# problems 11-20 (problems only)

These problems were posted as an ongoing contest on www.dansmath.com from late 1997 to late 2008, divided into 12 seasons.

# Impossible Twins?

Anna, the older twin, was born three hours before her younger sister Lana. On Lana's 21st birthday, they both went out to a club with some friends. The bouncer was checking ID's and let Lana go in.

But when the older Anna tried to follow her in, the bouncer said, "I'm sorry, your 21st birthday isn't for another two days."

How is this possible? Explain fully and correctly to win the contest.

# The ABC's of Cars

The student parking lot has 81 cars in it; all Acuras, Beetles, and Camrys. There are half as many Acuras as Beetles, and the number of Camrys is 80% of the number of Acuras and Beetles together.

How many of each kind of car is in the parking lot? Show all steps to win contest!

# The Spider and the Fly

A spider, in the top-left-front corner of a 10 x 10 x 10 foot room, sees a big fat fly in the bottom-right-back corner. Describe the shortest path, and the length of the path, that the spider can crawl to get the fly. That's crawl, not jump, fly or spider-web express!

Your explanation must be clear. Not affiliated with the squished fly from Problem #2  ;-}

# The Intersecting Bubbles

Two overlapping spherical soap bubbles, whose centers are 50 mm apart, have radii 40 mm and 30 mm. What is the diameter D of their circle of intersection?

# The Right Numbers

(a) The area and volume of a certain sphere are both 4-digit integers times π. What is the radius?

(b) The integers 1, 3, 8, and N have the property that the product of any two, when added to 1, gives a perfect square. What is the smallest positive integer value of N?

# The Magic Number

A certain six-digit number is split into two parts; the first three digits and the last three digits are added (as 3-digit numbers), the resulting sum is squared, and the answer is the original 6-digit number!

What was the number? (There might be more than one answer!)

# How Many Ants?

At least a dozen ants are marching through my kitchen! If the ants walk in rows of 7, 11, or 13, there are 2 ants left over, while in rows of 10, there are 6 left over.

What is the smallest number of ants there could be?

# What Time Was It?

A basketball playoff game started between 3pm and 4pm, and ended between 6pm and 7pm. The positions of the minute hand and the hour hand were reversed at the end of the game, compared to the beginning.

What was the exact time the game started and ended, and how long was the game? (Try to give exact times, not rounded to the nearest anything.)

# World Cup Soccer Standings

In this year's Coupe du Monde '98, there are 4 teams in each Groupe, and they each play each of the other 3 teams once. Here are the final "Pts standings" of Groupe C, with the W . L . T . PTS records (a win is 3 pts and a tie is 1 pt):

Team                W     L     T    PTS
France              3      0     0     9
Denmark           1      1      1      4
South Africa     0      1      2     2
Saudi Arabia    0      2     1      1

How many "Pts standings" are possible, and are any gettable in more than one way? (not counting order or particular teams) This one would be called 9-4-2-1.

# What's The Average Speed?

a)  Aaron rides his bicycle at 20 kph (kilometers per hour) to his job, and 30 kph back home along the same road. What was his average speed for the round trip?

b)  Andy walks 3 mph to his class. How fast does he have to run back home in order to average 6 mph for the round trip?