Download ITHACA COLLEGE 2010 MATH DAY COMPETITION

ITHACA COLLEGE 2010 MATH DAY COMPETITION
Please write your final answer in the box provided.
1.
The price of an item increases by 10% and then by another 10%. What is the overall price
increase of the item, in percent?
2.
An odd integer between 600 and 800 is divisible by 7 and also divisible by 9. What is the
sum of its digits?
3.
A quart of liquid contains 10% alcohol, and another three quart bottle full of liquid
contains 30% alcohol. They are mixed together. What is the percent of alcohol in the
mixture?
4.
What 5-digit number 32a1b is divisible by 156? (Here a and b represent digits.)
5.
John randomly selects an integer from 1 to 9, and so does Mary, independently of John’s
choice. What is the most likely value for the units digit of the sum of their numbers?
6.
If all people eat the same amount of pizza, and a pizza 12 inches in diameter serves two
people, how many inches in diameter should each of two pizzas be in order to serve three
people? (Pizzas are circular and are eaten entirely.)
7.
What is the value of
8.
Jill rides her bike around a course in the shape of an equilateral triangle. Her speed is 10
miles per hour on the first side of the course, 15 miles per hour on the second side of the
course, and 20 miles per hours on the third and final side of the course. What is Jill’s
average speed during her ride?
9.
Is the graph of
ellipses?
10.
How many positive odd integers n less than 1000 have the property that the product of the
digits of n is 252?
(
log 2 (log 81 3)
log3 81
)
?
x 4 + 1 = 2 x 2 + y 2 a pair of intersecting lines, circles, parabolas, or
11.
Given that P( x ) is a polynomial such that
P( x 2 + 1) = x 4 + 5 x 2 + 3 , what is
P( x 2 − 1) ?
12.
A fair coin is tossed repeatedly. What is the probability that we obtain a total of two tails
before we obtain a total of three heads?
13.
If a is a positive real number, what is the area of the region in the first quadrant that is
bounded above by the graph of y = x and below by the graph of y = 2 x − a ?
14.
If C is a right circular cylinder whose volume is 24 cubic inches, is there a smallest
possible value for the area of the curved surface of C, and, if so, what is this value?
15.
If A through F are the vertices of a regular hexagon listed in clockwise order, consider the
triangle ACE. What is the ratio of the area of the triangle to the area of the hexagon?
16.
 1 1 1


Consider the three by three matrix  2 3 x  . How many ordered pairs of positive


 4 9 y
integers (x, y) are there such that the determinant of this matrix is 0?
17.
What four-digit integer n has the property that the value of 9n is the four-digit integer
obtained by writing the digits of n in reverse order?
18.
What is the measure of the acute angle between the hour and minute hands of a correctly
working clock at 4:18?
19.
What is the largest power of two that divides
20.
How many composite (i.e. non-prime) integers from 1 to 2009 have a prime number of
(positive integral) divisors? [NOTE: 1 is not a prime.]
2 2008 + 10 2008 ?