For how many pairs of numbers \(x\) and \(y\) between \(1\) and \(100\) is the expression \(x^2 + y^2\) divisible by \(7\)?
Seven robbers are dividing a bag of coins of various denominations. It turned out that the sum could not be divided equally between them, but if any coin is set aside, the rest could be divided so that every robber would get an equal part. Prove that the bag cannot contain \(100\) coins.
Show that the equation \(x^2 +6x-1 = y^2\) has no solutions in integer \(x\) and \(y\).
Can there exist two functions \(f\) and \(g\) that take only integer values such that for any integer \(x\) the following relations hold:
a) \(f (f (x)) = x\), \(g (g (x)) = x\), \(f (g (x)) > x\), \(g (f (x)) > x\)?
b) \(f (f (x)) < x\), \(g (g (x)) < x\), \(f (g (x)) > x\), \(g (f (x)) > x\)?
The surface of a \(3\times 3\times 3\) Rubik’s Cube contains 54 squares. What is the maximum number of squares we can mark, so that no marked squares share a vertex or are directly adjacent to another marked square?
Cut the interval \([-1, 1]\) into black and white segments so that the integrals of any a) linear function; b) a square trinomial in white and black segments are equal.
For each pair of real numbers \(a\) and \(b\), consider the sequence of numbers \(p_n = \lfloor 2 \{an + b\}\rfloor\). Any \(k\) consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length \(k\) is a word of the sequence given by some \(a\) and \(b\) for \(k = 4\); when \(k = 5\)?
Note: \(\lfloor c\rfloor\) is the integer part, \(\{c\}\) is the fractional part of the number \(c\).
\(x_1\) is the real root of the equation \(x^2 + ax + b = 0\), \(x_2\) is the real root of the equation \(x^2 - ax - b = 0\).
Prove that the equation \(x^2 + 2ax + 2b = 0\) has a real root, enclosed between \(x_1\) and \(x_2\). (\(a\) and \(b\) are real numbers).
In the number \(a = 0.12457 \dots\) the \(n\)th digit after the decimal point is equal to the digit to the left of the decimal point in the number. Prove that \(\alpha\) is an irrational number.
The function \(f\) is such that for any positive \(x\) and \(y\) the equality \(f (xy) = f (x) + f (y)\) holds. Find \(f (2007)\) if \(f (1/2007) = 1\).