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: \[\left\{ \begin{aligned} x+y+z = a\\ x^2 + y^2+z^2 = a^2\\ x^3+y^3+z^3 = a^3 \end{aligned} \right.\]
Now there are three doors with statements on them:
There is nothing behind the third door.
There is a trap behind the first door.
There is nothing behind this door.
Your guide says: There is treasure behind one of the doors, trap
behind another one and there is nothing behind the third door. The sign
on the door leading to treasure is true, the sign on the door leading to
a trap is false, and the third sign might be true or false.
Which door will you open, if you really really want the treasure?