Section A
15 Multiple Choice Questions
(+3)
1
combinatorics
easy
Sam has six different fruits. He would like to pick five different fruits to make a cup of fruit juice. How many different flavors can he make if he always uses five different fruits in each flavor?
A
Five
B
Six
C
Twelve
D
Fifteen
E
Thirty
Show answer
Answer: B
Choosing 5 fruits out of 6 to include is the same as choosing 1 fruit out of 6 to leave out, so there are C(6,1) = 6 different flavors. The answer is B.
2
algebra
easy
The average of ten numbers is 12. If one of the numbers is replaced with 10, the average of the ten numbers becomes 9. What is the original value of the replaced number?
A
Not enough information
B
10
C
20
D
30
E
40
Show answer
Answer: E
The original sum of the ten numbers is 10×12 = 120. After replacing one number with 10, the new sum is 10×9 = 90. The sum decreased by 120−90 = 30, and since the new value inserted was 10, the original value removed was 10+30 = 40. The answer is E.
3
algebra
easy
After a 20% price cut, a sport shirt is selling for $36 before taxes. What percentage of an increase will restore the shirt to its original price?
A
20%
B
25%
C
40%
D
15%
E
36%
Show answer
Answer: B
The original price P satisfies 0.8P = 36, so P = 45. To go from 36 back to 45 requires an increase of 45−36 = 9, which is 9/36 = 25% of 36. The answer is B.
4
geometry
medium
In the figure below, ABCDEFGH is a cube, where AB is one inch long. A triangle ABG is constructed inside the cube. What is the perimeter of triangle ABG?
A
3.5 inches
B
5 inches
C
6 inches
D
(√3+√2+1) inches
E
7 inches
Show answer
Answer: D
Place the unit cube with A=(0,0,0), B=(1,0,0), C=(1,1,0), D=(0,1,0), E=(0,0,1), F=(1,0,1), G=(1,1,1), H=(0,1,1), matching the figure's labeling. AB=1 (a cube edge). BG is a face diagonal of the face BCGF, so BG=√2. AG is a space diagonal of the cube, so AG=√3. Perimeter = AB+BG+GA = 1+√2+√3 inches. The answer is D.
5
geometry
easy
In the figure below, the two circles share the same center point A. AB is 2 inches long. AC is 4 inches long. What is the ratio between the area of the shaded ring and the area of the small circle?
A
6:1
B
4:2
C
3:2
D
3:1
E
2:1
Show answer
Answer: D
The small circle has radius AB=2, so its area is π(2)²=4π. The large circle has radius AC=4, so its area is π(4)²=16π. The shaded ring (annulus) has area 16π−4π=12π. The ratio of the ring's area to the small circle's area is 12π:4π = 3:1. The answer is D.
6
coordinate-geometry
medium
In the Cartesian Coordinate Plane, there are three marked points A=(1,2), B=(4,4), and C=(4,1). If the three points are connected to form a triangle ABC, what is the area of the ∆ABC in unit squares?
A
3.5
B
4
C
4.5
D
5.6
E
6
Show answer
Answer: C
Using the shoelace formula, Area = ½|x_A(y_B−y_C)+x_B(y_C−y_A)+x_C(y_A−y_B)| = ½|1(4−1)+4(1−2)+4(2−4)| = ½|3−4−8| = ½(9) = 4.5. The answer is C.
7
statistics
easy
In a data analysis class, Mrs. Smith's students measured the diameter and circumference of various circular objects in the school building. Using the data collected, they made a scatter plot and further built a linear model: y = 2.6x−0.5 using computer software. The results are shown in the figure below, where Y represents the circumference and X represents the diameter, both in centimeters. Based on the data analysis, what is the circumference of a circle with a diameter of 50 centimeters?
A
129.5
B
157.5
C
314.5
D
78.5
E
628.5
Show answer
Answer: A
Substitute x=50 into the model: y = 2.6(50)−0.5 = 130−0.5 = 129.5. The answer is A.
8
probability
medium
In a dark room, there are two boxes, A and B, which contain some small toy balls. Box A has four green balls and six red balls. Box B has eight green balls and two red balls. Amy went to the room and randomly picked a box and then a ball. She came out of the room and found it was a green ball. What is the probability that Amy's ball was taken from Box A?
A
1/2
B
1/3
C
1/4
D
1/5
E
1/6
Show answer
Answer: B
By Bayes' theorem: P(green) = P(A)·P(green|A)+P(B)·P(green|B) = ½(4/10)+½(8/10) = 0.2+0.4 = 0.6. Then P(A|green) = P(A)·P(green|A)/P(green) = 0.2/0.6 = 1/3. The answer is B.
9
geometry
hard
A certain coffee filter is a circle with a radius of three inches. Linda cut out one-third of the circle as shown in the picture and rolled it into a perfect cone without any overlap. What is the radius of the base of the cone?
A
½ inch
B
3π/2 inches
C
π inches
D
1 inch
E
π/2 inches
Show answer
Answer: D
The one-third sector (120° of arc) cut from the radius-3 circle has arc length equal to one-third of the full circumference: ⅓×(2π×3) = 2π. When rolled into a cone, this arc becomes the circumference of the cone's circular base, so 2πr = 2π, giving r = 1 inch. The answer is D.
10
geometry
medium
Katrina has a rectangular piece of copy paper that measures 10 inches by 12 inches. To make an open-top box, she cuts four congruent squares off each corner and then folds the rest of the paper into a box as illustrated in the figure. If each square has a side length of 2 inches, what is the volume of the box in cubic inches?
A
240
B
64
C
84
D
1205
E
96
Show answer
Answer: E
After cutting 2-inch squares from each corner and folding up the sides, the box has height 2, length 12−2(2)=8, and width 10−2(2)=6. Volume = 8×6×2 = 96 cubic inches. The answer is E.
11
geometry
hard
In a math class, Lucy was looking at the cover of her notebook, which is a rectangle. She placed a small marble, represented by P in the figure. Then, she measured the distances from P to three corners, A, B, and C of the notebook, which are 2 inches, 4 inches, and 10 inches, respectively. She then posed a question to her class about the distance from P to D. What is the distance from P to D, based on the information given by Lucy?
A
Not enough information
B
2√30 inches
C
2√22 inches
D
2√23 inches
E
8 inches
Show answer
Answer: C
By the British Flag Theorem, for any point P and rectangle ABCD, PA²+PC² = PB²+PD² (pairing opposite corners A,C with B,D). So 2²+10² = 4²+PD², giving 4+100 = 16+PD², so PD² = 88 and PD = √88 = 2√22 inches. The answer is C.
12
geometry
medium
Parallelogram ABCD has an area of 24 square inches. P is a point on AD. What is the area of ∆BCP?
A
Not enough information
B
6 square inches
C
18 square inches
D
12 square inches
E
2√6 square inches
Show answer
Answer: D
Side BC and side AD are parallel and the same length in the parallelogram, and since P lies on AD, the perpendicular distance from P to line BC equals the full height of the parallelogram between AD and BC, regardless of where P sits on AD. So triangle BCP has the same base (BC) and same height as the parallelogram, giving Area(∆BCP) = ½×Area(ABCD) = ½×24 = 12 square inches. The answer is D.
13
geometry
medium
In the figure below, star S is dilated (proportionally scaled) from point A to star S'. AF=FF', AC=CC'. The area of star S is 3.8 square inches. What is the area of star S'?
A
Not enough information
B
7.6 square inches
C
11.4 square inches
D
15.2 square inches
E
19.0 square inches
Show answer
Answer: D
Since AF=FF', point F is the midpoint of segment AF', so AF' = 2·AF; likewise AC'=2·AC. The dilation from A therefore has scale factor k=2. Areas under a dilation scale by k²=4, so Area(S') = 4×3.8 = 15.2 square inches. The answer is D.
14
geometry
hard
In the figure below, triangle ABC is an equilateral triangle, where AB is one foot long. D and E are the midpoints of AC and BC, respectively. FG is half the length of AB. EF is ⁴√3/2 feet long. What is the length of DH?
A
Not enough information
B
⁴√3/4 feet
C
⁴√3·√3/2 feet
D
3 feet
E
⁴√3/4 feet
Show answer
Answer: E
Since D and E are midpoints of AC and BC, DE is a midsegment of the equilateral triangle, so DE=½AB=½ and DE is parallel to AB, at perpendicular distance equal to half the triangle's height, i.e. ½×(√3/2)=√3/4 above line AB. H is the foot of the point where segment EF crosses near D, and DH is the perpendicular distance from D to line EF. Using the area of triangle DEF two ways: Area(DEF) = ½×DE×(height from F to line DE) = ½×½×(√3/4) = √3/16 (the height from F, which lies on AB, up to the parallel line DE is the same fixed √3/4 regardless of exactly where F sits on AB). Also Area(DEF) = ½×EF×DH. So DH = 2×Area(DEF)/EF = 2×(√3/16)/(⁴√3/2) = (√3/8)×(2/⁴√3) = √3/(4·⁴√3) = 3^(1/2)/(4·3^(1/4)) = 3^(1/4)/4 = ⁴√3/4 feet. The answer is E.
15
number-theory
medium
Consider the hypotenuse of a right triangle with two legs that are two inches and one inch long, respectively. The length of the hypotenuse, in inches, is between which pair of the following fractions?
A
9/4 and 10/4
B
17/8 and 18/8
C
1/4 and 1/3
D
14/4 and 15/4
E
3/4 and 5/4
Show answer
Answer: B
The hypotenuse is √(2²+1²)=√5≈2.236. Checking option B: 17/8=2.125 and 18/8=2.25, and indeed 2.125<2.236<2.25, so √5 lies strictly between them. (Option A's range 2.25 to 2.5 does not contain 2.236.) The answer is B.
Section C
5 Open-Ended Questions
(+6)
21
algebra
medium
Simplify the following expression and find the value of W. W = 4096×(1+1/2+1/2²+1/2³+…+1/2⁹)×(1−½)
Show answer
Answer: 4092
The sum 1+1/2+1/2²+…+1/2⁹ is a geometric series with 10 terms, first term 1 and ratio ½, so it equals (1−(1/2)^10)/(1−½) = 2(1−1/1024) = 2×(1023/1024) = 1023/512. Multiplying by (1−½)=½ gives 1023/1024. So W = 4096×(1023/1024) = 4×1023 = 4092.
22
combinatorics
hard
Lacie is making bracelets for her friends. Each bracelet has 11 beads on a loop of elastic cord. For each bead, she has two colors, red or green, to choose from. For Lacie, a bracelet can be rotated without losing its uniqueness but cannot be flipped. How many different bracelets can Lacie make?
Show answer
Answer: 188
This counts binary necklaces of length 11 under rotation only (no reflections), via Burnside's lemma with the cyclic group C_11. Since 11 is prime, the identity rotation fixes all 2^11=2048 colorings, and each of the other 10 nontrivial rotations fixes only the 2 monochromatic colorings (all-red or all-green), since a coloring fixed by a rotation of order-11 group with gcd(k,11)=1 must be constant. Average number of fixed colorings = (2048 + 10×2)/11 = 2068/11 = 188.
23
algebra
medium
Jennifer has a new book to read in the summer. The book has 600 pages. She plans to read one page on the first day and then read three more pages every day than the previous day. In other words, she will read four pages on the second day, seven pages on the third day, and so on. How many days will it take Jennifer to finish the whole book?
Show answer
Answer: 21
The pages read on day n form an arithmetic sequence 1,4,7,… with first term 1 and common difference 3, so the cumulative total after n days is S(n) = n/2×(2×1+(n−1)×3) = n(3n−1)/2. We need the smallest n with S(n)≥600. S(20) = 20(59)/2 = 590 < 600. S(21) = 21(62)/2 = 651 ≥ 600. So it takes 21 days.
24
algebra
medium
Jake agreed to work for a local computer repair company for 100 hours during the summer. In return, he would get a tablet computer and $800. However, after working for 60 hours, he had to quit the job. The computer company was fair to him. They gave Jake the tablet computer and $240. How much was the tablet computer worth, assuming it was a fair deal?
Show answer
Answer: 600
Let T be the tablet's value. The full 100-hour deal is worth T+800 total. A fair payment for 60 out of 100 hours (60% of the work) should be 0.6×(T+800). This must equal what Jake actually received, T+240: T+240 = 0.6(T+800) = 0.6T+480. So 0.4T = 240, giving T = 600. The tablet was worth $600.
25
probability
hard
At the high school stadium, benches are 18 feet long. If two people are to sit randomly on a bench, the probability that they are more than six feet apart from each other is p. What is the value of 900p?
Show answer
Answer: 400
Let X and Y be the two people's positions, each uniform on [0,18], independently. The sample space is the square [0,18]×[0,18] with area 324. The event |X−Y|>6 consists of two corner triangles, each a right triangle with both legs of length 18−6=12, so each has area ½×12×12=72, for a combined area of 144. So p = 144/324 = 4/9. Then 900p = 900×(4/9) = 400.