Problems

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 $k$?

The sequence of numbers $a_1,a_2,\dots$ is given by the conditions $a_1=1$, $a_2=143$ and

for all $n\ge 2$.

Prove that all members of the sequence are integers.

Which term in the expansion $(1+\sqrt{3}{)}^{100}$ will be the largest by the Newton binomial formula?

Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.

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

Hannah placed 101 counters in a row which had values of 1, 2 and 3 points. It turned out that there was at least one counter between every two one point counters, at least two counters lie between every two two point counters, and at least three counters lie between every two three point counters. How many three point counters could Hannah have?

In a row there are 20 different natural numbers. The product of every two of them standing next to one another is the square of a natural number. The first number is 42. Prove that at least one of the numbers is greater than 16,000.

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?

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 4 or 5, and any two neighbouring terms differ by no more than 2. How many sequences will he have to write out?

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?