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A game with 25 coins. In a row there are 25 coins. For a turn it is allowed to take one or two neighbouring coins. The player who has nothing to take loses.

Peter bought an automatic machine at the store, which for 5 pence multiplies any number entered into it by 3, and for 2 pence adds 4 to any number. Peter wants, starting with a unit that can be entered free of charge to get the number 1981 on the machine number whilst spending the smallest amount of money. How much will the calculations cost him? What happens if he wants to get the number 1982?

In a set there are 100 weights, each two of which differ in mass by no more than 20 g. Prove that these weights can be put on two cups of weighing scales, 50 pieces on each one, so that one cup of weights is lighter than the other by no more than 20 g.

The White Rook pursues a black horse on a board of $3 \times 1969$ cells $($ they walk in turn according to the usual rules $)$. How should the rook play in order to take the horse? White makes the first move.

The triangle $C_1C_2O$ is given. Within it the bisector $C_2C_3$ is drawn, then in the triangle $C_2C_3O$ – bisector $C_3C_4$ and so on. Prove that the sequence of angles $γ_n$ = $C_{n + 1}C_nO$ tends to a limit, and find this limit if $C_1OC_2$ = α.

Gary drew an empty table of $50 \times 50$ and wrote on top of each column and to the left of each row a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down a product of numbers written at the top of its column and to the left of the row $($ the “multiplication table” $)$. What is the largest number of products in this table which could be rational numbers?

Fred chose 2017 $($not necessarily different$)$ natural numbers $a_1, a_2, …, a_{2017}$ and plays by himself in the following game. Initially, he has an unlimited supply of stones and 2017 large empty boxes. In one move Fred adds a1 stones to any box $($at his choice$)$, in any of the remaining boxes $($of his choice$)$ – $a_2$ stones, …, finally, in the remaining box – $a_{2017}$ stones. His purpose is to ensure that eventually all the boxes have an equal number of stones. Could he have chosen the numbers so that the goal could be achieved in 43 moves, but is impossible for a smaller non-zero number of moves?

There are 30 students in a class: excellent students, mediocre students and slackers. Excellent students answer all questions correctly, slackers are always wrong, and the mediocre students answer questions alternating one by one correctly and incorrectly. All the students were asked three questions: “Are you an excellent pupil?”, “Are you a mediocre student?”, “Are you a slacker?”. 19 students answered “Yes” to the first question, to the second 12 students answered yes, to the third 9 students answered yes. How many mediocre students study in this class?

Hannah Montana wants to leave the round room which has six doors, five of which are locked. In one attempt she can check any three doors, and if one of them is not locked, then she will go through it. After each attempt her friend Michelle locks the door, which was opened, and unlocks one of the neighbouring doors. Hannah does not know which one exactly. How should she act in order to leave the room?

100 children were each given a bowl with 100 pieces of pasta. However, these children did not want to eat and instead started to play. One of the children started to place one piece of her pasta into other children’s bowls $($to whomever she wants$)$. What is the least amount of transfers needed so that everyone has a different number of pieces of pasta in their bowl?

10 children were each given a bowl with 100 pieces of pasta. However, these children did not want to eat and instead started to play. One of the children started to place one piece of pasta into every other child’s bowl. What is the least amount of transfers needed so that everyone has a different number of pieces of pasta in their bowl?

One hundred cubs found berries in the forest: the youngest managed to grab 1 berry, the next youngest cub – 2 berries, the next – 4 berries, and so on, until the oldest who got $2^{99}$ berries. The fox suggested that they share the berries “fairly.” She can approach two cubs and distribute their berries evenly between them, and if this leaves an extra berry, then the fox eats it. With such actions, she continues, until all the cubs have an equal number of berries. What is the largest number of berries that the fox can eat?

Let p and q be nonzero real numbers and $p^2$ – 4q $>$ 0. Prove that the following sequences converge:

a$)$ $y_0$ = 0, $y_{n + 1} = \frac{q}{p-y_n} (n \geq 0)$;

b$)$ $z_0$ = 0, $z_{n + 1}$ = p – q/$z_n$ $( n \geq 0 )$.

Establish a connection between the limiting values of these sequences y *, z * and the roots of the equation $x^2$ – px + q = 0.

Old calculator II. The derivative of the function $\ln x$ for $x = 1$ is 1. Hence

\[\lim\limits_{x\to 0} \frac{\ln (1+x)}{x} = \lim\limits_{x\to0} \frac{\ln(1+x) – \ln 1}{(x+1)-1} = 1.\]

Use this fact to approximate the natural logarithm of the number N. As in Problem 61302, the standard arithmetic operations and the square root extraction operation are allowed.

a$)$ Using geometric considerations, prove that the base and the side of an isosceles triangle with an angle of 36 $^{\circ}$ at the vertex are incommensurable.

b$)$ Invent a geometric proof of the irrationality of $\sqrt{2}$.

Let f $(x)$ be a polynomial of degree n with roots $α_1$, …, $α_n$. We define the polygon M as the convex hull of the points $α_1$, …, $α_n$ on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon M.

Let f $(x)$ = $(x – a)$ $(x – b)$ $(x – c)$ be a polynomial of the third degree with complex roots a, b, c.

Prove that the roots of the derivative of this polynomial lie inside the triangle with vertices at the points a, b, c.

Using the theorem on rational roots of a polynomial $($ see Problem 61013 $)$, prove that if p / q is rational and cos $(p / q)$ $^{\circ}$ ≠ 0, ± 1, ± 1, then

cos $(p / q)$ $^{\circ}$ is irrational.

Prove that there are infinitely many composite numbers among the numbers [$2^k$ $\sqrt{2}$] $(k = 0, 1, …)$.