Annie found a prime number \(p\) to which you can add \(4\) to make it a perfect square. What is the value of \(p\)?
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 \((a+b)^3 = a^3 +3a^2b+3ab^2 +b^3\).
The real numbers \(a,b,c\) are non-zero and satisfy the following equations: \[\left\{ \begin{array}{l} a^2 +a = b^2 \\ b^2 +b = c^2 \\ c^2 +c = a^2. \end{array} \right.\] 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 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: \[\left\{ \begin{aligned} x+y = 2\\ xy-z^2 = 1 \end{aligned} \right.\]
Find all solutions of the equation: \(xy + 1 = x + y\).