Problems

There was a football match of 10 versus 10 players between a team of liars (who always lie) and a team of truth-tellers (who always tell the truth). After the match, each player was asked: “How many goals did you score?” Some participants answered “one”, Callum said “two”, some answered “three”, and the rest said “five”. Is Callum lying if it is known that the truth-tellers won with a score of 20:17?

Author: L.N. Vaserstein

For any natural numbers $a_1,a_2,\dots ,a_m$, no two of which are equal to each other and none of which is divisible by the square of a natural number greater than one, and also for any integers and non-zero integers $b_1,b_2,\dots ,b_m$ the sum is not zero. Prove this.

There are 13 weights, each weighing an integer number of grams. It is known that any 12 of them can be divided into two cups of weights, six weights on each one, which will come to equilibrium. Prove that all the weights have the same weight.

Numbers $1,2,3,\dots ,101$ are written out in a row in some order. Prove that one can cross out 90 of them so that the remaining 11 will be arranged in their magnitude (either increasing or decreasing).

Prove that if $x_0^4+a_1{x}_0^3+a_2{x}_0^2+a_3{x}_0+a_4$ and $4x_0^3+3a_1{x}_0^2+2a_2{x}_0+a_3=0$ then $x^4+a_1{x}^3+a_2{x}^2+a_{3}x+a_4$ is divisible by $(x-x_0{)}^2$.

Solve the equation $x^3-\lfloor x\rfloor =3$.

There is a system of equations $$\begin{array}{rl}\ast x+\ast y+\ast z& =0,\\ \ast x+\ast y+\ast z& =0,\\ \ast x+\ast y+\ast z& =0.\end{array}$$ Two people alternately enter a number instead of a star. Prove that the player that goes first can always ensure that the system has a non-zero solution.

There are two sets of numbers made up of 1s and $-1$s, and in each there are 2022 numbers. Prove that in some number of steps it is possible to turn the first set into the second one if for each step you are allowed to simultaneously change the sign of any 11 numbers of the starting set. (Two sets are considered the same if they have the same numbers in the same places.)

Two players play on a square field of size $99\times 99$, which has been split onto cells of size $1\times 1$. The first player places a cross on the center of the field; After this, the second player can place a zero on any of the eight cells surrounding the cross of the first player. After that, the first puts a cross onto any cell of the field next to one of those already occupied, etc. The first player wins if he can put a cross on any corner cell. Prove that with any strategy of the second player the first can always win.

The number $n$ has the property that when it is divided by $q^2$ the remainder is smaller than $q^2/2$, whatever the value of $q$. List all numbers that have this property.