Tasks

KMS

Tasks

6 light and 6 heavy coins are given. In appearance they are indistinguishable. In one weighing of any set of coins, you can find out how many heavy coins are in it. How to find two coins in two weighings, one of which is light and the other heavy? A. Shapovalov

For which natural numbers x and y does the equality $1 + 2^{1} + 2^{2} +_{K} + 2^{x} = 3^{y}$ hold? V. Senderov

Vasya thought of three natural numbers. For each pair of numbers, he found the difference between their product and their sum. It turned out that one of these differences is negative, and the other is positive. What is the sign of the third difference? B.Frenkin

In a country where every resident is either a knight or a liar (knights, as you know, always tell the truth, and liars always lie), a company of 19 aborigines gathered at a round table. Each of those gathered declared that both of his neighbors were liars. A scandal broke out, as a result of which part of the company left the feast. After this, each of those remaining announced with satisfaction that now both of his neighbors were knights. “And in fact, there is not a single liar among you now,” the last of those leaving the company agreed with them. Meanwhile, the “renegades” organized a new meeting, again at a round table. Each of those sitting at this table said that among his neighbors there is exactly one knight. How many people remained in their seats after the company split? V. Letsko

These tasks are primarily intended for students in grades 6–8.

Illustration by D. Grishukova

In a right triangle, one leg is twice as long as the other. Cut it into 5 identical triangles. R. Sarbash