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x and y are real numbers that satisfy this system of equations:
x^{2} + y^{2} =107
2xy+3x+3y =73
Find the value of x+y
Simplify this expression: 2^23  2^22  2^21  1

1 + 2 + 2^2 + ... + 2^20
Let N = 299...98200...029. There are n 9's and n 0's. Prove that N can be written as the sum of the squares of 3 consecutive natural numbers.
An addlargestdigit sequence in base 10 is defined for some n by the recursive rule a_{1}=n, a_{t}=a_{t1}+[the largest digit of a_{t1}] for t>1
For example, if n=165 the sequence begins 165, 171, 178, 186, 194, 203, 206, 212, 214, ...
What is the greatest number of successive odd terms that can be found in an addlargestdigit sequence?
Solve in reals the equation
x^{2}(2[x])=1{x}
Determine all possible values of a real number N that satisfy this equation:
N^{N1/√N = 2}
I have a sheet of stamps that I want to separate into individual stamps. How can I do it with the fewest possible tears?
Assume the stamps are all rectangular, all the same size, arranged in a uniform rectangular grid, each stamp is perforated along all 4 edges, and the sheet is very thin.
The stamps can be stacked and a square spiral tear counts as a long tear.
The Sine River is located between y = sin(x) and y = 1 + sin(x) (x in radians, not degrees).
Your starting point is on the shore, at location (0,0) and your destination is location (5,1).
Your running speed is 1 unit per minute.
Your swimming speed is 1/2 units per minute.
Your plan is to divide the trip into three legs (run, then swim, then run) each leg being a straight line. Assuming there is no current in the river (and no wind), what is the minimum travel time to the destination, and by what pathway?
If z is a complex number such that z=3, then find the maximum value of the expression zi/z+1.
A king wants his daughter to marry the smartest of 3 extremely intelligent young princes, and so the king's wise men devised an intelligence test.
The princes are gathered into a room and seated, facing one another, and are shown 2 black hats and 3 white hats. They are blindfolded, and 1 hat is placed on each of their heads, with the remaining hats hidden in a different room.
The king tells them that the first prince to deduce the color of his hat without removing it or looking at it will marry his daughter. A wrong guess will mean death. The blindfolds are then removed.
You are one of the princes. You see 2 white hats on the other prince's heads. After some time you realize that the other prince's are unable to deduce the color of their hat, or are unwilling to guess. What color is your hat?
Note: You know that your competitors are very intelligent and want nothing more than to marry the princess. You also know that the king is a man of his word, and he has said that the test is a fair test of intelligence and bravery.
The 19 numbers 472, 473, ..., 490 are juxtaposed in some order to form a 57digit number. Can any of the numbers thus obtained be prime?
Find a nontrivial integer solution to this equation: x^{2}  181y^{2} = 1.


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