Problems

During the ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced with during the previous dance, or more intelligent, but most of the men (at least 80%) – with a girl who was at the same time more beautiful and more intelligent. Could this happen? (There was an equal number of boys and girls at the ball.)

A $1\times 10$ strip is divided into unit squares. The numbers $1,2,\dots ,10$ are written into squares. First, the number 1 is written in one square, then the number 2 is written into one of the neighboring squares, then the number 3 is written into one of the neighboring squares of those already occupied, and so on (the choice of the first square is made arbitrarily and the choice of the neighbor at each step). In how many ways can this be done?

On a plane there is a square, and invisible ink is dotted at a point $P$. A person with special glasses can see the spot. If we draw a straight line, then the person will answer the question of on which side of the line does $P$ lie (if $P$ lies on the line, then he says that $P$ lies on the line).

What is the smallest number of such questions you need to ask to find out if the point $P$ is inside the square?

Let $n$ numbers are given together with their product $p$. The difference between $p$ and each of these numbers is an odd number.

Prove that all $n$ numbers are irrational.

Two play tic-tac-toe on a $10\times 10$ board according to the following rules. First they fill the whole board with noughts and crosses, putting them in turn (the first player puts crosses, their partner – noughts). Then two numbers are counted: $K$ is the number of five consecutively standing crosses and $H$ is the number of five consecutively standing zeros. (Five, standing horizontally, vertically and parallel to the diagonal are counted, if there are six crosses in a row, this gives two fives, if there are seven, then three, etc.). The number $K-H$ is considered to be the winnings of the first player (the losses of the second).

a) Does the first player have a winning strategy?

b) Does the first player have a non-losing strategy?

a) Could an additional $6$ digits be added to any $6$-digit number starting with a $5$, so that the $12$-digit number obtained is a complete square?

b) The same question but for a number starting with a $1$.

c) Find for each $n$ the smallest $k=k(n)$ such that to each $n$-digit number you can assign $k$ more digits so that the resulting $(n+k)$-digit number is a complete square.

A cube with side length of 20 is divided into 8000 unit cubes, and on each cube a number is written. It is known that in each column of 20 cubes parallel to the edge of the cube, the sum of the numbers is equal to 1 (the columns in all three directions are considered). On some cubes a number 10 is written. Through this cube there are three layers of $1\times 20\times 20$ cubes, parallel to the faces of the cube. Find the sum of all the numbers outside of these layers.

A game takes place on a squared $9\times 9$ piece of checkered paper. Two players play in turns. The first player puts crosses in empty cells, its partner puts noughts. When all the cells are filled, the number of rows and columns in which there are more crosses than zeros is counted, and is denoted by the number $K$, and the number of rows and columns in which there are more zeros than crosses is denoted by the number $H$ (18 rows in total). The difference $B=K-H$ is considered the winnings of the player who goes first. Find a value of B such that

1) the first player can secure a win of no less than $B$, no matter how the second player played;

2) the second player can always make it so that the first player will receive no more than $B$, no matter how he plays.

Two people are playing. The first player writes out numbers from left to right, randomly alternating between 0 and 1, until there are 2021 numbers in total. Each time after the first one writes out the next digit, the second switches two numbers from the already written row (when only one digit is written, the second misses its move). Is the second player always able to ensure that, after his last move, the arrangement of the numbers is symmetrical relative to the middle number?

In Conrad’s collection there are four royal gold five-pound coins. Conrad was told that some two of them were fake. Conrad wants to check (prove or disprove) that among the coins there are exactly two fake ones. Will he be able to do this with the help of two weighings on weighing scales without weights? (Counterfeit coins are the same in weight, real ones are also the same in weight, but false ones are lighter than real ones.)