Problems

There are 40 weights of weights of 1 g, 2 g, ..., 40 grams. Of these, 10 weights of even weight were chosen and placed on the left hand side of the scales. Then we selected 10 weights of odd weight and put it on the right hand side of the scales. The scales were balanced. Prove that on one of the bowls of the scales there are two weights with a mass difference of 20 g.

Seven coins are arranged in a circle. It is known that some four of them, lying in succession, are fake and that every counterfeit coin is lighter than a real one. Explain how to find two counterfeit coins from one weighing on scales without any weights. (All counterfeit coins weigh the same.)

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?

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).

Find all possible ways to represent $2025$ as the sum of four three-digit numbers. We have the restriction that we can use only two digits across the four numbers.

We are given a polynomial $P(x)$ and numbers $a_1$, $a_2$, $a_3$, $b_1$, $b_2$, $b_3$ such that $a_1{a}_2{a}_3\ne 0$. It turned out that $P(a_{1}x+b_1)+P(a_{2}x+b_2)=P(a_{3}x+b_3)$ for any real $x$. Prove that $P(x)$ has at least one real root.

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?