Applications of algebra
Lalo, Tyson, and Michael played a number of games of pick-up basketball. At the end, their combined points across all the games were:
They noticed that in every game, one of them scored x points, one of them scored y points, and one of them scored z points, where x > y > z and all three are distinct positive integers.
If Tyson got the highest score in the first game, who got the second highest score in the second game?
You drink half a cup of water. Then I drink half of what is left.
Then you drink half of what is left after that, and I drink half of what is left after you drink.
This continues until the cup of water is all gone.
What proportion of the cup of water did you drink by the end?
In a certain classroom, 75% of the students can speak English, 55% of the students can speak French, and 10% speak neither language.
What percentage of the students speak both languages?
Two ferry boats serve the same route on a river, but travel at different speeds. They depart from opposite ends of the river at the same time, meeting at a point 720 yards from the nearest shore.
When each boat reaches the other side, it takes 10 minutes to unload and load passengers, then begins the return trip. This time, the boats meet at a point 400 yards from the other shore.
How wide is the river?
The ferry boat problem is created by well-known puzzle author Sam Loyd.
Given the iterative formula:
xn+1 = sqrt(xn) + a
There are some values of a for which a positive starting value for x results in this formula converging to an integer.
For example, if a = 2, you would observe that xn+1 = sqrt(xn) + 2 eventually converges to 4 for any positive starting value.
What form must a take in order for this formula to converge to an integer?
Find a combination of three different positive integers x, y, and z such that:
x3 + y3 = z4
Hint: there’s a better way than brute force / trial & error.
If x is an even integer that can be written as the sum of two perfect squares, prove that x / 2 can also be written as the sum of two perfect squares.