Problems

Let $a$ and $b$ be positive real numbers. Using areas of rectangles and squares, show that $a^2-b^2=(a-b)\times (a+b)$.
Try to prove it in two ways, one geometric and one algebraic.

Let $a$ and $b$ be positive real numbers. Using volumes of cubes and parallelepipeds, show that $(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$.
Hint: Place the cubes with sides $a$ and $b$ along the same diagonal.

The real numbers $a,b,c$ are non-zero and satisfy the following equations: $$\{\begin{array}{l}{a}^2+a=b^2\\ b^2+b=c^2\\ c^2+c=a^2.\end{array}$$ Show that $(a-b)(b-c)(c-a)=1$.

A five-digit number is called indecomposable if it is not decomposed into the product of two three-digit numbers. What is the largest number of indecomposable five-digit numbers that can come in a row?

Find the representation of $(a+b{)}^n$ as the sum of $X_{n,k}{a}^k{b}^{n-k}$ for general $n$. Here by $X_{n,k}$ we denote coefficients that depend only on $k$ and $n$.

The positive real numbers $a,b,c,x,y$ satisfy the following system of equations: $$\{\begin{array}{r}{x}^2+xy+y^2=a^2\\ y^2+yz+z^2=b^2\\ x^2+xz+z^2=c^2\end{array}$$

Find the value of $xy+yz+xz$ in terms of $a,b,$ and $c.$

Find all solutions of the equation: $x^2+y^2+z^2+t^2=x(y+z+t)$.

Let $a$ and $b$ be real numbers. Find a representation of $a^3+b^3$ as a product.

• Find a representation of the number $117=121-4$ as a product.

• Let $a$ and $b$ be real numbers. Find a representation of $a^2-b^2$ as a product.

Solve the system of equations in real numbers: $$\{\begin{array}{r}x+y=2\\ xy-z^2=1\end{array}$$