Problem #33

A Senate committee has $5$ Democrats, $5$ Republicans, and $1$ Independent. In how many ways can they sit around a circular table if all the members of each party all sit next to each other? (Two seatings are considered equivalent if one is a rotation of the other.)

Correct answer gets 20 PP.

Problem #32-Solved by Walter Li

What is $123_5+123_5$?

Problem #31-Solved by Walter LI

How many zeros does $10!$ end with when expressed in base $11$?

Problem #30

How many rectangles are there in the Cartesian plane with center at the origin, all four vertices with integer coordinates, and a diagonal of length $20$?

Correct answer gets a grand prize of 50 PP.

Problem #29-Solved by Walter Li

Let $\log_4 3=x$. Then $\log_2 27=kx$. Find $k$.

Problem #28-Solved by Peter Dun

Find all values of $x$ that satisfy the equation $x=\sqrt{11-2x}+4$.

Problem #27

If $60^a=3$ and $60^b=5$, then what is $12^{[(1-a-b)/2(1-b)]$?

Correct answer gets a grand prize of 50 PP.

Problem #26

Find the area of $|ax+b|+|cy+d|\le e$ in terms of $a, b, c, d,$ and $e$.

Correct answer gets a grand prize of 50 PP.

Problem #25-Solved by Peter Dun

I went to my favorite taco stand with some friends. Between us we bought $6$ tacos and $2$ burritos, and I noticed that the ratio of the amount we spent on tacos to the amount we spent on burritos was $7:4$. Later we went back and bought $4$ tacos and $3$ burritos and spent $1$ dollar less. How much money did we spent that time?

Problem #24-Solved by Peter Dun

Find all positive integer values of $c$ such that the equation $x^2-7x+c=0$ only have roots that are real and rational.