Cut an arbitrary triangle into parts that can be used to build a triangle that is symmetrical to the original triangle with respect to some straight line (the pieces cannot be inverted, they can only be rotated on the plane).
The numbers from \(1\) to \(9\) are written in a row. Is it possible to write down the same numbers from \(1\) to \(9\) in a second row beneath the first row so that the sum of the two numbers in each column is an exact square?
On a Halloween night ten children with candy were standing in a row. In total, the girls and boys had equal amounts of candy. Each child gave one candy to each person on their right. After that, the girls had \(25\) more candy than they used to. How many girls are there in the row?
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 \((a-b)(b-c)(c-a)=1\).
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.
Find all solutions of the system of equations: \[\left\{ \begin{aligned} x+y+z = a\\ x^2 + y^2+z^2 = a^2\\ x^3+y^3+z^3 = a^3 \end{aligned} \right.\]