A function $f (x)$ is defined for all positive integers. If $f (a) + f (b) = f (ab)$ for any two positive integers $a$ and $b$ and $f (3) = 5$, what is $f (27)$?
Rectangle $ABCD$ is shown with $AB = 6$ units and $AD = 5$ units. If $AC$ is extended to point $E$ such that $AC$ is congruent to $CE$, what is the length of $DE$?
The digits of a 3-digit integer are reversed to form a new integer of greater value. The product of this new integer and the original integer is 91,567. What is the new integer?
Diagonal $XZ$ of rectangle $WXYZ$ is divided into three segments each of length 2 units by points $M$ and $N$ as shown. Segments $MW$ and $NY$ are parallel and are both perpendicular to $XZ$. What is the area of $WXYZ$? Express your answer in simplest radical form.
A spinner is divided into 5 sectors as shown. Each of the central angles of sectors 1 through 3 measures $60^\circ$ while each of the central angles of sectors 4 and 5 measures $90^\circ$. If the spinner is spun twice, what is the probability that at least one spin lands on an even number? Express your answer as a common fraction.
The student council at Round Junior High School has eight members who meet at a circular table. If the four officers must sit together in any order, how many distinguishable circular seating orders are possible? Two seating orders are distinguishable if one is not a rotation of the other.
Initially, a chip is placed in the upper-left corner square of a $n\times m$ grid of squares as shown. The chip can move in an $L$-shaped pattern, moving two squares in one direction (up, right, down or left) and then moving one square in a corresponding perpendicular direction. What is the minimum number of $L$-shaped moves needed to move the chip from its initial location to the square marked “$X$”?
On line segment $AE$, shown here, $B$ is the midpoint of segment $AC$ and $D$ is the midpoint of segment $CE$. If $AD = 17$ units and $BE = 21$ units, what is the length of segment $AE$? Express your answer as a common fraction.
There are twelve different mixed numbers that can be created by substituting three of the numbers $1$, $2$, $3$ and $5$ for $a$, $b$ and $c$ in the expression $a\frac{b}{c}$ , where $b < c$. What is the mean of these twelve mixed numbers? Express your answer as a mixed number.
If $2016$ consecutive integers are added together, where the $999^{th}$ number in the sequence is $1,244,584$, what is the remainder when this sum is divided by $6$?
Consider a coordinate plane with the points $A(−5, 0)$ and $B(5, 0)$. For how many points $X$ in the plane is it true that $XA$ and$XB$ are both positive integer distances, each less than or equal to 10?
The function $f (n) = a\cdot n! + b$, where $a$ and $b$ are positive integers, is defined for all positive integers. If the range of $f$ contains two numbers that differ by 20, what is the least possible value of $f (1)$?
In the list of numbers $1, 2, \cdots, 9999$, the digits $0$ through $9$ are replaced with the letters $A$ through $J$, respectively. For example, the number $501$ is replaced by the string $FAB$ and $8243$ is replaced by the string $ICED$. The resulting list of $9999$ strings is sorted alphabetically. How many strings appear before $CHAI$ in this list?
A 12-sided game die has the shape of a hexagonal bipyramid, which consists of two pyramids, each with a regular hexagonal base of side length 1 cm and with height 1 cm, glued together along their hexagons. When this game die is rolled and lands on one of its triangular faces, how high off the ground is the opposite face? Express your answer as a common fraction in simplest radical form.
Starting at the origin, a bug crawls 1 unit up, 2 units right, 3 units down and 4 units left. From this new point, the bug repeats this entire sequence of four moves 2015 more times, for a total of 2016 times. The coordinates of the bug’s final location are $(a, b)$. What is the value of $a + b$?
A rectangular piece of cardboard measuring 6 inches by 8 inches is trimmed identically on all four corners, as shown, so that each trimmed corner is a quarter circle of greatest possible area. What is the perimeter of the resulting figure? Express your answer in terms of $\pi$.
A fair coin is flipped four times. Written as a percent, what is the probability of getting two heads and two tails, in any order? Express your answer to the nearest tenth.
The areas of three faces of a rectangular prism are 54, 24 and 36 units . What is the length of the space diagonal of this prism? Express your answer in simplest radical form.
Two 8-inch by 10-inch sheets of paper are placed flat on top of a 2-foot by 3-foot rectangular table. Nothing else is on the table, and the area of the table not covered by the sheets of paper is $708 in^2$. In square inches, what is the area of the overlap between the two sheets of paper?
Each card in a particular deck of cards contains a number denoting its value from 2 to 6, inclusive. The deck is made up of four cards of each value for
a total of 20 cards. If two of these cards are chosen at random and without replacement, what is the probability that the sum of their values is less than 10? Express your answer as a common fraction.
If $(x^2 + 3x + 6)(x^2 + ax + b) = x^4 + mx^2 + n$ for integers $a, b, m$ and $n$, what is the product of $m$ and $n$?
The polygon shown here is constructed from two squares and six equilateral triangles, each of side length 6 units. This polygon may be folded into a polyhedron by creasing along the dotted lines and joining adjacent edges as indicated by the arrows. What is the volume of the resulting polyhedron? Express your answer in simplest radical form.
Billy wrote a sequence of five numbers on the board, each an integer between $0$ and $4$, inclusive. Penny then wrote a sequence of five numbers that measured some statistics about Billy’s sequence. In particular, Penny first wrote down the number of $0$s in Billy’s sequence. Then Penny wrote the number of $1$s in Billy’s sequence, and then the number of $2$s, the number of $3$s, and finally the number of $4$s. It turned out that Penny’s sequence was exactly the same as Billy’s! What was this sequence? Express your answer as an ordered $5$-tuple.
If $x$ is a number such that $3^x + 3^{x+2} = 9^x + 9^{x+2}$, then what is the value of $3^x$? Express your answer as a common fraction.
For each positive integer $n$, $a_n = 9n + 2$ and $b_n = 7n + 3$. If the values common to both sequences are written as a sequence, the $n^{th}$ term of that sequence can be expressed as $pn + q$. What is the value of $p − q$?