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.
Solve the equation: \((x^3 - 2) (2^{\sin x} - 1) + (2^{x^3} - 4) \sin x = 0\).
Replace \(a, b\) and \(c\) with integers not equal to \(1\) in the equality \((ay^b)^c = - 64y^6\), so it would become an identity.
A row of 4 coins lies on the table. Some of the coins are real and some of them are fake (the ones which weigh less than the real ones). It is known that any real coin lies to the left of any false coin. How can you determine whether each of the coins on the table is real or fake, by weighing once using a balance scale?
With a non-zero number, the following operations are allowed: \(x \rightarrow \frac{1+x}{x}\), \(x \rightarrow \frac{1-x}{x}\). Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?
Find all functions \(f (x)\) defined for all positive \(x\), taking positive values and satisfying the equality \(f (x^y) = f (x)^f (y)\) for any positive \(x\) and \(y\).
Seven triangular pyramids stand on the table. For any three of them, there is a horizontal plane that intersects them along triangles of equal area. Prove that there is a plane intersecting all seven pyramids along triangles of equal area.
Prove that for all \(x\), \(0 < x < \pi /3\), we have the inequality \(\sin 2x + \cos x > 1\).