Problems

Prove that for any natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_1,a_2,a_3,\dots$, for which $a_1^2+a_2^2+\cdots +a_k^2$ is divisible by $a_1+a_2+\cdots +a_k$ for all $k\ge 1$.

Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any $k=1,2,3,\dots$ the sum of the first $k$ terms of the sequence is divisible by $k$?

Author: I.I. Bogdanov

Peter wants to write down all of the possible sequences of 100 natural numbers, in each of which there is at least one 3, and any two neighbouring terms differ by no more than 1. How many sequences will he have to write out?

At a contest named “Ah well, monsters!”, 15 dragons stand in a row. Between neighbouring dragons the number of heads differs by 1. If the dragon has more heads than both of his two neighbors, he is considered cunning, if he has less than both of his neighbors – strong, the rest (including those standing at the edges) are considered ordinary. In the row there are exactly four cunning dragons – with 4, 6, 7 and 7 heads and exactly three strong ones – with 3, 3 and 6 heads. The first and last dragons have the same number of heads.

a) Give an example of how this could occur.

b) Prove that the number of heads of the first dragon in all potential examples is the same.

Author: G. Zhukov

The square trinomial $f(x)=a{x}^2+bx+c$ that does not have roots is such that the coefficient $b$ is rational, and among the numbers $c$ and $f(c)$ there is exactly one irrational.

Can the discriminant of the trinomial $f(x)$ be rational?

At what value of $k$ is the quantity $A_k=({19}^k+{66}^k)/k!$ at its maximum?

At what value of $k$ is the quantity $A_k=({19}^k+{66}^k)/k!$ at its maximum? You are given a number $x$ that is greater than 1. Is the following inequality necessarily fulfilled $\lfloor \sqrt{\sqrt{x}}\rfloor =\lfloor \sqrt{\sqrt{x}}\rfloor$?

$N$ points are given, no three of which lie on one line. Each two of these points are connected by a segment, and each segment is coloured in one of the $k$ colours. Prove that if $N>\lfloor k!e\rfloor$, then among these points one can choose three such that all sides of the triangle formed by them will be colored in one colour.

An iterative polyline serves as a geometric interpretation of the iteration process. To construct it, on the $Oxy$ plane, the graph of the function $f(x)$ is drawn and the bisector of the coordinate angle is drawn, as is the straight line $y=x$. Then on the graph of the function the points $$A_0(x_0,f(x_0)),A_1(x_1,f(x_1)),\dots ,A_n(x_n,f(x_n)),\dots$$ are noted and on the bisector of the coordinate angle – the points $$B_0(x_0,x_0),B_1(x_1,x_1),\dots ,B_n(x_n,x_n),\dots .$$ The polygonal line $B_0{A}_0{B}_1{A}_1\dots B_n{A}_n\dots$ is called iterative.

Construct an iterative polyline from the following information:

a) $f(x)=1+x/2$, $x_0=0$, $x_0=8$;

b) $f(x)=1/x$, $x_0=2$;

c) $f(x)=2x-1$, $x_0=0$, $x_0=\mathrm{1,125}$;

d) $f(x)=-3x/2+6$, $x_0=5/2$;

e) $f(x)=x^2+3x-3$, $x_0=1$, $x_0=0,99$, $x_0=1,01$;

f) $f(x)=\sqrt{1+x}$, $x_0=0$, $x_0=8$;

g) $f(x)=x^3/3-5x^2/x+25x/6+3$, $x_0=3$.

The sequence of numbers $a_n$ is given by the conditions $a_1=1$, $a_{n+1}=a_n+1/a_n^2$ ($n\ge 1$).

Is it true that this sequence is limited?