CAT — Miscellaneous
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Eight gymnastics players numbered 1 through 8 underwent a training camp where they were coached by three coaches - Xena, Yuki, and Zara. Each coach trained at least two players.Yuki trained only even numbered players, while Zara trained only odd numbered players. After the camp, the coaches evaluated the players and gave integer ratings to the respective players trained by them on a scale of 1 to 7, with 1 being the lowest rating and 7 the highest.
The following additional information is known.
1. Xena trained more players than Yuki.
2. Player-1 and Player-4 were trained by the same coach, while the coaches who trained Player-2, Player-3 and Player-5 were all different.
3. Player-5 and Player-7 were trained by the same coach and got the same rating. All other players got a unique rating.
4. The average of the ratings of all the players was 4.
5. Player-2 got the highest rating.
6. The average of the ratings of the players trained by Yuki was twice that of the player strained by Xena and two more than that of the players trained by Zara.
7. Player-4’s rating was double of Player-8's and less than Player-5’s.
What best can be concluded about the number of players coached by Zara?
Exactly 3
Either 2 or 3
Exactly 2
Either 2 or 3 or 4
Exactly 2
Step 1:
From conditions (3) and (4), exactly two players received the same rating.
Let the repeated rating be N.
Then,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + N) / 8 = 4
⇒ N = 4
Hence, Players 5 and 7 both received a rating of 4.
Since Xena trained more players than Yuki, there are two possible distributions:
- Possibility 1:
- Xena trains 3 players.
- Yuki trains 2 players.
- Zara trains 3 players.
- Possibility 2:
- Xena trains 4 players.
- Yuki trains 2 players.
- Zara trains 2 players.
For Possibility 1,
X = 38/13, which is not an integer and is therefore not feasible.
For Possibility 2,
X = 3, which satisfies the conditions.
Thus,
- Xena trained 4 players with an average score of 3, giving a total score of 12.
- Yuki trained 2 players with an average score of 6, giving a total score of 12.
- Zara trained 2 players with an average score of 4, giving a total score of 8.
Step 2:
From condition (5), Player 2 received a rating of 7.
Also, Player 4 received a rating of 2, and Player 8 received a rating of 1.
The known ratings are:
- Player 2 → 7
- Player 4 → 2
- Player 5 → 4
- Player 7 → 4
- Player 8 → 1
Yuki trained only two players, whose ratings add up to 12.
The only possible combination is 7 and 5.
Since 7 belongs to Player 2, the rating 5 must belong to Player 6.
As Yuki trained only even-numbered players and only Player 6 remains, Players 2 and 6 must have been trained by Yuki.
Zara's trainees have a total score of 8.
This total can be obtained either as 2 + 6 or 4 + 4.
Since the rating 2 belongs to Player 4, who is even-numbered, and Zara trained only odd-numbered players, the 2 + 6 combination is not possible.
Therefore, Zara must have trained Players 5 and 7, both of whom have a rating of 4.
The remaining players—1, 3, 4, and 8—were trained by Xena.
Thus, the final assignments are:
- Player 1 → Rating 3/6, Trained by Xena
- Player 2 → Rating 7, Trained by Yuki
- Player 3 → Rating 6/3, Trained by Xena
- Player 4 → Rating 2, Trained by Xena
- Player 5 → Rating 4, Trained by Zara
- Player 6 → Rating 5, Trained by Yuki
- Player 7 → Rating 4, Trained by Zara
- Player 8 → Rating 1, Trained by Xena
What was the rating of Player-7?
Step 1:
From conditions (3) and (4), exactly two players received the same rating.
Let the repeated rating be N.
Then,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + N) / 8 = 4
⇒ N = 4
Hence, Players 5 and 7 both received a rating of 4.
Since Xena trained more players than Yuki, there are two possible distributions:
- Possibility 1:
- Xena trains 3 players.
- Yuki trains 2 players.
- Zara trains 3 players.
- Possibility 2:
- Xena trains 4 players.
- Yuki trains 2 players.
- Zara trains 2 players.
For Possibility 1,
X = 38/13, which is not an integer and is therefore not feasible.
For Possibility 2,
X = 3, which satisfies the conditions.
Thus,
- Xena trained 4 players with an average score of 3, giving a total score of 12.
- Yuki trained 2 players with an average score of 6, giving a total score of 12.
- Zara trained 2 players with an average score of 4, giving a total score of 8.
Step 2:
From condition (5), Player 2 received a rating of 7.
Also, Player 4 received a rating of 2, and Player 8 received a rating of 1.
The known ratings are:
- Player 2 → 7
- Player 4 → 2
- Player 5 → 4
- Player 7 → 4
- Player 8 → 1
Yuki trained only two players, whose ratings add up to 12.
The only possible combination is 7 and 5.
Since 7 belongs to Player 2, the rating 5 must belong to Player 6.
As Yuki trained only even-numbered players and only Player 6 remains, Players 2 and 6 must have been trained by Yuki.
Zara's trainees have a total score of 8.
This total can be obtained either as 2 + 6 or 4 + 4.
Since the rating 2 belongs to Player 4, who is even-numbered, and Zara trained only odd-numbered players, the 2 + 6 combination is not possible.
Therefore, Zara must have trained Players 5 and 7, both of whom have a rating of 4.
The remaining players—1, 3, 4, and 8—were trained by Xena.
Thus, the final assignments are:
- Player 1 → Rating 3/6, Trained by Xena
- Player 2 → Rating 7, Trained by Yuki
- Player 3 → Rating 6/3, Trained by Xena
- Player 4 → Rating 2, Trained by Xena
- Player 5 → Rating 4, Trained by Zara
- Player 6 → Rating 5, Trained by Yuki
- Player 7 → Rating 4, Trained by Zara
- Player 8 → Rating 1, Trained by Xena
What was the rating of Player-6?
Step 1:
From conditions (3) and (4), exactly two players received the same rating.
Let the repeated rating be N.
Then,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + N) / 8 = 4
⇒ N = 4
Hence, Players 5 and 7 both received a rating of 4.
Since Xena trained more players than Yuki, there are two possible distributions:
- Possibility 1:
- Xena trains 3 players.
- Yuki trains 2 players.
- Zara trains 3 players.
- Possibility 2:
- Xena trains 4 players.
- Yuki trains 2 players.
- Zara trains 2 players.
For Possibility 1,
X = 38/13, which is not an integer and is therefore not feasible.
For Possibility 2,
X = 3, which satisfies the conditions.
Thus,
- Xena trained 4 players with an average score of 3, giving a total score of 12.
- Yuki trained 2 players with an average score of 6, giving a total score of 12.
- Zara trained 2 players with an average score of 4, giving a total score of 8.
Step 2:
From condition (5), Player 2 received a rating of 7.
Also, Player 4 received a rating of 2, and Player 8 received a rating of 1.
The known ratings are:
- Player 2 → 7
- Player 4 → 2
- Player 5 → 4
- Player 7 → 4
- Player 8 → 1
Yuki trained only two players, whose ratings add up to 12.
The only possible combination is 7 and 5.
Since 7 belongs to Player 2, the rating 5 must belong to Player 6.
As Yuki trained only even-numbered players and only Player 6 remains, Players 2 and 6 must have been trained by Yuki.
Zara's trainees have a total score of 8.
This total can be obtained either as 2 + 6 or 4 + 4.
Since the rating 2 belongs to Player 4, who is even-numbered, and Zara trained only odd-numbered players, the 2 + 6 combination is not possible.
Therefore, Zara must have trained Players 5 and 7, both of whom have a rating of 4.
The remaining players—1, 3, 4, and 8—were trained by Xena.
Thus, the final assignments are:
- Player 1 → Rating 3/6, Trained by Xena
- Player 2 → Rating 7, Trained by Yuki
- Player 3 → Rating 6/3, Trained by Xena
- Player 4 → Rating 2, Trained by Xena
- Player 5 → Rating 4, Trained by Zara
- Player 6 → Rating 5, Trained by Yuki
- Player 7 → Rating 4, Trained by Zara
- Player 8 → Rating 1, Trained by Xena
For how many players the ratings can be determined with certainty?
Step 1:
From conditions (3) and (4), exactly two players received the same rating.
Let the repeated rating be N.
Then,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + N) / 8 = 4
⇒ N = 4
Hence, Players 5 and 7 both received a rating of 4.
Since Xena trained more players than Yuki, there are two possible distributions:
- Possibility 1:
- Xena trains 3 players.
- Yuki trains 2 players.
- Zara trains 3 players.
- Possibility 2:
- Xena trains 4 players.
- Yuki trains 2 players.
- Zara trains 2 players.
For Possibility 1,
X = 38/13, which is not an integer and is therefore not feasible.
For Possibility 2,
X = 3, which satisfies the conditions.
Thus,
- Xena trained 4 players with an average score of 3, giving a total score of 12.
- Yuki trained 2 players with an average score of 6, giving a total score of 12.
- Zara trained 2 players with an average score of 4, giving a total score of 8.
Step 2:
From condition (5), Player 2 received a rating of 7.
Also, Player 4 received a rating of 2, and Player 8 received a rating of 1.
The known ratings are:
- Player 2 → 7
- Player 4 → 2
- Player 5 → 4
- Player 7 → 4
- Player 8 → 1
Yuki trained only two players, whose ratings add up to 12.
The only possible combination is 7 and 5.
Since 7 belongs to Player 2, the rating 5 must belong to Player 6.
As Yuki trained only even-numbered players and only Player 6 remains, Players 2 and 6 must have been trained by Yuki.
Zara's trainees have a total score of 8.
This total can be obtained either as 2 + 6 or 4 + 4.
Since the rating 2 belongs to Player 4, who is even-numbered, and Zara trained only odd-numbered players, the 2 + 6 combination is not possible.
Therefore, Zara must have trained Players 5 and 7, both of whom have a rating of 4.
The remaining players—1, 3, 4, and 8—were trained by Xena.
Thus, the final assignments are:
- Player 1 → Rating 3/6, Trained by Xena
- Player 2 → Rating 7, Trained by Yuki
- Player 3 → Rating 6/3, Trained by Xena
- Player 4 → Rating 2, Trained by Xena
- Player 5 → Rating 4, Trained by Zara
- Player 6 → Rating 5, Trained by Yuki
- Player 7 → Rating 4, Trained by Zara
- Player 8 → Rating 1, Trained by Xena
Who all were the players trained by Xena?
Player-1, Player-3, Player-4
Player-1, Player-3, Player-4, Player-6
Player-1, Player-3, Player-4, Player-8
Player-1, Player-4, Player-6, Player-8
Player-1, Player-3, Player-4, Player-8
Step 1:
From conditions (3) and (4), exactly two players received the same rating.
Let the repeated rating be N.
Then,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + N) / 8 = 4
⇒ N = 4
Hence, Players 5 and 7 both received a rating of 4.
Since Xena trained more players than Yuki, there are two possible distributions:
- Possibility 1:
- Xena trains 3 players.
- Yuki trains 2 players.
- Zara trains 3 players.
- Possibility 2:
- Xena trains 4 players.
- Yuki trains 2 players.
- Zara trains 2 players.
For Possibility 1,
X = 38/13, which is not an integer and is therefore not feasible.
For Possibility 2,
X = 3, which satisfies the conditions.
Thus,
- Xena trained 4 players with an average score of 3, giving a total score of 12.
- Yuki trained 2 players with an average score of 6, giving a total score of 12.
- Zara trained 2 players with an average score of 4, giving a total score of 8.
Step 2:
From condition (5), Player 2 received a rating of 7.
Also, Player 4 received a rating of 2, and Player 8 received a rating of 1.
The known ratings are:
- Player 2 → 7
- Player 4 → 2
- Player 5 → 4
- Player 7 → 4
- Player 8 → 1
Yuki trained only two players, whose ratings add up to 12.
The only possible combination is 7 and 5.
Since 7 belongs to Player 2, the rating 5 must belong to Player 6.
As Yuki trained only even-numbered players and only Player 6 remains, Players 2 and 6 must have been trained by Yuki.
Zara's trainees have a total score of 8.
This total can be obtained either as 2 + 6 or 4 + 4.
Since the rating 2 belongs to Player 4, who is even-numbered, and Zara trained only odd-numbered players, the 2 + 6 combination is not possible.
Therefore, Zara must have trained Players 5 and 7, both of whom have a rating of 4.
The remaining players—1, 3, 4, and 8—were trained by Xena.
Thus, the final assignments are:
- Player 1 → Rating 3/6, Trained by Xena
- Player 2 → Rating 7, Trained by Yuki
- Player 3 → Rating 6/3, Trained by Xena
- Player 4 → Rating 2, Trained by Xena
- Player 5 → Rating 4, Trained by Zara
- Player 6 → Rating 5, Trained by Yuki
- Player 7 → Rating 4, Trained by Zara
- Player 8 → Rating 1, Trained by Xena
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