The problem says: "increasing finite arithmetic progression of various integer negative integers" - what does this have to say to the solver?

Question

Programming

I'm going to enter a university, and it became necessary to have part 2 of the KIM USE in mathematics. I solve part 1 without problems, but the last tasks from part 2 cause difficulties. Decided to deal with this part first. From part two, the last 7 tasks are recorded in the answer sheet number 2, where, in addition to the answer, you must write a solution. How many points will I get given that I will solve the first 14 tasks and two or three of the last seven? What tasks should be addressed in the first place?
Provided that the most recent task may be the most difficult, because. it is necessary from the decisive knowledge of the theory of numbers (the concept of prime and composite numbers) and many others, how to determine that it is within the power of everyone?
Task 21 (most recent):

An increasing finite arithmetic progression consists of various negative integers. The mathematician calculated the difference between the square of the sum of all members of the progression and the sum and the sum of their squares. Then the mathematician added the next term to this progression and again calculated the same difference.
a) Give an example of such a progression if the difference was 48 more the second time than the first time.
b) The second time the difference was 1440 more than the first time. Could the progression have originally consisted of 12 terms?
c) The second time the difference was 1440 more than the first time. What is the largest number of members that could have been in progression at first?

What knowledge do I need in order to solve such tasks?

Answer 1

Points in the USE are measured first in some "parrots" (primary score), and then transferred to other parrots (test score).
The translation scale changes every year. In 2014 it was like this .
How many primary parrots cost which task is most likely written in the KIMs themselves.
To solve the problem, you will need:

• knowledge of the topic "arithmetic progression"
  
• mastery of the method of uncertain coefficients
  
• ability to solve equations and inequalities.
  

Answer 2

You don't need to know almost anything - only the formula of the nth term and the sum of an arithmetic progression.
In this problem, everything is expressed in terms of a new term added by the mathematician (let it be -b), the number of terms that were before (n) and the progression step d (a natural number).
Having played a little with the formulas, you get that the change in the difference is D=n*b*(2*b+(n+1)*d). It is easy to make sure that b>0 should be. All you need to do is to choose natural b,d,n to satisfy the condition.
Answers:
1) -5,-4,-3 (added -2)
2) doesn't happen: b*(2*b+13*d)=120, i.e. 120/b-2*b was positive and divisible by 13. It is clear that b must be less than 8 and be a divisor of 120. Going through the possible values ​​(these are all numbers from 1 to 6), we find that there are no solutions.
3) n=15, progression -19,-18,...,-5 (added -4).
In the latter case, iterate over the possible values ​​of b * d (the smaller it is, the larger n is possible). It turns out that there are no solutions for b*d=1,2,3, but there are solutions for b*d=4.
UPD. They said above about "Olympiad problems" - I'm afraid that this is true. Olympiad problems don't require special knowledge, but you need a system of knowledge to quickly find the right path in the labyrinth of possible approaches.

Answer 3

You need knowledge about arithmetic progressions. Definition, formulas, the concept of difference, and so on, in short, a school course.