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At an archaeological site, a level floor is marked with a square grid, as shown, where each square in the grid has dimensions $$5$$ feet by $$5$$ feet. Two small artifacts, $$A$$ and $$B$$, were found on the floor at the labeled points. Another small artifact, $$C$$, lies underground directly below $$B$$. If the distance between artifacts $$A$$ and $$C$$ is $$15$$ feet, how many feet below artifact $$B$$ is artifact $$C$$?

________ feet

Quantity A

$$3^{-8}-3^{-9}-3^{-10}$$

Quantity B

$$3^{-9}$$
$$2^{-1}+2^{-2}+2^{-3}$$=
Organizations F and G have 20,000 and 30,000 members, respectively. The combined membership of the two organizations is 45,000. If one member of organization F is to be randomly selected, what is the probability that the member selected will also be a member of organization G?
Digit $$x$$ is an integer between 1 and 8, inclusive, and $$y=x+1$$. If the 3-digit integer with hundreds digit $$x$$, tens digit $$y$$, and units digit 4 were subtracted from the 3 digit integer with hundreds digit $$y$$, tens digit $$x$$, and units digit 4, the result would be equal to
$$N$$ is a $$2$$-digit positive integer and the sum of its digits equal $$\frac{N}{4}$$.

Quantity A

$$N$$

Quantity B

$$36$$
If $$x$$ and $$y$$ are positive integers and $$xy=36$$, then $$(x+y)^2$$ CANNOT be
$$(x+h)(x+j)=x^2+kx+12$$ In the equation above, $$h$$, $$j$$, and $$k$$ are constants. If $$h$$ and $$j$$ are integers, which of the following could be the value of $$k$$?

Indicate all such values.
$$N$$ and $$P$$ are positive integers and $$7N+5P=39$$

Quantity A

$$5N+7P$$

Quantity B

$$39$$
If $$m$$ and $$n$$ are positive integers and $$4m+7n=39$$, how many values of m are possible?
$$x$$ and $$y$$ are positive integers less than $$10$$ and $$6x=y+29$$

Quantity A

$$x$$

Quantity B

$$y$$
A customer bought $$x$$ sweaters costing $30 each and $$y$$ skirts costing $40 each, for a total cost of $140.

Quantity A

$$x$$

Quantity B

$$y$$
A cord that is 20 meters long is cut into 4 pieces, no two of which are the same length.

Quantity A

The sum of the lengths of the longest and shortest pieces

Quantity B

9 meters


In the rectangular coordinate system shown, how many points on line segment PR have coordinates $$(x, y)$$, such that both $$x$$ and $$y$$ are integers?
$$a$$, $$b$$, $$c$$, and $$d$$ are nonnegative integers less than $$3$$.

$$a+3b+9c+27d=80$$

Quantity A

$$a+b+c+d$$

Quantity B

$$8$$
At a certain reception, 126 people are to be seated for dinner. The tables to be used for seating the people are available in two sizes, large and small, with seating capacities of 10 people and 8 people, respectively. If each table is to be filled to its respective seating capacity, what is the least number of small tables that must be used?
Inez cashed a check for $1,490 at a bank and received only $50 bills, $20 bills, and $5 bills. If Inez received at least one $50 bill, at least one $20 bill, and at least one $5 bill, what is the least possible number of bills she could have received?
$$B$$ is a positive integer, and $$A$$ is $$125$$ greater than $$B$$. Which of the following statements must be true?

Indicate all such statements.
In a sequence of integers, the first 4 terms are odd integers and each term after the $$4^{th}$$ term is the sum of the previous 4 terms. How many of the first 100 terms are even integers?
A certain triangle has an area of 7.5. If the length of a base and the corresponding height of the triangle are integers, which of the following values could be the length of the base?

Indicate all such values.

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