Problems

After Albert discovered the previous rule, he began looking at differences of squares of consecutive odd numbers. He found the difference between $1^2$ and $3^2$ is $8$, the difference between $3^2$ and $5^2$ is $16$, the difference between $5^2$ and $7^2$ is $24$, and that the difference between $7^2$ and $9^2$ is $32$. What is the rule now? Can you prove it?

What is the last digit of the number $7^4-3^4$?

A number $n$ is an integer such that $n$ is not divisible by $3$ or by $2$. Show that $n^2-1$ is divisible by $24$.

Show that for any two positive real numbers $x,y$ it is true that $x^2+y^2\ge 2xy$.

Find all pairs of integers $(x,y)$ so that the following equation is true $xy=y+x$.

Calculate the following squares in the shortest possible way (without a calculator or any other device):
a) ${1001}^2$ b) ${9998}^2$ c) ${20003}^2$ d) ${497}^2$

Real numbers $x,y$ are such that $x^2+x\le y$. Show that $y^2+y\ge x$.

Today we will solve some problems using algebraic tricks, mostly related to turning a sum into a product or using an identity involving squares.
The ones particularly useful are: $(a+b{)}^2=a^2+b^2+2ab$, $(a-b{)}^2=a^2+b^2-2ab$ and $(a-b)\times (a+b)=a^2-b^2$. While we are at squares, it is also worth noting that any square of a real number is never a negative number.

The evil warlock found a mathematics exercise book and replaced all the decimal numbers with the letters of the alphabet. The elves in his kingdom only know that different letters correspond to different digits $\{0,1,2,3,4,5,6,7,8,9\}$ and the same letters correspond to the same digits. Help the elves to restore the exercise book to study.

The perimeter of the triangle $ABC$ is 10. Let $D,E,F$ be the midpoint of the segments $AB,BC,AC$ respectively. What is the perimeter of the triangle $DEF$?