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Problems 11–15 drill

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Problems 11–15 drillAMC 10 Step by Step

Name: Date: September 13, 2026

30 problems · 45 minutes · 6 / 1.5 / 0

1.2010 AMC 10A #11
The length of the interval of solutions of the inequality $a \le 2x + 3 \le b$ is $10$ . What is $b - a$ ?
1. A)
  $6$
2. B)
  $10$
3. C)
  $15$
4. D)
  $20$
5. E)
  $30$
2.2010 AMC 10A #12
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?
1. A)
  $0.04$
2. B)
  $\frac{0.4}{\pi}$
3. C)
  $0.4$
4. D)
  $\frac{4}{\pi}$
5. E)
  $4$
3.2011 AMC 10B #12
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
1. A)
  $\frac{\pi}{3}$
2. B)
  $\frac{2\pi}{3}$
3. C)
  $\pi$
4. D)
  $\frac{4\pi}{3}$
5. E)
  $\frac{5\pi}{3}$
4.2011 AMC 10B #14
A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?
1. A)
  $52$
2. B)
  $58$
3. C)
  $62$
4. D)
  $68$
5. E)
  $70$
5.2014 AMC 10A #12
A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. What is the area of the shaded region?
1. A)
  $27\sqrt{3}-9\pi$
2. B)
  $27\sqrt{3}-6\pi$
3. C)
  $54\sqrt{3}-18\pi$
4. D)
  $54\sqrt{3}-12\pi$
5. E)
  $108\sqrt{3}-9\pi$
6.2014 AMC 10B #11
For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts:
(1) two successive $15\%$ discounts
(2) three successive $10\%$ discounts
(3) a $25\%$ discount followed by a $5\%$ discount
What is the smallest possible positive integer value of $n$ ?
1. A)
  $27$
2. B)
  $28$
3. C)
  $29$
4. D)
  $31$
5. E)
  $33$
7.2014 AMC 10A #11
A customer who intends to purchase an appliance has three coupons, only one of which may be used:
Coupon 1: $10\%$ off the listed price if the listed price is at least $\$50$
Coupon 2: $\$20$ off the listed price if the listed price is at least $\$100$
Coupon 3: $18\%$ off the amount by which the listed price exceeds $\$100$
For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$ ?
1. A)
  $179.95
2. B)
  $199.95
3. C)
  $219.95
4. D)
  $239.95
5. E)
  $259.95
8.2015 AMC 10A #11
The ratio of the length to the width of a rectangle is $4$ : $3$ . If the rectangle has diagonal of length $d$ , then the area may be expressed as $kd^2$ for some constant $k$ . What is $k$ ?
1. A)
  $\frac{2}{7}$
2. B)
  $\frac{3}{7}$
3. C)
  $\frac{12}{25}$
4. D)
  $\frac{16}{25}$
5. E)
  $\frac{3}{4}$
9.2021 AMC 10A #11
For which of the following integers $b$ is the base- $b$ number $2021_b - 221_b$ not divisible by $3$ ?
1. A)
  $3$
2. B)
  $4$
3. C)
  $6$
4. D)
  $7$
5. E)
  $8$
10.2022 AMC 10B #12
A pair of fair $6$ -sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$ ?
1. A)
  $2$
2. B)
  $3$
3. C)
  $4$
4. D)
  $5$
5. E)
  $6$
11.2010 AMC 10B #12
At the beginning of the school year, $50\%$ of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answered "No." Altogether, $x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x$ ?
1. A)
  $0$
2. B)
  $20$
3. C)
  $40$
4. D)
  $60$
5. E)
  $80$
12.2012 AMC 10A #12
A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$ ) or is divisible by $4$ but not $100$ (such as $2012$ ). The $200$ th anniversary of the birth of novelist Charles Dickens was celebrated on February $7$ , $2012$ , a Tuesday. On what day of the week was Dickens born?
1. A)
  $\text{Friday}$
2. B)
  $\text{Saturday}$
3. C)
  $\text{Sunday}$
4. D)
  $\text{Monday}$
5. E)
  $\text{Tuesday}$
13.2016 AMC 10B #15
All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$ . What is the number in the center?
1. A)
  $5$
2. B)
  $6$
3. C)
  $7$
4. D)
  $8$
5. E)
  $9$
14.2016 AMC 10A #13
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
1. A)
  $1$
2. B)
  $2$
3. C)
  $3$
4. D)
  $4$
5. E)
  $5$
15.2018 AMC 10B #11
Which of the following expressions is never a prime number when $p$ is a prime number?
1. A)
  $p^2+16$
2. B)
  $p^2+24$
3. C)
  $p^2+26$
4. D)
  $p^2+46$
5. E)
  $p^2+96$
16.2018 AMC 10B #14
A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list?
1. A)
  $202$
2. B)
  $223$
3. C)
  $224$
4. D)
  $225$
5. E)
  $234$
17.2018 AMC 10A #11
When $7$ fair standard $6$ -sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as $\frac{n}{6^{7}},$ where $n$ is a positive integer. What is $n$ ?
1. A)
  $42$
2. B)
  $49$
3. C)
  $56$
4. D)
  $63$
5. E)
  $84$
18.2019 AMC 10A #12
Melanie computes the mean $\mu$ , the median $M$ , and the modes of the $365$ values that are the dates in the months of $2019$ . Thus her data consists of $12$ $1\text{s}$ , $12$ $2\text{s}$ , . . . , $12$ $28\text{s}$ , $11$ $29\text{s}$ , $11$ $30\text{s}$ , and $7$ $31\text{s}$ . Let $d$ be the median of the modes. Which of the following statements is true?
1. A)
  $\mu < d < M$
2. B)
  $M < d < \mu$
3. C)
  $d = M =\mu$
4. D)
  $d < M < \mu$
5. E)
  $d < \mu < M$
19.2020 AMC 10A #15
A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
1. A)
  $3$
2. B)
  $5$
3. C)
  $12$
4. D)
  $18$
5. E)
  $23$
20.2020 AMC 10A #13
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$ , and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$ . What is the probability that the sequence of jumps ends on a vertical side of the square?
1. A)
  $\frac12$
2. B)
  $\frac 58$
3. C)
  $\frac 23$
4. D)
  $\frac34$
5. E)
  $\frac 78$
21.2020 AMC 10A #11
What is the median of the following list of $4040$ numbers $?$ $1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2$
1. A)
  $1974.5$
2. B)
  $1975.5$
3. C)
  $1976.5$
4. D)
  $1977.5$
5. E)
  $1978.5$
22.2021 AMC 10A #15
Values for $A,B,C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the choices $A=3, B=2, C=4, D=1$ is considered the same as the choices $A=4, B=1, C=3, D=2.$ )
1. A)
  $30$
2. B)
  $60$
3. C)
  $90$
4. D)
  $180$
5. E)
  $360$
23.2021 AMC Fall 10B #11
A regular hexagon of side length $1{ }$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs?
1. A)
  $\frac{5\sqrt{3}}{2} - \pi$
2. B)
  $3\sqrt{3}-\pi$
3. C)
  $4\sqrt{3}-\frac{3\pi}{2}$
4. D)
  $\pi - \frac{\sqrt{3}}{2}$
5. E)
  $\frac{\pi + \sqrt{3}}{2}$
24.2021 AMC 10B #13
Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $\underline{32d}$ in base $n$ equals $263$ , and the value of the numeral $\underline{324}$ in base $n$ equals the value of the numeral $\underline{11d1}$ in base six. What is $n + d ?$
1. A)
  $10$
2. B)
  $11$
3. C)
  $13$
4. D)
  $15$
5. E)
  $16$
25.2021 AMC 10A #12
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \text{ cm}$ and $6 \text{ cm}$ . Into each cone is dropped a spherical marble of radius $1 \text{ cm}$ , which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
1. A)
  $1:1$
2. B)
  $47:43$
3. C)
  $2:1$
4. D)
  $40:13$
5. E)
  $4:1$
26.2022 AMC 10B #13
The positive difference between a pair of primes is equal to $2$ , and the positive difference between the cubes of the two primes is $31106$ . What is the sum of the digits of the least prime that is greater than those two primes?
1. A)
  $8$
2. B)
  $10$
3. C)
  $11$
4. D)
  $13$
5. E)
  $16$
27.2022 AMC 10B #14
Suppose that $S$ is a subset of $\left\{ 1, 2, 3, \ldots , 25 \right\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S.$ What is the maximum number of elements $S$ may contain?
1. A)
  $12$
2. B)
  $13$
3. C)
  $14$
4. D)
  $15$
5. E)
  $16$
28.2024 AMC 10B #13
Positive integers $x$ and $y$ satisfy the equation $\sqrt{x} + \sqrt{y} = \sqrt{1183}$ . What is the minimum possible value of $x+y$ ?
1. A)
  $585$
2. B)
  $595$
3. C)
  $623$
4. D)
  $700$
5. E)
  $791$
29.2025 AMC 10A #12
Carlos uses a $4$ -digit passcode to unlock his computer. In his passcode, exactly one digit is even, exactly one (possibly different) digit is prime, and no digit is $0$ . How many $4$ -digit passcodes satisfy these conditions?
1. A)
  $176$
2. B)
  $192$
3. C)
  $432$
4. D)
  $464$
5. E)
  $608$
30.2025 AMC 10A #13
In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is $k$ , where $0 < k < 1.$ The spaces between squares are alternately shaded as shown in the figure (which is not necessarily drawn to scale).
The area of the shaded portion of the figure is $64\%$ of the area of the original square. What is $k$ ?
1. A)
  $\frac{3}{5}$
2. B)
  $\frac{16}{25}$
3. C)
  $\frac{2}{3}$
4. D)
  $\frac{3}{4}$
5. E)
  $\frac{4}{5}$

Problems 11–15 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	D	Absolute Value & Inequalities — Solving for x squeezes the interval by a factor of 2: the x-interval has length (b - a)/2, so b - a = 20.
2	C	Solid Geometry — Volume scales as the cube of length: a volume ratio of one to a million means a length ratio of one to 100, so 40/100 = 0.4.
3	A	Circles — The straight sides are the same length on both edges; only the two semicircles differ, and their radii differ by 6, adding exactly 12 pi meters.
4	C	Algebraic Manipulation — Diagonal gives l^2 + w^2 = 625, area gives lw = 168, so (l + w)^2 = 625 + 336 = 961 and l + w = 31.
5	C	Circles — Each sector spans the hexagon's 120-degree interior angle, so the six sectors total two full circles of radius 3; subtract 18 pi from the hexagon area 54 root 3.
6	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%.
7	C	Ratios, Percents & Averages — Coupon 1 beats coupon 2 when 0.1P > 20 (P > 200) and beats coupon 3 when 0.1P > 0.18(P - 100) (P < 225); only 219.95 fits.
8	C	Triangles: Area & Pythagorean — Sides 4x and 3x make the diagonal 5x, so d = 5x and the area 12x^2 becomes 12(d/5)^2 = (12/25)d^2.
9	E	Bases & Digits — Expand in base b: the difference is 2b^3 - 2b^2 = 2b^2(b-1), divisible by 3 exactly when b or b-1 is a multiple of 3.
10	C	Basic Probability — A sum of 7 has probability 1/6 per roll, so avoid it n times with probability (5/6)^n; find the first n with (5/6)^n below 1/2.
11	D	Sets, Estimation & Miscellaneous — The net shift is 20% from No to Yes; switchers come in that 20% plus matched pairs who swap in opposite directions, and at most 30% can swap Yes-to-No.
12	A	Clocks, Calendars & Time — Every year shifts the weekday by one, plus one per leap day; 200 years and 49 leap days give 249, which is 4 mod 7: back 4 from Tuesday.
13	C	Paths & Grids — Checkerboard-color the grid: consecutive numbers alternate colors, so the five odd numbers fill the five corner-and-center squares; 25 - 18 = 7.
14	B	Logic Puzzles — Seat numbers always total 15 and the other four moved a net +1, so Ada moved one seat left, from seat 2 to end seat 1.
15	C	Primes — For any prime p other than 3, p^2 is 1 mod 3, so p^2 + 26 is divisible by 3; p = 3 gives 35 = 5*7.
16	D	Statistics & Data — Every non-mode value appears at most 9 times, so the other 2008 entries need at least ceil(2008/9) = 224 values, plus the mode makes 225.
17	E	Distributions & Stars and Bars — Every die shows at least 1, so subtract 1 from each: distribute the 3 leftover pips among 7 dice, which is C(9,3) = 84 with no cap ever reached.
18	E	Statistics & Data — Modes are 1 through 28, so d = 14.5; the 183rd value is 16, so M = 16; missing 29s, 30s, 31s pull the mean just below 16.
19	E	Divisibility & Factors — Factor 12! = 2^10 3^5 5^2 7 11; divisors number 116322 = 792, square divisors need even exponents: 63211 = 36, ratio 1/22.
20	B	Conditional Probability & States — Use symmetry to collapse the nine interior points to three unknowns (center is 1/2 by symmetry), then solve a small linear system for the start point.
21	C	Statistics & Data — With 4040 entries the median averages the 2020th and 2021st; exactly m + 44 entries are at most m (for m near 2000), so m = 1976.
22	C	Basic Counting — Parabolas meet iff the larger leading coefficient pairs with the larger intercept; swapping B and D exchanges good and bad configurations, so exactly half of 180 work.
23	B	Circles — Each reflected arc bulges into the hexagon, so the region is the hexagon minus six circular segments, each a 60-degree sector minus an equilateral triangle.
24	B	Bases & Digits — Both base-n numerals start 3n²: 3n² ≤ 263 forces n ≤ 9, while 324_n ≥ 1101_6 = 253 forces n ≥ 9, so n = 9.
25	E	Solid Geometry — Equal volumes with radii 3 and 6 force heights in ratio 4:1; adding equal volumes scales both heights by the same factor, so the rises are also 4:1.
26	E	Primes — Expand (p+2)³ − p³ = 6p² + 12p + 8 and complete the square: (p+1)² = 5184 = 72², so the primes are 71 and 73.
27	B	Number Properties — {13, …, 25} works (smallest sum 26); if M is the largest element, pairing k with M − k caps the size at (M+1)/2 ≤ 13.
28	B	Exponents, Logarithms & Radicals — sqrt(1183) = 13 sqrt(7) forces sqrt(x) = m sqrt(7), sqrt(y) = n sqrt(7), m + n = 13; then 7(m^2 + n^2) is least at 6, 7.
29	D	Basic Counting — Sort the digits 1-9 into even only {4,6,8}, prime only {3,5,7}, both {2}, and neither {1,9}, then split on whether one digit does both jobs or two digits do.
30	D	Sequences & Series — The shaded rings form a geometric series with first term 1 - k^2 and ratio k^4, and the sum collapses to (1 - k^2)/(1 - k^4) = 1/(1 + k^2).