12 Days of Christmas GMAT Competition - Day 5: At a Football Champions

Ans: B
At a Football Championship, 5 players from the Spanish team each scored at least one goal. Did any of them score at least 5 goals?

(1) Together, the 5 players scored a total of 20 goals during the tournament.

P1	P2	P3	P4	P5	Players
4	4	4	4	4	Option-1
4	4	4	5	6	Option-2

(Not Suff)

(2) No two of the 5 players scored the same number of goals.

P1	P2	P3	P4	P5	Players
1	2	3	4	5	Atleast
1	2	3	4	8	Option-2

(Suff)

All the five players have at least one goal

Statement 1: 5 players scored 20 goals, so the goals can be 4 each or 1,2,4,3,10. hence this statement is insufficient

Statement 2: no two players have same number of goals. this information is also not sufficient

Combining both statements: we know that to score 20 goals by 5 players at least one has to score 5 or more goals, hence both statements are required.

Answer = C

Statement (1):
The 5 players scored a total of 20 goals.
- The goals could be distributed as 4, 4, 4, 4, 4 (none scored 5+ goals) or 1, 2, 3, 4, 10 (one scored 10 goals).
- Since both possibilities exist, A is insufficient.

Statement (2):
No two players scored the same number of goals and we also know each player scored at least 1 goal
- The distinct scores even with the least possible goals are 1, 2, 3, 4, 5. So one player scored at least 5 goals.
- Sufficient

Answer : B

Atleast one goal by each of the 5 players.
Did any of them score atleast 5 goals?
(1)
When we consider a case where all of them scored equal number of goals where the total goals are 20.
4 4 4 4 4
Just this case only none of the players have scored atleast 5 goals
So (1) is not sufficient.
(2)
The sum of five different numbers is 20, and at least one of the players must score more than 5 goals. Additionally, each player scores a different number of goals.
So (2) alone is not sufficient
But both together are sufficient

At a Football Championship, 5 players from the Spanish team each scored at least one goal. Did any of them score at least 5 goals?

(1) Together, the 5 players scored a total of 20 goals during the tournament.
(2) No two of the 5 players scored the same number of goals.

By Statement A
S1, S2, S3, S4, S5
S1+S2+S3+S4+S5 = 20

Assuming All Goals 4 goal, so None Scored 5 Goals
Assuming 1 Scored 6 goals, so 1 Scored 5 Goals

So, Insufficient

By Statement B
No two players scored Same Goals

So We can infer that from it

hence answer is C

1. 4,4,4,4,4 ..No
2,3,4,5,6 ..Yes
NOT SUFFICIENT

2. Minimum 1 goal. Every one different with minimum goals
1,2,3,4,5 ...so atleast 1 person with atleast 5 goals
SUFFICIENT

Answer B

Let’s analyze the question and solve step by step.

Restate the Problem

• There are 5 players, each scored at least 1 goal.
• Total goals scored by the team: 20.
• Question: Did any player score at least 5 goals?

[hr]
Statement (1): Together, the 5 players scored 20 goals.

• This only provides the total number of goals scored, but it does not specify the distribution.
• Example 1: If the distribution is 4,4,4,4,4, no player scored 5 goals.
• Example 2: If the distribution is 5,5,5,3,2 at least one player scored 5 goals.

Statement (1) alone is insufficient.

AD can be eliminated, Now BCE
[hr]
Statement (2): No two players scored the same number of goals.

• If no two players scored the same number of goals, the scores must be distinct integers.
• Since each player scored at least 1 goal, the possible scores are 1,2,3,4,51, 2, 3, 4, 5.
• Adding these: 1+2+3+4+5=15, which is less than 20.
• To make the total 20, we must increase the highest score(s). For example:
  
  • Scores could be 2,3,4,5,6 (sum = 20).
  • Here, at least one player (6 goals) scored more than 5 goals. Hence always there will be 1 player who score 5 or more...

Statement (2) alone is sufficient.


Hence IMO B

Number of players = 5 players

Each player score atleast one goal.
Question : Did any of them score atleast 5 goals ?

Statement 1 :

Total number of goals = 20
Minimum number of goals by all 5 = 5

Rest 15 goals could have been distributed by the players in any order

Statement 1 alone is not sufficient

Statement 2 : No two of the 5 players scored the same number of goals

Throughout the championship, the 5 players could have had any number of goals - statement 2 is not sufficient alone

Statement 1 and Statement 2 together :

15 goals to be distributed between 5 players that nobody gets the same number of goals.

Possible ways to distribute these 15 goals :

(1, 2, 3, 4, 5) or (0, 1, 2, 3, 9)

Which makes goals scored by players as (2, 3, 4, 5, 6) or (1, 2, 3, 4, 10) - counting the atleast one goal they had scored

This answers the question if any of the players scored atleast 5 goals - yes

Statement 1 and statement 2 together is sufficient (OPTION C)

Using 1) If the total goals were 20, then each player could have scored 4 each or one player alone could have scored 16 and the rest could only have scored 1 each. Hence not sufficient


Using 2) If no two players scored same, then the possible distribution can be- Let A to E be the players

A-1
B-2
C-3
D-4
E-5

No matter how will distribute, we will have someone score at least 5, Hence sufficient

Answer (B) IMO

Statement (1) : Together, the 5 players scored a total of 20 goals during the tournament.
This statement alone is not sufficient. while we know the total no of goals, we don't know how they were distributed among the players. It's possible that one player scored 5 or more goals, but it's also possible that the goals were more evenly distributed ( e.g., 4 goals each for all 5 players)

Statement (2) : No two of the 5 players scored the same number of goals.
This statement alone is not sufficient. while we know each player scored a unique no of goals, we don't know the total no of goals or the range of goals scored by individual players

Combining both statements:
When we combine both statements we can determine the answer:
i) we know there are 20 total goals scored by 5 players
ii) each player scored a different no of goals
iii) each player scored at least one goal
The only possible combination that satisfy these condition is:
1, 2, 3, 4, and 10 goals
Any other combination would either violate the condition that no two players scored the same number of goals, or it would exceeds the total of 20 goals
Therefore we conclude that one player must have score at least 5 goals( in fact 10 goals)
Conclusion: The statement together are sufficient to answer the question

If G = avg of goals we want to know if G1-5 >= 5 goals.
We also know that min(G1-5) = 1

(1) 5G = 20 GOALS total but this could be either G1 = 1, G2 =1, G3=1, G4=1, G5=16 or G1 = G2 =G3=G4 =G5 = 4 goals each one.
Not sufficient

(2) If no two of them scored the same goals -> Knowing that min(G1-5) = 1
G1 = 1
G2 =2
G3 =3
G4=4
G5=5

Statement (2) alone is sufficient. (B)

IMO C

Statement (1): Together, the 5 players scored a total of 20 goals during the tournament.
If the total number of goals scored by the 5 players is 20, we can explore the distribution of these goals among the players. Let's denote the number of goals scored by the players as g1,g2,g3,g4,g5g1​,g2​,g3​,g4​,g5​ where g1+g2+g3+g4+g5=20g1​+g2​+g3​+g4​+g5​=20 and each gi≥1gi​≥1.
To see if at least one player scored 5 or more goals, consider the scenario where no player scores 5 or more goals. The maximum each player could score would be 4 goals. If each of the 5 players scored the maximum of 4 goals, the total would be 4×5=204×5=20 goals. This scenario is possible, so statement (1) alone does not definitively tell us if any player scored at least 5 goals.
Statement (2): No two of the 5 players scored the same number of goals.
This statement implies that each player scored a unique number of goals. Given that there are 5 players, the possible unique goal counts could be 1, 2, 3, 4, and 5 (or higher). Since the goal counts must be unique and there are only 5 players, the minimum and maximum goal counts must span at least 5 different values.
Combining both statements:
From statement (1), we know the total number of goals is 20. From statement (2), we know that no two players scored the same number of goals. Let’s consider the smallest possible unique values that add up to 20:
1 + 2 + 3 + 4 + 5 = 15 (this sum is too low).
To reach a sum of 20 with unique values, we need to use larger numbers. Let's try:
1 + 2 + 3 + 4 + 10 = 20.
In this case, one player scored 10 goals, which is more than 5 goals. This confirms that at least one player scored 5 or more goals.
Thus, combining both statements, we can definitively conclude that at least one player scored at least 5 goals.
Answer: C. Both statements together are sufficient to answer the question, but neither statement alone is sufficient.

We know at least each and eveyone one of them scored one goal, thus minimum value of total goals is 5. To get to the answer we need more than that.


(1) T = 20 but this total can be obtained by different combinations without a unique answer. Not Sufficient.
(2) A,B,C,D,E are the scorers and every integer is different, thus starting with the minimum scored goals of 1, we get the consectuvie minimum numbers 2,3,4 & 5. Making it a yes and definite answer. 2 is sufficient and thus, Answer (B)

5 players -> Atleast one goal each. Did any one of them score at least 5 goals?

(1) a + b + c + d + e = 20

If, 4 + 4 + 4 + 4 + 4 = 20 then No
If, 2 + 2 + 3 + 4 + 9 = 20 then Yes

Since both answers are possible, this statement is not sufficient.

(2) No two of the 5 players scored the same number of goals.
And as per the premise, the minimum number of goals each player has scored is 1.

1, 2, 3, 4, 5 then Yes
2, 3, 4, 5, 6 then Yes
and for any other case as well, it will be Yes.

This statement is sufficient.

[b] is the correct answer.[/b]

At a Football Championship, 5 players from the Spanish team each scored at least one goal.

This means at least 5 goals have been scored.

(1) Together, the 5 players scored 20 goals during the tournament.
From this, we can say that 15 goals are unaccounted for since each one has scored one goal. Not Sufficient.

(2) No two of the 5 players scored the same number of goals.
This info can't provide anything alone. Not Sufficient.

But together you can calculate something.

Those 15 goals should be scored such that no two players have the same no of goals, and that way you surely can say that, yes some player has scored at least 5 goals.

(1) If the number of goals of each player are {4,4,4,4,4} the answer is no.
If the number of goals of each player are {5,3,4,4,4} the answer is yes.

INSUFFICIENT

(2)
There is no way to answer no to the question because choosing the minimum possible values we have {1,2,3,4,5}. And the answer is yes.

SUFFICIENT

IMO B

The correct answer is B

Given: 5 players from the Spanish team each scored at least one goal.
Question: Did any of them score at least 5 goals?

Statement 1:
5 players scored 20 goals together during the tournament - This is insufficient as any player can score either less than 5 or more than 5 goals out of the remaining 15 goals. We know each player has scored at least 1 goal out of the 20.

Statement 2:
No two of the 5 players scored the same number of goals - This is sufficient as it is given that no two players can score the same number of goals. If we start with 1 goal, the remaining players can score only distinct numbers - e.g., 2,3,4,5, or any combination.