# Task 4. Maximum 20 points

## Option 1

At a school, the remote stage of a team geometry tournament is taking place, where participants' results are evaluated based on the number of points earned for a fully solved problem. A complete solution to a planimetry problem is worth 7 points, and a problem in stereometry is worth 12 points. The team that scores the highest number of points wins the tournament. Andrey is organizing his team of 3 people, where he will be the captain. He is considering whether to invite Volodya and Zhanna or Petya and Galina. Therefore, he asked all the candidates to honestly provide information about their capabilities in solving problems within the allotted time for this stage of the tournament. It is known that the opportunity costs for each student in solving planimetry and stereometry problems are always constant.

| Name | Maximum number of stereometry problems if solving only them | Maximum number of planimetry problems if solving only them |
| :--- | :---: | :---: |
| Andrey | 7 | 7 |
| Volodya | 6 | 3 |
| Zhanna | 3 | 18 |
| Petya | 12 | 3 |
| Galina | 7 | 14 |

Help Andrey decide which pair of students to take into his team if the team's only goal is to win the tournament.
# Solution:

Let's find out what the maximum result the team of Andrey, Volodya, and Zhanna could achieve.

Andrey, instead of solving 1 problem in planimetry, can solve 1 problem in stereometry. Since a problem in stereometry is more valuable, he should specialize in stereometry problems, earning $12 * 7 = 84$ points for the team.

Volodya, instead of solving 1 problem in planimetry, can solve 2 problems in stereometry. Since 1 problem in planimetry is less valuable than 2 problems in stereometry, he should specialize in stereometry problems, earning $12 * 6 = 72$ points for the team.

Zhanna, instead of solving 1 problem in stereometry, can solve 6 problems in planimetry. Since 1 problem in stereometry is less valuable than 6 problems in planimetry, she should specialize in planimetry problems, earning $7 * 18 = 126$ points for the team. In total, the team of Andrey, Volodya, and Zhanna can earn a maximum of 84 + 72 + 126 = 282 points.

Let's find out what the maximum result the team of Andrey, Petya, and Galina could achieve.

Petya, instead of solving 1 problem in planimetry, can solve 4 problems in stereometry. Since 1 problem in planimetry is less valuable than 4 problems in stereometry, he should specialize in stereometry problems, earning $12 * 12 = 144$ points for the team. Galina, instead of solving 1 problem in stereometry, can solve 2 problems in planimetry. Since 1 problem in stereometry is less valuable than 2 problems in planimetry, she should specialize in planimetry problems, earning $7 * 14 = 98$ points for the team.

In total, the team of Andrey, Petya, and Galina can earn a maximum of $84 + 144 + 98 = 326$ points.

Thus, Andrey should invite Petya and Galina to his team.

## Grading Criteria:

To compare the benefits of forming each team, it is sufficient to compare the contributions of only four participants: Volodya, Zhanna, Petya, and Galina.

A correct calculation of the points each of them can bring to the team, conducted in any correct way - 4 points.

A correct calculation of the total number of points (for the team as a whole or for the pair of participants that Andrey will invite to the team) - 2 points.

A comparison of the benefits of each pair of participants and the correct answer - 2 points.

Any arithmetic error that did not lead to a significant distortion of the results is penalized by 1 point.

Any arithmetic error that led to a significant distortion of the results is penalized by 5 points.

## Variant 2

A school is holding the remote stage of a team tournament in physics, where the results of the participants are evaluated based on the number of points earned for a fully solved problem. A fully solved problem in kinematics is worth 14 points, and a problem in thermodynamics is worth 24 points. The team that scores the most points wins the tournament. Volodya is forming his team of 3 people, in which he will be the captain. He is considering whether to invite Andrey and Tatyana, or Semyon and Maria. Therefore, he asked all the candidates to honestly indicate in the table their capabilities for solving problems during the allocated time for this stage of the tournament. It is known that the opportunity costs of each student for solving problems in kinematics and thermodynamics are always constant.

| Name | Maximum number of kinematics problems if solving only them | Maximum number of thermodynamics problems if solving only them |
| :--- | :---: | :---: |
| Volodya | 7 | 7 |
| Semyon | 4 | 8 |
| Maria | 8 | 2 |
| Andrey | 2 | 12 |
| Tatyana | 2 | 1 |

Help Volodya decide which pair of students to invite to his team if the team's only goal is to win the tournament.

## Solution:

Let's find out what the maximum result the team of Volodya, Semyon, and Maria could achieve.

Volodya, instead of solving 1 problem in thermodynamics, can solve 1 problem in kinematics. Since a problem in thermodynamics is more valuable, he should specialize in thermodynamics problems, earning $24 * 7 = 168$ points for the team.

Semyon, instead of solving 1 problem in kinematics, can solve 2 problems in thermodynamics. Since 1 problem in kinematics is less valuable than 2 problems in thermodynamics, he should specialize in thermodynamics problems, earning $24 * 8 = 192$ points for the team. Maria, instead of solving 1 problem in thermodynamics, can solve 4 problems in kinematics. Since 1 problem in thermodynamics is less valuable than 4 problems in kinematics, she should specialize in kinematics problems, earning $14 * 8 = 112$ points for the team. In total, the team of Volodya, Semyon, and Maria can earn a maximum of $168 + 192 + 112 = 472$ points.

Let's find out what the maximum result the team of Volodya, Andrey, and Tatyana could achieve.

Andrey, instead of solving 1 problem in kinematics, can solve 6 problems in thermodynamics. Since 1 problem in kinematics is less valuable than 6 problems in thermodynamics, he should specialize in thermodynamics problems, earning $24 * 12 = 288$ points for the team.

Tatyana, instead of solving 1 problem in thermodynamics, can solve 2 problems in kinematics. Since 1 problem in thermodynamics is less valuable than 2 problems in kinematics, she should specialize in kinematics problems, earning $14 * 2 = 28$ points for the team.

In total, the team of Volodya, Andrey, and Tatyana can earn a maximum of $168 + 288 + 28 = 484$ points.

Thus, Volodya should invite Andrey and Tatyana to his team.

## Grading Criteria:

To compare the benefits of forming each team, it is sufficient to compare the contributions of only four participants: Semyon, Maria, Andrey, and Tatyana.

A correct calculation of the points each of them can bring to the team, conducted in any correct way - 4 points.

A correct calculation of the total number of points (for the team as a whole or for the pair of participants that Volodya will invite to the team) - 2 points.

A comparison of the benefits of each pair of participants and the correct answer - 2 points.

Any arithmetic error that did not lead to a significant distortion of the results is penalized by 1 point.

Any arithmetic error that led to a significant distortion of the results is penalized by 5 points.