In how many ways can you rearrange the numbers 1, 2, ..., 100 so the neighbouring numbers differ by not more than 1?
All the positive fractions smaller than \(1\) with denominators not more than \(100\) are written in a row. Isley and Ella put signs \("+"\) or \("-"\) in front of any fraction, which does not yet have a sign before it. They write signs in turns, but it is known that Isley has to make the last move and calculate the resulting sum. If the total sum turns out to be an integer number, then Ella will give her a chocolate bar. Will Isley be able to get a chocolate bar regardless of Ella’s actions?
Using areas of squares and rectangles, show that for any positive real numbers \(a\) and \(b\), \((a+b)^2 = a^2+2ab+b^2\).
The identity above is true for any real numbers, not necessarily positive, in fact in order to prove it the usual way one only needs to remember that multiplication is commutative and the distributive property of addition and multiplication:
\(a\times b = b\times a\);
\((a+b)\times c = a\times c + b\times c\).
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 \((a+b)^3 = a^3 +3a^2b+3ab^2 +b^3\).
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: \[\left\{ \begin{array}{l} a^2 +a = b^2 \\ b^2 +b = c^2 \\ c^2 +c = a^2. \end{array} \right.\] Show that \((a-b)(b-c)(c-a)=1\).
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.
Find all solutions of the equation: \(x^2 + y^2 + z^2 + t^2 = x(y + z + t)\).