Jami picked three equally spaced integers on the number line. The sum of the first and second is 40, and the sum of the second and third is 60. What is the sum of all three numbers?
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For equally spaced numbers, the middle one is the average of the other two.
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Add the two given sums and see how many times the middle number appears.
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Since the numbers are equally spaced, first + third = 2 × second. Adding the two given sums: 40 + 60 = first + 2·second + third = 4·second, so the middle is 25.
A cube's two shaded faces share an edge, so both must be glued to neighbors to hide them.
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Look for an arrangement where every cube has two glued faces that meet at an edge.
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To hide a cube's two shaded faces — which meet at an edge — it must be glued to neighbors on two faces sharing an edge.
Four cubes arranged in a 2 × 2 square give each cube exactly two such adjacent glued faces; with three or fewer, some cube has only one glued face (or two opposite ones), so a shaded face shows.
The biggest circle that fits is limited by whichever inward corners of the region poke closest to the center.
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Each inward corner sits 1 unit horizontally and 2 units vertically from the center. Use the Pythagorean theorem to get the radius.
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By symmetry, the largest inscribed circle is centered at the region's center. Its radius is the distance from there to the nearest inward-poking corner.
Each such corner is 1 unit across and 2 units up (or down) from the center: distance = √(12 + 22) = √5.
Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of 6 hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)
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Each sequence needs 3 ups and 3 downs. But Buzz can never go below the ground — the running count of downs can never exceed ups.
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Start with U, end with D. Enumerate carefully without breaking the rule.
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The sequence has 3 U's and 3 D's, with D's never exceeding U's at any point (otherwise Buzz goes below the ground).
All valid sequences: UUUDDD, UUDUDD, UUDDUD, UDUUDD, UDUDUD.
NASA's Perseverance Rover was launched on July 30, 2020. After traveling 292,526,838 miles, it landed on Mars in Jezero Crater about 6.5 months later. Which of the following is closest to the Rover's average interplanetary speed in miles per hour?
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The choices span orders of magnitude, so round generously. Distance ≈ 3 × 108 miles.
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6.5 months ≈ 200 days ≈ 5000 hours. Then speed ≈ distance ÷ time.
Sum the shaded areas, subtract the white circles that sit inside the big shaded disk, divide by the area of the outer circle.
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Outer circle area = 9π. Three small shaded circles (radius 1/2): total 3π/4. Big shaded disk (radius 2) minus 2 inner whites (radius 1 each): 4π − 2π = 2π.
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Outer white circle area: π · 32 = 9π.
Three small shaded circles (radius 12): each has area π4; together 3π4.
Big shaded disk (radius 2): area 4π, minus two white inner circles (radius 1 each, total 2π): net 2π.
Number Theorycomplementary-countingcareful-counting
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of 5-cent, 10-cent, and 25-cent stamps, with exactly 20 of each type. What is the greatest number of stamps Nicolas can use to make exactly $7.10 in postage?
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Maximizing stamps used = minimizing stamps removed from his whole collection. What does the whole collection total?
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Total of all 60 stamps = $8. He needs to remove $0.90. Minimize the number of stamps that sum to $0.90.
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Total value of all 60 stamps: 20·($0.05 + $0.10 + $0.25) = 20 · $0.40 = $8.00.
He needs to make $7.10, so he removes $8.00 − $7.10 = $0.90 worth. Maximizing stamps used ≡ minimizing stamps removed.
Minimum stamps summing to $0.90: three 25¢ (75¢) + one 10¢ + one 5¢ = $0.90 in 5 stamps.
