MATHCOUNTS
®
2012
State Competition
Team Round
Problems 1–10
School
Chapter
Team
Members
, Captain
DO NOT BEGIN UNTIL YOU ARE INSTRUCTED
TO DO SO.
This section of the competition consists of 10 problems which the team
has 20 minutes to complete. Team members may work together in any
way to solve the problems. Team members may talk to each other during
this section of the competition. This round assumes the use of calculators,
and calculations also may be done on scratch paper, but no other aids are
allowed. All answers must be complete, legible and simplified to lowest
terms. The team captain must record the team’s official answers on his/her
own competition booklet, which is the only booklet that will be scored. If
the team completes the problems before time is called, use the remaining
time to check your answers.
Total Correct
Scorer’s Initials
National Sponsors
Raytheon Company * U.S. Department of Defense *
Northrop Grumman Foundation *
National Society of Professional Engineers *
Bezos Family Foundation * ConocoPhillips *
CNA Foundation * Texas Instruments Incorporated *
ThinkFun * 3M Foundation
2012 MATHCOUNTS
National Competition Sponsor
Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation
Copyright MATHCOUNTS, Inc. 2011. All rights reserved.
03-S12TEA
tickets
1. _____________
The Student Council sold tickets to the school’s annual
carnival. Adult tickets were $5 and student tickets
were $2. They sold 5 times as many student
tickets as adult tickets and raised $1125. How
many tickets were sold for the carnival? $
2. _____________
Every CD at Bargain Warehouse is sold for one of three different,
whole‑number dollar amounts. Three customers each bought three CDs. The
first customer spent $4, the second customer spent $9 and the third customer
spent $12. No customer purchased three CDs of the same price. What is the
price at which Bargain Warehouse sells the most expensive CD?
3. _____________
When one integer is removed from a list of five integers the mean of the
remaining four integers is 3 less than the mean of the original five integers.
What is the positive difference between the mean of the original five integers
and the integer that was removed?
4. _____________
ft2
The legs of a right triangle are in the ratio 3:4. One of its altitudes is 30 ft. In
square feet, what is the greatest possible area of this triangle? Express your
answer as a decimal to the nearest tenth.
5. _____________
The positive difference of the cubes of two consecutive positive integers is 111
less than five times the product of the two consecutive integers. What is the
sum of the two consecutive integers? Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Team Round
ways
6. _____________
In how many ways can 18 be written as the sum of four distinct positive
integers? Note: 1 + 3 + 5 + 9 and 5 + 1 + 3 + 9 are counted as different ways.
( , )
7. _____________
Four small towns are located at A(0, 0), B(2, 12), C(12, 8) and D(7, 2). A
warehouse serving these towns is to be built at point P so that the sum of the
distances PA + PB + PC + PD is minimized. What are the coordinates of
point P?
minutes
8. _____________
A hot-air balloon will slowly start to descend toward
the ground at a constant rate of 15 ft per minute from an
initial height of 1200 ft above ground at the same time a
small helium‑filled balloon, being released at an initial height
of 10 ft above ground, will start to ascend toward the sky at a
constant rate of 5 ft per second. In how many minutes will the
two balloons be at the same height above the ground? Express
your answer as a decimal to the nearest hundredth.
9. _____________
There is more than one four-digit positive integer in which the thousands digit
is the number of 0s in the four-digit number, the hundreds digit is the number
of 1s, the tens digit is the number of 2s and the units digit is the number of 3s.
What is the sum of all such integers?
cm3
10._____________
In the frustum of a right cone, shown here, segments
AD and BC are the radii of the top and bottom
bases, respectively. If AD = 8 cm, BC = 12 cm and
AC = 15 cm, what is the volume of the frustum? Express your answer in terms of π. A
8
D
15
B
12
C
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Team Round