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.
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?
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?
In front of a gnome there lie three piles of diamonds: one with 17, one with 21 and one with 27 diamonds. In one of the piles lies one fake diamond. All the diamonds have the same appearance, and all real diamonds weigh the same, and the fake one differs in its weight. The gnome has a cup weighing scale without weights. The dwarf must find with one weighing a pile, in which all the diamonds are real. How should he do it?
Ladybirds gathered in a sunny clearing. If the ladybird has \(6\) spots, then it always speaks the truth, and if it has \(4\) spots, then it always lies. There are no other types of ladybirds in the meadow. The first ladybird said: “We each have the same number of spots on our backs.” The second one said: “Everyone has \(30\) spots on their backs in total.” “No, we all have \(26\) spots on their backs in total,” the third objected. “Of these three, exactly one told the truth,” – said each of the other ladybirds. How many ladybugs were gathered in the meadow?
Thirty girls – 13 in red dresses and 17 in blue dresses – led a dance around the Christmas tree. Subsequently, each of them was asked if her neighbour on the right was in a blue dress. It turned out that those girls which answered correctly were only those who stood between two girls in dresses of the same color. How many girls could have said yes?
Here’s a rather simple rebus:
\(EX\) is four times larger than \(OJ\).
\(AJ\) is four times larger than \(OX\).
Find the sum of all four numbers.
Prove that a graph with \(n\) vertices, the degree of each of which is at least \(\frac{n-1}{2}\), is connected.