Math Challenge Problems
Monthly Math Challenge Problems
Submit your solutions to Mr. Kenyon in room 114. The overall winner can receive a medallion & also qualify to attend the US National High School Mathematics Championships.
New questions will appear around the first of each month & solutions to previous problems will be posted, along with the names of students submitting correct solutions.
1. A square is divided into three pieces of equal area by two parallel cuts, as shown. The distance between the parallel lines is 6 inches. What is the area of the square in square inches?
(Extra credit if you get this one!)
2. A student bought 3 kinds of candy. One kind costs 50 cents per piece, one costs 20 cents per piece, and the thrid costs 1 cent each. He bought 120 pieces of candy for $3.43. How many of each kind did he buy?
1. How many of the positive factors of the number 36,000,000 are not perfect squares?
2. Consider 352 to be a base 7 number. If you write that value as both a base 3 and a base 4 number, and then consider those values to be base 10 numbers, what is the positive difference between them?
(Helpful hint: the number system you're used to is base 10; there are 10 digits (counting 0). Computers use base 2 (also known as binary). There are only 2 possible digits in binary: 0's and 1's.
We use base 10 mostly because we have 10 fingers. If aliens had 6 fingers, then they most likely would have chosen base 6 to use. They would count like this: 1,2,3,4,5,10,11,12,13,14,15,20,21,22,23,24,25,30,31,32,33,34,35,40,41,42,43,44,45,50,51,52,53,54,55,100,101,...
In base 10, the number 42 means: "4 of all of our fingers, plus 2 more." In base 6, it means the same thing: "4 of all of our fingers, plus 2 more." Which, if we were looking at those aliens, we would realize meant 4*6 + 2.
In base 10, the number 5234 means "5 of all of our fingers to the 3rd power, plus 2 of all of our fingers squared, plus 3 of all of our fingers, plus 4." Or, 5*10^3 + 2*10^2 + 3*10 + 4
For the 6 fingered aliens, it also would mean "5 of all of our fingers to the 3rd power, plus 2 of all of our fingers squared, plus 3 of all of our fingers, plus 4." BUT, this means 5 * 6^3 + 2*6^2 + 3*6 + 4, which in base 10 would be 1174.
Problem 1: Two Poles: Two poles are placed vertically in the ground 50 meters apart. On a certain day at a certain time, the shadow of the taller pole falls directly on the shadow of the shorter pole, with the tips of the shadows coinciding. At that moment, the tip of the shadow is 30 meters from the shorter pole. The angle of the sun at that time is 46 degrees. What is the distance between the tops of the poles to the nearest meter?
Problem 2: Farmer Jones has 65 hens. If he had one more solid colored hen, then exactly one thrid of his hens would be speckled. From years of experience, Farmer Jones knows that one-half of the speckled hens will lay speckled eggs and that each hen and a half will lay an egg and a half in a day and a half After how many full days will Farmer Jones have 4 dozen speckled eggs to sell?