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.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Linear Equations & Word Problems | Adding one nickel would make the counts equal, so there are 7 pennies and 6 nickels among the 13 coins. | |
| 2 | E | Exponents, Logarithms & Radicals | Each sum is just twice a power of 2, so the fraction is 2^4 divided by 2^{-2}, which is 2^6. | |
| 3 | E | Ratios, Percents & Averages | Gravel and dirt take 1/3 + 1/5 = 8/15 of the trip, so the 20 paved miles are the remaining 7/15. | |
| 4 | B | Linear Equations & Word Problems | Calvin's bill equals twice Susie's, so 2m + 16b = 8m + 6b, which collapses to 10b = 6m. | |
| 5 | A | Linear Equations & Word Problems | With panes 2x by 5x in a 4-by-2 grid, width 8x + 10 must equal height 10x + 6, forcing x = 2. | |
| 6 | C | Ratios, Percents & Averages | Under the sale a pair of balloons costs 1 + 2/3 = 5/3 of a regular price, so 30 prices buy 18 pairs. | |
| 7 | A | Ratios, Percents & Averages | Percent greater is measured relative to the smaller base B: x/100 = (A - B)/B. | |
| 8 | E | Linear Equations & Word Problems | Convert to a rate of b/(6t) feet per second, multiply by 180 seconds, then divide by 3 for yards. | |
| 9 | A | Algebraic Manipulation | Multiply top and bottom by wz: the given fraction becomes (w+z)/(z-w), which is the negative of what is asked. | |
| 10 | C | Bases & Digits | The tens and thousands columns force C = 0 with no carries, so D = A + B for distinct nonzero A, B, giving D from 3 to 9. | |
| 11 | C | Ratios, Percents & Averages | Successive discounts multiply: the three options leave 72.25%, 72.9%, and 71.25% of the price, so n must beat 28.75%. | |
| 12 | C | Divisibility & Factors | Large divisors pair with small ones: the fifth largest divisor is N divided by the fifth smallest, and the smallest divisors are 1, 2, 4, 5, 8. | |
| 13 | B | Triangles: Area & Pythagorean | Each vertex of triangle ABC is 2 units from the central hexagon's center, so ABC is equilateral with circumradius 2 and side 2*sqrt(3). | |
| 14 | D | Bases & Digits | Reversing a 3-digit number changes it by 99(c - a); equating with 55h forces c - a to be a multiple of 5, so (a, c) = (1, 6). | |
| 15 | A | Triangles: Area & Pythagorean | Trisecting the right angle at D creates 30-60-90 triangles: with height 1, AE = 1/sqrt(3) and AF = sqrt(3), so EF = 2/sqrt(3). | |
| 16 | B | Basic Probability | Split into exactly three alike and all four alike; exactly three alike is 6 values times 4 positions for the odd die times 5 values for it. | |
| 17 | D | Divisibility & Factors | Pull out the common 2^1002, then find how many 2s divide 5^1002 - 1 by factoring it as (5^501 - 1)(5^501 + 1). | |
| 18 | E | Statistics & Data | Minimize the other ten entries: use few 8s so small numbers like 1 can fill the low slots, while the median pins 9 in the middle. | |
| 19 | D | Geometric Probability | Fix the first point and draw its two tangents to the inner circle; they cut off a 120-degree arc of the outer circle, so the probability is 1/3. | |
| 20 | C | Quadratics | Treat it as a quadratic in x^2: it factors as (x^2 - 1)(x^2 - 50), negative exactly when 1 < x^2 < 50. | |
| 21 | B | Quadrilaterals & Polygon Areas | Drop both altitudes: the overhangs satisfy x + y = 12 and y^2 - x^2 = 96, giving x = 2, y = 10, h^2 = 96. | |
| 22 | B | Circles | The tangent circle is centered at the square's center; its radius plus 1/2 equals the distance to a semicircle center, which is sqrt(5)/2. | |
| 23 | E | Solid Geometry | In the axial cross-section the incircle forces slant = R + r, hence sphere radius squared = Rr; the volume condition then becomes R^2 - 3Rr + r^2 = 0. | |
| 24 | B | Arrangements with Restrictions | A consecutive block and its complement are both blocks summing to n and 15 - n, so only sums 6 and 7 can fail; check adjacency conditions. | |
| 25 | C | Conditional Probability & States | Consecutive differences of the escape probabilities satisfy d_N = N/(10-N) d_{N-1}, so d_N = d_0 / C(9,N), and the differences sum to 1. |
Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic tags, difficulty ratings and key insights on this page are original to this site. No problem statements are reproduced here — each links to its own page.