Problems

In a physics club, the teacher created the following experiment. He spread out 16 weights of weight 1, 2, 3, ..., 16 grams onto weighing scales, so that one of the bowls outweighed the other. Fifteen students in turn left the classroom and took with them one weight each, and after each student’s departure, the scales changed their position and outweighed the opposite bowl of the scales. What weight could remain on the scales?

There is a \(5\times 9\) rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?

The bank of the Nile was approached by a group of six people: three Bedouins, each with his wife. At the shore is a boat with oars, which can withstand only two people at a time. A Bedouin can not allow his wife to be without him whilst in the company of another man. Can the whole group cross to the other side?

Cut the interval \([-1, 1]\) into black and white segments so that the integrals of any a) linear function; b) a square trinomial in white and black segments are equal.

Solve problem number 108736 for the inscription \(A\), \(BC\), \(DEF\), \(CGH\), \(CBE\), \(EKG\).

\(x_1\) is the real root of the equation \(x^2 + ax + b = 0\), \(x_2\) is the real root of the equation \(x^2 - ax - b = 0\).

Prove that the equation \(x^2 + 2ax + 2b = 0\) has a real root, enclosed between \(x_1\) and \(x_2\). (\(a\) and \(b\) are real numbers).

A row of 4 coins lies on the table. Some of the coins are real and some of them are fake (the ones which weigh less than the real ones). It is known that any real coin lies to the left of any false coin. How can you determine whether each of the coins on the table is real or fake, by weighing once using a balance scale?

With a non-zero number, the following operations are allowed: \(x \rightarrow \frac{1+x}{x}\), \(x \rightarrow \frac{1-x}{x}\). Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

Members of the State parliament formed factions in such a way that for any two factions \(A\) and \(B\) (not necessarily different)

– also a faction (through

the set of all parliament members not included in \(C\) is denoted). Prove that for any two factions \(A\) and \(B\), \(A \cup % \includegraphics{https://problems-static.s3.eu-west-2.amazonaws.com/production/task_images/700/109909-3.png} B\) is also a faction.

A set of weights has the following properties: It contains \(5\) weights, which are all different in weight. For any two weights, there are two other weights of the same total weight. What is the smallest number of weights that can be in this set?