Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of 10 hours per week helping around the house for 6 weeks. For the first 5 weeks she helps around the house for 8, 11, 7, 12 and 10 hours. How many hours must she work for the final week to earn the tickets?
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Answer: D — 12 hours.
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Hint 1
Target total = 6 · 10 = 60. Subtract what she's already done.
Chandler wants to buy a 500 dollar mountain bike. For his birthday, his grandparents send him 50 dollars, his aunt sends him 35 dollars and his cousin gives him 15 dollars. He earns 16 dollars per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike?
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Answer: B — 25 weeks.
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Hint 1
Subtract birthday total from 500, then divide by 16.
The average cost of a long-distance call in the USA in 1985 was 41 cents per minute, and the average cost of a long-distance call in the USA in 2005 was 7 cents per minute. Find the approximate percent decrease in the cost per minute of a long-distance call.
Look for numbers that appear on only one tile — those edges must sit on the outside boundary.
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Hint 2 of 2
Once you anchor a tile, propagate by matching shared-edge numbers to its neighbors.
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Approach: anchor on outside-only numbers, then propagate matches
Some edge numbers appear on only one tile (no other tile carries that number), so those edges must lie on the outer boundary of the 2 × 2 arrangement. Pinning those tiles into their forced corners removes most of the freedom.
From the anchored tile, walk to its neighbors by matching the shared edge number. The chain forces tile IV into rectangle C.
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?
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Answer: C — 40%.
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Hint 1
Yellow: 0.30 · 30 = 9 L. After adding 5: 14 L out of 35 L.
The product of the two 99-digit numbers 303,030,303,…,030,303 and 505,050,505,…,050,505 has thousands digit A and units digit B. What is the sum of A and B?
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Answer: D — 8.
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Hint 1
The last 4 digits of each factor are 0303 and 0505. Multiply those mod 10000.
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Approach: compute the last four digits
303 · 505 = 153015.
Last 4 digits: 3015 ⇒ thousands digit A = 3, units digit B = 5.
Pick two consecutive positive integers whose sum is less than 100. Square both of those integers and then find the difference of the squares. Which of the following could be the difference?
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Answer: C — 79.
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Hint 1
(x+1)2 − x2 = 2x + 1 = sum of the two integers. The sum < 100 and is odd.
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Approach: factor the difference
(x+1)2 − x2 = 2x + 1 = (x) + (x+1).
Difference equals the sum of the two integers — less than 100, and odd.
Before district play, the Unicorns had won 45% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?
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Answer: A — 48 games.
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Hint 1
Let pre-district games be x. Pre-district wins: 0.45x. Final wins: 0.45x + 6 = (x + 8)/2.
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Approach: set up an equation
0.45x + 6 = (x + 8)/2.
Multiply by 10: 4.5x + 60 = 5x + 40 ⇒ 0.5x = 20 ⇒ x = 40.
Two cards are dealt from a deck of four red cards labeled A, B, C, D and four green cards labeled A, B, C, D. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?
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Answer: D — 4/7.
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Hint 1
Fix the first card. Of the 7 remaining cards, count those that win against it: 3 of the same color and 1 of the same letter (different color).
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Approach: fix one card and count winners
Same color: 3 of the remaining 7.
Same letter (different color): 1 of the remaining 7.
A lemming sits at a corner of a square with side length 10 meters. The lemming runs 6.2 meters along a diagonal toward the opposite corner. It stops, makes a 90° right turn and runs 2 more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?
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Answer: C — 5.
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Hint 1
For any point inside a square of side 10, distance to opposite sides always sums to 10.
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Approach: use the inside-the-square invariant
Lemming stays inside the square.
Distance to left + right walls = 10. Distance to top + bottom walls = 10. Total: 20.
A bag contains four pieces of paper, each labeled with one of the digits 1, 2, 3, or 4, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of 3?
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Answer: C — 1/2.
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Hint 1
Divisible by 3 iff digit-sum divisible by 3. The chosen 3 digits' sum is what matters; the order is irrelevant for divisibility.
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Approach: count 3-element subsets with sum divisible by 3
Subsets of size 3 from {1, 2, 3, 4}: 4 total ({1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}).
Digit sums: 6 ✓, 7, 8, 9 ✓ ⇒ 2 subsets give a multiple of 3.
