2007 AMC 10A
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 | Ratios, Percents & Averages | A 25% discount leaves 75% of the price and a 30% discount leaves 70%; compute each total and subtract. | |
| 2 | A | Functions | A custom operation is just a formula; substitute a = 6 and b = 2 into each definition separately, then divide. | |
| 3 | D | Solid Geometry | The submerged brick displaces its own volume, so the rise equals brick volume divided by the aquarium's base area. | |
| 4 | A | Linear Equations & Word Problems | Consecutive odd integers differ by 2, so write them as x and x + 2 and solve x + 2 = 3x. | |
| 5 | B | Systems of Equations | Two purchases give two linear equations in the unit prices; eliminate one variable, find both prices, then price the third order. | |
| 6 | A | Ratios, Percents & Averages | Percent increase is the change divided by the starting value; 6/60 = 10% beats every other year's ratio, which are all under 10%. | |
| 7 | D | Ratios, Percents & Averages | The state tax is 10% of the remaining 80%, so total tax is 20% + 8% = 28% of the inheritance. | |
| 8 | D | Angles & Polygons | Both triangles share base AC, so angle BAD is the difference of the two base angles: 70 - 20 = 50. | |
| 9 | E | Exponents, Logarithms & Radicals | Write 81 as 3^4 and 125 as 5^3 so each equation becomes an equality of exponents, a linear system in a and b. | |
| 10 | E | Ratios, Percents & Averages | Convert each average into a total: the whole family sums to 20(n+2) and the mother plus children sum to 16(n+1); the difference is the father's 48. | |
| 11 | C | Sets, Estimation & Miscellaneous | Two opposite faces use every vertex exactly once, so two face sums equal 1 + 2 + ... + 8 = 36 and each face sums to 18. | |
| 12 | D | Basic Counting | Each tourist independently picks one of two guides, 2^6 ways; subtract the 2 assignments in which one guide gets nobody. | |
| 13 | B | Linear Equations & Word Problems | Measure time in walking units: walking b directly equals walking a home plus biking a + b at one-seventh the cost, so 7b = 8a. | |
| 14 | A | Circles | A 3-4-5 triangle is right, so its hypotenuse is a diameter: 6 = (6/5)*5, and area scales by (6/5)^2 from the 3-4-5 area of 6. | |
| 15 | B | Circles | Along a diagonal of the square, center to small-circle center is 2 + 1 = 3 and small-circle center to corner is sqrt(2), so the half-diagonal is 3 + sqrt(2). | |
| 16 | E | Basic Probability | ad - bc is even exactly when ad and bc have the same parity; a product is odd with probability 1/4, so the answer is (1/4)^2 + (3/4)^2. | |
| 17 | D | Number Properties | Since 75 = 3 * 5^2, m must supply 3^2 * 5 to make every exponent a multiple of 3, giving m = 45, n = 15. | |
| 18 | C | Quadrilaterals & Polygon Areas | ABCM is triangle ABG minus triangle CMG; similar triangles AMH and GMC (ratio 12 : 8) put M at horizontal distance 8/5 from line BG. | |
| 19 | C | Quadrilaterals & Polygon Areas | The unpainted half is four isosceles right triangles, each with hypotenuse s/sqrt(2) along a side; the leftover painted edge is two corner legs of length w/sqrt(2). | |
| 20 | D | Algebraic Manipulation | Squaring a + 1/a produces a^2 + 1/a^2 plus 2; square twice and subtract 2 each time: 16 - 2 = 14, then 196 - 2 = 194. | |
| 21 | C | Solid Geometry | The sphere's diameter is both the outer edge 2 and the inner cube's space diagonal s*sqrt(3), so s^2 = 4/3 and surface area is 8. | |
| 22 | D | Bases & Digits | Because the digits wrap around cyclically, every digit appears once in each of the three places, so S = 111 times the digit total and 111 = 3 * 37. | |
| 23 | B | Diophantine Equations | Factor (m+n)(m-n) = 96; m + n and m - n share parity, so both factors are even: 2*48, 4*24, 6*16, 8*12. | |
| 24 | B | Circles | Draw AE and BF: the shaded region is rectangle ABFE minus two isosceles right triangles (legs 2) and two 45-degree sectors of radius 2. | |
| 25 | D | Bases & Digits | S(n) + S(S(n)) is at most 38, so n is at least 1969; mod 9 forces n to be a multiple of 3, leaving 13 candidates to test. |
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.