In some country there are 101 cities, and some of them are connected by roads. However, every two cities are connected by exactly one path.
How many roads are there in this country?
Determine all integer solutions of the equation \(yk = x^2 + x\). Where \(k\) is an integer greater than \(1\).
\(a_1, a_2, a_3, \dots\) is an increasing sequence of natural numbers. It is known that \(a_{a_k} = 3k\) for any \(k\). Find a) \(a_{100}\); b) \(a_{2022}\).
\(f(x)\) 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?
For which natural \(n\) does the number \(\frac{n^2}{1.001^n}\) reach its maximum value?
The function \(F\) is given on the whole real axis, and for each \(x\) the equality holds: \(F (x + 1) F (x) + F (x + 1) + 1 = 0\).
Prove that the function \(F\) can not be continuous.
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.