It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.
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) \(n = 5\)?
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.
Prove that multiplying the polynomial \((x + 1)^{n-1}\) by any polynomial different from zero, we obtain a polynomial having at least \(n\) nonzero coefficients.
Determine all solutions of the equation \((n + 2)! - (n + 1)! - n! = n^2 + n^4\) in natural numbers.
In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.
Prove that the equation \[a_1 \sin x + b_1 \cos x + a_2 \sin 2x + b_2 \cos 2x + \dots + a_n \sin nx + b_n \cos nx = 0\] has at least one root for any values of \(a_1 , b_1, a_2, b_2, \dots, a_n, b_n\).
Solve the equation \(2x^x = \sqrt {2}\) for positive numbers.
Let \(f\) be a continuous function defined on the interval \([0; 1]\) such that \(f (0) = f (1) = 0\). Prove that on the segment \([0; 1]\) there are 2 points at a distance of 0.1 at which the function \(f 4(x)\) takes equal values.