Question 1
The ratio is closest to which of the following numbers?
Solution
Question 2
Given that and are non-zero real numbers, define . Find .
Solution
Question 3
According to the standard convention for exponentiation,
.
If the order in which the exponentiations are performed is changed, how many other values are possible?
Solution
Question 4
For how many positive integers is there at least 1 positive integer such that ?
infinitely many
Solution
Question 5
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
Solution
Question 6
From a starting number, Cindy was supposed to subtract 3, and then divide by 9, but instead, Cindy subtracted 9, then divided by 3, getting 43. If the correct instructions were followed, what would the result be?
Solution
Question 7
A arc of circle A is equal in length to a arc of circle B. What is the ratio of circle A's area and circle B's area?
Solution
Question 8
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let be the total area of the blue triangles, the total area of the white squares, and the area of the red square. Which of the following is correct?
Solution
Question 9
There are 3 numbers A, B, and C, such that , and . What is the average of A, B, and C?
Not uniquely determined
Solution
Question 10
What is the sum of all of the roots of ?
Solution
Question 11
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB each, 12 of the files take up 0.7 MB each, and the rest take up 0.4 MB each. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
Solution
Question 12
Mr. Earl E. Bird leaves home every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?
Solution
Question 13
Given a triangle with side lengths 15, 20, and 25, find the triangle's smallest height.
Solution
Question 14
Both roots of the quadratic equation are prime numbers. The number of possible values of is
Solution
What this means is that the sum of the 2 roots is 63, and the product of the 2 roots is k. Since both roots are primes, 1 must be an even number 2 to make the sum odd. This means the other root must be 63-2=61. Therefore there can be only 1 value of k.
Question 15
Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?
Solution
Question 16
Let . What is ?
Solution
1. x=a + 1 = b + 2 = c + 3 = d + 4
Obviously,
2. x=a + b + c + d + 5.
From 1:
4x=(a+1)+(b+2)+(c+3)+(d+4)=a+b+c+d+10
From 2:
4x=4(a+b+c+d)+20, ie
a+b+c+d+10 = 4(a+b+c+d)+20. Rearranging, we have 3(a+b+c+d)=-10, so a+b+c+d=-10/3.
Question 17
Sarah pours 4 ounces of coffee into a cup that can hold 8 ounces. Then she pours 4 ounces of cream into a second cup that can also hold 8 ounces. She then pours half of the contents of the first cup into the second cup, completely mixes the contents of the second cup, then pours half of the contents of the second cup back into the first cup. What fraction of the contents in the first cup is cream?
Solution
Question 18
A 3x3x3 cube is made of 27 normal dice. Each die's opposite sides sum to 7. What is the smallest possible sum of all of the values visible on the 6 faces of the large cube?
Solution
Question 19
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach?
Solution
Question 20
Points and lie, in that order, on , dividing it into five segments, each of length 1. Point is not on line . Point lies on , and point lies on . The line segments and are parallel. Find .
Solution
Question 21
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is
Solution
Suppose the largest integer is 15, then the smallest is 15-8=7. Since mean is 8, sum is 8*8=64. 64-15-8-8-7 = 26, which should be the sum of missing 4 numbers. But if 7 is the smallest number, then the sum of the missing numbers must be at least 4*7=28, which is a contradiction.
If largest is 14, then smallest is 14-8=6. 64-14-8-8-6= 28, larger than 4*6=24, so no contradiction here.
We have thus far honored constraints of range, unique mode, and mean. Let's how to honor the median = 8.
Only the following sequences can honor the constraint of median=8:
6,x,x,8,8,x,x,14
You can try putting 6,7,and 8 on the left, and 8-13 on the right and try to make their sum to be 64. The choice we get is:
6,6,6,8,8,8,8,14. So the largest integer is 14.
Question 22
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
Solution
Question 23
Points and lie on a line, in that order, with and . Point is not on the line, and . The perimeter of is twice the perimeter of . Find .
Solution
Question 24
Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, ..., 10}. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?
Solution
Question 25
In trapezoid with bases and , we have , , , and (diagram not to scale). The area of is
Solution
Answer Keys
Question 1: D
Question 2: C
Question 3: B
Question 4: E
Question 5: C
Question 6: A
Question 7: A
Question 8: A
Question 9: B
Question 10: A
Question 11: B
Question 12: B
Question 13: B
Question 14: B
Question 15: E
Question 16: B
Question 17: D
Question 18: D
Question 19: E
Question 20: D
Question 21: D
Question 22: C
Question 23: D
Question 24: A
Question 25: C