In the figure, the outer equilateral triangle has area 16, the inner equilateral triangle has area 1, and the three trapezoids are congruent. What is the area of one of the trapezoids?
Barney Schwinn notices that the odometer on his bicycle reads 1441, a palindrome, because it reads the same forward and backward. After riding 4 more hours that day and 6 the next, he notices that the odometer shows another palindrome, 1661. What was his average speed in miles per hour?
In 2005 Tycoon Tammy invested 100 dollars for two years. During the first year her investment suffered a 15% loss, but during the second year the remaining investment showed a 20% gain. Over the two-year period, what was the change in Tammy's investment?
The average age of the 6 people in Room A is 40. The average age of the 4 people in Room B is 25. If the two groups are combined, what is the average age of all the people?
Show hint
Combined average = total ages / total people. Each room's total = avg × count.
Each of the 39 students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and 26 students have a cat. How many students have both a dog and a cat?
Show hint
|A ∪ B| = |A| + |B| − |A ∩ B|. Everyone has at least one, so |A ∪ B| = 39.
A ball is dropped from a height of 3 meters. On its first bounce it rises to a height of 2 meters. It keeps falling and bouncing to 23 of the height it reached in the previous bounce. On which bounce will it rise to a height less than 0.5 meters?
Show hint
After the nth bounce, height = 3 · (2/3)n. Test small n until it drops below 1/2.
Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than 100 pounds or more than 150 pounds. So the boxes are weighed in pairs in every possible way. The results are 122, 125 and 127 pounds. What is the combined weight in pounds of the three boxes?
Show hint
Each box is in 2 of the 3 pair-sums. Adding all three pair-sums double-counts each weight.
Show solution
Sum of pair-sums: 122 + 125 + 127 = 374 = 2(a + b + c).
Three A's, three B's, and three C's are placed in the nine spaces so that each row and column contain one of each letter. If A is placed in the upper left corner, how many arrangements are possible?
Show hint
Place B in the second row (2 choices for column) and then in the third row (constrained). C is then forced.
Show solution
Row 1 is fixed up to permutation of B, C (2 ways). Row 2 starts with B or C (2 choices), then is determined. Each row 2 case constrains row 3 to one arrangement.
In Theresa's first 8 basketball games, she scored 7, 4, 3, 6, 8, 3, 1 and 5 points. In her ninth game, she scored fewer than 10 points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than 10 points and her points-per-game average for the 10 games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
Show hint (soft nudge)
Sum of first 8: 37. After game 9 (score < 10), total is between 37 and 47; must be a multiple of 9 (mean integer).
Show hint (sharpest)
Then after game 10 the total is < 56 and a multiple of 10.
Show solution
Sum after 8: 37.
After 9: total in [38, 47], divisible by 9 ⇒ 45. Game 9 = 8.
After 10: total in [46, 55], divisible by 10 ⇒ 50. Game 10 = 5.
Ms. Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of 50 units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
Show hint
l + w = 25, both positive integers. Area = l(25 − l); max near l = 12 or 13, min at l = 1.
Two circles that share the same center have radii 10 meters and 20 meters. An aardvark runs along the path shown, starting at A and ending at K. How many meters does the aardvark run?
Show hint
Decompose the path: quarter-arcs of the big circle, quarter-arcs of the small circle, the diameter of the small circle, and radial segments between the two circles.
Show solution
Big-circle quarter-arc (one piece): (1/4)(2π)(20) = 10π. Two such arcs ⇒ 20π.
Small-circle quarter-arc (one piece): (1/4)(2π)(10) = 5π. Two such arcs ⇒ 10π... but the path uses just enough small arcs and a small-circle diameter to traverse the inner ring.
Per the figure: 2 big-quarter arcs (total 20π) + 2 radial segments of length 10 (total 20) + a diameter of the small circle of length 20.
Eight points are spaced around at intervals of one unit around a 2 × 2 square, as shown. Two of the 8 points are chosen at random. What is the probability that the two points are one unit apart?
Show hint
Each point has exactly 2 neighbors at distance 1 (its left and right neighbors on the perimeter).
Show solution
Fix one point. Of the remaining 7, exactly 2 are 1 unit away.
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and 3/4 of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
Show hint (soft nudge)
Let p be the common count passing. Boys = (3/2)p, girls = (4/3)p; total = (17/6)p.
Show hint (sharpest)
Total must be a positive integer; smallest p making it integer is p = 6.
Show solution
Boys = (3/2)p, girls = (4/3)p. Total = (3/2 + 4/3)p = (17/6)p.
Smallest positive integer total requires p = 6 ⇒ total = 17.
Jerry cuts a wedge from a 6-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
Show hint
The wedge is exactly half the cylinder. Cylinder volume = πr2h.
Ten tiles numbered 1 through 10 are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
Show hint
Case on the die roll d; for each, count tiles t in 1–10 with dt a perfect square.
Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Approximately what percent of the design is black?
Show hint
Black regions: disk of radius 2, annulus 4–6, annulus 8–10. Use area = π(R2 − r2) for each annulus.