Harry and Terry are each told to calculate 8 − (2 + 5). Harry gets the correct answer. Terry ignores the parentheses and calculates 8 − 2 + 5. If Harry's answer is H and Terry's answer is T, what is the difference H − T?
Paul owes Paula 35 cents and has a pocket full of 5-cent coins, 10-cent coins, and 25-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
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Answer: E — 5 coins difference.
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Hint 1
Max coins → all 5-cent coins. Min coins → use the biggest coins possible.
Isabella had a week to read a book for a school assignment. She read an average of 36 pages per day for the first three days and an average of 44 pages per day for the next three days. She then finished the book by reading 10 pages on the last day. How many pages were in the book?
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Answer: B — 250 pages.
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Hint 1
Average × count gives the total for each three-day block. Then add the last day.
Eleven members of the Middle School Math Club each paid the same integer amount for a guest speaker to talk about problem solving at their math club meeting. In all, they paid their guest speaker $1A2. What is the missing digit A of this 3-digit number?
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Answer: D — A = 3.
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Hint 1 of 2
The total is 11 × (integer), so 1A2 must be divisible by 11.
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Hint 2 of 2
Divisibility rule for 11: alternating sum of digits must be a multiple of 11.
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Approach: divisibility test for 11
1A2 divisible by 11 ⇒ alternating sum 1 − A + 2 = 3 − A is a multiple of 11.
Single digit A ∈ {0, …, 9}: only A = 3 works (giving 0).
The first AMC 8 was given in 1985 and it has been given annually since that time. Samantha turned 12 years old the year that she took the seventh AMC 8. In what year was Samantha born?
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Answer: A — 1979.
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Hint 1
The n-th AMC 8 was given in 1985 + (n − 1). Subtract her age to get her birth year.
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Approach: compute year of 7th contest, then subtract age
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
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Answer: A — 4 ways.
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Hint 1 of 2
Total paths from (0,0) to (3,2) with E/N steps: C(5, 2) = 10. Subtract the ones that pass through the bad corner (1,1).
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Hint 2 of 2
Paths through (1,1) = (paths to (1,1)) × (paths from (1,1) to (3,2)) = 2 × 3 = 6.
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?
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Answer: B — 1/6.
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Hint 1
How many ways can 3 baby photos be ordered? Only one ordering matches the celebrities.
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Approach: 3! arrangements, one correct
3 baby photos can be assigned in 3! = 6 ways. Exactly 1 is the correct matching.
The "Middle School Eight" basketball conference has 8 teams. Every season, each team plays every other conference team twice (home and away), and each team also plays 4 games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?
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Answer: B — 88 games.
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Hint 1
Conference games: pairs of teams, each pair plays twice. Non-conference: each of 8 teams plays 4 extras — those involve only one MSE team, so don't divide by 2.
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Approach: count conference pairs × 2, plus non-conference
Conference pairs: C(8, 2) = 28. Each pair plays 2 games ⇒ 56 games.
Non-conference: 8 teams × 4 games each = 32 games (the opponent is outside MSE, so no double-counting).
George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first 12 mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last 12 mile in order to arrive just as school begins today?
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Answer: B — 6 mph.
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Hint 1
Total time on a normal day = 1/3 hour. How much of that did the slow first half use? Whatever's left must cover the second half-mile.
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Approach: subtract used time from total time
Normal total time = 1 / 3 hr.
First half-mile at 2 mph took (1/2) / 2 = 1/4 hr.
Time remaining for the second half = 1/3 − 1/4 = 1/12 hr.
Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?
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Answer: D — 3 of one gender and 1 of the other is most likely.
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Hint 1 of 2
All 16 sequences of BBBB … GGGG are equally likely. Count how many sequences each described outcome covers.
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Hint 2 of 2
C(4, k) for k = 0, 1, 2, 3, 4 gives 1, 4, 6, 4, 1. Note "3 of one, 1 of other" covers both k=1 and k=3.
A cube with 3-inch edges is to be constructed from 27 smaller cubes with 1-inch edges. Twenty-one of the cubes are colored red and 6 are colored white. If the 3-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?
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Answer: A — 5/54.
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Hint 1 of 2
The 27 unit cubes break into types by how many faces are exposed: 1 interior (0 exposed), 6 face-centers (1 each), 12 edges (2 each), 8 corners (3 each). Hide white cubes in the lowest-exposure positions first.
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Hint 2 of 2
Total surface = 6 · 9 = 54. Compute the white area; divide.
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Approach: place white cubes where the fewest faces show
Hide one white cube in the very center (0 faces showing).
Place the remaining 5 white cubes at the 6 face-centers (1 face showing each) ⇒ 5 white faces visible.
Each circle contributes a quarter-circle inside the rectangle (radius fits in the rectangle without overlapping the others). Compute the rectangle's area minus the three quarter-circles.
Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all 2-digit primes. Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of your two uniform numbers is today's date. What number does Caitlin wear?
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Answer: A — 11.
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Hint 1 of 3
All three pairwise sums of the uniform numbers are dates (1–31), so the three primes are small two-digit primes. List candidates.
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Hint 2 of 3
Pairwise sums ≤ 31 force the primes to be in {11, 13, 17, 19}. Find a triple whose three pairwise sums are all distinct.
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Hint 3 of 3
Then Bethany's date is the smallest sum (earlier in the month), Caitlin's is the largest (later), today is between.
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Approach: list primes, find a workable triple, assign by date order
Each pairwise sum is a date 1–31, so each is ≤ 31. Two-digit primes: 11, 13, 17, 19, 23, 29. Pairs that sum to ≤ 31: only those from {11, 13, 17, 19} (any pair including 23 or 29 exceeds 31 for the larger sums).
Among the triples in {11, 13, 17, 19} we need three distinct pairwise sums. {11, 13, 17}: 24, 28, 30 — distinct. ✓
Bethany's birthday = smallest sum = 24, so Ashley + Caitlin = 24 ⇒ {A, C} = {11, 13}.
Caitlin's birthday = largest sum = 30, so Ashley + Bethany = 30 ⇒ {A, B} = {13, 17}.
Ashley is in both sets, so Ashley = 13. Then Caitlin = 11, Bethany = 17. Today = Bethany + Caitlin = 28 (between 24 and 30 — consistent).
One day the Beverage Barn sold 252 cans of soda to 100 customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?
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Answer: C — 3.5.
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Hint 1 of 3
Order the counts smallest to largest. Median = average of 50th and 51st values. To make those big, push the first 49 down to the minimum (which is 1).
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Hint 2 of 3
After setting the first 49 to 1, 203 cans remain for 51 customers; each from 51st onward must be ≥ 51st value.
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Hint 3 of 3
Let 50th = a, 51st = b, with a ≤ b. Make the last 50 all equal to b. Constraint: a + 50b ≤ 203.
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Approach: minimize the first 49, push the median pair as high as possible
Set customers 1–49 to 1 can each: uses 49 cans, leaves 252 − 49 = 203 for the last 51.
Let the 50th value be a and the 51st value be b (a ≤ b). Make customers 51–100 all equal to b: total of last 51 = a + 50b ≤ 203.
Maximize (a + b)/2 over integers with a ≤ b. Try b = 4: a + 200 ≤ 203 ⇒ a ≤ 3, so a = 3 (still ≤ b). Median = (3 + 4)/2 = 3.5.
Try b = 5: a + 250 ≤ 203 — impossible. So 3.5 is the max.
Each semicircle replaces a straight diameter d with a half-circumference (π/2)d. So the bike path is π/2 times the straight 1-mile distance.
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Hint 2 of 2
Then time = distance / speed.
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Approach: compare the semicircular path length to the straight distance
Each semicircle's diameter lies along the highway and the curve just reaches the edge, so for any diameter d the half-circumference is (π/2)d. Stacking semicircles end-to-end multiplies the total length by π/2.
