What is the maximum number of pairwise non-parallel segments with endpoints at the vertices of a regular \(n\)-gon?
Numbers \(a, b, c\) are integers with \(a\) and \(b\) being coprime. Let us assume that integers \(x_0\) and \(y_0\) are a solution for the equation \(ax + by = c\).
Prove that every solution for this equation has the same form \(x = x_0 + kb\), \(y = y_0 - ka\), with \(k\) being a random integer.
There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.
Prove the mass of all the weights is the same, if it is known that:
a) the mass of each weight in grams is an integer;
b) the mass of each weight in grams is a rational number;
c) the mass of each weight could be any real (not negative) number.
Can there exist two functions \(f\) and \(g\) that take only integer values such that for any integer \(x\) the following relations hold:
a) \(f (f (x)) = x\), \(g (g (x)) = x\), \(f (g (x)) > x\), \(g (f (x)) > x\)?
b) \(f (f (x)) < x\), \(g (g (x)) < x\), \(f (g (x)) > x\), \(g (f (x)) > x\)?
The student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be seven times bigger than their product. Determine these numbers.
The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.
A cherry which is a ball of radius r is dropped into a round glass whose axial section is the graph of the function \(y = x^4\). At what maximum r will the ball touch the most bottom point of the bottom of the glass? (In other words, what is the maximum radius r of a circle lying in the region \(y \geq x^4\) and containing the origin?).
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.
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 \(1/x\).