Seventeen intelligent people all correspond with one another. Each person writes “letters” to each of the 16 others (emails, texts, facebook messages, carrier pigeons, whatever). In their letters, one of only three topics is discussed: love, death, or dragons. Each pair of correspondents always writes about the same one of these three topics. Prove that there is a group of at least three people who write to each other about the same topic.
Six points are scattered around in space so that no three of them are collinear and no four of them are coplanar. The points are named 1, 2, 3, 4, 5, and 6. Each point is connected to each other point by a tubular neon light that is either red or blue. Show that there is some neon triangle that has all of its sides the same color.
Six people who all know each other are in a room. Every two people either like each other or dislike each other. Show that there is a group of three people who either all like each other or all dislike each other.
A fireman stood on the middle rung of a ladder pouring water on a burning building. As the smoke cleared, he stepped up three rungs. But a sudden flare-up forced him to go down five rungs. He later climbed seven rungs and worked until the fire was out. At that point, he climbed the last six rungs and entered the building. How many rungs were on the ladder?
Suppose every point in a plane is colored either red or blue. Show that there is some line segment such that its two end points and its midpoint are all the same color.
Dyani and Tim are each walking at a constant speed along Front and Main Streets, respectively, which are perpendicular. When Dyani is at the intersection, Tim is still 500 yards away. In two minutes they are equidistant from the intersection. In eight more minutes they are again equidistant from the intersection. What’s the ratio of Dyani’s speed to Tim’s speed?
A freight train that is 1 mile long goes through a tunnel that is 2 miles long. If the train is traveling at 15 miles per hour, how long does it take it to pass completely through the tunnel?
Part 2: Generalize: the train is n miles long; the tunnel is m miles long, and the train is going r miles per hour. The answer this time will of course be a formula.
A cow and a goat can eat the entire contents of a pasture, including an abandoned 1958 Edsel, in 40 days. The cow and a goose can do it in 90 days, and the goose and the goat take 60 days. How long should it take all three of them chomping away together?
The value(s) of y for which the following pair of equations
may have a real, common solution are:
- 4 only
- -7, 4
- 0, 4
- no y
- all y.