5. The sum of the terms of an arithmetic sequence without the last term is 77, the sum of the terms of the same sequence without the first term is 119. The sum of the first and last term is 14. Determine the first term, the difference, and the number of terms.

(6 points)

## You have 120 minutes to solve the problems.

Solve the problems independently on the provided paper, with each problem on a separate page. Do not sign the sheets, write only your code. Write your answers in ink, clearly and neatly. Draw graphs with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, cross it out.

The National Competition Committee wishes you great success.

11th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools National Competition, April 16, 2011

# Solutions to Problems and Scoring Guide

## A contestant who arrives at the solution by any correct method (even if not anticipated by the scoring guide) receives all possible points.

A correct method is one that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the contestant generally receives all the points anticipated. However, if the solution is guessed (if it cannot be arrived at without calculation), even if it is correct but without a written check or proof, such a solution is scored with 0 points.

A contestant who has only partially solved a problem, with correct procedures but no clear path to the final solution, cannot receive more than half of the possible points.

## First Year
5. We find that the difference between the first sum and the second sum is the difference between $a_{n}$ and $a_{1}$, which is 42. We set up the system of equations $a_{n}-a_{1}=42, a_{n}+a_{1}=14$ and solve it. We get the solution $a_{n}=28$ and $a_{1}=-14$. We calculate the sum $s_{n}=119+a_{1}=105$. We use the formula for the sum $s_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)$ and calculate $n=15$. We use the relationship for $a_{n}=a_{1}+(n-1) d$ and calculate $d=3$.

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