Printable set

Qualifying zone 3

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

Print without answers Answer sheet

Qualifying zone 3AMC 10 Step by Step

Name: Date: September 13, 2026

15 problems · 45 minutes · 6 / 1.5 / 0

1.2011 AMC 10B #19
What is the product of all the roots of the equation $\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.$
1. A)
  $-64$
2. B)
  $-24$
3. C)
  $-9$
4. D)
  $24$
5. E)
  $576$
2.2012 AMC 10B #20
Bernardo and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$ ?
1. A)
  $7$
2. B)
  $8$
3. C)
  $9$
4. D)
  $10$
5. E)
  $11$
3.2012 AMC 10A #21
Let points $A = (0 ,0 ,0)$ , $B = (1, 0, 0)$ , $C = (0, 2, 0)$ , and $D = (0, 0, 3)$ . Points $E$ , $F$ , $G$ , and $H$ are midpoints of line segments $\overline{BD},\text{ } \overline{AB}, \text{ } \overline {AC},$ and $\overline{DC}$ respectively. What is the area of $EFGH$ ?
1. A)
  $\sqrt{2}$
2. B)
  $\frac{2\sqrt{5}}{3}$
3. C)
  $\frac{3\sqrt{5}}{4}$
4. D)
  $\sqrt{3}$
5. E)
  $\frac{2\sqrt{7}}{3}$
4.2014 AMC 10A #15
David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home?
1. A)
  $140$
2. B)
  $175$
3. C)
  $210$
4. D)
  $245$
5. E)
  $280$
5.2017 AMC 10A #17
Distinct points $P$ , $Q$ , $R$ , $S$ lie on the circle $x^2+y^2=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$ ?
1. A)
  $3$
2. B)
  $5$
3. C)
  $3\sqrt{5}$
4. D)
  $7$
5. E)
  $5\sqrt{2}$
6.2017 AMC 10A #19
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?
1. A)
  $12$
2. B)
  $16$
3. C)
  $28$
4. D)
  $32$
5. E)
  $40$
7.2018 AMC 10B #16
Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that $a_1+a_2+\cdots+a_{2018}=2018^{2018}.$ What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$ ?
1. A)
  $0$
2. B)
  $1$
3. C)
  $2$
4. D)
  $3$
5. E)
  $4$
8.2018 AMC 10B #20
A function $f$ is defined recursively by $f(1)=f(2)=1$ and $f(n)=f(n-1)-f(n-2)+n$ for all integers $n \geq 3$ . What is $f(2018)$ ?
1. A)
  $2016$
2. B)
  $2017$
3. C)
  $2018$
4. D)
  $2019$
5. E)
  $2020$
9.2019 AMC 10A #19
What is the least possible value of $(x+1)(x+2)(x+3)(x+4)+2019$ where $x$ is a real number?
1. A)
  $2017$
2. B)
  $2018$
3. C)
  $2019$
4. D)
  $2020$
5. E)
  $2021$
10.2020 AMC 10B #15
Steve wrote the digits $1$ , $2$ , $3$ , $4$ , and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$ rd, $6$ th, $9$ th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$ th, $8$ th, $12$ th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$ ?
1. A)
  $7$
2. B)
  $9$
3. C)
  $10$
4. D)
  $11$
5. E)
  $12$
11.2023 AMC 10B #11
Suzanne went to the bank and withdrew $\$800$ . The teller gave her this amount using $\$20$ bills, $\$50$ bills, and $\$100$ bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?
1. A)
  $45$
2. B)
  $21$
3. C)
  $36$
4. D)
  $28$
5. E)
  $32$
12.2023 AMC 10B #14
How many ordered pairs of integers $(m,n)$ satisfy the equation $m^2+mn+n^2 = m^2n^2$ ?
1. A)
  $7$
2. B)
  $1$
3. C)
  $3$
4. D)
  $6$
5. E)
  $5$
13.2024 AMC 10A #16
All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length $AB$ ? $\newline$
1. A)
  $4+4\sqrt5$
2. B)
  $10\sqrt2$
3. C)
  $5+5\sqrt5$
4. D)
  $10\sqrt[4]{8}$
5. E)
  $20$
14.2014 AMC 10A #19
Four cubes with edge lengths $1$ , $2$ , $3$ , and $4$ are stacked as shown. What is the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$ ?
1. A)
  $\dfrac{3\sqrt{33}}5$
2. B)
  $2\sqrt3$
3. C)
  $\dfrac{2\sqrt{33}}3$
4. D)
  $4$
5. E)
  $3\sqrt2$
15.2022 AMC 10B #19
Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:
- Any filled square with two or three filled neighbors remains filled.
- Any empty square with exactly three filled neighbors becomes a filled square.
- All other squares remain empty or become empty.
A sample transformation is shown in the figure below. Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
1. A)
  $14$
2. B)
  $18$
3. C)
  $22$
4. D)
  $26$
5. E)
  $30$

Qualifying zone 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	A	Absolute Value & Inequalities — Square both sides and set u = \|x\|: u^2 - 5u - 24 = 0 forces \|x\| = 8, so the roots 8 and -8 multiply to -64.
2	A	Games & Processes — Bernardo wins exactly when one of his outputs 2N, 4N+100, 8N+300, 16N+700 lies in [950, 999], because Silvia adds only 50; the smallest N is 16.
3	C	Coordinate Geometry — EF and GH are midlines parallel to AD, FG is parallel to BC, and AD is perpendicular to BC, so EFGH is a rectangle with sides AD/2 and BC/2.
4	C	Linear Equations & Word Problems — Compare the two plans for the remaining distance d - 35: at 35 mph versus 50 mph the times differ by exactly 1.5 hours.
5	D	Coordinate Geometry — Only 12 lattice points lie on the circle; the longest irrational chord is sqrt(98) = 7 sqrt(2) and the shortest is sqrt(2), so the ratio is 7.
6	C	Arrangements with Restrictions — Alice's neighbors must be Derek and Eric; place Alice first, and only when she sits at an end can Derek and Eric become adjacent, which costs a few arrangements.
7	E	Modular Arithmetic — n^3 - n = (n-1)n(n+1) is always divisible by 6, so the cube sum is congruent mod 6 to the plain sum 2018^2018, which is 4 mod 6.
8	B	Sequences & Series — Adding consecutive recursions gives f(n) = f(n-6) + 6, so f(2018) = f(2) + 6 * 336 = 2017.
9	B	Algebraic Manipulation — Pair outer and inner factors: (x+1)(x+4) = u+4 and (x+2)(x+3) = u+6 with u = x^2+5x, so the product is (u+5)^2 - 1, at least -1.
10	D	Modular Arithmetic — Each erasure keeps the list periodic: 30 original digits become 20, then 15, then 12, so the final list repeats every 12 digits.
11	B	Diophantine Equations — Reduce to 2a + 5b + 10c = 80; b is even, so b = 2k gives a = 40 - 5(k+c) with 2 <= k+c <= 7.
12	C	Diophantine Equations — Add mn to both sides: (m+n)^2 = (mn)^2 + mn, which is strictly between consecutive squares unless mn is 0 or -1.
13	D	Similar & Congruent Triangles — With long-to-short ratio r, a rectangle of area S has short side √(S/r); rectangle 8 spanning 9 and 1 forces r = √2, then AB² = 200r.
14	A	Solid Geometry — XY joins opposite 3-D corners of the stack, (0,0,10) to (4,4,0); the part at heights 4 to 7 is 3/10 of its total length 2 root 33.
15	C	Paths & Grids — Only 2 or 3 outer cells can be filled; translate 'everything else ends empty' into adjacency rules on the 8-cell ring and count by how many corners are used.