Problems

What has a greater value: $300!$ or ${100}^{300}$?

Find the locus of points whose coordinates $(x,y)$ satisfy the relation $\sin (x+y)=0$.

The equations $$\begin{array}{}\text{(1)}& a{x}^2+bx+c=0\end{array}$$ and $$\begin{array}{}\text{(2)}& -a{x}^2+bx+c\end{array}$$ are given. Prove that if $x_1$ and $x_2$ are, respectively, any roots of the equations (1) and (2), then there is a root $x_3$ of the equation $\frac{1}{2}a{x}^2+bx+c$ such that either $x_1\le x_3\le x_2$ or $x_1\ge x_3\ge x_2$.

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 $x^3-\lfloor x\rfloor =3$.

A table of $4\times 4$ cells is given, in some cells of which a star is placed. Show that you can arrange seven stars so that when you remove any two rows and any two columns of this table, there will always be at least one star in the remaining cells. Prove that if there are fewer than seven stars, you can always remove two rows and two columns so that all the remaining cells are empty.

Prove that in a group of 11 arbitrary infinitely long decimal numbers, it is possible to choose two whose difference contains either, in decimal form, an infinite number of zeroes or an infinite number of nines.

At what value of $k$ is the quantity $A_k=({19}^k+{66}^k)/k!$ at its maximum?

In draughts, the king attacks by jumping over another draughts-piece. What is the maximum number of draughts kings we can place on the black squares of a standard $8\times 8$ draughts board, so that each king is attacking at least one other?

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.