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GMAT Club Olympics 2025 (Day 2): Anton, Beatrice, and Carl

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Bunuel

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Bunuel

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Bunuel

GMAT Club Official Explanation:

Anton, Beatrice, and Carl, each running at a constant rate, competed in a 30-kilometer race. The combined time Anton and Beatrice took to finish the race was exactly 3 hours longer than the time Carl took. Did Carl win the race?

Let A, B, and C be the times, in hours, that Anton, Beatrice, and Carl took to finish the race, respectively. So, we are given that:

A + B = C + 3

and are asked to find whether C is the least of the three.

(1) None of the three ran faster than 6 kilometers per hour.

This implies that the minimum time anyone could take to complete the race is 30/6 = 5 hours. Thus, A + B ≥ 10.

Let's see if Carl could have won running at this maximum rate. If he ran at the fastest rate, he would finish in 5 hours, making A + B equal to C + 3 = 5 + 3 = 8 hours. However, this is impossible since we know that A + B ≥ 10. So, if Carl cannot win even running at the fastest rate, he could not have won at all. Sufficient.

(2) Anton finished before Beatrice.

This implies that Beatrice did not win. However, either Anton or Carl could have won. For example, if A = 1, B = 10, and C = 8, Anton won; but if A = 1.5, B = 2.5, and C = 1, then Carl won. Not sufficient.

Answer: A.

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Let Ta, Tb, and Tc be the finishing times for Anton, Beatrice, and Carl.
The question is: Is Tc the shortest time?
The given equation is: Ta + Tb = Tc + 3.

(1) None of the three ran faster than 6 kilometers per hour.
The race is 30 km. This means the minimum time for any runner is 30 km / 6 km/h = 5 hours.
So, Ta, Tb, and Tc are all at least 5.

Let's rearrange the given equation: Ta = Tc + 3 - Tb.
Since Tb is at least 5, the term (3 - Tb) must be -2 or less.
This means Ta <= Tc - 2.
Anton's time was at least 2 hours shorter than Carl's time, so Anton beat Carl. Carl did not win.
This gives a definite "No" answer. Statement (1) is SUFFICIENT.

(2) Anton finished before Beatrice.
This means Ta < Tb.
Let's test two scenarios with the equation Ta + Tb = Tc + 3.
Scenario A (Carl wins): Let Tc=2. Then Ta+Tb=5. If Ta=2.4 and Tb=2.6, Carl's time is the shortest. The answer is "Yes".
Scenario B (Carl loses): Let Tc=4. Then Ta+Tb=7. If Ta=3 and Tb=4, Anton's time is the shortest. The answer is "No".
Since the answer could be Yes or No, statement (2) is INSUFFICIENT.

The correct answer is (A).

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Answer
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Explanation

Let t_a, t_b, and t_c be the times for Anton, Beatrice, and Carl.

The Question: Did Carl win? This means, is t_c the shortest time? (t_c < t_a and t_c < t_b?)

From the prompt, we get the core equation:
t_a + t_b = t_c + 3

Statement (1): "None of the three ran faster than 6 kilometers per hour."

Time = Distance / Rate. The race is 30 km.
If the maximum speed is 6 km/h, the minimum time for anyone is 30 km / 6 km/h = 5 hours.
So, t_a ≥ 5 and t_b ≥ 5.
This means their combined time t_a + t_b must be at least 5 + 5 = 10 hours.
Now use the core equation: t_c + 3 = t_a + t_b.
Since t_a + t_b ≥ 10, then t_c + 3 ≥ 10, which means t_c ≥ 7.
Carl's time is at least 7 hours, while Anton's and Beatrice's times are at least 5 hours. It's impossible for Carl's time to be the shortest.
The answer to "Did Carl win?" is a definite NO. This statement is SUFFICIENT.

Statement (2): "Anton finished before Beatrice."

This just tells us t_a < t_b. It doesn't give us any specific numbers.
Case 1 (Carl wins): Let t_c = 2. Then t_a + t_b = 5. We can have t_a = 2.4 and t_b = 2.6. Here, Carl won. The answer is YES.
Case 2 (Carl loses): Let t_c = 10. Then t_a + t_b = 13. We can have t_a = 6 and t_b = 7. Here, Carl lost. The answer is NO.
Since we can get both a "Yes" and a "No", this statement is INSUFFICIENT.

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01 Jul 2025, 08:22

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D=30km
Time by A and B = Time by C + 3
Did Carl win the race?

Or Time by C least?

S1
None of the three ran faster than 6 kilometers per hour.
This means each took time more than 5hrs (=30/6)

Say Time by A=Time by B=5hrs, then
Time by A and B = Time by C + 3
Time by C = 5+5-3 = 7
In this scenario C didn't win

Say Time by C is 5
then
Time by A and B = Time by C + 3
Time by A and B = 5+3 = 8
However, since each took more than 5 hours 8 isn't possible
In this scenario as well C didn't win

For the Time by A and B to be just 3 hrs more than Time by C, one has to be more than C and one has to be less than C

Sufficient, C cannot win

S2
Anton finished before Beatrice.
This provides no info on time taken by C
Insufficient

Answer A

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01 Jul 2025, 08:26

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* Anton, Beatrice, and Carl each ran a 30-kilometer race at constant rates
* Time_Anton + Time_ Beatrice = Time_Carl + 3 hours
* We need to determine if Carl won the race (i.e., if Carl finished before both Anton and Beatrice)
Let's denote the speeds as r_A, r_B, and r_C for Anton, Beatrice, and Carl respectively.
Using the distance formula (time - distance/rate
* Time_Anton = 30/г_A
* Time_Beatrice = 30/г_B
* Time_Carl = 30/г_C
From the given condition:
30/г_А + 30/г_B = 30/г_C+ 3
Now, let's evaluate the statements:
Statement 1: None of the three ran faster than 6 kilometers per hour.
This means r_As 6, r_B ≤ 6, and r_C ≤ 6.
From our equation: 30/_A + 30/г_B = 30/г_C+ 3
If r_C = 6 (maximum possible speed for Carl), then Time_Carl = 30/6 = 5 hours.
If both Anton and Beatrice also ran at 6 km/h, then Time_Anton + Time_Beatrice = 5 + 5 = 10 hours.
But our equation requires Time_Anton + Time_Beatrice = 5 + 3 = 8 hours.
This contradiction means Anton and Beatrice cannot both run at 6 km/h if Carl does. At least one must run slower.
However, we still can't determine if Carl won. If Anton runs at 6 km/h and Beatrice runs slower, Carl could win. But if both Anton and Beatrice run slower, we can't be sure who finished first.
Statement 1 alone is insufficient.
Statement 2: Anton finished before Beatrice.
This tells us r_A > r_B, but gives no information about Carl's speed relative to either Anton or Beatrice.
Statement 2 alone is insufficient.
Statements 1 and 2 together:
We know r_A > r_B and all speeds are ≤ 6 km/h.
From our equation: 30/r_A + 30/г_B = 30/r_C+ 3
Even with both statements, we still can't determine if Carl finished before both Anton and Beatrice. Cari could be faster than both, faster than just Beatrice, or slower than both.
Both statements together are insufficient.
The answer is E Neither statement is sufficient, even when combined.

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01 Jul 2025, 08:34

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Distance= 30 km

A+B = C+3
1) no one ran faster than 6 km/hr
To min speed = 5 hrs
C=7, A=B=5
Carl wins
Sufficient
2) A finishes before B
If C=1, A=1, B=3
Then C doesnt win
If C= 7, A=B=5
Then C wins
Not sufficient
Hence A

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Combined time was 3 hours longer than carl (A + B = C +3). 30km

1) fastest time someone could have done would be 5hours - we dont know their times though.
2) Still unable to determine what times anyone had as they could of taken 10 hours. Need a max time

Neither E

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(A) is sufficient.
Considering the speed is less than 6m/hr, in all cases, Carl's speed will be less than that of Anton and Beatrice.

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01 Jul 2025, 08:40

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Information given:
- Anton, Beatrice and Carl each run a 30-kilometer race at constant speeds
- Combined time for Anton + Beatrice is 3 hours longer than Carl's time

Question:
- Did Carl win the race? (Was Carl the one to finish first?)

Solution:
- Statement 1: None of the three ran faster than 6 kilometers per hour
- We know that A + B = C + 3
- Carl must have taken at least 30 / 6 = 5 hours to finish the race
- Anton and Beatrice together take 3 hours more, so at least 5 + 3 = 8 hours
- However, each individual could be much faster or slower, so we don't know if Carl was the fastest
- Statement 1 is insufficient

- Statement 2: Anton finished before Beatrice
- This only tells us A < B
- Does not give us any information about Carl's time relative to either
- Statement 2 is insufficient

- The statements together tell us that Carl must have taken at least 5 hours, and that Anton and Beatrice together must have taken at least 8 hours
- The statements together also tell us Anton was faster than Beatrice
- However, either could still be faster or slower than Carl
- Statement 1 and 2 together are therefore insufficient

Answer: E, the statements together are insufficient

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01 Jul 2025, 08:43

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Bunuel

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Ie Anton+Beatrice is greater than Carl by 3 hours
Anton time is less than Beatrice time
If Anton run by 6km per hour implies 5 hours then Beatrice can be 5.1 hours ie 10.1 hours implies Carl is 7.1 hours but Anton finishes before Beatrice therefore we take Beatrice to be 7.2hours then (12.2)-3=9.2 hours is Anton
therefore the statement is not sufficient

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Bunuel

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Time taken = A+B = C+3

Statement 1 tells us max speed is 6km/h
30/6 = 5 Hours = minimum time ran by any of them.

try 5:
5+5 = C+3
C = 7
Carl loses.

try A = 20, B = 5:
20+5 = C+3
C= 22

Carl loses. There is no scenario where he wins, statement 1 is sufficient

statement 2 tells us that A<B

If A = 2, B=3
5 = C + 3
C=2
is sufficient, Carl could only ever tie with Anton but not win.

Statement 2 is sufficient
D: each sufficient on their own.

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01 Jul 2025, 08:46

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Statement 1 - This tells us nothing about the winner and only gives us the information that each participant will minimum take 5 hours and nothing about their position so not sufficent.

Statement 2 - let the speed of each participant respectively be a,b,c then we know a+b=c+3
we know a<b
a+b<2b
clearly c>a and c>b
hence c is last

Statement 2 is sufficient

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Anton, Beatrice, and Carl, each running at a constant rate, competed in a 30-kilometer race. The combined time Anton and Beatrice took to finish the race was exactly 3 hours longer than the time Carl took.

Did Carl win the race?

Let us assume the time taken by Anton , Beatrice and Carl to finish the race be a, b & c hours respectively

a + b = c + 3
Is c < {a, b} ?

(1) None of the three ran faster than 6 kilometers per hour.
a, b, c <= 6
c = a + b - 3 <= 6
a + b <= 9
Case 1: a = b = 2; c = 1; Carl won the race
Case 2: a = b= 4; c = 5; Carl did not win the race
NOT SUFFICIENT

(2) Anton finished before Beatrice.
a < b
Case 1: a = 2 ; b = 4 ; c = 3; Carl did not win the race
Case 2: a = 2.5; b = 2.75; c = 2.25; Carl won the race
NOT SUFFICIENT

(1) + (2)
(1) None of the three ran faster than 6 kilometers per hour.
a, b, c < 6
(2) Anton finished before Beatrice.
a < b
Case 1: a = 2 ; b = 2.5; c = 1.5 ; Carl won the race
Case 2: a = 4; b = 5; c = 6; Carl did not win the race
NOT SUFFICIENT

IMO E

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01 Jul 2025, 08:48

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So we know Tc+3 = Ta+Tb
we need to find if Tc<Ta and Tc<Tb?

Statement 1:
all T>=5
lets say
Tc = 10 then Ta+Tb = 13 now it can be 5,8 or 5.1 7.9
Tc = 7 then Ta+ Tb = 10 then 5,5 so this is sufficient
Statement 2:
Ta = 5 Tb = 6 Tc = 8 Tc does not win
Ta = 2.1 Tb=2.9 Tc=2 Tc wins
Not Sufficient

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The prompt gives me the following info and asks me whether Carl wins the race:

A + B -> (rA + rB) x (t + 3hrs) = 30km
C -> rC x t = 30km

Using statement (1):
Let's assume Carl managed to run 6 km/h, then rC = 6 -> t = 5 hrs
This means that A and B together took 8 hrs. Whether one of A or B took 1 hour and the other took 7, or whether they both took 4 hours each, one person (or both) will always be quicker then Carl. Therefore, Carl did not win the race and statement (1) alone is sufficient.

Using statement (2):
Anton finished before Beatrice meaning rA > rB. However, we still have too many unknowns (rC and t) and therefore, statement (2) alone is not sufficient.

Answer: A)

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distance 30 km
30/Sa + 30/Sb= 3+ 30/Sc
target is Sc >Sa & Sb
#1
None of the three ran faster than 6 kilometers per hour.
speed can be 0.5 to 6
Time of A & B can be from 60 to 5
only when speed of A & B is 4 time and speed of C is 2.5 the condition is correct
also at speed of A & B is 2.5 , 5 , speed of c is 2 then also carl is slower than a & b
combined time is 15 on LHS & RHS
Carl is slower than A & B
sufficient
#2

Anton finished before Beatrice.
we can have many possibility cannot determine
OPTION A is correct

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(1) None of the three ran faster than 6 kilometers per hour.

The minimum time required by each runner = 30/6 = 5 hours

A + B = C + 3

A, B , and C are time taken by Anton, Beatrice, and Carl respectively.

As the minimum value of A + B = 10, the minimum possible time taken by C is 7, but that's not the case. Hence, we can say that Carl didn't win the race as even if he ran the fastest.

(2) Anton finished before Beatrice.

There could be multiple possibilities as we know A < B

5 + 3 = 1 + 7

In this case Carl didn't win the race

0.5 + 3 = 1 + 2.5

In this case, carl wins.

The statement is not sufficient.

Option A.

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01 Jul 2025, 08:56

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Ans: A (Statement 1 alone is Sufficient)

Anton, Beatrice, and Carl, each running at a constant rate, competed in a 30-kilometer race.
The combined time Anton and Beatrice took to finish the race was exactly 3 hours longer than the time Carl took. A + B = 3 + C
Did Carl win the race?

(1) None of the three ran faster than 6 kilometers per hour.
So, A, B, C <= 6 km/hr
With a 30 km race, Time taken by A, ,B and C has to be >= 5 Hrs
so time taken by A and B would be >= 10 hrs
and we know that C = A+B-3
C = 10 - 3 = 7 atleast (as A+B increases C increases)
That means C is the slowest, and C did not win.
[Sufficient]

(2) Anton finished before Beatrice.

lets say A takes 5 hrs and B takes 7 hrs
5+7 = 3+ C ==> C = 9 hence C is the slowest

lets say A takes 1 hrs and B takes 4 hrs
1+4 = 3+ C ==> C = 2 hence C is not the slowest

A finished before B doesn't give us any information regarding their speed relativity to C. [Not Sufficient]

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* Statement (1):
None of the three ran faster than 6 km/h.
Since the race is 30 km, time = distance / speed.
So each person’s time is at least 30/6=530/6=5 hours.
Therefore:
TA≥5TA≥5, TB≥5TB≥5
So TA+TB≥10TA+TB≥10
Then TC=TA+TB−3≥10−3=7TC=TA+TB−3≥10−3=7
So Carl’s time is at least 7 hours, and each of the others is at least 5 hours.
But this doesn’t tell us if Carl’s time is less than both of theirs.
Statement (1) is NOT sufficient.

* Statement (2):
Anton finished before Beatrice → TA<TBTA<TB
We still know TC=TA+TB−3TC=TA+TB−3, but knowing the order of Anton and Beatrice doesn’t tell us if Carl was faster than both.
Example:
If TA=4TA=4, TB=5TB=5, then TC=6TC=6, so Carl is slower than Anton and Beatrice.
Try other examples and you’ll find you still can’t be certain.
Statement (2) is NOT sufficient.

* Statements (1) and (2) together:
From (1): All times ≥ 5
From (2): TA<TBTA<TB
Still, this doesn’t tell us whether Carl’s time is less than both of theirs. We can create examples where Carl is slower or faster than either.
Statements together are also NOT sufficient.

Final Answer: E
Statements (1) and (2) TOGETHER are NOT sufficient.

Bunuel

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