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Technique: plug in values

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Technique: plug in valuesAMC 10 Step by Step

Name: Date: September 13, 2026

12 problems · 40 minutes · 6 / 1.5 / 0

1.2013 AMC 10A #11
A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly 10 ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?
1. A)
  $10$
2. B)
  $12$
3. C)
  $15$
4. D)
  $18$
5. E)
  $25$
2.2014 AMC 10A #7
Nonzero real numbers $x$ , $y$ , $a$ , and $b$ satisfy $x < a$ and $y < b$ . How many of the following inequalities must be true?
$\textbf{(I)}\ x+y < a+b\qquad$
$\textbf{(II)}\ x-y < a-b\qquad$
$\textbf{(III)}\ xy < ab\qquad$
$\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}$
1. A)
  $0$
2. B)
  $1$
3. C)
  $2$
4. D)
  $3$
5. E)
  $4$
3.2016 AMC 10B #6
Isaac added two three-digit positive integers. All six digits in these numbers are different. Isaac's sum is a three-digit number $S$ . What is the smallest possible value for the sum of the digits of $S$ ?
1. A)
  $1$
2. B)
  $4$
3. C)
  $5$
4. D)
  $15$
5. E)
  $20$
4.2000 AMC 10 #21
If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?
$\textrm{I. All alligators are creepy crawlers.}$ $\textrm{II. Some ferocious creatures are creepy crawlers.}$ $\textrm{III. Some alligators are not creepy crawlers.}$
1. A)
  $\text{I only}$
2. B)
  $\text{II only}$
3. C)
  $\text{III only}$
4. D)
  $\text{II and III only}$
5. E)
  $\text{None must be true}$
5.2005 AMC 10B #19
On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam?
1. A)
  $0$
2. B)
  $1$
3. C)
  $2$
4. D)
  $4$
5. E)
  $5$
6.2012 AMC 10B #18
Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a $2\%$ false positive rate--in other words, for such people, $98\%$ of the time the test will turn out negative, but $2\%$ of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let $p$ be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Which of the following is closest to $p$ ?
1. A)
  $\frac{1}{98}$
2. B)
  $\frac{1}{9}$
3. C)
  $\frac{1}{11}$
4. D)
  $\frac{49}{99}$
5. E)
  $\frac{98}{99}$
7.2015 AMC 10A #13
Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have?
1. A)
  $3$
2. B)
  $4$
3. C)
  $5$
4. D)
  $6$
5. E)
  $7$
8.2024 AMC 10A #13
Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:
- a translation $2$ units to the right,
- a $90^{\circ}$ -rotation counterclockwise about the origin,
- a reflection across the $x$ -axis, and
- a dilation centered at the origin with scale factor $2.$
Of the $6$ pairs of distinct transformations from this list, how many commute?
1. A)
  $1$
2. B)
  $2$
3. C)
  $3$
4. D)
  $4$
5. E)
  $5$
9.2011 AMC 10A #19
In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period?
1. A)
  $42\%$
2. B)
  $47\%$
3. C)
  $52\%$
4. D)
  $57\%$
5. E)
  $62\%$
10.2016 AMC 10A #23
A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$ . (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. What is $p+q?$
1. A)
  $109$
2. B)
  $201$
3. C)
  $301$
4. D)
  $3049$
5. E)
  $33,601$
11.2021 AMC 10A #24
The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$ , where $a$ is a positive real number. What is the area of this region in terms of $a$ , valid for all $a > 0$ ?
1. A)
  $\frac{8a^2}{(a+1)^2}$
2. B)
  $\frac{4a}{a+1}$
3. C)
  $\frac{8a}{a+1}$
4. D)
  $\frac{8a^2}{a^2+1}$
5. E)
  $\frac{8a}{a^2+1}$
12.2023 AMC 10B #24
What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$ , $0\le v\le1,$ and $0\le w\le1$ ?
1. A)
  $10\sqrt{3}$
2. B)
  $13$
3. C)
  $12$
4. D)
  $18$
5. E)
  $16$

Technique: plug in values — answersAMC 10 Step by Step

Name: Date: September 13, 2026

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Answers

#	Answer	Topic and the idea it turns on
1	A	Basic Counting — C(n,2) = 10 forces n = 5 council members, and C(5,3) = C(5,2) = 10 by the symmetry of binomial coefficients.
2	B	Absolute Value & Inequalities — Only addition preserves both inequalities; a single negative example such as x = y = -2, a = b = -1 breaks II, III and IV at once.
3	B	Bases & Digits — Distinct nonzero hundreds digits force S >= 300, ruling out digit sums 1, 2, 3; the example 143 + 257 = 400 achieves digit sum 4.
4	B	Logic Puzzles — Only statement II follows: the creepy crawlers that are alligators are automatically ferocious; I and III each fail in a small made-up world.
5	B	Statistics & Data — The mean is the percent-weighted average, 86; the median is the score where the cumulative percent first passes 50, which is 85.
6	C	Conditional Probability & States — Picture 500 people: 1 true positive and about 0.02 * 499 ≈ 10 false positives, so roughly 1 in 11 positive tests is real.
7	C	Basic Counting — With at least one nickel every multiple of 5 up to the total is attainable, so the count of values is 12 + n, forcing n = 5.
8	C	Transformations & Symmetry — Write each map as a formula on (x, y): the dilation commutes with the rotation and the reflection, the translation commutes only with the reflection; the other three pairs fail.
9	E	Number Properties — The 2001 condition gives b^2 - a^2 = 141 = 1141 = 347, so a is 70 or 22; only a = 22 makes a^2 + 300 a perfect square.
10	A	Functions — Substituting c = b and then b = a in the rules shows a diamond b = a/b, so the equation is 336x = 100.
11	D	Coordinate Geometry — Each equation is a pair of parallel lines, and the two directions are perpendicular, so the region is a rectangle whose sides are the distances between the parallel pairs.
12	E	Coordinate Geometry — The region is the sum of three segments (2,0), (0,1), (-3,4); such a region is a centrally symmetric hexagon whose sides are these segments, each used twice.