Determine all the integer solutions for the equation \(21x + 48y = 6\).
Prove there are no integer solutions for the equation \(x^2 + 1990 = y^2\).
The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?
An adventurer is travelling to the planet of liars and truth tellers with an official guide and is introduced to a local. “Are you a truth teller?” asked the adventurer. The alien answers “Yrrg,” which means either “yes” or “no”. The adventurer asks the guide for a translation. The guide says “"yrrg" means "yes". I will add that the local is actully a liar.” Is the local alien liar or truth teller?
Prove that in a game of noughts and crosses on a \(3\times 3\) grid, if the first player uses the right strategy then the second player cannot win.
Vincent makes small weights. He made 4 weights which should have masses (in grams) of 1, 3, 4 and 7, respectively. However, he made a mistake and one of these weights has the wrong mass. By weighing them twice using balance scales (without the use of weights other than those mentioned) can he find which weight has the wrong mass?
There are some coins on a table. One of these coins is fake (has a different weight than a real coin). By weighing them twice using balance scales, determine whether the fake coin is lighter or heavier than a real coin (you don’t need to find the fake coin) if the number of coins is: a) 100; b) 99; c) 98?
Prove that if the irreducible rational fraction \(p/q\) is a root of the polynomial \(P (x)\) with integer coefficients, then \(P (x) = (qx - p) Q (x)\), where the polynomial \(Q (x)\) also has integer coefficients.
On a function \(f (x)\), defined on the entire real line, it is known that for any \(a>1\) the function \(f (x) + f (ax)\) is continuous on the whole line. Prove that \(f (x)\) is also continuous on the whole line.
Prove that the infinite decimal \(0.1234567891011121314 \dots\) (after the decimal point, all of the natural numbers are written out in order) is an irrational number.