Question 1
What is
Solution
Question 2
At the theater children get in for half price. The price for adult tickets and child tickets is . How much would adult tickets and child tickets cost?
Solution
Question 3
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
Solution
Question 4
Suppose that cows give gallons of milk in days. At this rate, how many gallons of milk will cows give in days?
Solution
Question 5
On an algebra quiz, of the students scored points, scored points, scored points, and the rest scored points. What is the difference between the mean and median score of the students' scores on this quiz?
Solution
Question 6
The difference between a two-digit number and the number obtained by reversing its digits is times the sum of the digits of either number. What is the sum of the two digit number and its reverse?
Solution
Question 7
The first three terms of a geometric progression are , , and . What is the fourth term?
Solution
Question 8
A customer who intends to purchase an appliance has three coupons, only one of which may be used:
Coupon 1: off the listed price if the listed price is at least
Coupon 2: dollars off the listed price if the listed price is at least
Coupon 3: off the amount by which the listed price exceeds
For which of the following listed prices will coupon offer a greater price reduction than either coupon or coupon ?
Solution
Question 9
Five positive consecutive integers starting with have average . What is the average of consecutive integers that start with ?
Solution
(5(a+2)+10)/5=a+4
Question 10
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length . The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?
Solution
Question 11
David drives from his home to the airport to catch a flight. He drives miles in the first hour, but realizes that he will be hour late if he continues at this speed. He increases his speed by miles per hour for the rest of the way to the airport and arrives minutes early. How many miles is the airport from his home?
Solution
Question 12
Two circles intersect at points and . The minor arcs measure on one circle and on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?
Solution 1
Solution 2
Let O1, O2 be the centre of the 2 circles.
We can see that triangle AO2B is equilateral. Therefore, AB=y.
In triangle AO1B, apply the Law of Cosines:
square of y = x2+x2-2x*x*cos30 = (2 - square root of 3) * square of x
Note that x is radius of larger circle and y is radius of smaller circle.
Question 13
A fancy bed and breakfast inn has rooms, each with a distinctive color-coded decor. One day friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than friends per room. In how many ways can the innkeeper assign the guests to the rooms?
Solution
Question 14
Let be three integers such that is an arithmetic progression and is a geometric progression. What is the smallest possible value of ?
Solution
Question 15
A five-digit palindrome is a positive integer with respective digits , where is non-zero. Let be the sum of all five-digit palindromes. What is the sum of the digits of ?
Solution
10001+99999=110000
10101+99899=110000
......
10901+99199=110000
11011+98089=110000
......
So the question is: how many 5-digit palindromes do we have? We have:
9 * 10 * 10 = 900
So pairing the 900 palindromes based on above pattern, how many pairs of 5-digit palindromes do we have?
We have 900/2 = 450 pairs.
Sum up all these 450 pairs: 450 * 11000 = 4950000
Question 16
The product , where the second factor has digits, is an integer whose digits have a sum of . What is ?
Solution
Question 17
A rectangular box contains a sphere of radius and eight smaller spheres of radius . The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is ?
Solution
Question 18
The domain of the function is an interval of length , where and are relatively prime positive integers. What is ?
Solution
Question 19
There are exactly distinct rational numbers such that and has at least one integer solution for . What is ?
Solution
Question 20
In , , , and . Points and lie on and respectively. What is the minimum possible value of ?
Solution
Question 21
For every real number , let denote the greatest integer not exceeding , and let The set of all numbers such that and is a union of disjoint intervals. What is the sum of the lengths of those intervals?
Solution
Question 22
The number is between and . How many pairs of integers are there such that and
Solution
Question 23
The fraction where is the length of the period of the repeating decimal expansion. What is the sum ?
Solution
Question 24
Let , and for , let . For how many values of is ?
Solution
Question 25
The parabola has focus and goes through the points and . For how many points with integer coordinates is it true that ?
Solution
Answer Keys
Question 1: C
Question 2: B
Question 3: B
Question 4: A
Question 5: C
Question 6: D
Question 7: A
Question 8: C
Question 9: B
Question 10: B
Question 11: C
Question 12: D
Question 13: B
Question 14: C
Question 15: B
Question 16: D
Question 17: A
Question 18: C
Question 19: E
Question 20: D
Question 21: A
Question 22: B
Question 23: B
Question 24: C
Question 25: B