Printable set

Algebra set

20 problems, difficulty 2–4, one per topic where possible

Print without answers Answer sheet

Algebra setAMC 10 Step by Step

Name: Date: September 13, 2026

20 problems · 60 minutes · 6 / 1.5 / 0

1.2011 AMC 10B #15
Let $@$ denote the "averaged with" operation: $a @ b = (a+b)/2$ . Which of the following distributive laws hold for all numbers $x, y,$ and $z$ ? $\text{I. x @ (y + z) = (x @ y) + (x @ z)}$ $\text{II. x + (y @ z) = (x + y) @ (x + z)}$ $\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}$
1. A)
  $\text{I only}$
2. B)
  $\text{II only}$
3. C)
  $\text{III only}$
4. D)
  $\text{I and III only}$
5. E)
  $\text{II and III only}$
2.2012 AMC 10B #13
It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?
1. A)
  $36$
2. B)
  $40$
3. C)
  $42$
4. D)
  $48$
5. E)
  $52$
3.2012 AMC 10A #8
The sums of three whole numbers taken in pairs are $12$ , $17$ , and $19$ . What is the middle number?
1. A)
  $4$
2. B)
  $5$
3. C)
  $6$
4. D)
  $7$
5. E)
  $8$
4.2013 AMC 10B #11
Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x - 6y - 34$ . What is $x+y$ ?
1. A)
  $1$
2. B)
  $2$
3. C)
  $3$
4. D)
  $6$
5. E)
  $8$
5.2013 AMC 10B #14
Define $a\clubsuit b=a^2b-ab^2$ . Which of the following describes the set of points $(x, y)$ for which $x\clubsuit y=y\clubsuit x$ ?
1. A)
  $\text{a finite set of points}$
2. B)
  $\text{one line}$
3. C)
  $\text{two parallel lines}$
4. D)
  $\text{two intersecting lines}$
5. E)
  $\text{three lines}$
6.2013 AMC 10B #8
Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?
1. A)
  $10$
2. B)
  $16$
3. C)
  $25$
4. D)
  $30$
5. E)
  $40$
7.2016 AMC 10A #9
A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$ th row. What is the sum of the digits of $N$ ?
1. A)
  $6$
2. B)
  $7$
3. C)
  $8$
4. D)
  $9$
5. E)
  $10$
8.2010 AMC 10B #11
A shopper plans to purchase an item that has a listed price greater than $\$100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\$30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds $\$100$ .
Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or Coupon C. What is $y - x$ ?
1. A)
  $50$
2. B)
  $60$
3. C)
  $75$
4. D)
  $80$
5. E)
  $100$
9.2011 AMC 10A #16
Which of the following is equal to $\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}$ ?
1. A)
  $3\sqrt2$
2. B)
  $2\sqrt6$
3. C)
  $\frac{7\sqrt2}{2}$
4. D)
  $3\sqrt3$
5. E)
  $6$
10.2017 AMC 10A #14
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda?
1. A)
  $9\%$
2. B)
  $19\%$
3. C)
  $22\%$
4. D)
  $23\%$
5. E)
  $25\%$
11.2018 AMC 10A #14
What is the greatest integer less than or equal to $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?$
1. A)
  $80$
2. B)
  $81$
3. C)
  $96$
4. D)
  $97$
5. E)
  $625$
12.2021 AMC Fall 10A #20
For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions?
1. A)
  $4$
2. B)
  $6$
3. C)
  $8$
4. D)
  $12$
5. E)
  $16$
13.2021 AMC Fall 10B #12
Which of the following conditions is sufficient to guarantee that integers $x$ , $y$ , and $z$ satisfy the equation $x(x-y)+y(y-z)+z(z-x) = 1?$
1. A)
  x>y $and$ y=z
2. B)
  x=y-1 $and$ y=z-1
3. C)
  x=z+1 $and$ y=x+1
4. D)
  x=z $and$ y-1=x
5. E)
  $x+y+z=1$
14.2021 AMC Fall 10A #16
The graph of $f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|$ is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$ .)
1. A)
  $\text{the }y\text{-axis}$
2. B)
  $\text{the line }x = 1$
3. C)
  $\text{the origin}$
4. D)
  $\text{ the point }\left(\dfrac12, 0\right)$
5. E)
  $\text{the point }(1,0)$
15.2021 AMC Fall 10A #14
How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \begin{align} x^2+3y&=9 \\ (|x|+|y|-4)^2 &= 1 \end{align}
1. A)
  $1$
2. B)
  $2$
3. C)
  $3$
4. D)
  $5$
5. E)
  $7$
16.2023 AMC 10B #13
What is the area of the region in the coordinate plane defined by
$| | x | - 1 | + | | y | - 1 | \le 1?$
1. A)
  $2$
2. B)
  $8$
3. C)
  $4$
4. D)
  $15$
5. E)
  $12$
17.2025 AMC 10A #19
An array of numbers is constructed beginning with the numbers $-1\qquad3\qquad1$ in the top row. Each adjacent pair of numbers is summed to produce a number in the next row. Each row begins and ends with $-1$ and $1$ , respectively.
$\begin{array}{ccccccccc} &&-1&&3&&1&&\\ &-1&&2&&4&&1&\\ -1&&1&&6&&5&&1\\ \end{array}$
If the process continues, one of the rows will sum to $12,288$ . In that row, what is the third number from the left?
1. A)
  $-29$
2. B)
  $-21$
3. C)
  $-14$
4. D)
  $-8$
5. E)
  $-3$
18.2025 AMC 10B #10
Let $f(n)=n^3-5n^2+2n+8$ and $g(n)=n^3-6n^2+5n+12.$ What is the sum of all integers $n$ such that $\tfrac{f(n)}{g(n)}$ is an integer?
1. A)
  $2$
2. B)
  $3$
3. C)
  $4$
4. D)
  $5$
5. E)
  $6$
19.2017 AMC 10A #24
For certain real numbers $a$ , $b$ , and $c$ , the polynomial $g(x) = x^3 + ax^2 + x + 10$ has three distinct roots, and each root of $g(x)$ is also a root of the polynomial $f(x) = x^4 + x^3 + bx^2 + 100x + c.$ What is $f(1)$ ?
1. A)
  $-9009$
2. B)
  $-8008$
3. C)
  $-7007$
4. D)
  $-6006$
5. E)
  $-5005$
20.2024 AMC 10B #25
Each of $27$ bricks (right rectangular prisms) has dimensions $a \times b \times c$ , where $a$ , $b$ , and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28$ th brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is $1$ unit taller, $1$ unit wider, and $1$ unit deeper than the old one. What is $a + b + c$ ?
1. A)
  $88$
2. B)
  $89$
3. C)
  $90$
4. D)
  $91$
5. E)
  $92$

Algebra set — 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	E	Functions — Expand each side by the definition: I fails because its right side carries an extra x/2, while II and III both simplify to matching expressions.
2	B	Linear Equations & Word Problems — Rates add: walking covers 1/60 of the escalator per second and walking-plus-riding 1/24, so riding alone covers 1/24 - 1/60 = 1/40 per second.
3	D	Systems of Equations — Adding all three pair sums counts each number twice, so the total is 24; subtracting each pair sum from 24 recovers the missing number.
4	B	Quadratics — Move everything to one side and complete the square in x and y: (x-5)^2 + (y+3)^2 = 0 forces x = 5, y = -3.
5	E	Algebraic Manipulation — Both sides factor as xy(x-y) and -xy(x-y), so 2xy(x-y) = 0: the three lines x = 0, y = 0, x = y.
6	B	Ratios, Percents & Averages — Combined mileage is total miles over total gallons; with equal distances, choose a convenient distance like 40 miles and count gallons.
7	D	Sequences & Series — The rows total N(N+1)/2, so N(N+1) = 4032 = 63 * 64, giving N = 63 and digit sum 9.
8	A	Linear Equations & Word Problems — Express each coupon's savings in terms of the price p; A beats B when p >= 200 and beats C when p <= 250.
9	B	Exponents, Logarithms & Radicals — Square the whole sum: the cross term is 2 sqrt((9 - 6 sqrt 2)(9 + 6 sqrt 2)) = 2 sqrt(9) = 6, so the square is 24.
10	D	Ratios, Percents & Averages — Turn the percent statements into 5m = A - s and 20s = A - m; solving gives m = 19A/99 and s = 4A/99, total 23A/99.
11	A	Exponents, Logarithms & Radicals — The numerator equals 81 times the denominator minus 65 * 2^96; that leftover is positive yet smaller than the denominator, so the quotient is just below 81.
12	B	Quadratics — Both discriminants must be nonpositive: b^2 <= 4c and c^2 <= 4b, which force b, c <= 4; checking the 16 small pairs leaves 6.
13	D	Algebraic Manipulation — The left side equals half the sum (x-y)^2 + (y-z)^2 + (z-x)^2, which is 1 exactly when two of the integers are equal and the third differs by 1.
14	D	Functions — Replacing x by 1 - x swaps the two floor terms, so f(1 - x) = -f(x): the graph is symmetric about the point (1/2, 0).
15	D	Absolute Value & Inequalities — The parabola y = 3 - x^2/3 meets the diamond \|x\|+\|y\| = 3 only at its three vertices and cuts the lower half of \|x\|+\|y\| = 5 twice.
16	B	Absolute Value & Inequalities — By symmetry in both axes, restrict to x, y >= 0 where the region is the diamond \|x-1\| + \|y-1\| <= 1 of area 2; multiply by 4.
17	A	Sequences & Series — Each row sum doubles, since every entry is used twice except the fixed ends that restore it, so S_n = 3 * 2^(n-1) and 12,288 is row 13.
18	A	Polynomials — f and g differ by (n-4)(n+1), which is a factor of both, so the quotient collapses to (n-2)/(n-3) = 1 + 1/(n-3).
19	C	Polynomials — Three shared distinct roots force f(x) = g(x)(x - r); the x coefficient gives r = -90, then a = -89 and f(1) = g(1)(1 - r).
20	E	Systems of Equations — 2p or 7p never equals 3p+1, so each new dimension comes from a different edge: 7p = 3q+1, 2q = 3r+1, 2r = 3p+1.