Prove that, for $k \in {\mathbb Z^+}$ $$k(k+1)(k+2)(k+3)$$is not a perfect square.

$100$ points at a circle with radius $1$ $cm$. Show that, we find an another point such that, this point's distance to other $100$ points is greater than $100$ $cm$.

Let $x,y,z$ be non-negative numbers such that $x+y+z \leq 3$. Prove that $$\frac{2}{1+x}+\frac{2}{1+y}+\frac{2}{1+z} \geq 3$$

Let $ABC$ be an equilateral triangle with side lenght is $1$ $cm$.Let $D \in [AB]$ is a point. Perpendiculars from $D$ to $[AC]$ and $[BC]$ intersects with $[AC]$ and $[BC]$ at points $E$ and $F$ respectively. Perpendiculars from $E$ and $F$ to $[AB]$ intersects with $[AB]$ at points $E_1$ and $F_1$. Prove that $$[E_1F_1]=\frac{3}{4}$$

Solve the equation system for real numbers: $$x_1+x_2=x_3^2$$$$x_2+x_3=x_4^2$$$$x_3+x_4=x_1^2$$$$x_4+x_1=x_2^2$$

Whichever $3$ odd numbers is given. Prove that we can find a $4.$ odd number such that, sum of squares of the these numbers is a perfect square.

We have $1998$ polygon such that sum of the areas is $1997,5$ $cm^2$. These polygons placing inside a square with side lenght $1$ $cm$. (Polygons no overflow). Prove that we can find a point such that, all polygons have this point.

Let $x,y,z$ be real numbers such that, $x \geq y \geq z >0$. Prove that $$\frac{x^2-y^2}{z}+\frac{z^2-y^2}{x}+\frac{x^2-z^2}{y} \geq 3x-4y+z$$

A floor has $2 \times 11$ dimension, and this floor covering with $1 \times 2$ rectangles. (No two rectangles overlap). How many cases we done this job?

Let $ABCD$ a convex quadrilateral with $[BC]$ and $[CD]$'s midpoint is $P$ and $N$ respectively. If $$[AP]+[AN]=d$$Show that, area of the $ABCD$ is less then $\frac{1}{2}d^2$