sofia has a car without a trailer and a small green one with a trailer, and Freia has a big car and a small blue car with a trailer. What kind of car (in terms of colour, size and availability of a trailer) did all of the girls bring to the kindergarten? Explain the answer.
The \(KUB\) is a cube. Prove that the ball, \(CIR\), is not a cube. (\(KUB\) and \(CIR\) are three-digit numbers, where different letters denote different numbers).
Can I replace the letters with numbers in the puzzle \(RE \times CTS + 1 = CE \times MS\) so that the correct equality is obtained (different letters need to be replaced by different numbers, and the same letters must correspond to the same digits)?
Of the four inequalities \(2x > 70\), \(x < 100\), \(4x > 25\) and \(x > 5\), two are true and two are false. Find the value of \(x\) if it is known that it is an integer.
Pinocchio correctly solved a problem, but stained his notebook. \[(\bullet \bullet + \bullet \bullet+1)\times \bullet= \bullet \bullet \bullet\]
Under each blot lies the same number, which is not equal to zero. Find this number.