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”?
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).
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?
On a board there are written four three-digit numbers, totaling 2012. To write them all, only two different digits were used.
Give an example of such numbers.
We are given a polynomial \(P(x)\) and numbers \(a_1\), \(a_2\), \(a_3\), \(b_1\), \(b_2\), \(b_3\) such that \(a_1a_2a_3 \ne 0\). It turned out that \(P (a_1x + b_1) + P (a_2x + b_2) = P (a_3x + b_3)\) for any real \(x\). Prove that \(P (x)\) has at least one real root.
Matt, Conrad and Louie ate some sweets. Their surnames are Smith, Jones and Cooper. Smith ate 2 sweets fewer than Matt, Jones – 2 sweets fewer than Conrad, and Conrad ate more than anyone. Which of them has which last name?
A monkey, donkey and goat decided to play a game. They sat in a row, with the monkey on the right. They started to play the violin, but very poorly. They changed places and then the donkey was in the middle. However the violin trio still didn’t sound as they wanted it to. They changed places once more. After changing places 3 times, each of the three “musicians” had a chance to sit in the left, middle and right of the row. Who sat where after the third change of seats?
There is a group of 5 people: Alex, Beatrice, Victor, Gregory and Deborah. Each of them has one of the following codenames: V, W, X, Y, Z. We know that:
Alex is 1 year older than V,
Beatrice is 2 years older than W,
Victor is 3 years older than X,
Gregory is 4 years older than Y.
Who is older and by how much: Deborah or Z?
Going to school, Michael found everything he needed under the pillow, under the sofa, on the table or under the table. The items he needed to find were a notebook, a cheat sheet, an mp3 player and sneakers. Under the table, he did not find a notebook or an mp3 player. His cheat sheet never lies on the floor. The mp3 player was neither on the table nor under the sofa. What was lying where, if there was only one object in each of the places?
2012 pine cones lay under the fir-tree. Winnie the Pooh and the donkey Eeyore play a game: they take turns picking up these pine cones. Winnie-the-Pooh takes either one or four cones in each of his turns, and Eeyore – either one or three. Winnie the Pooh goes first. The player who cannot make a move loses. Which of the players can be guaranteed to win, no matter how their opponent plays?