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Mini mock 2

15 problems matching a real contest's position mix, 45 minutes

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Mini mock 2AMC 10 Step by Step

Name: Date: September 13, 2026

15 problems · 45 minutes · 6 / 1.5 / 0

1.2016 AMC 10A #1
What is the value of $\dfrac{11!-10!}{9!}$ ?
1. A)
  $99$
2. B)
  $100$
3. C)
  $110$
4. D)
  $121$
5. E)
  $132$
2.2013 AMC 10A #2
Alice is making a batch of cookies and needs $2\frac{1}{2}$ cups of sugar. Unfortunately, her measuring cup holds only $\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar?
1. A)
  $8$
2. B)
  $10$
3. C)
  $12$
4. D)
  $16$
5. E)
  $20$
3.2015 AMC 10B #5
David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$ -person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$ th place. Who finished in $8$ th place?
1. A)
  $\text{David}$
2. B)
  $\text{Hikmet}$
3. C)
  $\text{Jack}$
4. D)
  $\text{Rand}$
5. E)
  $\text{Todd}$
4.2018 AMC 10B #5
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?
1. A)
  $128$
2. B)
  $192$
3. C)
  $224$
4. D)
  $240$
5. E)
  $256$
5.2020 AMC 10A #6
How many $4$ -digit positive integers (that is, integers between $1000$ and $9999$ , inclusive) having only even digits are divisible by $5?$
1. A)
  $80$
2. B)
  $100$
3. C)
  $125$
4. D)
  $200$
5. E)
  $500$
6.2018 AMC 10B #7
In the figure below, $N$ congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the small semicircles. The ratio $A:B$ is $1:18$ . What is $N$ ?
1. A)
  $16$
2. B)
  $17$
3. C)
  $18$
4. D)
  $19$
5. E)
  $36$
7.2017 AMC 10A #11
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$ . What is the length $\textit{AB}$ ?
1. A)
  $6$
2. B)
  $12$
3. C)
  $18$
4. D)
  $20$
5. E)
  $24$
8.2017 AMC 10B #11
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
1. A)
  $10\%$
2. B)
  $12\%$
3. C)
  $20\%$
4. D)
  $25\%$
5. E)
  $33\frac{1}{3}\%$
9.2011 AMC 10A #14
A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
1. A)
  $\frac{1}{36}$
2. B)
  $\frac{1}{12}$
3. C)
  $\frac{1}{6}$
4. D)
  $\frac{1}{4}$
5. E)
  $\frac{5}{18}$
10.2011 AMC 10A #17
In the eight-term sequence $A,B,C,D,E,F,G,H$ , the value of $C$ is 5 and the sum of any three consecutive terms is 30. What is $A+H$ ?
1. A)
  $17$
2. B)
  $18$
3. C)
  $25$
4. D)
  $26$
5. E)
  $43$
11.2012 AMC 10A #18
The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$ , where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$ . What is the area enclosed by the curve?
1. A)
  $2\pi+6$
2. B)
  $2\pi+4\sqrt{3}$
3. C)
  $3\pi+4$
4. D)
  $2\pi+3\sqrt{3}+2$
5. E)
  $\pi+6\sqrt{3}$
12.2012 AMC 10A #20
A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
1. A)
  $\frac{49}{512}$
2. B)
  $\frac{7}{64}$
3. C)
  $\frac{121}{1024}$
4. D)
  $\frac{81}{512}$
5. E)
  $\frac{9}{32}$
13.2013 AMC 10B #20
The number $2013$ is expressed in the form $2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!},$ where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$ ?
1. A)
  $1$
2. B)
  $2$
3. C)
  $3$
4. D)
  $4$
5. E)
  $5$
14.2024 AMC 10B #23
The Fibonacci numbers are defined by $F_1 = 1, F_2 = 1,$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3.$ What is ${\frac{F_2}{F_1}} + {\frac{F_4}{F_2}} + {\frac{F_6}{F_3}} + ... + {\frac{F_{20}}{F_{10}}}?$
1. A)
  $318$
2. B)
  $319$
3. C)
  $320$
4. D)
  $321$
5. E)
  $322$
15.2022 AMC 10B #25
Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$ . For each positive integer $n$ , define $S_n = \sum_{k=0}^{n-1} x_k 2^k$ Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$ . What is the value of the sum $x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?$
1. A)
  $6$
2. B)
  $7$
3. C)
  $12$
4. D)
  $14$
5. E)
  $15$

Mini mock 2 — answersAMC 10 Step by Step

Name: Date: September 13, 2026

Keep this page separate from the problems.

Answers

#	Answer	Topic and the idea it turns on
1	B	Algebraic Manipulation — Factor 10! out of the numerator: 11! - 10! = 10!(11 - 1), and 10!/9! = 10.
2	B	Fractions & Decimals — The number of fills is the total amount divided by the cup size: (5/2) divided by (1/4) equals 10.
3	B	Logic Puzzles — Marta's 6th place anchors a chain: Jack 5th, Todd 3rd, Rand 2nd, then Hikmet is 6 behind Rand in 8th.
4	D	Basic Counting — Count all 2^8 subsets and subtract the 2^4 subsets built only from the four non-primes 4, 6, 8, 9.
5	B	Basic Counting — Divisible by 5 with an even last digit forces the units digit to be 0; the thousands digit has 4 even choices and each middle digit 5.
6	D	Circles — With small radius r the large radius is Nr, so A:B = N : (N^2 - N) = 1 : (N-1), giving N - 1 = 18.
7	D	Solid Geometry — Points within 3 of a segment form a cylinder of radius 3 capped by two hemispheres, i.e. a cylinder plus one full sphere of radius 3.
8	D	Ratios, Percents & Averages — With 100 students, 12 like dancing but say they dislike it and 36 dislike it and say so; the requested fraction is 12 out of 48.
9	B	Basic Probability — Area below circumference means pi d^2/4 < pi d, i.e. d < 4, so the dice sum must be 2 or 3: three of the 36 outcomes.
10	C	Sequences & Series — Subtracting overlapping triple sums gives A = D = G and B = E = H, so A + H = A + B = 30 - 5.
11	E	Circles — Each arc has radius 1 and spans 120 degrees; three vertices carry 240-degree outward bulges and three carry 120-degree dents, so hexagon plus 2pi minus pi.
12	A	Basic Probability — The rotation splits the grid into the center and two 4-cycles; a square stays white only if its predecessor was white, so each 4-cycle needs no adjacent whites.
13	B	Primes — 2013 = 31161 forces a_1 >= 61; then 61! contains the prime 59, which must be cancelled, so b_1 >= 59, and 61 and 59 work.
14	B	Sequences & Series — The ratios F_{2n}/F_n are the Lucas numbers 1, 3, 4, 7, 11, ... with the same recurrence, so just add ten small terms.
15	A	Modular Arithmetic — S_n is the inverse of 7 mod 2^n; because 7 divides 2^{3m} − 1, the bits are 1 then 1,1,0 repeating: x_k = 0 iff 3 divides positive k.