Problems

Is the number ${25}^3-{11}^4$ a prime number?

Find all pairs of whole numbers $(x,y)$ so that this equation is true: $xy=y+1$.

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 $y^2+y\ge x$.

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 $DEF$?