Each of the 1994 deputies in parliament slapped exactly one of his colleagues. Prove that it is possible to draw up a parliamentary commission of 665 people whose members did not clarify the relationship between themselves in the manner indicated above.
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.
Eight schoolchildren solved \(8\) tasks. It turned out that \(5\) schoolchildren solved each problem. Prove that there are two schoolchildren, who solved every problem at least once.
If each problem is solved by \(4\) pupils, prove that it is not necessary to have two schoolchildren who would solve each problem.
Peter has 28 classmates. Each 2 out of these 28 have a different number of friends in the class. How many friends does Peter have?
To each pair of numbers \(x\) and \(y\) some number \(x * y\) is placed in correspondence. Find \(1993 * 1935\) if it is known that for any three numbers \(x, y, z\), the following identities hold: \(x * x = 0\) and \(x * (y * z) = (x * y) + z\).
\(x_1\) 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.
We are given a \(100\times 100\) square grid and \(N\) counters. All of the possible arrangements of the counters on the grid which follow the following rule are considered: no two counters lie in adjacent squares.
What is the largest value of \(N\) for which, in every single possible arrangement of counters following this rule, it is possible to find at least one counter such that moving it to an adjacent square does not break the rule. Squares are considered adjacent if they share a side.
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 \(y\).