### Author Topic: Test 19 corrections  (Read 4572 times)

#### yongcheng3315

• USAMO Winner
• Posts: 2585
##### Test 19 corrections
« on: March 17, 2012, 07:18:05 PM »
Test 19 corrections

(1)

Question 8:  It takes Alex 15 minutes to catch Betsy the first time and to run a total of 16 rounds more..."  means

Alex uses 15 minutes to catch Betsy the first time. When he catches Betsy the first time, Betsy runs x rounds and he runs (16 + x) rounds.

(2)

Test 19 #5
D is not labeled on diagram

#### yongcheng3315

• USAMO Winner
• Posts: 2585
##### Re: Test 19 corrections
« Reply #1 on: March 17, 2012, 07:26:04 PM »
#8 Solution:

Method 1:

Let d be the circumference of the circle and x be the number of rounds Betsy runs:

(25/60)[(16+x)d/15 + xd/15] = d

Solve for x: x = 10.

Method 2:

Let Va and Vb be the speeds of Alex and Betsy, respectively.

(25/60)(Va + Vb) = d    (1)
15(Va - Vb) = 16d        (2)

Solve for Va/Vb.

Then Va/Vb = (16+x)/x.

x = 10.

« Last Edit: March 17, 2012, 07:27:09 PM by A 2015 »