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Qualifying zone 2

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

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Qualifying zone 2AMC 10 Step by Step

Name: Date: September 13, 2026

15 problems · 45 minutes · 6 / 1.5 / 0

1.2010 AMC 10B #18
Positive integers $a$ , $b$ , and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$ . What is the probability that $abc + ab + a$ is divisible by $3$ ?
1. A)
  $\dfrac{1}{3}$
2. B)
  $\dfrac{29}{81}$
3. C)
  $\dfrac{31}{81}$
4. D)
  $\dfrac{11}{27}$
5. E)
  $\dfrac{13}{27}$
2.2015 AMC 10B #16
Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$ , inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?
1. A)
  $\dfrac{9}{1000}$
2. B)
  $\dfrac{1}{90}$
3. C)
  $\dfrac{1}{80}$
4. D)
  $\dfrac{1}{72}$
5. E)
  $\dfrac{2}{121}$
3.2016 AMC 10B #14
How many squares whose sides are parallel to the axis and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$ , the line $y=-0.1$ and the line $x=5.1?$
1. A)
  $30$
2. B)
  $41$
3. C)
  $45$
4. D)
  $50$
5. E)
  $57$
4.2017 AMC 10A #20
Let $S(n)$ equal the sum of the digits of positive integer $n$ . For example, $S(1507) = 13$ . For a particular positive integer $n$ , $S(n) = 1274$ . Which of the following could be the value of $S(n+1)$ ?
1. A)
  $1$
2. B)
  $3$
3. C)
  $12$
4. D)
  $1239$
5. E)
  $1265$
5.2017 AMC 10B #15
Rectangle $ABCD$ has $AB=3$ and $BC=4$ . Point $E$ is the foot of the perpendicular from $B$ to diagonal $\overline{AC}$ . What is the area of $\triangle AED$ ?
1. A)
  $1$
2. B)
  $\frac{42}{25}$
3. C)
  $\frac{28}{15}$
4. D)
  $2$
5. E)
  $\frac{54}{25}$
6.2019 AMC 10A #17
A child builds towers using identically shaped cubes of different colors. How many different towers with a height $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)
1. A)
  $24$
2. B)
  $288$
3. C)
  $312$
4. D)
  $1,260$
5. E)
  $40,320$
7.2019 AMC 10A #18
For some positive integer $k$ , the repeating base- $k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$ . What is $k$ ?
1. A)
  $13$
2. B)
  $14$
3. C)
  $15$
4. D)
  $16$
5. E)
  $17$
8.2021 AMC 10B #14
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$ . What is the distance between two adjacent parallel lines?
1. A)
  $5\frac{1}{2}$
2. B)
  $6$
3. C)
  $6\frac{1}{2}$
4. D)
  $7$
5. E)
  $7\frac{1}{2}$
9.2021 AMC Fall 10B #16
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?
1. A)
  $1.6$
2. B)
  $1.8$
3. C)
  $2.0$
4. D)
  $2.2$
5. E)
  $2.4$
10.2021 AMC 10B #15
The real number $x$ satisfies the equation $x+\frac{1}{x} = \sqrt{5}$ . What is the value of $x^{11}-7x^{7}+x^3?$
1. A)
  $-1$
2. B)
  $0$
3. C)
  $1$
4. D)
  $2$
5. E)
  $\sqrt{5}$
11.2021 AMC 10A #18
Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$ . Furthermore, suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$ . For which of the following numbers $x$ is $f(x)<0$ ?
1. A)
  $\frac{17}{32}$
2. B)
  $\frac{11}{16}$
3. C)
  $\frac{7}{9}$
4. D)
  $\frac{7}{6}$
5. E)
  $\frac{25}{11}$
12.2021 AMC 10A #14
All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$ ?
1. A)
  ${-}88$
2. B)
  ${-}80$
3. C)
  ${-}64$
4. D)
  ${-}41$
5. E)
  ${-}40$
13.2025 AMC 10B #18
What is the ones digit of the sum $\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \dots + \lfloor \sqrt{2025} \rfloor?$ (Recall that $\lfloor x \rfloor$ represents the greatest integer less than or equal to $x$ .)
1. A)
  $1$
2. B)
  $2$
3. C)
  $3$
4. D)
  $5$
5. E)
  $8$
14.2020 AMC 10B #17
There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to him or her, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ pairs so that the members of each pair know each other?
1. A)
  $11$
2. B)
  $12$
3. C)
  $13$
4. D)
  $14$
5. E)
  $15$
15.2021 AMC 10A #17
Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$ , $BC = CD = 43$ , and $\overline{AD} \perp \overline{BD}$ . Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$ , and let $P$ be the midpoint of $\overline{BD}$ . Given that $OP = 11$ , the length $AD$ can be written in the form $m\sqrt{n}$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m + n$ ?
1. A)
  $65$
2. B)
  $132$
3. C)
  $157$
4. D)
  $194$
5. E)
  $215$

Qualifying zone 2 — 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	Modular Arithmetic — Factor as a(b(c+1)+1): divisible by 3 when a is, or else when b(c+1) is 2 mod 3; residues are uniform since 3 divides 2010.
2	C	Basic Probability — Distinct numbers in a divisor chain strictly increase, so Cal's number is at most 2; listing the chains c \| b \| a gives 9 of 720 outcomes.
3	D	Coordinate Geometry — Only the top-left corner can cross y = pi x, so a square with lower-left (a, b) and side s fits iff b + s <= floor(pi a).
4	D	Bases & Digits — Adding 1 to a number ending in k nines changes the digit sum by 1 - 9k, so S(n+1) is 1275 minus a multiple of 9; only 1239 fits.
5	E	Similar & Congruent Triangles — Similar right triangles give AE = AB^2/AC = 9/5, and triangles AED and ACD share the height from D, so [AED] = (AE/AC)[ACD] = (9/25)(6).
6	D	Basic Counting — Arrange all 9 cubes in a line, 9!/(2! 3! 4!) = 1260 ways, and remove the top one; each tower of 8 arises from exactly one such line.
7	D	Bases & Digits — A two-digit repeating block in base k equals (2k+3)/(k^2-1); setting this to 7/51 forces 51 to divide k^2-1, which only k = 16 satisfies.
8	B	Circles — Equal chords are equidistant from the center, so the center lies midway between the 38-chords: r² = 19² + (d/2)² = 17² + (3d/2)².
9	D	Expected Value — By linearity, the answer is 5 times the chance one fixed ball ends home: untouched twice (3/5 * 3/5) or swapped away and swapped straight back (2/5 * 1/5).
10	B	Algebraic Manipulation — Squaring twice gives x⁴ + 1/x⁴ = 7; multiply by x⁷ and the target expression x¹¹ − 7x⁷ + x³ is exactly zero.
11	E	Functions — Multiplicativity makes f(n) the sum of n's prime factors with multiplicity and f(m/n) = f(m) - f(n); only 25/11 gives 10 - 11 < 0.
12	A	Polynomials — Vieta gives root sum 10 and product 16; six positive integers with product 16 are powers of 2, and only 1,1,2,2,2,2 sums to 10.
13	D	Sequences & Series — The value n is taken by exactly 2n+1 consecutive terms, so the sum is sum n(2n+1) for n up to 44 plus the single leftover term 45.
14	C	Basic Counting — Split by how many pairs are 'across' pairs; the leftover people must pair with neighbors along the circle, which forces even-length arcs.
15	D	Similar & Congruent Triangles — CP is perpendicular to BD since CBD is isosceles, so right triangles CPD and ADB are similar with ratio 1:2, giving AB = 86 and BD = 66.