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.
On the dining room table, there is a choice of six dishes. Every day Valentina takes a certain set of dishes (perhaps, she does not take a single dish), and this set of dishes should be different from all of the sets that she took in the previous days. What is the maximum number of days that Valentina will be able to eat according to such rules and how many meals will she eat on average during the day?
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.
Are there any irrational numbers \(a\) and \(b\) such that the degree of \(a^b\) is a rational number?
Three people play table tennis, and the player who lost the game gives way to the player who did not participate in it. As a result, it turned out that the first player played 10 games and the second played 21 games. How many games did the third player play?
Construct a function defined at all points on a real line which is continuous at exactly one point.
Find all functions \(f (x)\) defined for all real values of \(x\) and satisfying the equation \(2f (x) + f (1 - x) = x^2\).
10 numbers are written around the circle, the sum of which is equal to 100. It is known that the sum of every three numbers standing side by side is not less than 29.
Specify the smallest number \(A\) such that in any such set of numbers each of the numbers does not exceed \(A\).
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 \(a_1 , b_1, a_2, b_2, \dots, a_n, b_n\).