Which one of the following options can be folded from the diagram below?
A–D: figures in the preserved source image.
E. None of the above.
**Editorial note for this self-study exercise:** Use every rectangular panel, fold only along the dotted lines, and do not cut or conceal an extra panel by overlapping it. The six box faces form a closed thin cuboid; retain the seventh panel as a projecting flap. The drawing is schematic, not a scale measurement. Enter one option letter A–E. Letter case and outside whitespace are ignored.
A television set has a 30-inch screen size, i.e. the length of the diagonal of the screen is 30 inches. The aspect ratio of the screen is 4 : 3. Find the height of the screen. Leave your answer to the nearest whole number if necessary. (Note: The aspect ratio of the screen refers to the ratio of the width of the screen to its height.)
**Editorial note for this self-study exercise:** Treat the screen as a rectangle. Enter its height in inches, without units.
**Editorial note for this self-study exercise:** Use the source rule: each outer number is the sum of the two inner sectors immediately adjoining the opposite end of its spoke. This supplied rule fixes the intended pattern; the diagram alone need not determine a unique abstract rule. Enter the missing number.
A rectangle was altered by increasing its length by 10 percent and decreasing its width by p percent. If these alterations decreased the area of the rectangle by 12 percent, what is the value of p?
A. 12
B. 15
C. 20
D. 22
E. None of the above
**Editorial note for this self-study exercise:** Enter p as a number, without a percent sign or option letter.
The 7 × 7 chessboard contains one shaded square as shown. How many squares of any size do not include the shaded square?
**Editorial note for this self-study exercise:** Count squares whose sides lie on the grid lines. The shaded cell is row 4 from the top and column 6 from the left. A square includes it when that entire cell lies inside the square. Enter the count.
Town A and Town B are 435km apart. At 9 : 15 a.m, a car left Town A and traveled toward Town B at a constant speed of 80km/h. At the same time, a lorry left B and traveled toward Town A at a constant speed. When they met, the car traveled a distance of 240km. Find the speed of the lorry.
**Editorial note for this self-study exercise:** Both vehicles use the same route, travel without stopping, and meet between A and B. Enter the lorry speed in km/h, without units.
In the table below, each letter represents the number of students in that category. Which of the following must be equal to z?
STUDENTS IN ADVANCED BIOLOGY CLASS
Category | Juniors | Seniors | Total
--- | --- | --- | ---
Boys | k | n | m
Girls | r | s | t
Total | w | x | z
A. k + s
B. m + x
C. r + s
D. r + s + t
E. k + n + r + s
**Editorial note for this self-study exercise:** The row and column categories partition the students: every student is counted in exactly one of the four inner cells. Enter one option letter A–E. Letter case and outside whitespace are ignored.
Larry’s 3-digit number satisfies the following conditions:
• The number is not a multiple of 3.
• Exactly one of the digits is a prime number.
• Another digit is a square number.
• The other digit is neither prime nor square.
What is the largest possible value of Larry’s number?
**Editorial note for this self-study exercise:** Use prime digits 2, 3, 5, 7 and square digits 0, 1, 4, 9 (including zero as 0 squared). The third digit must be from 6, 8. Enter the three-digit number.
A car and a motorcycle started traveling toward each other at the same instant, from cities A and B respectively. 72 minutes later, they met along the road and continued to travel toward their destinations. Given that the speed of the car is $1\frac13$ times that of the motorcycle. How many minutes after the car reached the city B would the motorcycle reach the city A?
**Editorial note for this self-study exercise:** Both speeds stay constant for the whole trip, both follow the same route, and neither stops. Enter the difference between their arrival times in minutes, without units.
Find the perimeter and area of the shaded figure below which made up of different circles.
**Editorial note for this self-study exercise:** The scan has no clear shading. Use the region inside the large semicircle and outside the three smaller semicircles, whose diameters lie end-to-end on the same 56 cm segment; the end diameters are 14 cm each. Keep π exact. Write P = aπ cm and A = bπ cm². Enter a then b separated by a semicolon, with no spaces, for example 12;34. Only outside whitespace and letter case are ignored; spaces adjacent to punctuation are not removed.
Lamp poles are installed along the road side on every 8m. Markings are made on the road side on every 12m. How many markings are made beside the lamp poles along a 240m long road?
**Editorial note for this self-study exercise:** Assume a pole and a marking both occur at the start, and count coincidences at both endpoints, 0 m and 240 m. This endpoint convention is required by the source solution. Enter the number of coincident markings.
Jack’s salary in June was $\$4000$. He spent 40% of his salary on rent, 15% on transport, 20% on food and saved the rest. In July his salary was increased. The amount of money he spent on rent and transport for July remained the same. However, he spent $\$400$ more on food. If Jack saved the remaining 20% of his salary in July, find the percentage increase in his salary.
**Editorial note for this self-study exercise:** Enter the percentage increase as a number without a percent sign. Use a decimal point if entering a decimal, for example 1.25.
In the diagram, PQRS represents a rectangular piece of paper. The paper is folded along a line VW so that ∠VWQ = 125°. When the folded paper is flattened, points R and Q have moved to points R′ and Q′, respectively, and R′V crosses PW at Y. What is the measure of ∠PYV?
**Editorial note for this self-study exercise:** The fold is an exact reflection across VW. P, Y, W, Q lie on the original top side in that order, and S, V, R on the bottom side. Enter the angle in degrees as a number, without the degree sign.
