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Problems 1–10 drill

30 early problems (2010+): the zero-error zone

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Problems 1–10 drillAMC 10 Step by Step

Name: Date: September 13, 2026

30 problems · 30 minutes · 6 / 1.5 / 0

1.2010 AMC 10A #5
The area of a circle whose circumference is $24\pi$ is $k\pi$ . What is the value of $k$ ?
1. A)
  $6$
2. B)
  $12$
3. C)
  $24$
4. D)
  $36$
5. E)
  $144$
2.2011 AMC 10A #6
Set $A$ has 20 elements, and set $B$ has 15 elements. What is the smallest possible number of elements in $A \cup B$ , the union of $A$ and $B$ ?
1. A)
  $5$
2. B)
  $15$
3. C)
  $20$
4. D)
  $35$
5. E)
  $300$
3.2012 AMC 10A #1
Cagney can frost a cupcake every $20$ seconds and Lacey can frost a cupcake every $30$ seconds. Working together, how many cupcakes can they frost in $5$ minutes?
1. A)
  $10$
2. B)
  $15$
3. C)
  $20$
4. D)
  $25$
5. E)
  $30$
4.2012 AMC 10B #3
The point in the xy-plane with coordinates $(1000, 2012)$ is reflected across the line $y = 2000$ . What are the coordinates of the reflected point?
1. A)
  $(998,2012)$
2. B)
  $(1000,1988)$
3. C)
  $(1000,2024)$
4. D)
  $(1000,4012)$
5. E)
  $(1012,2012)$
5.2013 AMC 10B #4
When counting from $3$ to $201$ , $53$ is the $51^\mathrm{st}$ number counted. When counting backwards from $201$ to $3$ , $53$ is the $n^\mathrm{th}$ number counted. What is $n$ ?
1. A)
  $146$
2. B)
  $147$
3. C)
  $148$
4. D)
  $149$
5. E)
  $150$
6.2015 AMC 10B #1
What is the value of $2-(-2)^{-2}$ ?
1. A)
  $-2$
2. B)
  $\dfrac{1}{16}$
3. C)
  $\dfrac{7}{4}$
4. D)
  $\dfrac{9}{4}$
5. E)
  $6$
7.2016 AMC 10A #5
A rectangular box has integer side lengths in the ratio $1: 3: 4$ . Which of the following could be the volume of the box?
1. A)
  $48$
2. B)
  $56$
3. C)
  $64$
4. D)
  $96$
5. E)
  $144$
8.2018 AMC 10A #4
How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$ -period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.)
1. A)
  $3$
2. B)
  $6$
3. C)
  $12$
4. D)
  $18$
5. E)
  $24$
9.2018 AMC 10B #3
In the expression $\left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained?
1. A)
  $2$
2. B)
  $3$
3. C)
  $4$
4. D)
  $6$
5. E)
  $24$
10.2019 AMC 10A #4
A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn $?$
1. A)
  $75$
2. B)
  $76$
3. C)
  $79$
4. D)
  $84$
5. E)
  $91$
11.2021 AMC 10B #5
The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$ , while the other two multiply to $30$ . What is the sum of the ages of Jonie's four cousins?
1. A)
  $21$
2. B)
  $22$
3. C)
  $23$
4. D)
  $24$
5. E)
  $25$
12.2023 AMC 10B #5
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$ . Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$ . How many numbers are written on the blackboard?
1. A)
  $10$
2. B)
  $5$
3. C)
  $6$
4. D)
  $8$
5. E)
  $9$
13.2023 AMC 10B #4
Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint?
1. A)
  $162{,}500$
2. B)
  $162.5$
3. C)
  $1{,}625$
4. D)
  $1{,}625{,}000$
5. E)
  $16{,}250$
14.2012 AMC 10A #9
A pair of six-sided dice are labeled so that one die has only even numbers (two each of $2$ , $4$ , and $6$ ), and the other die has only odd numbers (two each of $1$ , $3$ , and $5$ ). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is $7$ ?
1. A)
  $\frac{1}{6}$
2. B)
  $\frac{1}{5}$
3. C)
  $\frac{1}{4}$
4. D)
  $\frac{1}{3}$
5. E)
  $\frac{1}{2}$
15.2013 AMC 10A #6
Joey and his five brothers are ages 3, 5, 7, 9, 11, and 13. One afternoon two of his brothers whose ages sum to 16 went to the movies, two brothers younger than 10 went to play baseball, and Joey and the 5-year-old stayed home. How old is Joey?
1. A)
  $3$
2. B)
  $7$
3. C)
  $9$
4. D)
  $11$
5. E)
  $13$
16.2013 AMC 10B #6
The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and fifth-graders?
1. A)
  $22$
2. B)
  $23.25$
3. C)
  $24.75$
4. D)
  $26.25$
5. E)
  $28$
17.2014 AMC 10B #8
A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes?
1. A)
  $\frac{b}{1080t}$
2. B)
  $\frac{30t}{b}$
3. C)
  $\frac{30b}{t}$
4. D)
  $\frac{10t}{b}$
5. E)
  $\frac{10b}{t}$
18.2016 AMC 10B #10
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?
1. A)
  $14.0$
2. B)
  $16.0$
3. C)
  $20.0$
4. D)
  $33.3$
5. E)
  $55.6$
19.2017 AMC 10B #9
A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?
1. A)
  $\frac{1}{27}$
2. B)
  $\frac{1}{9}$
3. C)
  $\frac{2}{9}$
4. D)
  $\frac{7}{27}$
5. E)
  $\frac{1}{2}$
20.2019 AMC 10A #10
A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?
1. A)
  $17$
2. B)
  $25$
3. C)
  $26$
4. D)
  $27$
5. E)
  $28$
21.2021 AMC Fall 10B #6
The least positive integer with exactly $2021$ distinct positive divisors can be written in the form $m \cdot 6^k$ , where $m$ and $k$ are integers and $6$ is not a divisor of $m$ . What is $m+k?$
1. A)
  $47$
2. B)
  $58$
3. C)
  $59$
4. D)
  $88$
5. E)
  $90$
22.2021 AMC Fall 10A #8
A two-digit positive integer is said to be $\emph{cuddly}$ if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
1. A)
  $0$
2. B)
  $1$
3. C)
  $2$
4. D)
  $3$
5. E)
  $4$
23.2021 AMC Fall 10B #10
Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?
1. A)
  $27$
2. B)
  $37$
3. C)
  $47$
4. D)
  $57$
5. E)
  $67$
24.2022 AMC 10B #3
How many three-digit positive integers have an odd number of even digits?
1. A)
  $150$
2. B)
  $250$
3. C)
  $350$
4. D)
  $450$
5. E)
  $550$
25.2024 AMC 10B #10
Quadrilateral $ABCD$ is a parallelogram, and $E$ is the midpoint of the side $\overline{AD}$ . Let $F$ be the intersection of lines $EB$ and $AC$ . What is the ratio of the area of quadrilateral $CDEF$ to the area of $\triangle CFB$ ?
1. A)
  $5:4$
2. B)
  $4:3$
3. C)
  $3:2$
4. D)
  $5:3$
5. E)
  $2:1$
26.2025 AMC 10A #7
Suppose $a$ and $b$ are real numbers. When the polynomial $x^3+x^2+ax+b$ is divided by $x-1$ , the remainder is $4$ . When the polynomial is divided by $x-2$ , the remainder is $6$ . What is $b-a$ ?
1. A)
  $14$
2. B)
  $15$
3. C)
  $16$
4. D)
  $17$
5. E)
  $18$
27.2025 AMC 10B #2
Jerry wrote down the ones digit of each of the first $2025$ positive squares: $1,4,9,6,5,6,\dots$ . What is the sum of all the numbers Jerry wrote down?
1. A)
  $9025$
2. B)
  $9070$
3. C)
  $9090$
4. D)
  $9115$
5. E)
  $9160$
28.2025 AMC 10B #7
Frances stands $15$ meters directly south of a locked gate in a fence that runs east-west. Immediately behind the fence is a box of chocolates, located $x$ meters east of the locked gate. An unlocked gate lies $9$ meters east of the box, and another unlocked gate lies $8$ meters west of the locked gate. Frances can reach the box by walking toward an unlocked gate, passing through it, and walking toward the box. It happens that the total distance Frances would travel is the same via either unlocked gate. What is the value of $x$ ?
1. A)
  $3\frac27$
2. B)
  $3\frac37$
3. C)
  $3\frac47$
4. D)
  $3\frac57$
5. E)
  $3\frac67$
29.2025 AMC 10A #5
Consider the sequence of positive integers
$1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,2,3,4,5,6,5,4,3,2,1,2,\dots$
What is the $2025$ th term in this sequence?
1. A)
  $5$
2. B)
  $15$
3. C)
  $16$
4. D)
  $44$
5. E)
  $45$
30.2025 AMC 10A #6
In an equilateral triangle each interior angle is trisected by a pair of rays. The intersection of the interiors of the middle $20^{\circ}$ -angle at each vertex is the interior of a convex hexagon. What is the degree measure of the smallest angle of this hexagon?
1. A)
  $80$
2. B)
  $90$
3. C)
  $100$
4. D)
  $110$
5. E)
  $120$

Problems 1–10 drill — answersAMC 10 Step by Step

Name: Date: September 13, 2026

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Answers

#	Answer	Topic and the idea it turns on
1	E	Circles — Circumference 2 pi r = 24 pi gives r = 12, so the area is pi r^2 = 144 pi.
2	C	Sets, Estimation & Miscellaneous — The union is smallest when the smaller set sits entirely inside the larger one, so it cannot have fewer than 20 elements and 20 is attainable.
3	D	Linear Equations & Word Problems — Convert each frosting time to a rate per minute (3 and 2 cupcakes per minute), add the rates, and multiply by 5 minutes.
4	B	Coordinate Geometry — Reflecting over a horizontal line keeps x and sends y to 2(2000) - y, so 2012 goes to 1988.
5	D	Number Properties — Counting backwards from 201, the number 53 is the (201 - 53 + 1)th, because inclusive counts need the extra 1.
6	C	Exponents, Logarithms & Radicals — A negative exponent means reciprocal, and squaring kills the sign: (-2)^(-2) = 1/4, so the value is 2 - 1/4.
7	D	Number Properties — Sides k, 3k, 4k give volume 12k^3, so the volume must be 12 times a perfect cube; only 96 = 12 * 8 qualifies.
8	E	Arrangements with Restrictions — Only four sets of three non-adjacent periods exist among six; each set can be filled with the three courses in 3! = 6 orders.
9	B	Basic Counting — The value depends only on how the four digits are split into two pairs, and there are just three pairings: 14, 11 and 10.
10	B	Sets, Estimation & Miscellaneous — Worst case: draw 14 of each color that has at least 14 balls and every ball of the small colors, 75 in all; the next ball forces a 15th.
11	B	Divisibility & Factors — 30 has only one single-digit factor pair, 5 × 6, so 6 is taken and 24 must be 3 × 8.
12	A	Linear Equations & Word Problems — Adding 3 to each of n numbers adds 3n to the sum, while tripling each number triples the sum, so S + 3n = 45 = 3S.
13	C	Sets, Estimation & Miscellaneous — Convert both dimensions to centimeters first: 6.5 mm = 0.65 cm and 25 m = 2500 cm, then multiply to get 1625 square centimeters.
14	D	Basic Probability — Each die shows one of three values with equal probability, so there are 9 equally likely value pairs, and exactly three of them (2+5, 4+3, 6+1) sum to 7.
15	D	Logic Puzzles — With the 5-year-old home, the movie pair is 3 and 13 or 7 and 9; only 3 and 13 leaves two under-10 brothers for baseball, so Joey is 11.
16	C	Statistics & Data — A combined average is total age over total people: (3311 + 5533)/(33 + 55), not the average of 11 and 33.
17	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.
18	D	Similar & Congruent Triangles — Same wood and thickness means weight is proportional to area, and area of similar triangles scales by the square of the side ratio: 12 * (5/3)^2 = 100/3.
19	D	Basic Probability — Winning means exactly 2 right or all 3 right; count 3 ways for exactly two (each 1/27 times 2) plus 1 way for three, out of 27.
20	C	Paths & Grids — Each time the diagonal crosses a grid line it enters a new tile; it crosses 16 vertical and 9 horizontal lines, never at a corner since gcd(10,17) = 1.
21	B	Divisibility & Factors — 2021 = 43 * 47, so the exponents are 46 and 42; put the bigger exponent on the smaller prime to get 2^46 * 3^42 = 16 * 6^42.
22	B	Bases & Digits — The condition 10a + b = a + b^2 becomes 9a = b(b-1); checking digits b, only b = 9 gives a valid tens digit a = 8.
23	A	Logic Puzzles — Alice's uncertainty rules out 1 and 40; Bob then knows only if he holds 1, 2, 39, or 40; prime forces 2, and 200 + a = 225.
24	D	Basic Counting — Fix the first two digits; among the ten possible units digits exactly five flip the parity of the even-digit count, so exactly half of all 900 numbers work.
25	A	Similar & Congruent Triangles — Triangles AFE and CFB are similar with ratio 1:2, so [AFE] = 1 gives [CFB] = 4, and every other region follows from area ratios.
26	E	Polynomials — The Remainder Theorem turns the two divisions into P(1) = 4 and P(2) = 6, a two-by-two linear system in a and b.
27	D	Modular Arithmetic — The ones digit of n^2 depends only on the ones digit of n, so the list repeats with period 10 and each block of ten sums to 45.
28	C	Coordinate Geometry — Put the locked gate at the origin: the western route is 17 + (x + 8) by the 8-15-17 triple, and the eastern route is sqrt(15^2 + (x+9)^2) + 9.
29	E	Sequences & Series — The sequence runs up to n and back to 1; the first n runs use n(n-1) + 1 terms, and 45*44 + 1 = 1981 falls just short of 2025.
30	C	Angles & Polygons — Each hexagon vertex is where two trisectors from different corners cross, and the triangle cut off has base angles 20 and 20, or 40 and 40.