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2016 Mathcounts State Sprint Problems

Note to all: all figures can be found here (Don't have time. If you can do an asymptote, please do)

Problem 1

Let $a@b$ = $\frac{a}{2a+b}$. What is the value of $5@3$? Express your answer as a common fraction.

Solution

Problem 2

How many rectangles of any size are in the grid shown here?

Solution

Problem 3

Given $7x+13=328$, what is the value of $14x+13$?

Solution

Problem 4

What is the median of the positive perfect squares less than $250$?

Solution

Problem 5

If $\frac{x+5}{x-2}=\frac{2}{3}$, what is the value of $x$?

Solution

Problem 6

In rectangle $TUVW$, shown here, $WX=4$ units, $XY=2$ units, $YV=1$ unit and $UV=6$ units. What is the absolute difference between the areas of triangles $TXZ$ and $UYZ$?

Solution

Problem 7

A bag contains $4$ blue, $5$ green and $3$ red marbles. How many green marbles must be added to the bag so that $75$ percent of the marbles are green?

Solution

Problem 8

MD rides a three wheeled motorcycle called a trike. MD has a spare tire for his trike and wants to occasionally swap out his tires so that all four will have been used for the same distance as he drives $25000$ miles. How many miles will each tire drive?

Solution

Problem 9

Lucy and her father share the same birthday. When Lucy turned $15$ her father turned $3$ times her age. On their birthday this year, Lucy’s father turned exactly twice as old as she turned. How old did Lucy turn this year?

Solution

Problem 10

The sum of three distinct $2$-digit primes is $53$. Two of the primes have a units digit of $3$, and the other prime has a units digit of $7$. What is the greatest of the three primes?

Solution

Problem 11

Ross and Max have a combined weight of $184$ pounds. Ross and Seth have a combined weight of $197$ pounds. Max and Seth have a combined weight of $189$ pounds. How many pounds does Ross weigh?

Solution

Problem 12

What is the least possible denominator of a positive rational number whose repeating decimal representation is $0.\overline{AB}$, where $A$ and $B$ are distinct digits?

Solution

Problem 13

A taxi charges $3.25 for the first mile and $0.45 for each additional $\frac{1}{4}$ mile thereafter. At most, how many miles can a passenger travel using $13.60? Express your answer as a mixed number.

Solution

Problem 14

Kali is mixing soil for a container garden. If she mixes $2$ $m^3$ of soil containing $35\%$ sand with $6$ $m^3$ of soil containing $15\%$ sand, what percent of the new mixture is sand?

Solution

Problem 15

Alex can run a complete lap around the school track in $1$ minute, $28$ seconds, and Becky can run a complete lap in $1$ minute, $16$ seconds. If they begin running at the same time and location, how many complete laps will Alex have run when Becky passes him for the first time?

Solution

Problem 16

The Beavers, Ducks, Platypuses and Narwhals are the only four basketball teams remaining in a single-elimination tournament. Each round consists of the teams playing in pairs with the winner of each game continuing to the next round. If the teams are randomly paired and each has an equal probability of winning any game, what is the probability that the Ducks and the Beavers will play each other in one of the two rounds? Express your answer as a common fraction.

Solution

Problem 17

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 $f(27)$?

Solution

Problem 18

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 $DE$?

Solution

Problem 19

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 $91567$. What is the new integer?

Solution

Problem 20

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.

Solution

Problem 21

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.

Solution

Problem 22

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.

Solution

Problem 23

Initially, a chip is placed in the upper-left corner square of a 15 × 10 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 “$X$”?

Solution

Problem 24

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.

Solution

Problem 25

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.

Solution

Problem 26

If $738$ consecutive integers are added together, where the $178^{\text{th}}$ number in the sequence is $4256815$, what is the remainder when this sum is divided by $6$?

Solution

Problem 27

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 $10$?

Solution

Problem 28

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 $f(1)$?

Solution

Problem 29

In the list of numbers $1,2,\ldots,9999$, 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?

Solution

Problem 30

A $12$-sided game die has the shape of a hexagonal bi-pyramid, 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 of the ground is the opposite face? Express your answer as a common fraction in simplest radical form.

Solution

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