Some squares on a chess board contain a chess piece. It is known that each row contains at least one chess piece, but that different rows all have different numbers of pieces. Prove that it is always possible to mark 8 pieces so that each row and each column of the board contains exactly one marked piece.
Consider the powers of the number five: 1, 5, 25, 125, 625, ... We form the sequence of their first digits: 1, 5, 2, 1, 6, ...
Prove that any part of this sequence, written in reverse order, will occur in the sequence of the first digits of the powers of the number two (1, 2, 4, 8, 1, 3, 6, 1, ...).
Three functions are written on the board: \(f_1 (x) = x + 1/x\), \(f_2 (x) = x^2, f_3 (x) = (x - 1)^2\). You can add, subtract and multiply these functions (and you can square, cube, etc. them). You can also multiply them by an arbitrary number, add an arbitrary number to them, and also do these operations with the resulting expressions. Therefore, try to get the function \(1/x\).
Prove that if you erase any of the functions \(f_1, f_2, f_3\) from the board, it is impossible to get \(1/x\).
Is it possible to find natural numbers \(x\), \(y\) and \(z\) which satisfy the equation \(28x+30y+31z=365\)?
A continuous function \(f\) has the following properties:
1. \(f\) is defined on the entire number line;
2. \(f\) at each point has a derivative (and thus the graph of f at each point has a unique tangent);
3. the graph of the function \(f\) does not contain points for which one of the coordinates is rational and the other is irrational.
Does it follow that the graph of \(f\) is a straight line?
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 \(x * (y * z) = (x * y) + z\).
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 \(c\).
Prove that any convex polygon contains not more than \(35\) vertices with an angle of less than \(170^\circ\).
On the sides \(AB\), \(BC\) and \(AC\) of the triangle \(ABC\) points \(P\), \(M\) and \(K\) are chosen so that the segments \(AM\), \(BK\) and \(CP\) intersect at one point and \[\vec{AM} + \vec{BK}+\vec{CP} = 0\] Prove that \(P\), \(M\) and \(K\) are the midpoints of the sides of the triangle \(ABC\).