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GMAT Club Olympics 2024 (Day 4): A cross-country runner trains by runn

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Kattu404

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11 Jul 2024, 11:07

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Route 1: City Centre – Market Square – Songbird Park – City Centre (3 miles)
Route 2: City Centre – Laurel Lane – Market Square – City Centre (3 miles)
Route 3: City Centre – City Hall – Laurel Lane – City Centre (3 miles)
Route 4: City Centre – City Hall – Market Square – Laurel Lane – City Centre (4 miles)
Route 5: City Centre – Songbird Park – Market Square – Laurel Lane – City Centre (4 miles)

The possible conditions he could take any route:

Case 1:
D1: R1+R2+R4; D2: R1+R3+R5; D3: R2+R3+R4
Case 2:
D1: R1+R2+R5; D2: R1+R3+R4; D3: R2+R3+R5

There are other cases too possible, but the point to note is: Among first three routes, there will be two routes each day and among fourth and fifth route, there will be one route each day.

With this, we can see that in each case he enter's City Centre during each route. (Therefore 9); In case of Laurel Lane, we see that the only case he can't enter Laurel Lane is route 1. (Therefore two instances need to be deducted, so 7).

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11 Jul 2024, 11:33

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10 miles can be travelled in 3,3,4 miles combination. Under this, City Centre is in all the routes, so will come thrice each day,hence 9 times total. Similarly, the train will move towards Laural Lane 7 times.

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11 Jul 2024, 11:41

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Quote:

Route 1: City Centre – Market Square – Songbird Park – City Centre (3 miles)
Route 2: City Centre – Laurel Lane – Market Square – City Centre (3 miles)
Route 3: City Centre – City Hall – Laurel Lane – City Centre (3 miles)
Route 4: City Centre – City Hall – Market Square – Laurel Lane – City Centre (4 miles)
Route 5: City Centre – Songbird Park – Market Square – Laurel Lane – City Centre (4 miles)

A cross-country runner trains by running three predetermined routes every day. Each day, her total distance travelled is 10 miles. She does not run any of the routes more than two days in a row and never avoids a route two days in a row.

For the City Centre, select the number of times the runner will run towards the city centre over the span of any 3-day period. For Laurel Lane, select the number of times the runner will run towards Laurel Lane over the span of any 3-day period. Make only two selections, one in each column.

The runner will go towards city center in every route
3 routes each day for 3 days = 9 times
City Center = 9

For Laurel Lane, we have to make sure that all rules are followed.
always select two 3 miles routes and one 4 mile route.
Now make sure that while selecting second day's route, make sure that we only repeat one of the 3 mile routes and take a new 3 mile route so that we dont end up violating the rules on the third day
For the third day follow the same strategy while selecting 3 mile routes.

day 1: route 1, route 2, route 4
day 2: route 2, route 3, route 4
day 3: route 1, route 3, route 5

calculating the number of times runner visits Laurel Lane
2 + 3 + 2 = 7 times
Laurel Lane = 7

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11 Jul 2024, 11:53

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Route 1: City Centre – Market Square – Songbird Park – City Centre (3 miles)
Route 2: City Centre – Laurel Lane – Market Square – City Centre (3 miles)
Route 3: City Centre – City Hall – Laurel Lane – City Centre (3 miles)
Route 4: City Centre – City Hall – Market Square – Laurel Lane – City Centre (4 miles)
Route 5: City Centre – Songbird Park – Market Square – Laurel Lane – City Centre (4 miles)

Criteria:
- three predetermined routes every day
- Each day total distance travelled = 10 miles
- Does not run any of the routes more than two days in a row
- Never avoids a route two days in a row
Possible routes:
Day 1: 1-2-4
Day 2: 2-3-5
Day 3: 1-3-4
number of times the runner will run towards the city centre over the span of any 3-day period: 3+3+3=9
number of times the runner will run towards Laurel Lane over the span of any 3-day period: 2+3+2 =7

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11 Jul 2024, 11:56

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11 Jul 2024, 12:16

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Bunuel

Route 1: City Centre – Market Square – Songbird Park – City Centre (3 miles)
Route 2: City Centre – Laurel Lane – Market Square – City Centre (3 miles)
Route 3: City Centre – City Hall – Laurel Lane – City Centre (3 miles)
Route 4: City Centre – City Hall – Market Square – Laurel Lane – City Centre (4 miles)
Route 5: City Centre – Songbird Park – Market Square – Laurel Lane – City Centre (4 miles)

A cross-country runner trains by running three predetermined routes every day. Each day, her total distance travelled is 10 miles. She does not run any of the routes more than two days in a row and never avoids a route two days in a row.

For the City Centre, select the number of times the runner will run towards the city centre over the span of any 3-day period. For Laurel Lane, select the number of times the runner will run towards Laurel Lane over the span of any 3-day period. Make only two selections, one in each column.

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Since the runner runs exactly 10 miles per day, she needs to choose routes that give her that (so 3+3+4). That means that each day, she must choose two options from 1-3 and one from 4-5 ($3C2*2C1=6 options$). The options are as follows:

1) 124
2) 125
3) 134
4) 135
5) 234
6) 235

I chose to use a visual approach for this problem (days 1, 2, and 3 represent what we will we use, but I added 4, 5, and 6 to show that the pattern repeats and to show that regardless of which 3-day ssection is chosen, the result will be the same):

Route Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1 x x x x
2 x x x x
3 x x x x
4 x x x
5 x x x

She will always go through city center, so as she goes on 9 runs, she will go through it 9 times. She will go through Laurel Lane in each route except for 1, so we can see she will go through there 7 times.

Answer:
City Center - 9 times
Laurel Lanes - 7 times

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11 Jul 2024, 12:32

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A cross-country runner trains by running three predetermined routes every day.

Understand these conditions carefully.

Each day, her total distance travelled is 10 miles.
She does not run any of the routes more than two days in a row (If on first day if she is running on route R1 she can continue on route R1 on next day)
and never avoids a route two days in a row. (If she is not taking route R1 on first day then on next day she should take route R1 on next day

For the City Centre (CC), select the number of times the runner will run towards the city centre over the span of any 3-day period. For Laurel Lane (LL), select the number of times the runner will run towards Laurel Lane over the span of any 3-day period. Make only two selections, one in each column.

Now if you observe the given routes. For every route that she takes, will eventually lead to city centre. Lets list down the counts in front of every route for both CC and LL
Route 1: City Centre – Market Square – Songbird Park – City Centre (3 miles) - Towards - (CC 1, LL-0)
Route 2: City Centre – Laurel Lane – Market Square – City Centre (3 miles) - Towards - (CC 1, LL-1)
Route 3: City Centre – City Hall – Laurel Lane – City Centre (3 miles) - Towards - (CC 1, LL-1)
Route 4: City Centre – City Hall – Market Square – Laurel Lane – City Centre (4 miles) - Towards - (CC 1, LL-1)
Route 5: City Centre – Songbird Park – Market Square – Laurel Lane – City Centre (4 miles) - Towards - (CC 1, LL-1)

Now, Each day, her total distance travelled is 10 miles.
This count of 10 is only possible with combination of 3 + 3 + 4 = 10
Therefore for each day she is taking 3 routes in any order.
lets form possible routes with given conditions 2 and 3. -
for 3 miles there are 3 options and for 4miles there are 2 options
number of possible routes = 3C2 * 2C1 = 6 routes.
Lets list down those routes.
1 - R1>R2>R4, 2-R2>R3>R4, 3-R1>R3>R4, 4-R1>R2>R5, 5-R2>R3>R5, 6-R1>R3>R5
well these 3 routes can arrange within themseleves but its not going to change anything that we need for answering the question.
Now lets take conditions for 3 days,
If she takes 1 - R1>R2>R4 on first day then on second day she could either take 5-R2>R3>R5 or 6-R1>R3>R5 and depending on selection of 2nd day we can make selection for 3rd day from options 3-R1>R3>R4 or 2-R2>R3>R4 but make sure you are not violating condition 2nd and 3rd.
Day 1 -------------------------- Day 2 ------------------------------ Day 3
1 - R1>R2>R4 5-R2>R3>R5 3-R1>R3>R4
1 - R1>R2>R4 6-R1>R3>R5 2-R2>R3>R4
you can form other configurations like this if you want to.

lets now move on to calculation part for these two configurations that we have made for 3 days
1st - 1 - R1>R2>R4 ---------------- 5-R2>R3>R5 ---------------- 3-R1>R3>R4
Total CC, each route has 1 run toward cc so for this route there will be 9 running towards CC = 9
Total LL = 1 - R1>R2>R4 (0+1+1) 5-R2>R3>R5 (1+1+1) 3-R1>R3>R4 (0+1+1) = Total LL = 2+3+2 = 7

For confirming again lets check for 2nd configuration of 3 day route.
2nd - 1 - R1>R2>R4 ---------------- 6-R1>R3>R5 ---------------- 2-R2>R3>R4
Again Total CC, each route has 1 run toward cc so for this route there will be 9 running towards CC = 9
Total LL = 1 - R1>R2>R4 (0+1+1) 6-R1>R3>R5 (0+1+1) 2-R2>R3>R4 (1+1+1) = Total LL = 2+2+3 = 7

Therefore, For the City Centre, number of times the runner will run towards the city centre over the span of any 3-day period = 9.
and For Laurel Lane, number of times the runner will run towards Laurel Lane over the span of any 3-day period = 7

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11 Jul 2024, 13:24

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All the routes go towards city centre. So per day 3 routes, for 3 days= 3*3 = 9 times
probable route combinations are: 124, 134,235,215,134,235,125 etc.
For any three consecutive days, 1 come twice. except 1 all routes have Lauren in them. So 9-2 = 7 times

Answer 9 & 7

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11 Jul 2024, 13:25

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Bunuel

Route 1: City Centre – Market Square – Songbird Park – City Centre (3 miles)
Route 2: City Centre – Laurel Lane – Market Square – City Centre (3 miles)
Route 3: City Centre – City Hall – Laurel Lane – City Centre (3 miles)
Route 4: City Centre – City Hall – Market Square – Laurel Lane – City Centre (4 miles)
Route 5: City Centre – Songbird Park – Market Square – Laurel Lane – City Centre (4 miles)

A cross-country runner trains by running three predetermined routes every day. Each day, her total distance travelled is 10 miles. She does not run any of the routes more than two days in a row and never avoids a route two days in a row.

For the City Centre, select the number of times the runner will run towards the city centre over the span of any 3-day period. For Laurel Lane, select the number of times the runner will run towards Laurel Lane over the span of any 3-day period. Make only two selections, one in each column.

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In order to travel 10 miles daily, She has to take 2 six miles routes & 1 four miles routes

In each route she runs from city centre & runs towards city centre, means she makes a circle & then starts from City centre & again run towards city centre but takes different route, so she runs 3 times towards city centre daily
9 times in 3 days

For Laurel, she has to run daily towards Laurel as it's common in 4 mile route so, this gives 3 times in 3 days & for 6 mile route she takes
Day1: Route 1 & Route 2, 1 times towards Laurel lane
Day2: Route 1 & Route 3, 1 times through Laurel lane
Day3: Route 2 & Route 3, 2 times through Laurel lane

Total: 3+1+1+2 = 7

She runs 9 times towards City centre & 7 times towards Laurel lane.

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11 Jul 2024, 14:08

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ANS=EC

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11 Jul 2024, 14:13

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ery route starts and ends at the City Centre. Therefore, regardless of which routes are run, the runner will always start and end each route at the City Centre.

The runner runs three predetermined routes each day, and each route starts and ends at the City Centre.

Thus, for each route, the runner will head towards the City Centre twice (once at the end of the route and once at the beginning of the next route). Over three routes in one day, the runner heads towards the City Centre 3×2=63 \times 2 = 63×2=6 times.

Over a span of three days, the runner will run towards the City Centre 6×3=186 \times 3 = 186×3=18 times. However, the City Centre is counted twice for the start and end of each day, hence the runner heads towards City Centre 9 times.

Laurel Lane Analysis

Now, let's analyze how many times the runner will run towards Laurel Lane over a span of any 3-day period.

Routes involving Laurel Lane:

Route 2: City Centre – Laurel Lane – Market Square – City Centre
Route 3: City Centre – City Hall – Laurel Lane – City Centre
Route 4: City Centre – City Hall – Market Square – Laurel Lane – City Centre
Route 5: City Centre – Songbird Park – Market Square – Laurel Lane – City Centre

Routes without Laurel Lane:

Route 1: City Centre – Market Square – Songbird Park – City Centre

The runner must run three routes each day to cover 10 miles. To achieve this distance, she must mix and match routes, but considering the requirement of not running more than 2 days in a row and not avoiding any route for 2 days in a row:

For 3-day span (assuming a good mix of rules): Let's assume in most balanced combinations following the rules avoiding and not avoiding in 3 days at least say:

Using those routes involving Laurel Lane (assuming best typical adherence mix), average 2/3 repetition frequency minimally reasonable given constraints repeat rules balancing:

Typically reasonable: Route-2, 4, 5 with Route 1 (avoiding each at least once two days then say covering full three varied) i.e 2 routes at least have Laurel typical, hence averaging 7

Thus runner towards running such realistic should average typically 6 thus for city center typical runs totals above balanced constraints:

Final Conclusion So the number times typically runner will run:

City Centre: 9
Laurel Lane: 7

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11 Jul 2024, 14:35

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Route 1: City Centre – Market Square – Songbird Park – City Centre (3 miles)
Route 2: City Centre – Laurel Lane – Market Square – City Centre (3 miles)
Route 3: City Centre – City Hall – Laurel Lane – City Centre (3 miles)
Route 4: City Centre – City Hall – Market Square – Laurel Lane – City Centre (4 miles)
Route 5: City Centre – Songbird Park – Market Square – Laurel Lane – City Centre (4 miles)

A cross-country runner trains by running three predetermined routes every day. Each day, her total distance travelled is 10 miles. She does not run any of the routes more than two days in a row and never avoids a route two days in a row.

10 miles could be done 3 miles + 4 miles + 3 miles
for 3-days, she travelled 30 miles.
3*(3 miles + 4 miles + 3 miles)
or
2*(3 miles + 4miles + 3miles) + 1*(3 miles +4miles +3miles)
as she can travel max two times on same routes.

2*(Route 1 + Route 4 + Route 2) + (Route 3 + Route 5 + Route 3)
Route 1 + Route 1 + Route 4 + Route 4 + Route 2 + Route 2+ Route 3 + Route 5 + Route 3

Now, count the City centre on these routes.
9 City Centres.

Count the Laurel Line on these routes
0 + 0 + 1 + 1+ 1+ 1+ 1 + 1 + 1
7 Laurel lines.

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11 Jul 2024, 15:23

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According to the informations given, for a given day (10miles), we necessarily should choose 2 routes from those consisting of 3 miles and 1 route from those of 4 miles,
We cannot chose 2 routes from those of 4 miles, since 4+4 is 8 and we have only 3 miles

So the number of possible routes is 3C2 x 2C1 = 3 x 2 = 6
According to the rules given, the possible routes are
124
135
234
125
134
235
if we take that the number is the number of the route given in the passage,

If we select any 3-day , we can count and find that
City Centre will be driven toward 3 times and
Laurel Lane will be driven towards 7 times

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11 Jul 2024, 16:01

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the runner runs to the city centre once in each route

in order to travel 10 miles she needs to run 2 3 miles routes and 1 4 mile route,

running 3 routes in one day, 9 routes in 3 days

she runs 9 times towards the city centre

say that the 3 miles routes are x y z

if she runs x y one day she need to run z the next day in order not to avoid a route two days in a row, so there are two possibilities:

a) 1. x y ; 2. x z

b) 1. x y ; 2. z y

and to avoid running a route two days in a row she needs to run the 3 mile route she didn´t run in day two instead of the route ran in day 1. and 2.:

a) 1. x y ; 2. x z ; 3. y z

b) 1. x y ; 2. z y ; 3. z x

repeating the logic for the next days

a) 1. x y ; 2. x z ; 3. y z ; 4. y x ; 5. z x ; 6. z y ; 7. x y 8. x z and repeats

b) 1. x y ; 2. z y ; 3. z x ; 4. y x ; 5. y z ; 6. x z ; 7. x y ; 8. z y and repeats

any three consecutive days you choose you get 2 of each route (2x 2y 2z)

so in 3 days running she´s going to have run 2 times route 1, 2 times route 2 and 2 times route 3 and 3 times either route 4 or 5.

all routes run towards Laurel Lane but route 1.

2 + 2 + 3 = 7

she runs 7 times towards Laurel Lane

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11 Jul 2024, 16:01

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The day-wise routes that can be planned are -
R1-R2-R4
R2-R3-R5
R1-R3-R4

i) Towards city centre the runner can run - 3 times per day i.e. 9 times in 3 days
ii) Towards Laurel Lane the runner can run - 2 times per day i.e. 6 times in 3 days

IMO e & b

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11 Jul 2024, 16:05

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Meeting the constrains
-10 miles per day
-does not run any of the routes more than two days in a row
-never avoids a route two days in a row
Possible combinations is
day 1 R1 R2 R4 (3+3+4) days
day 2 R2 R3 R5 (3+3+4) days
day 3 R1 R3 R4 (3+3+4) days
City Centre 9 times
DAY 1
R1 R2 R4
(R1) Songbird Park – City Centre
(R2)Market Square – City Centre
(R4) Laurel Lane – City Centre

day 2 R2 R3 R5
(R2) Market Square – City Centre
(R3) Laurel Lane – City Centre
(R4) Laurel Lane – City Centre

day 3 R1 R3 R4
(R1) Songbird Park – City Centre
(R3) Market Square – City Centre
(R4) Laurel Lane – City Centre

[b]Laurel Lane/b] 7 times
DAY 1
R1 R2 R4
(R1) NA#
(R2) City Centre – Laurel Lane
(R4) Market Square – Laurel Lane

day 2 R2 R3 R5
(R2) City Centre – Laurel Lane
(R3) City Hall – Laurel Lane
(R4) Market Square – Laurel Lane

day 3 R1 R3 R4
(R1) NA#
(R3) City Hall – Laurel Lane
(R4) Market Square – Laurel Lane

Route 1: City Centre – Market Square – Songbird Park – City Centre (3 miles)
Route 2: City Centre – Laurel Lane – Market Square – City Centre (3 miles)
Route 3: City Centre – City Hall – Laurel Lane – City Centre (3 miles)
Route 4: City Centre – City Hall – Market Square – Laurel Lane – City Centre (4 miles)
Route 5: City Centre – Songbird Park – Market Square – Laurel Lane – City Centre (4 miles)

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11 Jul 2024, 16:26

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Bunuel

Route 1: City Centre – Market Square – Songbird Park – City Centre (3 miles)
Route 2: City Centre – Laurel Lane – Market Square – City Centre (3 miles)
Route 3: City Centre – City Hall – Laurel Lane – City Centre (3 miles)
Route 4: City Centre – City Hall – Market Square – Laurel Lane – City Centre (4 miles)
Route 5: City Centre – Songbird Park – Market Square – Laurel Lane – City Centre (4 miles)

A cross-country runner trains by running three predetermined routes every day. Each day, her total distance travelled is 10 miles. She does not run any of the routes more than two days in a row and never avoids a route two days in a row.

For the City Centre, select the number of times the runner will run towards the city centre over the span of any 3-day period. For Laurel Lane, select the number of times the runner will run towards Laurel Lane over the span of any 3-day period. Make only two selections, one in each column.

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A cross-country runner trains by running three predetermined routes every day. Each day, her total distance travelled is 10 miles. She does not run any of the routes more than two days in a row and never avoids a route two days in a row.

For the City Centre, select the number of times the runner will run towards the city centre over the span of any 3-day period
10 miles can be achieved by selecting any 2 routes out of Route 1 , Route 2 and Route 3 while any one route out of Route 4 or Route 5
However regardless each route has running towards city centre . so each day 3 routes for city centre aand for span of any 3 day result in 9 days

For Laurel Lane, select the number of times the runner will run towards Laurel Lane over the span of any 3-day period.
10 miles can be achieved by selecting any 2 routes out of Route 1 , Route 2 and Route 3 while any one route out of Route 4 or Route 5
Laurel lane is available on 4 routes except Route 1.
and runner does not run any of the routes more than 2 days in a row and never avoids a route two days in a row
hence R1 can come twice in 3 routes , and rest of the routes have Laurel lane. So 7 routes will go to Laurel lane alwaysfor a span of any 3 days

Cobination wise it can be thought like this (R1 is Route 1, R2 is Route 2 , R3 is Route 3 - all with 3 points each while R4 is Route 4, R5 is Route 5 - all with 4 points each). Here any 3 days be selected R 1 always appears twice and hence twice it will not cover Laurel Lane out of 9 times.
R1-R2- -R4-
R1- -R3- -R5
-R2-R3-R4
R1-R2- - -R5
R1- -R3-R4
-R2-R3- -R5
R1-R2- -R4
R1- -R3- -R5
-R2-R3-R4
R1-R2- - -R5
R1- -R3-R4

jairovx

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11 Jul 2024, 17:15

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Given routes:

Route 1: 3 miles
Route 2: 3 miles
Route 3: 3 miles
Route 4: 4 miles
Route 5: 4 miles
Key observations:
To total 10 miles, the runner must run either two 3-mile routes and one 4-mile route or one 3-mile route and two 4-mile routes.
Running towards the City Centre:
Each route ends at the City Centre. For every 3-day period, she will run towards the City Centre three times each day for three days:
3 days×3 times per day=9

Running towards Laurel Lane:
Routes 2, 3, 4, and 5 include running toward Laurel Lane.
Let's consider possible scenarios in a 3-day period:

If she runs Route 2, 3, and 4 (or any combination that totals 10 miles).
To count the number of times she runs towards Laurel Lane, she must run one of these routes. Given she cannot avoid a route for two days, she will likely run towards Laurel Lane each day.

Hence, in any 3-day period, assuming a balanced rotation of routes:

If she runs Route 2 or 3 each day, she will run towards Laurel Lane at least once each day.
Routes 4 and 5 also include Laurel Lane, making it sure.

Conclusion:
She runs towards the City Centre: 9 times in any 3-day period.
She runs towards Laurel Lane: at least 3 times in any 3-day period.
So, the answers are:

City Centre: 9
Laurel Lane: 3

wwcd

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11 Jul 2024, 17:26

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to cover 10 miles, she must choose any two 3-mile routes and one 4 mile route.

All routes lead to the city centre. So, she runs towards the city centre three times a day. Total = 9.

Only route 1 does not lead to Laurel Lane. And since she does not avoid any route for more than 2 days in a row, she'll take route 1 on two days. So, she runs 7 times towards laurel Lane

BGbogoss

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11 Jul 2024, 18:08

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For City Center : all routes go towards city center so the answer is 3 routes * 3 days = 9

For Laurel Lane:
Since she runs 10 miles a day, she will take 2 routes from 1,2,3 and 1 route from 4,5 each day
Since she does not run any of the routes more than two days in a row and never avoids a route two days in a row she will skip 1 route from 1,2,3 and one route from 4,5 each day and run the rest.
In 3 days she will have run twice each of the routes 1,2,3 and 3 times route 4 or 5.
Since only route 1 doesn't have Laurel Lane, the answer is 2*2+3=7

Answer : 9,7

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