Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from 10, doubles his answer, and then adds 2. Thuy doubles 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from 10, adds 2 to his number, and then doubles the result. Who gets the largest final answer?
The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array). The first 8 numbers go in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be
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Answer: A — 130.
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Hint 1 of 2
The top row is 1–8 and the bottom row is 57–64.
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Hint 2 of 2
The corners are the first and last entry of those two rows.
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Approach: identify the four corner values
The corners are 1 and 8 (top row) and 57 and 64 (bottom row).
What is the smallest result that can be obtained from the following process? Choose three different numbers from the set {3, 5, 7, 11, 13, 17}, add two of them, then multiply their sum by the third number.
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Answer: C — 36.
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Hint 1 of 2
To make a product small, multiply by the smallest number.
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Hint 2 of 2
Then add the two next-smallest numbers.
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Approach: multiply by the smallest, add the next two
Use 3 as the multiplier and add the next two smallest, 5 and 7: (5 + 7) × 3 = 36.
Any other choice (such as (3 + 7) × 5 = 50) is larger, so the smallest is 36.
Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, in how many months from that time will they have the same number of goldfish?
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Answer: B — 5 months.
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Hint 1 of 2
Write both counts as powers of 2 and match the exponents.
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Hint 2 of 2
Brent: 4 · 4^m; Gretel: 128 · 2^m.
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Approach: equate powers of 2
After m months Brent has 4 · 4^m = 2^(2m+2) and Gretel has 128 · 2^m = 2^(m+7).
Points A and B are 10 units apart. Points B and C are 4 units apart. Points C and D are 3 units apart. If A and D are as close as possible, then the number of units between them is
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Answer: B — 3 units.
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Hint 1 of 2
Walk from A: out 10, then come back as much as possible.
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Hint 2 of 2
The best you can backtrack is 4 + 3 = 7.
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Approach: go out 10, backtrack 4 then 3
Place B 10 from A, then step C back 4 and D back another 3: A is at 0, D ends at 10 − 4 − 3 = 3.
You can't reach 0 (10 ≠ 4 + 3), so the closest is 3 units.
When Walter drove up to the gasoline pump, his tank was 1/8 full. He bought 7.5 gallons, after which the tank was 5/8 full. How many gallons does the tank hold when it is full?
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Answer: D — 15 gallons.
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Hint 1 of 2
The 7.5 gallons raised the level from 1/8 to 5/8 — what fraction of the tank is that?
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Hint 2 of 2
Then scale up to a full tank.
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Approach: fraction added gives the whole
Going from 1/8 to 5/8 is 4/8 = half the tank, filled by 7.5 gallons.
Let x be the number 0.00…01, where there are 1996 zeros after the decimal point before the 1. Which of the following expressions represents the largest number?
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Answer: D — 3/x.
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Hint 1 of 2
x is an extremely tiny positive number.
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Hint 2 of 2
Dividing by something tiny makes the result enormous.
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Approach: compare sizes when x is tiny
Since x is a tiny positive number, 3 + x, 3 − x, and 3·x are all near 3 (or near 0), and x/3 is tiny.
But 3/x divides by something tiny, giving a gigantic number, so 3/x is largest.
In the fall of 1996, 800 students took part in an annual school clean-up day. The organizers expect that in each of 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants expected in the fall of 1999 is
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Answer: E — 2700.
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Hint 1 of 2
A 50% increase multiplies by 1.5 each year.
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Hint 2 of 2
Apply that three times to reach 1999.
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Approach: multiply by 1.5 three times
Three yearly increases multiply by 1.5 three times: 800 × (3/2)³ = 800 × 27/8.
Ana's monthly salary was $2000 in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was
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Answer: A — 1920 dollars.
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Hint 1 of 2
A 20% raise then a 20% cut do NOT cancel out.
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Hint 2 of 2
Multiply by 1.2, then by 0.8.
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Approach: apply both percent changes in turn
After the raise: 2000 × 1.2 = 2400. After the cut: 2400 × 0.8 = 1920.
A special key on a calculator replaces the displayed number x with 1 ÷ (1 − x). (For example, from 2 it gives 1 ÷ (1 − 2) = −1.) If the calculator shows 5 and the key is pressed 100 times in a row, the calculator will display
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Answer: A — −0.25.
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Hint 1 of 2
Press the key a few times and watch for a repeating cycle.
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Hint 2 of 2
Once you know the cycle length, reduce 100 by it.
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Approach: find the repeating cycle
Starting at 5: 5 → −0.25 → 0.8 → 5, a cycle of length 3.
Since 100 = 3·33 + 1, after 100 presses the display matches one press: −0.25.
How many subsets containing three different numbers can be selected from the set {89, 95, 99, 132, 166, 173} so that the sum of the three numbers is even?
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Answer: D — 12.
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Hint 1 of 2
Only the parity (odd/even) of each number matters: there are 4 odds and 2 evens.
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Hint 2 of 2
A sum of three is even only when an even count of them are odd — here that means exactly 2 odds and 1 even.
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Approach: count by parity
The set has 4 odd numbers (89, 95, 99, 173) and 2 even (132, 166). Three numbers sum to even only with 2 odds and 1 even (zero odds would need 3 evens, impossible).
The manager of a company planned to give a $50 bonus to each employee from the company fund, but the fund was $5 short of what was needed. Instead the manager gave each employee a $45 bonus and kept the remaining $95 in the fund. How much money was in the company fund before any bonuses were paid?
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Answer: E — 995 dollars.
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Hint 1 of 2
Let n be the number of employees and write the fund two ways.
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Hint 2 of 2
Fund = 50n − 5 (just short of $50 each) and fund = 45n + 95 (after the $45 bonuses).
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Approach: set two expressions for the fund equal
The fund is 50n − 5 (5 short of $50 each) and also 45n + 95 (gave $45 each, kept $95).
Setting them equal: 50n − 5 = 45n + 95 gives n = 20, so the fund was 45·20 + 95 = $995.
A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?
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Answer: A — 1/4.
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Hint 1 of 2
A point at distance r from the center is r from the center and R − r from the boundary.
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Hint 2 of 2
Closer to the center means r < R − r, i.e. r < R/2.
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Approach: compare distances, then take an area ratio
Being closer to the center than the boundary means r < R − r, so r < R/2 — the point lies in the inner circle of radius R/2.
Its area is a fraction (R/2)² / R² = 1/4 of the whole.
