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Each month we will post the answer to another one of the following problems. Thanks to Elgin Johnston for creating these delights:

Problem 1

Maria recently attended a going a way party for a friend. When asked the next day how many people were at the party she said that she did not remember. However, she did remember that she was in charge of cutting the rectangular cake. She related “I made 20 straight cuts, each from from one side of the cake to another side, and I remember that there were 9 points where cut lines intersected and that no intersection point was on more than two cut lines. Everybody at the party got one piece of cake there was no cake left.”

How many people were at the party?

Problem 2

Let N be the infinite set consisting of the ordered paird (-k, k2), k=1, 2, 3, … and P the infinte set considting of the orxer pairs (k, k2, k=1,2,3, … Take a point QN from set N, a point QP from set P and construct the segment ${\over Q_NQ_P}$.

How many of these segments inersect the y-axis in the point (0, 1000)?

Problem 3

It is known that a polynomial $${P(x)}$$ of degree 19 has the form

$$P(x) = x^{19} – 17x^{18} + … + 11,$$

and that all the zeros of $${P(x)}$$ integers. White out the complete factorization of $${P(x)}$$.

Problem 4

Given that

$$sec\ \theta\ + tan\ \theta\ = 2017,$$

find the value of csc $$ \theta\ + $$ cot $$\theta.$$ The answer must be exact (no decimals).

Problem 5

For real number x the fractional part of x is denoted {x} and is defined by

$${x} = x – [x]$$

Where $$[x]$$ is the greatest integer less than or equal to $$x$$. Find a positive number $$r$$ such that

$${r} + {1/r} = 1.$$

For extra fame and fortune, find all positive solutions to this equation.