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Technique: casework

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Technique: caseworkAMC 10 Step by Step

Name: Date: September 13, 2026

12 problems · 40 minutes · 6 / 1.5 / 0

1.2018 AMC 10A #7
For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer?
1. A)
  $3$
2. B)
  $4$
3. C)
  $6$
4. D)
  $8$
5. E)
  $9$
2.2020 AMC 10A #5
What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$
1. A)
  $12$
2. B)
  $15$
3. C)
  $18$
4. D)
  $21$
5. E)
  $25$
3.2025 AMC 10A #8
Agnes writes the following four statements on a blank piece of paper.
$\bullet$ At least one of these statements is true.
$\bullet$ At least two of these statements are true.
$\bullet$ At least two of these statements are false.
$\bullet$ At least one of these statements is false.
Each statement is either true or false. How many false statements did Agnes write on the paper?
1. A)
  $0$
2. B)
  $1$
3. C)
  $2$
4. D)
  $3$
5. E)
  $4$
4.2025 AMC 10B #4
The value of the two-digit number $\underline{a}~\underline{b}$ in base seven equals the value of the two-digit number $\underline{b}~\underline{a}$ in base nine. What is $a+b?$
1. A)
  $7$
2. B)
  $9$
3. C)
  $10$
4. D)
  $11$
5. E)
  $14$
5.2019 AMC 10A #14
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$ ?
1. A)
  $14$
2. B)
  $16$
3. C)
  $18$
4. D)
  $19$
5. E)
  $21$
6.2021 AMC Fall 10B #7
Call a fraction $\frac{a}{b}$ , not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$ . How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
1. A)
  $9$
2. B)
  $10$
3. C)
  $11$
4. D)
  $12$
5. E)
  $13$
7.2017 AMC 10A #23
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$ , inclusive?
1. A)
  $2128$
2. B)
  $2148$
3. C)
  $2160$
4. D)
  $2200$
5. E)
  $2300$
8.2021 AMC 10A #19
The area of the region bounded by the graph of $x^2+y^2 = 3|x-y| + 3|x+y|$ is $m+n\pi$ , where $m$ and $n$ are integers. What is $m + n$ ?
1. A)
  $18$
2. B)
  $27$
3. C)
  $36$
4. D)
  $45$
5. E)
  $54$
9.2021 AMC 10A #20
In how many ways can the sequence $1, 2, 3, 4, 5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
1. A)
  $10$
2. B)
  $18$
3. C)
  $24$
4. D)
  $32$
5. E)
  $44$
10.2022 AMC 10B #23
Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$ th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$ th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $1$ ?
1. A)
  $\frac{1}{3}$
2. B)
  $\frac{1}{2}$
3. C)
  $\frac{2}{3}$
4. D)
  $\frac{3}{4}$
5. E)
  $\frac{5}{6}$
11.2024 AMC 10B #24
Let $P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}$ How many of the values $P(2022)$ , $P(2023)$ , $P(2024)$ , and $P(2025)$ are integers?
1. A)
  $0$
2. B)
  $1$
3. C)
  $2$
4. D)
  $3$
5. E)
  $4$
12.2024 AMC 10B #15
A list of $9$ real numbers consists of $1$ , $2.2$ , $3.2$ , $5.2$ , $6.2$ , and $7$ , as well as $x$ , $y$ , and $z$ with $x$ $\le$ $y$ $\le$ $z$ . The range of the list is $7$ , and the mean and the median are both positive integers. How many ordered triples ( $x$ , $y$ , $z$ ) are possible?
1. A)
  $1$
2. B)
  $2$
3. C)
  $3$
4. D)
  $4$
5. E)
  $\text{infinitely many}$

Technique: casework — 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	Exponents, Logarithms & Radicals — Write 4000 = 2^5 * 5^3; positive n costs factors of 5 (at most 3), negative n costs factors of 2 (at most 5), and n = 0 works.
2	C	Absolute Value & Inequalities — Split into x^2-12x+34 = 2 (roots 4 and 8) and x^2-12x+34 = -2 (double root 6); sum the distinct roots 4 + 8 + 6.
3	B	Logic Puzzles — Each statement is a condition on T, the number of true statements, so test T = 0, 1, 2, 3, 4 and keep the value that reproduces itself.
4	A	Bases & Digits — Place value turns the condition into 7a + b = 9b + a, so 3a = 4b, and a must be a base-seven digit.
5	D	Angles & Polygons — Six pairs of lines give at most 6 points; parallel pairs and concurrent triples remove points. Every count from 0 to 6 is achievable except 2.
6	C	Fractions & Decimals — Reduce the 14 special fractions; two of them sum to an integer only if both are integers or both have the same denominator 2 or 4.
7	B	Basic Counting — Count all C(25,3) triples and subtract collinear ones, organized by the line's direction: rows, columns, both diagonal directions, and slopes of plus or minus 1/2 and 2.
8	E	Coordinate Geometry — In the sector \|y\| <= x the equation is x^2 + y^2 = 6x, a circle through the origin; four such arcs bound a 6-by-6 square with four half-disks attached.
9	D	Arrangements with Restrictions — The sequence must zigzag; for the up-down shape, 5 sits at a peak (position 2 or 4), and each choice leaves 8 completions, so 16 total, doubled by reflection.
10	C	Geometric Probability — She always takes two steps and stops there iff t₁ + t₂ > 1 (probability 1/2); positions are independent of times, so P = ½·½ + ½·⅚.
11	E	Modular Arithmetic — Over denominator 8, P(m) is an integer iff 8 divides 4m+2m^2+m^4+m^8; even m is trivial, odd m gives 4(m+1).
12	C	Statistics & Data — The median must be one of x, y, z; the mean forces x+y+z = 11.2 or 20.2; the range forces x = 0, z = 8, or z = x+7.