If 10 g = 16, 100 g = 196, 4 g = 4 and 12 r = 5; 300 r = 101, 30 r = 11, then what is n if n g r = 9?
Number Sense and Logical Reasoning
Function as Average
Let xk = (-1)k for any positive integer k. Let f(n) = (x1 + x2 + … + xn)/n, where n is a positive integer. Give the range of this function.
- 0
- 1/n (where n is any positive integer)
- 0 and -1/n (where n is any odd positive integer)
- 0 and 1/n (where n is any positive integer)
- 1 and 1/n (where n is any odd positive integer).
Big f(x), Find f(6)
If f(n) is a function such that f(1) = f(2) = f(3) = 1, and such that:
for n > 3, then f(6) is equal to:
- 2
- 3
- 7
- 11
- 26.
Find f(x+1) – f(x)
Hard Functional Equation, f(n) = n
The function f satisfies the functional equation
f(x) + f(y) = f(x + y) – xy – 1
for every pair x, y of real numbers. If f(1) = 1, then the number of positive integers n for which f(n) = n is:
- 0
- 1
- 2
- 3
- infinite
Sequence Formula
Consider the numbers Fn defined by the following formula:
where n may be any positive integer. Calculate F1 through F5. Do you notice anything suprising?
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)|.
Defined Operations 1; Do Composition
Suppose the operations “#” and “t” are defined so that:
6 # = 20; 2 # = 4; 10 # = 36; and 5 t = 4.5; 10 t = 7; 8 t = 6.
Then what is n if 3 # # t = n?
Euler Fortune
A father leaves at his death several children who share in his property in the following manner:
- The first child receives 100 crowns and a tenth part of what remains.
- The second child receives 200 crowns and a tenth part of what remains.
- The third child receives 300 crowns and a tenth part of what remains.
- And so on.
Now, it is found in the end that the property has been divided equally among all the children. How much was the fortune, how many children were there, and how much did each child receive?
Newton’s Cows
This problem has been attributed to Sir Isaac Newton. Three cows eat in two weeks all the grass on two acres of land, together with all the grass that grows there in the two weeks. Two cows eat in four weeks all the grass on two acres of land, together with all the grass that grows there in the four weeks. How many cows, then, will eat in six weeks all the grass on six acres of land together with all the grass that grows there in the six weeks? Assume that the quantity of grass on each acre is the same when the cows begin to graze, that the rate of growth is uniform during the time of grazing, and that the cows eat the same amount of grass each week.