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Qualifying zone 1

15 problems of difficulty 3–4 in 45 minutes: the score-deciding zone

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Qualifying zone 1AMC 10 Step by Step

Name: Date: September 13, 2026

15 problems · 45 minutes · 6 / 1.5 / 0

1.2001 AMC 10 #13
A telephone number has the form $\text{ABC-DEF-GHIJ}$ , where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$ , $D > E > F$ , and $G > H > I > J$ . Furthermore, $D$ , $E$ , and $F$ are consecutive even digits; $G$ , $H$ , $I$ , and $J$ are consecutive odd digits; and $A + B + C = 9$ . Find $A$ .
1. A)
  $4$
2. B)
  $5$
3. C)
  $6$
4. D)
  $7$
5. E)
  $8$
2.2014 AMC 10A #21
Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$ -axis at the same point. What is the sum of all possible $x$ -coordinates of these points of intersection?
1. A)
  ${-20}$
2. B)
  ${-18}$
3. C)
  ${-15}$
4. D)
  ${-12}$
5. E)
  ${-8}$
3.2014 AMC 10B #19
Two concentric circles have radii $1$ and $2$ . Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
1. A)
  $\frac{1}{6}$
2. B)
  $\frac{1}{4}$
3. C)
  $\frac{2-\sqrt{2}}{2}$
4. D)
  $\frac{1}{3}$
5. E)
  $\frac{1}{2}$
4.2015 AMC 10A #18
Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$ . Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$ ?
1. A)
  $17$
2. B)
  $18$
3. C)
  $19$
4. D)
  $20$
5. E)
  $21$
5.2016 AMC 10B #18
In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?
1. A)
  $1$
2. B)
  $3$
3. C)
  $5$
4. D)
  $6$
5. E)
  $7$
6.2017 AMC 10B #21
In $\triangle ABC$ , $AB=6$ , $AC=8$ , $BC=10$ , and $D$ is the midpoint of $\overline{BC}$ . What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC$ ?
1. A)
  $\sqrt{5}$
2. B)
  $\frac{11}{4}$
3. C)
  $2\sqrt{2}$
4. D)
  $\frac{17}{6}$
5. E)
  $3$
7.2019 AMC 10A #16
The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$
1. A)
  $4 \pi \sqrt{3}$
2. B)
  $7 \pi$
3. C)
  $\pi\left(3\sqrt{3} +2\right)$
4. D)
  $10 \pi \left(\sqrt{3} - 1\right)$
5. E)
  $\pi\left(\sqrt{3} + 6\right)$
8.2020 AMC 10B #11
Ms. Carr asks her students to read any $5$ of the $10$ books on a reading list. Harold randomly selects $5$ books from this list, and Betty does the same. What is the probability that there are exactly $2$ books that they both select?
1. A)
  $\frac{1}{8}$
2. B)
  $\frac{5}{36}$
3. C)
  $\frac{14}{45}$
4. D)
  $\frac{25}{63}$
5. E)
  $\frac{1}{2}$
9.2020 AMC 10B #13
Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$ -direction). Andy moves $1$ unit and then turns $90^{\circ}$ left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ left. He then moves $3$ units (west) and again turns $90^{\circ}$ left. Andy continues his progress, increasing his distance each time by $1$ unit and always turning left. What is the location of the point which Andy makes the $2020$ left turn?
1. A)
  $(-1030, -994)$
2. B)
  $(-1030, -990)$
3. C)
  $(-1026, -994)$
4. D)
  $(-1026, -990)$
5. E)
  $(-1022, -994)$
10.2022 AMC 10B #15
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence that has a common difference of $2$ . The quotient $\frac{S_{3n}}{S_n}$ does not depend on $n$ . What is $S_{20}$ ?
1. A)
  $340$
2. B)
  $360$
3. C)
  $380$
4. D)
  $400$
5. E)
  $420$
11.2025 AMC 10A #11
The sequence $1,x,y,z$ is arithmetic. The sequence $1,p,q,z$ is geometric. Both sequences are strictly increasing and contain only integers, and $z$ is as small as possible. What is the value of $x+y+z+p+q$ ?
1. A)
  $66$
2. B)
  $91$
3. C)
  $103$
4. D)
  $132$
5. E)
  $149$
12.2025 AMC 10B #11
On Monday, $6$ students went to the tutoring center at the same time, and each one was randomly assigned to one of the $6$ tutors on duty. On Tuesday, the same $6$ students showed up, the same $6$ tutors were on duty, and the students were again randomly assigned to the tutors. What is the probability that exactly $2$ students met with the same tutor both Monday and Tuesday?
1. A)
  $\frac{1}{16}$
2. B)
  $\frac{3}{16}$
3. C)
  $\frac{1}{4}$
4. D)
  $\frac{3}{8}$
5. E)
  $\frac{1}{2}$
13.2013 AMC 10B #21
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term, beginning with the third, is the sum of the previous two terms, and the seventh term of each sequence is $N$ . What is the smallest possible value of $N$ ?
1. A)
  $55$
2. B)
  $89$
3. C)
  $104$
4. D)
  $144$
5. E)
  $273$
14.2020 AMC 10A #19
As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in empty space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?
1. A)
  $125$
2. B)
  $250$
3. C)
  $405$
4. D)
  $640$
5. E)
  $810$
15.2021 AMC Fall 10B #18
Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$ -sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$ , where $a$ , $b$ , and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$
1. A)
  $75$
2. B)
  $93$
3. C)
  $96$
4. D)
  $129$
5. E)
  $147$

Qualifying zone 1 — 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	Logic Puzzles — All ten digits are used, so ABC is whatever the odd run GHIJ and even run DEF leave behind; only 9753 and 642 leave digits summing to 9.
2	E	Diophantine Equations — Equal x-intercepts -5/a = -b/3 means ab = 15, so a runs over the divisors of 15 and the intercepts are -5, -5/3, -1, -1/3.
3	D	Geometric Probability — Fix the first point and draw its two tangents to the inner circle; they cut off a 120-degree arc of the outer circle, so the probability is 1/3.
4	E	Bases & Digits — Since 1000 = 3E8 in hex, every all-numeric hex string up to 399 (decimal 921) is in range: 41010 - 1 = 399 numbers, digit sum 21.
5	E	Divisibility & Factors — Consecutive integers a, ..., a+k-1 sum to k(2a+k-1)/2, so k(2a+k-1) = 690; count factor pairs of 690 whose smaller factor k is at least 2.
6	D	Triangle Centers & Cevians — The median to the hypotenuse equals half of it, so AD = 5; each half-triangle has area 12, and r = Area/semiperimeter gives 12/8 + 12/9.
7	A	Circles — Connect centers: two tangent steps of length 2 bending 120 degrees put the top center 2 sqrt 3 from the middle, so R = 2 sqrt 3 + 1.
8	D	Basic Probability — Fix Harold's five books; Betty must choose 2 of those 5 and 3 of the other 5 out of C(10,5) equally likely selections.
9	B	Coordinate Geometry — Every block of four moves (east, north, west, south) shifts Andy by (-2, -2); 2020 turns is exactly 505 blocks.
10	D	Sequences & Series — With d = 2, S_n = n(a + n − 1), so S_{3n}/S_n is constant only when a = 1; then S_n = n² and S_20 = 400.
11	E	Sequences & Series — The geometric sequence starts at 1, so z = p^3; the arithmetic one forces 3 \| z - 1, and p^3 = p mod 3 makes p = 4 smallest.
12	B	Basic Probability — Relabelling the tutors by Monday's assignment turns 'met the same tutor twice' into 'fixed point of a uniformly random permutation of the six students'.
13	C	Sequences & Series — Seventh term is 5a+8b; two starts with equal N differ by (+8,-5), so b >= a+13 and (0,13), (8,8) give 104.
14	E	Basic Counting — Routes factor into stages: entry face on the top ring (5), walk along it (9), step down (2), walk the bottom ring (9): 810.
15	E	Quadrilaterals & Polygon Areas — The 24-gon has 12-fold symmetry: its 24 vertices alternate between square corners (every 30 degrees) and side intersections exactly halfway between, so it splits into 24 congruent triangles.