2014 AMC 10B
All 25 problems, 75 minutes, scored 6 / 1.5 / 0. The answer key and the idea behind each problem are on this page too, folded away until you ask for them.
The problems
Show the answer key, topics and key insights
This gives away all 25 answers and the idea behind each one. Sit the paper first if you mean to.
The same key as a PDF 127 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Linear Equations & Word Problems | One more nickel would make the two piles equal, so right now there is one more penny than nickel: 7 pennies and 6 nickels. | |
| 2 | E | Exponents, Logarithms & Radicals | The top is 8 + 8 = 16 and the bottom is 1/8 + 1/8 = 1/4; dividing by 1/4 is multiplying by 4. | |
| 3 | E | Ratios, Percents & Averages | Gravel and dirt together are 1/3 + 1/5 = 8/15 of the trip, so the 20 paved miles are the remaining 7/15 of it. | |
| 4 | B | Linear Equations & Word Problems | Calvin's bill is twice Susie's, so 2 muffins and 16 bananas cost the same as 8 muffins and 6 bananas; cancel what they share. | |
| 5 | A | Linear Equations & Word Problems | Call the panes 2x wide and 5x tall. Across there are 4 panes and 5 borders, down there are 2 panes and 3 borders; the two totals must match. | |
| 6 | C | Ratios, Percents & Averages | Measure money in regular balloon prices. On sale, a pair costs 1 + 2/3 = 5/3 prices, so 30 prices buy 18 pairs, which is 36 balloons. | |
| 7 | A | Ratios, Percents & Averages | Percent greater is measured against the number you start from, B: find how much bigger A is (A - B), divide by B, and multiply by 100. | |
| 8 | E | Linear Equations & Word Problems | Two conversions are needed: 3 minutes is 180 seconds, and every 3 feet is one yard. Find feet per second first, then scale. | |
| 9 | A | Algebraic Manipulation | Multiply top and bottom by wz: the given fraction becomes (z + w)/(z - w), which is the asked fraction with its sign flipped. | |
| 10 | C | Bases & Digits | Columns where the answer digit repeats an added digit force C = 0 and no carrying, so D = A + B for different nonzero digits A and B. | |
| 11 | C | Ratios, Percents & Averages | Discounts one after another multiply what is left of the price. The three deals take off 27.75%, 27.1% and 28.75%, so n must be more than 28.75. | |
| 12 | C | Divisibility & Factors | Divisors come in pairs that multiply to N, so the fifth largest divisor is N divided by the fifth smallest. The smallest divisors are 1, 2, 4, 5, 8. | |
| 13 | B | Triangles: Area & Pythagorean | Measure from the center of the middle hexagon: each corner of triangle ABC is 2 units away, and the three corners are evenly spaced, so the triangle is equilateral. | |
| 14 | D | Bases & Digits | Reversing a 3-digit number changes it by 99 times (c - a). That must equal 55 times the hours, forcing c - a = 5. | |
| 15 | A | Triangles: Area & Pythagorean | Splitting the right angle at D into three equal parts gives 30 and 60 degree angles, so the 30-60-90 triangles fix where E and F land. | |
| 16 | B | Basic Probability | Count two separate cases: all four dice match (6 ways), and exactly three match (6 values, 4 spots for the odd die, 5 values for it). | |
| 17 | D | Divisibility & Factors | Pull out the obvious 2^1002. Then count the 2s hiding in 5^1002 - 1 by splitting it into (5^501 - 1)(5^501 + 1). | |
| 18 | E | Statistics & Data | To make the biggest number large, make the other ten as small as possible. Using only three 8s lets two 1s into the low spots. | |
| 19 | D | Geometric Probability | Fix the first point. The chords from it that just touch the inner circle end 120 degrees away on each side, leaving a 120-degree arc that works. | |
| 20 | C | Quadratics | Treat x^2 as a single number. The expression factors as (x^2 - 1)(x^2 - 50), which is negative only when x^2 is between 1 and 50. | |
| 21 | B | Quadrilaterals & Polygon Areas | Drop both heights. The two end pieces of the long side add to 12; the Pythagorean theorem on each side gives pieces 2 and 10, height squared 96. | |
| 22 | B | Circles | The middle circle sits at the square's center. Its radius plus 1/2 equals the distance from the center to a semicircle's center, which is half of root 5. | |
| 23 | E | Solid Geometry | Slice through the middle: a circle inside a trapezoid. Equal touching segments make the slanted side R + r, so sphere radius squared equals Rr. | |
| 24 | B | Arrangements with Restrictions | What is left after removing a run of neighbors is also a run, and the two sums add to 15. So only sums 6 and 7 can fail. | |
| 25 | C | Conditional Probability & States | Look at the gaps between escape chances on neighboring pads: each gap is the previous one times N/(10 - N), and all gaps add up to 1. |
2014 AMC 10B. Problems © Mathematical Association of America, American Mathematics Competitions (maa.org/amc), licensed CC BY-NC-SA 4.0. Typeset and annotated by AMC 10 Step by Step (amcstep.com) and shared under the same licence. Not endorsed by the MAA.