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

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abhimaster

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Bunuel

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Interesting. This is a nice question. Let's say speed be A, B and C as their first name alphabet respectively.
If we consider just condition A, then if we have speed like 4, 3 for A,B then C is 4 so it is a tie. and if we have 5,4 for A,B then C is 6 so here C wins but if we consider speed be 4,1 then C be 2 so C will loose which makes Situation A not sufficient.

Simillarly, for Situation B if we consider Anton finished before Bea, which means A is faster but from main question equation of respective speed is

A + B = 3+ C

Which does not depend who out of A or B finishes first and even in that situation lets say we keep speed 6,5 for A , B then C is 8 , but if we keep speed 3,1 for A,B then speed for C is 1 so this situation is also not sufficient to answer.

Also, if we consider the situation together then also since A + B = 3 + C and we have speeds like (3,1) and (5,4) for A,B which makes result of C uncertain so both equation together is not sufficient to answer.
So, E is correct

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Let Anton, Beatrice, and Carl = A, B and C

Derived from Question = Time by A + Time by B = Time by C + 3 Hours

Statement 1 :

Assuming they all run at 6Km/H, Minimum possible time taken by them = 5 Hours.

Minimum possible time taken by A + B = 5 + 5 = 10 Hours.

Time by C = 10 - 3 = 7 Hours (Carl Did Not win), Sufficient

Statement 2 :

We get Time taken by A is less than Time taken by B. Does not prove anything here. Not Sufficient.

Hence, Answer is (A)

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Bunuel

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Given:

Anton (a), Beatrice (b), Carl (c)
Track= 30 km race
Time taken by a+ Time Taken by b = Time Taken by c + 3

Question:
Did carl Win? (Yes or No)
or we can also say was he the fastest? or did he spend the least time completing the race.

Answers can match:
either definitely first or not first (2nd 3rd we dont need to think about it)

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

So we have a floor on speed i.e 6km/h

if speed is 6km/ hr, time taken to complete the race is 30/6 i.e 5 hours. So each will take minimum 5 hours to complete the race.

Lets assume A and B both ran at speed of 6 km/hr that means both will take 5 hrs each
so we can see 5+5=10 therefore carl will take 10-3=7 hours (so carl will not win definitely)

Lets assume A ran at 6 km/ hr and B ran at 3km/hr i.e A will take 5 hrs and B will take 10 hours to complete
Total time taken by both is 5+10=15 hours i.e carl will take 15-3=12 hours. (here also carl will not win)

So we can conclude by saying Statement 1 alone is sufficient since we can definitely say Carl will not win.

Statement 2: Anton finished before Beatrice.

from this we know A faster than B but no idea about their speed related to C.
It is possible C is faster than A or Not. No definite answer can be be made whether he won or not

So Statement 2 alone is not sufficient.

Answer is A.

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If A, B, and C are the times that each one took, we know from the question that A+B=C+3
We want to know if Carl won the race.

If we assume that Carl won, C<A "and" C<B are correct simultaneously
We can add the inequalities: C+C < A+B
We know that A+B=C+3, so we can rewrite the inequality as follows: 2C < C+3 , which simplifies as C < 3
So, one condition for Carl to win is that he finishes the race in under 3 hours.

Statement 1: If the speed is less than 6 km/h, then the least amount of time for finishing the race is 5 hours, which contradicts the C < 3
So this statement is sufficient to say Carl did not win.

Statement 2: If Anton finished before Beatrice, we have A< B. This does not give any information about the amount of C., so this statement is not sufficient.

The answer is A.

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Based on given conditions,

distance of the race = D =30 km
constant rates (speeds) of Anton, Beatrice, and Carl, respectively, in km/hour = r_a, r_b,r_c
Anton and Beatrice took was exactly 3 hours longer than Carl's time : t_a +t_b = t_c +3
Anton's time to finish the race : t_a = D / r_a
Beatrice's time to finish the race : t_b = D / r_b
Carl's time to finish the race : t_c = D / r_c

We have to check, Carl win the race which mean t_c < t_a and t_b.

statment 1:None of the three ran faster than 6 kilometers per hour.
Let analyze statment 1 :
As per above statment, we can understand speed as 6kmph and distance as 30km, hence time taken by each player would be less than or equal to 5

when we consider t_a + t_b, both combined would less than or equal to 10
t_c + 3 = 10
t_c = 7, which is not less than 5.It means time taken by Carl is greator than Anton and Beatrice. Hence, Carl cannot win

Therefore, we can consider statment(1) is sufficient

statment 2:Anton finished before Beatrice.
Let's analyze statment 2, we can observe that relative finishing times of Anton and Beatrice, but it provides no information about Carl's time in relation to theirs, or about their actual speeds or times.

Hence we cannot provide finite condition where carl cannot win.

Therefore, we can consider statment(2) is not sufficient

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Anton and Beatrice together took 3 hrs longer than Carl. We need to check if Carl won?

Statement 1: None of the three ran faster than 6 kilometers per hour.
That means the fastest anyone could finish the 30 km race is: 30 ÷ 6 = 5 hrs
So Anton and Beatrice each took >= 5 hrs. Their combined time >= 10 hrs.
Since Anton and Beatrice together took 3 hrs more than Carl :
-> Carl’s time = (Anton + Beatrice) − 3 => >=7 hrs
That means Carl took at least 7 hrs.

Now, let’s think about Anton’s time:
-> Anton = (Carl’s time + 3) − Beatrice’s time
Since Beatrice took at least 5 hrs, and Carl took at least 7:
->Anton ≤ (7 + 3) − 5 = 5 hrs
But we also know Anton took at least 5 hrs, due to the speed limit.
So Anton = 5 hrs. Same for Beatrice.

Therefore, Carl did not win. He actually finished last. Sufficient

Statement 2: Anton finished before Beatrice.
That only tells us about Anton and Beatrice but nothing about Carl. Not sufficient.

Answer: A

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Bunuel

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Reading we setup the formula c = a +b - 3. so (1) means no one ran faster than 5 hours. well if both a and b ran at 5, then c is slower at 7, but what if a was 10 and B 15 then c = 22 so again no. There isnt a combination because unless the running speed is sub 3 hours than c is longer so sufficient c+3= a+b so if a or b is greater than 3 then c must increase to make the equation match. (2) tells us nothing, so A

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sA = speed of Anton
sB = speed of Beatrice
sC = speed of Carl

tA = time of Anton
tB = time of Beatrice
tC = time of Carl

tA + tB = tC + 3 -> tC = tA + tB - 3

Is tC < tA and tC < tB?

Combining the equation and the two inequalities:

tC = tA + tB - 3 < tA -> tB < 3
tC = tA + tB - 3 < tB -> tA < 3

So, it must happen that tA < 3 and tB < 3

For tA < 3 -> sA = 30/tA > 10 km/h
For tB < 3 -> sB = 30/tB > 10 km/h

Is sA > 10 and sB > 10?

(1)
As sA is not greater than 10 (neither sB is), Carl did not win the race.

SUFFICIENT

(2)
If tA = 2 and tB = 2.5 then tC = 1.5 and the answer is yes.
If tA = 2 and tB = 3.5 then tC = 2.5 and the answer is no.

INSUFFICIENT

IMO A

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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?

(1) None of the three ran faster than 6 kilometers per hour.
(2) Anton finished before Beatrice.

The correct answer to this question is (D) - either statement alone is sufficient to answer the question.

Based on the information given, Anton + Beatrice's time together took more time than Carl, which means Carl automatically wins the race of the 3 of them.

Statement 1 provides more specific information about the speeds of the participants and confirms that Carl won the race. Even if he is the fastest at 6 km per hour, the minimum time for 30 km would be 5 hours. According to the information given, the combined race time of the other two would be 5+3 = 8 hours at least. We could further calculate and approximate speeds, but to answer the question about Carl winning the race, this is enough and does not refute anything we already know.

Statement 2 provides general information about the fact that Anton beat Beatrice. It really does not provide more information about Carl winning the race, as that is already clear from the base information given.

Therefore both statements don't need to be used in conjunction and neither of them refute the fact that Carl wins the race, so the answer should be D, that the statements alone or together can be used.

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a,b,c = time for A,B,C to finish the race, respectively. We have: a + b = c + 3
For c to win, we have to see if c < a,b or not

From S1: 30/a, 30/b, 30/c <= 6 --> a,b,c ≥ 5
c+3 = a+b --> c = a + (b-3). (b - 3) > 0 --> c > a --> c never wins. S1 is sufficient.

From S2: a < b --> 2a < a+b < 2b --> (2a-3) < c < (2b-3)
Take a = 2 and b = 3 --> 1 < c < 3. c could be < 2 or > 2. We cannot answer whether c < a or not. S2 is insufficient.

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Bunuel

Given: Distance = 30; ta + tb = tc + 3; is tc < ta & tb ?;

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

Analysis: if a and b ran at Maximum Speed = 6 km/hour; Min Time = 30/6 = 5 Hours

5 + 5 = 3 + tc; tc = 7;

Let's say c ran at max speed then ta + tb = 3 + 5 = 8; this can't be since ta & tb must be greater than or equal to 5

Now, we can say that tc would always be less than ta or tb; hence, tc hasn't won the race. Sufficient.

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(1) None of the three ran faster than 6 kilometers per hour.
If none of them ran faster than 6km/h, they took more than or equal 5 hours/each. In every combination that Carl could win the difference between them would be more than 3 hours (more than 5 hours to be more precisely). Sufficient

(2) Anton finished before Beatrice.
It Ddoesn't matter who finished first, we are considering both times together, so this information can't be usefull. Insufficient

Letter A

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Time A = time Anton (A) took
Time B = time Beatrice (B) took
Time C = time Carl (C) took

Time A + Time B = Time C + 3

Carl win the race only when time Carl took was the least and smaller than Anton and Beatrice => Time C<Time A; Time C<Time B

(1) All speed of A, B, C ≤ 6km/h
Distance = 30km => Time A, B, C ≥ 5 hours (30/6=5)
Don't know actual time of each => Not sufficient

(2) Anton finished before Beatrice. => Time A < Time B
Let:
Time A = 1
Time B = 2.5
=> Time C = 1+2.5 -3 = 0.5 (< Time A & Time B)
or
Time A = 2
Time B = 4.5
=> Time C = 2+4.5-3= 3.5 (> Time A & Time B)

Carl can win or not win => Not sufficient

(1) + (2):

Time A + Time B = Time C + 3
All speeds ≤ 6 km/h => Time of each ≥ 5 hours
Time A < Time B

Let:
Time A = 5, Time B = 5.5 => Time C = 5+5.5-3=7.5
or Time A = 10, Time B 11 => Time C = 10+11-3=18

Time C in both cases is bigger than Time A & Time B. The bigger A and B are, the bigger C is => Carl could not win.

(1) + (2) Sufficient

Answer: C

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Statement 1

Speed limit = 6 km/h.

30 km ÷ 6 km/h = 5 h, so each runner took ≥ 5 h.

Plug the minimums in:
tc = ta + tb − 3 ≥ 5 + 5 − 3 = 7 h.
Carl’s time is always longer than both others → he never wins.
Sufficient

Statement 2

Anton finished before Beatrice (ta < tb).
We can pick numbers that fit the 3-hour rule where:

Carl wins: ta = 2 h, tb = 2.5 h ⇒ tc = 1.5 h
Carl loses: ta = 4 h, tb = 5 h ⇒ tc = 6 h
Both are possible → Not sufficient.

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timeA + timeB = timeC + 3 -> timeC = timeA + timeB - 3

To win Carl, timeC must be less than time A and less than timeB

(1)
time = 30/speed
As speed is less than or equal to 6, time is greater than or equal to 5.

Taking minimum values for timeA and timeB:
timeC = timeA + timeB - 3 = 5 + 5 - 3 = 7

timeC is, at least, 7.
Taking greater values for timeA and timeB is even worse. So Carl didn't win.

Statement (1) alone is sufficient.

(2)
timeA=1, timeB=2:30 -> timeC=0:30. Carl won
timeA=1, timeB=4 -> timeC=2. Carl didn't win

Statement (2) alone is insufficient.

Answer A

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

the minimum time that could be taken to finish the race in this case would be 30/6 = 5 hours.
Let's suppose Carl (C) won the race by completing it in 5 hours.
Then Anton & Beatrice (A & B) would take 5+3 = 8 hours.
Now for any possible combination of A& B, atleast one of A or B would take lesser time than C
>> 4,4 -> both A and B take lesser time than C > C did not win
>> 3,5 -> A takes lesser time than C > C did not win
>> 5,3 -> B takes lesser time than C > C did not win
...and so on
Clearly all cases will contradict our assumption that C won the race.
So, we can see there is no way Carl will win this race. Hence statement 1 is sufficient.

Statement (2) Anton finished before Beatrice.

There could be cases where A took lesser time than B, but C could have taken lesser/more time than A. We don't know for sure if Carl will win or lose in this case.

For example, let C take 1 hour to complete, then A+B take 4 hours
>here A could take 1.5 hrs, while B could take 2.5 hours. In this case, C would win
>or A could take 0.5 hours, while B could take 3.5 hours. In this case, A would win
So we are really not sure - C could win or lose.

Statement B is not sufficient.

Answer: A

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Assume C ran at 6kmph then it will take 5hr; A + B = 8 hours => each may take 6 and 2 hours resp. giving the situation C may not win. (6 is considered) because we are assuming the limit to 5.99 hours.

Now if C ran at 5kmph then it will take 6 hours; A+B = 9 hours => each may take 4.5 hours resp; C did not win.

hence C did not win.

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Anton's time + Beatrice's time = Carl's time + 3 hours

(1)
No one ran faster than 6 km/h, meaning everyone took at least 5 hours to finish (since 30 km / 6 km/h = 5 hours).
Using the previous equation, there is no way of doing Carl's time less than Anton's time and Beatrice's time if all the times are, at least, 5.

Sufficient

(2)
Anton's time = 1 hour 30 minutes, Beatrice's time = 2 hours, Carl's time = 30 minutes, Carl wins
Anton's time = 1 hour 30 minutes, Beatrice's time = 3 hours 30 minutes, Carl's time = 2 hours, Carl doesn't win

Insufficient

Correct answer is A

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From the question: a, b, c are the respective rates.

30/a+30/b = 3+ 30/c
1/a + 1/b = 3/10 + 1/c

I) inssuficient, It doesnt give me info to arrive to the conclusion that c is the max

II) a is greater than b. We can try choices

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Bunuel

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Let
Anton's speed be 'A' and time be T(A)
Beatrice's speed be 'B' and time be T(B)
Carl's speed be 'C' and time be T(C)

The race is 30 km

As per the question stem
$\frac{30}{A} + \frac{30}{B} = \frac{30}{C} + 3$

(1) None of the three ran faster than 6 kilometers per hour.
=> A, B, C <= 6 kmph
=> T(A), T(B), T(C) >= 5 hours to complete the race
If we assume C completed the race in 5 hours we get

T(A) + T(B) = T(C) + 3
=> T(A) + T(B) = 5 + 3
=> T(A) + T(B) = 8
=> T(A) = 4 and T(B) = 4 or T(A) = 3 and T(B) = 5
This is not true as either T(A) or T(B) or both may be below 5

Going higher if T(C) = 8 we get
T(A) + T(B) = 8 + 3
T(A) + T(B) = 11
=> T(A) = 5.5 and T(B) = 5.5 or T(A) = 4 and T(B) = 7
Here, also we have one value or both below T(C)

Similarly for T(A) + T(B) = 9 or 10 or 11........
However, in all these cases the value of either T(A) or T(B) or both is below T(C)
Hence Carl never wins the race.
For Carl to win the race his time has to be lower than 5 hours and that is not possible as per the statement.

Sufficient.

(2) Anton finished before Beatrice.
T(A) + T(B) = T(C) + 3
If T(C) = 2, then
=> T(A) + T(B) = 2 + 3
=> T(A) + T(B) = 5
Possibility of T(A) = 2.5 and T(B) = 2.5

But it is given that
T(A) < T(B) hence we can say
T(A) = 2.4 and T(B) = 2.6
Carl wins in this scenario

Let's try another example
T(A) + T(B) = T(C) + 3
If T(C) = 6, then
=> T(A) + T(B) = 6 + 3
=> T(A) + T(B) = 9
Possibility of T(A) = 4.5 and T(B) = 4.5
But it is given that
T(A) < T(B) hence we can say
T(A) = 4 and T(B) = 5
However, Carl loses here.
So, Carl can win or lose as per this statement
Not Sufficient

IMHO Option A

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Top GMAT Focus Debriefs

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards