Download haymath - Art of Problem Solving

MSM Competition
Test #1
■ 2005 ■
Sprint Round
Problems 1-30
_______________________
Name _________________________________________
School ________________________________________
Chapter _______________________________________
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO
This round of the competition consists of 30 problems.
You will have 40 minutes to complete the problems. You
are not allowed to use calculators, slide rules, books, or
any other aids during this round. If you are wearing a
calculator wrist watch, please give it to your proctor now.
Calculations may be done on scratch paper. All answers
must be complete, legible, and simplified to lowest terms.
Record only final answers in the blanks in the right-hand
column of the competition booklet. If you complete the
problems before time is called, use the remaining time to
check your answers.
_______________________
Total Correct
Scorer’s initials
1.
If y is equal to 7x – 3, what is the multiplicative inverse of y when x
is equal to ½?
1. _________________
2.
In Pascal’s triangle, how many odd numbers will be in the 2049th
row?
2. _________________
3.
Solve for x: The sum of the first x odd positive integers equals 2500.
3. _________________
4.
What is the maximum number of points of intersection when two
circles and five lines intersect each other? (Assume these are all
coplanar, no collinear/concentric.)
4. _________________
5.
What is the sum of the infinite convergent series 1/2 + 1/4 +1/8 +
1/16 +1/32……?
5. _________________
6.
What is half the area of the circle defined by the equation
(x – 2)2 + (y + 6)2 = 12? Express your answer in terms of π.
6. _________________
7.
A square is divided into six segments with 3 horizontal lines equally
spaced, and 1 diagonal of the square. If the area of the blue segment
in the diagram below is 121.5, what is the perimeter of the square?
7. _________________
8.
What is the repeating decimal .765656565… expressed as a common
fraction in lowest terms?
8. _________________
9.
What is the number of distinct ways of arranging the letters in the
word TELEPHONE?
9. _________________
10. In a particular primitive Pythagorean triple (19, y, z), y and z differ
by 1. What is y + z?
10. ________________
11. How many terms are in the arithmetic sequence 28, 37, 46, …... 964?
11. ________________
12. What is two times the coefficient of x in the slope-intercept form of a
line that passes through (2,5) and (6,7)?
12. ________________
13. Sue wanted to sew the pattern below. The shaded area is a new
fabric. How many square meters of the new fabric does she need?
13. ________________
14. Joe gives Nick and Tom as many pennies as each already has. Then
Nick gives Joe and Tom as many pennies as each of them then has.
Finally, Tom gives Nick and Joe as many pennies as each has. If at
the end each has sixteen pennies, how many dollars did Tom start out
with? Express your answer as a decimal to the nearest hundredth.
14. ________________
15. A right triangle in Quadrant I is bounded by lines y = x, y = 0, and
y = - x + 5. Find its area in square units.
15. ________________
16. A parabola with vertex (2, 0) and an axis of symmetry parallel to the
y-axis passes through (3, 1) and (-3, t). Find the value of t.
16. ________________
17. What is the sum of all the digits in the sequence 1, 2, 3,….99, 100?
17. ________________
18. What is the third root of 970299?
18. ________________
19. What is the difference between 9982 and 10022?
19. ________________
20. Three wealthy men and three robbers are traveling together. They
come to a river that they must cross. The only boat available carries
two people at a time. The wealthy men must be careful that there are
never more robbers than wealthy men on the same side of the river or
they will be robbed. How many trips will it take for them all to cross
safely?
20. ________________
21. Place the digits 1-9 in the boxes so that each diagonal adds up to 26,
21. ________________
and that the four corners add up to 26 as well. What is the digit in the
middle square?
22. My brother and I are over 9 years old, and under 100 years old. If my 22. ________________
age is a palindrome, and my older brother’s age is a palindrome, and
the difference between our ages is a palindrome, what is the average
of all my possible ages, rounded to the nearest whole number?
23. Solve for y: 1/2005 + 1/y = 1/2000
23. ________________
24. If you write down every integer 1 through 1,000,000, what would be
the one millionth digit you write?
24. ________________
25. Five numbers are in geometric progression. Their sum is 6 and the
sum of their reciprocals is 3. What is their product?
25. ________________
26. What is the maximum number of points of intersection created by 10
lines?
26. ________________
27. If I have 2 pies, and Beth has x pies, and Tom has x+2 pies, how
many pies does Beth have if Tom has double the number of pies that
I have?
27. ________________
28. Evaluate 74 – 73
28. ________________
29. What is the coefficient of a2b2 in the expanded form of:
(a+b)4
29. ________________
30. The sum of three children's ages is 23. The product of their ages is
113 more than the product of their ages exactly one year ago. What is
the sum of the squares of the children's ages?
30. ________________
Copyrights, Credits, etc:
© 2005 Mysmartmouth
Credits:
Problem # 7: Image from The Elias Saab MathCounts Page
<http://mathcounts.saab.org/mc.cgi>
Problem # 13: Image by TheAnswerIsPi
Thanks to Tarquin for the layout, it was created to resemble a MathCounts test!