Problems

Decipher the following puzzle. All the numbers indicated by the letter E, are even (not necessarily equal); all the numbers indicated by the letter O are odd (also not necessarily equal).

In a vase, there is a bouquet of 7 white and blue lilac branches. It is known that 1) at least one branch is white, 2) out of any two branches, at least one is blue. How many white branches and how many blue are there in the bouquet?

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).

There are 68 coins, and it is known that any two coins differ in weight. With 100 weighings on a two-scales balance without weights, find the heaviest and lightest coin.

a) Give an example of a positive number $a$ such that $\{a\}+\{1/a\}=1$.

b) Can such an $a$ be a rational number?

A cat tries to catch a mouse in labyrinths A, B, and C. The cat walks first, beginning with the node marked with the letter “K”. Then the mouse (from the node “M”) moves, then again the cat moves, etc. From any node the cat and mouse go to any adjacent node. If at some point the cat and mouse are in the same node, then the cat eats the mouse.

Can the cat catch the mouse in each of the cases A, B, C?

For which natural $n$ does the number $\frac{n^2}{{1.001}^n}$ reach its maximum value?