# 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)

## Contents

- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30

## Problem 1

Let = . What is the value of ? Express your answer as a common fraction.

## Problem 2

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

## Problem 3

Given , what is the value of ?

## Problem 4

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

## Problem 5

If , what is the value of ?

## Problem 6

In rectangle , shown here, units, units, unit and units. What is the absolute difference between the areas of triangles and ?

## Problem 7

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

## 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 miles. How many miles will each tire drive?

## Problem 9

Lucy and her father share the same birthday. When Lucy turned her father turned 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?

## Problem 10

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

## Problem 11

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

## Problem 12

What is the least possible denominator of a positive rational number whose repeating decimal representation is , where and are distinct digits?

## Problem 13

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

## Problem 14

Kali is mixing soil for a container garden. If she mixes of soil containing sand with of soil containing sand, what percent of the new mixture is sand?

## Problem 15

Alex can run a complete lap around the school track in minute, seconds, and Becky can run a complete lap in minute, 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?

## 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.

## Problem 17

A function is defined for all positive integers. If for any two positive integers and and , what is ?

## Problem 18

Rectangle is shown with units and units. If is extended to point such that is congruent to , what is the length of ?

## Problem 19

The digits of a -digit integer are reversed to form a new integer of greater value. The product of this new integer and the original integer is . What is the new integer?

## Problem 20

Diagonal of rectangle is divided into three segments each of length units by points and as shown. Segments and are parallel and are both perpendicular to . What is the area of ? Express your answer in simplest radical form.

## Problem 21

A spinner is divided into sectors as shown. Each of the central angles of sectors through measures while each of the central angles of sectors and measures . 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.

## 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.

## 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 “”?

## Problem 24

On line segment , shown here, is the midpoint of segment and is the midpoint of segment . If units and units, what is the length of segment ? Express your answer as a common fraction.

## Problem 25

There are twelve different mixed numbers that can be created by substituting three of the numbers , , and for , and in the expression , where . What is the mean of these twelve mixed numbers? Express your answer as a mixed number.

## Problem 26

If consecutive integers are added together, where the number in the sequence is , what is the remainder when this sum is divided by ?

## Problem 27

Consider a coordinate plane with the points and . For how many points in the plane is it true that and are both positive integer distances, each less than or equal to ?

## Problem 28

The function , where and are positive integers, is defined for all positive integers. If the range of f contains two numbers that differ by , what is the least possible value of ?

## Problem 29

In the list of numbers , the digits through are replaced with the letters through , respectively. For example, the number is replaced by the string “” and is replaced by the string “”. The resulting list of strings is sorted alphabetically. How many strings appear before “” in this list?

## Problem 30

A -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 cm and with height 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.