Suppose every point in a plane is colored either red or blue. Show that there is an equilateral triangle somewhere in the plane whose vertices are all the same color.

## Set 17

## Robot Stones

To test a new robot, two hundred smooth white stones are placed 3 feet apart in a long straight line on the ground. A basket is placed at the beginning of the line, 3 feet away from the first stone. The robot starts at the basket and collects the stones one by one, putting each stone in the basket before getting the next stone. The robot never moves the basket. It takes awhile. How far does the robot travel? (What is the robot’s name?)

## Sums of Series

The initial term of an arithemetic series is 1. If the sum of the first twenty terms is four times the sum of the first twelve terms, then the second term is:

## Fractions Equal 1/3

## Logs Base x and Base x2

If log_{x}10 + log_{x2} 10 = 10, and if *x* = 10^{k}, what’s *k*?

- 1/10
- 1/5
- 3/20
- 2/5
- None of these

## Given Graphs,Find |f – g|

Suppose that the functions *f(x)* and *g(x)* have the following graphs:

Find the graph which most resembles the graph of |*f(x) – g(x)*|.

## Monkey and Rope

A piece of rope is hanging over a pulley. A weight is at one end of the rope and a monkey is hanging at the other end, scratching himself. The monkey and the weight are exactly level with each other. The monkey’s age and the age of his mother total seven years. The monkey weighs as many pounds as his mother is years old. Four feet of rope weighs one pound.

The monkey’s mother is one third again as old as the monkey would be if the monkey’s mother were half as old as the monkey will be when the monkey is three times as old as the monkey’s mother was when the monkey’s mother was three times as old as the monkey was then.

The weight of the weight and the rope together come to twice the difference between the sum of the weight of the weight plus the weight of the monkey and the weight of the weight.

How long is the rope?

## Stella Racing Jasper

Stella, in spite of her age, still jogs regularly and can keep up a speed of *k* meters per second. Her young friend Jasper can jog at a speed *x* times as fast, with *x* > 1 (definitely). Suppose Jasper gives Stella a head start of y meters, and then they both start jogging at the same time in the same direction. How many meters will Jasper jog before he catches up with Stella? Your answer will be some sort of formula involving the variables in this problem.

## Maggie & Aggie Driving

Maggie and Aggie synchronize their watches, rehearse their plan one last time, and start driving their cars at exactly midnight. Maggie heads due east, and Aggie heads due north, traveling 15 mph faster than Maggie. At exactly 1:20 a.m. the two ladies are exactly 100 miles apart, as observed from an inconspicuous police glider piloted by Sulphronia the super agent, who is watching through infra-red binoculars. At what speeds are the two sisters driving?

## Race to Bus

Dino, Kyle, and Nina race the 100 meters from school to the bus each day. They have found that Dino and Kyle will reach the bus together if Dino is given a head start of 20 meters. Kyle and Nina will arrive together if Kyle is given a head start of 25 meters. If Dino and Nina want to arrive at the bus at the same time, who should start where?