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Mini mock 3

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

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Mini mock 3AMC 10 Step by Step

Name: Date: September 13, 2026

15 problems · 45 minutes · 6 / 1.5 / 0

1.2012 AMC 10A #4
Let $\angle ABC = 24^\circ$ and $\angle ABD = 20^\circ$ . What is the smallest possible degree measure for angle $CBD$ ?
1. A)
  $0$
2. B)
  $2$
3. C)
  $4$
4. D)
  $6$
5. E)
  $12$
2.2022 AMC 10B #4
A donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $700$ th hiccup occur?
1. A)
  $15 \text{ seconds after } 4:58$
2. B)
  $20 \text{ seconds after } 4:58$
3. C)
  $25 \text{ seconds after } 4:58$
4. D)
  $30 \text{ seconds after } 4:58$
5. E)
  $35 \text{ seconds after } 4:58$
3.2017 AMC 10A #4
Mia is “helping” her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time?
1. A)
  $13.5$
2. B)
  $14$
3. C)
  $14.5$
4. D)
  $15$
5. E)
  $15.5$
4.2012 AMC 10B #5
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?
1. A)
  $18$
2. B)
  $20$
3. C)
  $21$
4. D)
  $22$
5. E)
  $24$
5.2025 AMC 10B #5
In $\triangle ABC$ , $AB=10$ , $AC=18$ , and $\angle B=130^\circ.$ Let $O$ be the center of the circle containing $A$ , $B$ , and $C.$ What is the degree measure of $\angle CAO$ ?
1. A)
  $20$
2. B)
  $30$
3. C)
  $40$
4. D)
  $50$
5. E)
  $60$
6.2024 AMC 10A #7
The product of three integers is $60$ . What is the least possible positive sum of the three integers?
1. A)
  $2$
2. B)
  $3$
3. C)
  $5$
4. D)
  $6$
5. E)
  $13$
7.2015 AMC 10B #13
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
1. A)
  $20$
2. B)
  $\dfrac{360}{17}$
3. C)
  $\dfrac{107}{5}$
4. D)
  $\dfrac{43}{2}$
5. E)
  $\dfrac{281}{13}$
8.2021 AMC Fall 10A #13
Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?
1. A)
  $\frac{1}{64}$
2. B)
  $\frac{1}{6}$
3. C)
  $\frac{1}{4}$
4. D)
  $\frac{5}{16}$
5. E)
  $\frac{1}{2}$
9.2014 AMC 10B #14
Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \ge 1$ and $a + b + c \le 7$ . At the end of the trip, the odometer showed $cba$ miles. What is $a^2 + b^2 + c^2$ ?
1. A)
  $26$
2. B)
  $27$
3. C)
  $36$
4. D)
  $37$
5. E)
  $41$
10.2018 AMC 10B #17
In rectangle $PQRS$ , $PQ=8$ and $QR=6$ . Points $A$ and $B$ lie on $\overline{PQ}$ , points $C$ and $D$ lie on $\overline{QR}$ , points $E$ and $F$ lie on $\overline{RS}$ , and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$ , where $k$ , $m$ , and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$ ?
1. A)
  $1$
2. B)
  $7$
3. C)
  $21$
4. D)
  $92$
5. E)
  $106$
11.2024 AMC 10A #18
There are exactly $K$ positive integers $5 \leq b \leq 2024$ such that the base- $b$ integer $2024_{b}$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$ ?
1. A)
  $16$
2. B)
  $17$
3. C)
  $18$
4. D)
  $20$
5. E)
  $21$
12.2013 AMC 10B #19
The real numbers $c,b,a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$ . The quadratic $ax^2+bx+c$ has exactly one root. What is this root?
1. A)
  $-7-4\sqrt{3}$
2. B)
  $-2-\sqrt{3}$
3. C)
  $-1$
4. D)
  $-2+\sqrt{3}$
5. E)
  $-7+4\sqrt{3}$
13.2021 AMC Fall 10B #20
In a particular game, each of $4$ players rolls a standard $6{ }$ -sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game?
1. A)
  $\frac{61}{216}$
2. B)
  $\frac{367}{1296}$
3. C)
  $\frac{41}{144}$
4. D)
  $\frac{185}{648}$
5. E)
  $\frac{11}{36}$
14.2021 AMC Fall 10A #23
For each positive integer $n$ , let $f_1(n)$ be twice the number of positive integer divisors of $n$ , and for $j \ge 2$ , let $f_j(n) = f_1(f_{j-1}(n))$ . For how many values of $n \le 50$ is $f_{50}(n) = 12?$
1. A)
  $7$
2. B)
  $8$
3. C)
  $9$
4. D)
  $10$
5. E)
  $11$
15.2023 AMC 10B #23
An arithmetic sequence of positive integers has $n\ge3$ terms, initial term $a$ , and common difference $d>1$ . Carl wrote down all the terms in this sequence correctly except for one term, which was off by $1$ . The sum of the terms he wrote down was $222$ . What is $a+d+n$ ?
1. A)
  $24$
2. B)
  $20$
3. C)
  $22$
4. D)
  $28$
5. E)
  $26$

Mini mock 3 — 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	C	Angles & Polygons — Rays BC and BD can lie on the same side of ray BA or on opposite sides; the same side gives 24 - 20 = 4, the smaller option.
2	A	Clocks, Calendars & Time — The 700th hiccup is 699 gaps after the first, so it occurs 699 × 5 = 3495 seconds, i.e. 58 minutes 15 seconds, after 4:00.
3	B	Games & Processes — Each 30-second cycle nets one toy, but the box reaches 30 the instant Mom drops in her third batch, before Mia can react, so 27 cycles plus one more deposit.
4	D	Ratios, Percents & Averages — Tax and tip are both percentages of the same base price, so the total is 1.25 times the meal: 27.50 / 1.25 = 22.
5	C	Circles — Angle B is obtuse, so it subtends the major arc: the central angle AOC is 360 - 260 = 100 degrees, and OA = OC gives 40.
6	B	Divisibility & Factors — Allow two negative factors: (−1)(−6)(10) = 60 has sum 3, and checking the factor triples shows a positive sum of 1 or 2 is impossible.
7	E	Triangles: Area & Pythagorean — The intercepts give a 5-12-13 right triangle; two altitudes are the legs, and the third is (leg times leg)/hypotenuse = 60/13.
8	D	Basic Probability — A black ball needs at least 3 white balls and a white ball needs at least 3 black, so the split must be exactly 3-3: C(6,3)/2^6 = 20/64.
9	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).
10	B	Triangles: Area & Pythagorean — With side s, the corner legs are (8-s)/2 and (6-s)/2, and the Pythagorean theorem gives s^2 + 14s - 50 = 0, so s = -7 + 3 sqrt(11).
11	D	Bases & Digits — Because 2024_b = 2(b³+b+2), the condition is b³+b+2 ≡ 0 (mod 8), which holds exactly when b ≡ 3, 6, 7 (mod 8).
12	D	Quadratics — One root means b^2 = 4ac with b = (a+c)/2; dividing by a^2 gives a quadratic in t = c/a, and the root is -b/(2a) = -(1+t)/4.
13	C	Conditional Probability & States — P(Hugo wins) is 1/4 by symmetry, and given a first roll of 5 he wins with probability sum over k other 5s of C(3,k)(1/6)^k(4/6)^(3-k)/(k+1).
14	D	Divisibility & Factors — Since f_1(n) = 2d(n) <= 20 here, iterating small even numbers shows only 12, 18, 20 reach 12; so count n <= 50 with d(n) in {6, 9, 10}.
15	B	Sequences & Series — The true sum is 221 or 223, so n divides 442 or 446; d >= 2 forces n <= 14, leaving n = 13, d = 2, a = 5.