After a circus came back from its country-wide tour, relatives of the animal tamer asked him questions about which animals travelled with the circus.
“Where there tigers?”
“Yes, in fact, there were seven times more tigers than non-tigers.”
“What about monkeys?”
“Yes, there were seven times less monkeys than non-monkeys.”
“Where there any lions?”
What is the answer he gave to this last question?
There are 100 boxes numbered from 1 to 100. In one box there is a prize and the presenter knows where the prize is. The spectator can send the presented a pack of notes with questions that require a “yes” or “no” answer. The presenter mixes the notes in a bag and, without reading out the questions aloud, honestly answers all of them. What is the smallest number of notes you need to send to know for sure where the prize is?
Solve this equation: \[(x+2010)(x+2011)(x+2012)=(x+2011)(x+2012)(x+2013).\]
There are a thousand tickets with numbers 000, 001, ..., 999 and a hundred boxes with the numbers 00, 01, ..., 99. A ticket is allowed to be dropped into a box if the number of the box can be obtained from the ticket number by erasing one of the digits. Is it possible to arrange all of the tickets into 50 boxes?
The nonzero numbers \(a\), \(b\), \(c\) are such that every two of the three equations \(ax^{11} + bx^4 + c = 0\), \(bx^{11} + cx^4 + a = 0\), \(cx^{11} + ax^4 + b = 0\) have a common root. Prove that all three equations have a common root.
At a round table, 30 people are sitting – knights and liars (knights always tell the truth, and liars always lie). It is known that each of them at that table has exactly one friend, and for each knight this friend is a liar, and for a liar this friend is a knight (friendship is always mutual). To the question “Does your friend sit next to you?” those in every other seat answered “yes”. How many of the others could also have said “Yes”?
The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and
for all \(n \geq 2\).
Prove that all members of the sequence are integers.
In the equality \(TIME + TICK = SPIT\), replace the same letters with the same numbers, and different letters with different digits so that the word \(TICK\) is as small as possible (there are no zeros among the digits).
Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?
Four children said the following about each other.
Mary: Sarah, Nathan and George solved the problem.
Sarah: Mary, Nathan and George didn’t solve the problem.
Nathan: Mary and Sarah lied.
George: Mary, Sarah and Nathan told the truth.
How many of the children actually told the truth?