Arithmetic and algebraic problems

It turned out that carbon concentrations changed little over the years. However, great difficulties will arise in determining the age of buildings. The house can be built from wood that has already been used in another structure. In addition, the age of the tree itself in some cases plays a big role. Somehow scientists decided on the “age” of roadside grass. The counters registered a very small proportion of radiocarbon in the substance of roadside plants. The calculations led to an appreciating result - the “age” of the grass turned out to be equal to several years! A quick explanation was found. Exhaust gases automatically passing along the road contain carbon without the heavy isotope $^{14}$ C. The age of the oil from which gasoline is obtained is measured in millions of years, and naturally, all the radiocarbon in it has decayed.

Determining the age of a shoe warehouse of acceptable times is one of the first archaeological problems solved by physics using the radiocarbon method. This method helped determine the timing of ancient Vykanichic eruptions and establish the dates of ice ages.

Advancing on the continent, the ice destroyed plant life, people went south, marking their path with the coals of cooled ochars. The trunks of ancient trees and human sites have been preserved to this day under a layer of glacial sediment. But only after the advent of the radiocarbon method was a “schedule” of ice movement established.

Measurements have shown that over the past 40 thousand years there have been three ice ages on Earth. The latest is about 11,400 years ago. Since then, the Earth has been relatively warm.

Arithmetic problems

1. Find all numbers $x$ such that $x^3=7p+1$, where $p$ is a prime number.

2. Odd numbers are arranged in ascending order 1, 3, 5, 7, …; after which they are divided into groups containing 1, 2, 3, 4, … numbers, that is, like this: (1), (3,5), (7, 9, 11), (13, 15, 17, 19), … Prove that the sum of the numbers belonging to a group of $r$ numbers is equal to $r^3$.

Next, let the numbers be divided into groups consisting of 1, 1, 4, 3, 9, 6, …, $r^2$, $\frac{r(r+1)}{2}$, $(r+1)^2$, … numbers.

Prove that the sum of the numbers in a group of $r^2$ numbers is equal to $r^5$.

3. Prove that there exists an infinite quadruple of natural numbers $a, b, c, d$ such that $(a^3+b^3)(c^3+d^3)$ can be represented as $M^3+N^3$, where $M, N$ are natural numbers.

4. How many numbers divisible by 37 can be obtained from the number 3333777 by rearranging its digits?

5. Find rational numbers $x$ and $y$ such that $0<x<3$, $y>4$, and $x^3+y^2=3^3+4^3$.

… and algebraic

6. Write an equation whose roots are the ratios of the roots of the equation $y^3+py+q=0$.

7. Let us denote by $t_k$ the arithmetic mean of the $k$ powers of positive numbers $a_1, ..., a_n$. Prove that the sequence $t_1, t_2^{1/2}, ..., t_k^{1/k}, ...$ does not kill.