Determine all integer solutions of the equation \(yk = x^2 + x\). Where \(k\) is an integer greater than \(1\).
\(f(x)\) is an increasing function defined on the interval \([0, 1]\). It is known that the range of its values belongs to the interval \([0, 1]\). Prove that, for any natural \(N\), the graph of the function can be covered by \(N\) rectangles whose sides are parallel to the coordinate axes so that the area of each is \(1/N^2\). (In a rectangle we include its interior points and the points of its boundary).
a) Give an example of a positive number \(a\) such that \(\{a\} + \{1 / a\} = 1\).
b) Can such an \(a\) be a rational number?
Two players in turn increase a natural number in such a way that at each increase the difference between the new and old values of the number is greater than zero, but less than the old value. The initial value of the number is 2. The winner is the one who can create the number 1987. Who wins with the correct strategy: the first player or his partner?
a) The vertices (corners) in a regular polygon with 10 sides are colored black and white in an alternating fashion (i.e. one vertex is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same color. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?
b) The same problem, but for a regular polygon with 12 sides.
We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the \(k\)-th word is obtained from the \((k-1)\)-th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB...
a) Where in this sequence will the 1000th letter “A” be?
b) Prove that this sequence is non-periodic.
Solve the equation \(x + \frac{1}{(y + 1/z)}= 10/7\) in natural numbers.
A numerical sequence is defined by the following conditions: \[a_1 = 1, \quad a_{n+1} = a_n + \lfloor \sqrt{a_n}\rfloor .\]
Prove that among the terms of this sequence there are an infinite number of complete squares.
The function \(f(x)\) on the interval \([a, b]\) is equal to the maximum of several functions of the form \(y = C \times 10^{- | x-d |}\) (where \(d\) and \(C\) are different, and all \(C\) are positive). It is given that \(f (a) = f (b)\). Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.
Let \(n\) numbers are given together with their product \(p\). The difference between \(p\) and each of these numbers is an odd number.
Prove that all \(n\) numbers are irrational.