Question 1
Sandwiches at Joe's Fast Food cost dollars each and sodas cost dollars each. How many dollars will it cost to purchase sandwiches and sodas?
Solution
Question 2
Define . What is ?
Solution
Question 3
The ratio of Mary's age to Alice's age is . Alice is years old. How old is Mary?
Solution
Question 4
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?
Solution
Question 5
Doug and Dave shared a pizza with equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was dollars, and there was an additional cost of dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
Solution
Question 6
The rectangle is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is ?
Solution
Question 7
Mary is older than Sally, and Sally is younger than Danielle. The sum of their ages is years. How old will Mary be on her next birthday?
Solution
Question 8
How many sets of two or more consecutive positive integers have a sum of ?
Solution
Question 9
Oscar buys pencils and erasers for . A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?
Solution
Question 10
For how many real values of is an integer?
Solution
Question 11
Which of the following describes the graph of the equation ?
Solution
Question 12
A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?
Solution
Question 13
The vertices of a right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?
Solution
Question 14
Two farmers agree that pigs are worth dollars and that goats are worth dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
Solution
Question 15
Suppose and . What is the smallest possible positive value of ?
Solution
Question 16
Circles with centers and have radii and , respectively. A common internal tangent intersects the circles at and , respectively. Lines and intersect at , and . What is ?
Solution
Question 17
Square has side length , a circle centered at has radius , and and are both rational. The circle passes through , and lies on . Point lies on the circle, on the same side of as . Segment is tangent to the circle, and . What is ?
Solution
Question 18
The function has the property that for each real number in its domain, is also in its domain and
What is the largest set of real numbers that can be in the domain of ?
Solution
Question 19
Circles with centers and have radii and , respectively. The equation of a common external tangent to the circles can be written in the form with . What is ?
Solution
http://www.homesweetlearning.com/resources/math/math910/geometry/inner_cirle_of_triangle.html
Then you draw the line that cuts through the centers of the 2 circles.
You can derive that line's equation
You need to know of the trigonometry "double angle formula"
https://www.homesweetlearning.com/resources/math/math910/trigonometry/functions_on_x-y_plane.html
Then you can derive the equation for the tangent line, then you can figure out value of b.
Question 20
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?
Solution
Now, figure out of all the possible paths, how many cover all 8 vertexes exactly once.
Draw the cube, and letter-mark each vertex. There are 2 scenarios:
1. Finish top face, then go down to bottom face
2. Finish only 3 vertexes on top face, then go down to bottom face.
In #1, there are 3*2*2=12 good paths
In #2, there are 3*2*1=6 good paths
Therefore, the possibility is (12+6)/2187.
Question 21
Let
and
.
What is the ratio of the area of to the area of ?
Solution
Question 22
A circle of radius is concentric with and outside a regular hexagon of side length . The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is . What is ?
Solution
Question 23
Given a finite sequence of real numbers, let be the sequence
of real numbers. Define and, for each integer , , define . Suppose , and let . If , then what is ?
Solution
Question 24
The expression
is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
Solution
Question 25
How many non-empty subsets of have the following two properties?
No two consecutive integers belong to .
If contains elements, then contains no number less than .
Solution
Answer Keys
Question 1: A
Question 2: C
Question 3: B
Question 4: E
Question 5: D
Question 6: A
Question 7: B
Question 8: C
Question 9: A
Question 10: E
Question 11: C
Question 12: B
Question 13: E
Question 14: C
Question 15: A
Question 16: B
Question 17: B
Question 18: E
Question 19: E
Question 20: C
Question 21: E
Question 22: D
Question 23: B
Question 24: D
Question 25: E