Printable set

Trap problems

20 problems whose most common wrong answer is a designed trap

Print without answers Answer sheet

Trap problemsAMC 10 Step by Step

Name: Date: September 13, 2026

20 problems · 60 minutes · 6 / 1.5 / 0

1.2001 AMC 10 #16
The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$ . What is their sum?
1. A)
  $5$
2. B)
  $20$
3. C)
  $25$
4. D)
  $30$
5. E)
  $36$
2.2010 AMC 10B #9
Lucky Larry's teacher asked him to substitute numbers for $a$ , $b$ , $c$ , $d$ , and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$ , $b$ , $c$ , and $d$ were $1$ , $2$ , $3$ , and $4$ , respectively. What number did Larry substitute for $e$ ?
1. A)
  $-5$
2. B)
  $-3$
3. C)
  $0$
4. D)
  $3$
5. E)
  $5$
3.2011 AMC 10A #12
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make?
1. A)
  $13$
2. B)
  $14$
3. C)
  $15$
4. D)
  $16$
5. E)
  $17$
4.2012 AMC 10B #11
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
1. A)
  $729$
2. B)
  $972$
3. C)
  $1024$
4. D)
  $2187$
5. E)
  $2304$
5.2013 AMC 10B #12
Let $S$ be the set of sides and diagonals of a regular pentagon. A pair of elements of $S$ are selected at random without replacement. What is the probability that the two chosen segments have the same length?
1. A)
  $\frac{2}5$
2. B)
  $\frac{4}9$
3. C)
  $\frac{1}2$
4. D)
  $\frac{5}9$
5. E)
  $\frac{4}5$
6.2013 AMC 10A #10
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
1. A)
  $15$
2. B)
  $30$
3. C)
  $40$
4. D)
  $60$
5. E)
  $70$
7.2014 AMC 10B #6
Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?
1. A)
  $33$
2. B)
  $34$
3. C)
  $36$
4. D)
  $38$
5. E)
  $39$
8.2015 AMC 10A #8
Two years ago Pete was three times as old as his cousin Claire. 2 years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ?
1. A)
  $2$
2. B)
  $4$
3. C)
  $5$
4. D)
  $6$
5. E)
  $8$
9.2020 AMC 10B #9
How many ordered pairs of integers $(x,y)$ satisfy the equation $x^{2020} + y^2 = 2y?$
1. A)
  $1$
2. B)
  $2$
3. C)
  $3$
4. D)
  $4$
5. E)
  $\text{infinitely many}$
10.2024 AMC 10B #6
A rectangle has integer length sides and an area of 2024. What is the least possible perimeter of the rectangle?
1. A)
  $160$
2. B)
  $180$
3. C)
  $222$
4. D)
  $228$
5. E)
  $390$
11.2014 AMC 10A #20
The product $(8)(888\dots8)$ , where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$ . What is $k$ ?
1. A)
  $901$
2. B)
  $911$
3. C)
  $919$
4. D)
  $991$
5. E)
  $999$
12.2017 AMC 10A #12
Let $S$ be a set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3,~x+2,$ and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for $S?$
1. A)
  $\text{a single point}$
2. B)
  $\text{two intersecting lines}$
3. C)
  $\text{ three lines whose pairwise intersections are three distinct points}$
4. D)
  $\text{a triangle}$
5. E)
  $\text{three rays with a common endpoint}$
13.2020 AMC 10B #18
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
1. A)
  $\frac16$
2. B)
  $\frac15$
3. C)
  $\frac14$
4. D)
  $\frac13$
5. E)
  $\frac12$
14.2021 AMC Fall 10A #21
Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$ ?
1. A)
  $1$
2. B)
  $4$
3. C)
  $8$
4. D)
  $12$
5. E)
  $16$
15.2024 AMC 10B #19
In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the 12 entries will be "Possible"?
1. A)
  $4$
2. B)
  $5$
3. C)
  $6$
4. D)
  $7$
5. E)
  $9$
16.2024 AMC 10A #11
How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$ ?
1. A)
  $1$
2. B)
  $2$
3. C)
  $3$
4. D)
  $4$
5. E)
  $\text{infinitely many}$
17.2000 AMC 10 #23
When the mean, median, and mode of the list
$10,2,5,2,4,2,x$
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$ ?
1. A)
  $3$
2. B)
  $6$
3. C)
  $9$
4. D)
  $17$
5. E)
  $20$
18.2010 AMC 10B #20
Two circles lie outside regular hexagon $ABCDEF$ . The first is tangent to $\overline{AB}$ , and the second is tangent to $\overline{DE}$ . Both are tangent to lines $BC$ and $FA$ . What is the ratio of the area of the second circle to that of the first circle?
1. A)
  $18$
2. B)
  $27$
3. C)
  $36$
4. D)
  $81$
5. E)
  $108$
19.2012 AMC 10B #24
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?
1. A)
  $108$
2. B)
  $132$
3. C)
  $671$
4. D)
  $846$
5. E)
  $1105$
20.2022 AMC 10B #22
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$ , $x^{2}+y^{2}=64$ , and $(x-5)^{2}+y^{2}=3$ . What is the sum of the areas of all circles in $S$ ?
1. A)
  $48\pi$
2. B)
  $68\pi$
3. C)
  $96\pi$
4. D)
  $102\pi$
5. E)
  $136\pi$

Trap problems — 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	D	Statistics & Data — Write the least and greatest as m - 10 and m + 15 in terms of the mean m; with median 5 the sum 2m + 10 must equal 3m.
2	D	Algebraic Manipulation — Simplifying gives a-b+c-d-e, while Larry computed a-b-c-d+e; the two agree exactly when e = c.
3	A	Linear Equations & Word Problems — Let y be the number of two-pointers; then three-pointers also contribute 2y points and free throws contribute y + 1, so 5y + 1 = 61.
4	A	Arrangements with Restrictions — Fix Friday as cake first; every other day then has 3 options once its neighbor on the Friday side is chosen, giving 3^6.
5	B	Basic Probability — A regular pentagon has 5 equal sides and 5 equal diagonals, so a matching pair is two sides or two diagonals: 2*C(5,2) out of C(10,2).
6	E	Ratios, Percents & Averages — Take 100 flowers: 60 pink of which 40 are carnations, 40 red of which 30 are carnations, so 70 carnations.
7	C	Ratios, Percents & Averages — Under the sale a pair of balloons costs 1 + 2/3 = 5/3 of a regular price, so 30 prices buy 18 pairs.
8	B	Linear Equations & Word Problems — Set up both past conditions in terms of current ages, solve to get Pete 20 and Claire 8, then find x with 20 + x = 2(8 + x).
9	D	Diophantine Equations — Complete the square: x^2020 + (y-1)^2 = 1, a sum of two nonnegative integers equal to 1, so one is 0 and the other 1.
10	B	Divisibility & Factors — 2024 = 45^2 - 1 = 44 * 46, a factor pair as close to sqrt(2024) as integers allow, so the perimeter is 2(44 + 46) = 180.
11	D	Bases & Digits — Multiplying out 8 times 8, 88, 888, 8888 shows the product is 7, then k-2 ones, then 04, so the digit sum is k + 9.
12	E	Coordinate Geometry — Each pair of equal quantities gives a line, and the 'third is no greater' condition trims each line to a ray; all three rays start at (1, 7).
13	B	Conditional Probability & States — Every sequence of two red and two blue draws has the same probability 1/30, because the denominators are always 2,3,4,5 and the numerators are 1,2 for each color.
14	E	Basic Counting — Both probabilities share the 5^20 denominator, so p/q is a ratio of multinomials: 5 * 4 arrangements of bins times 4!4!/(3!5!) = 20 * 4/5 = 16.
15	C	Coordinate Geometry — Two lattice points force a rational slope, and one lattice point plus a rational slope forces infinitely many; so each row allows exactly two of the four counts.
16	D	Diophantine Equations — Square to get n² − m² = (n−m)(n+m) = 49 with m ≥ 0; the factor pairs give (n, m) = (±25, 24) and (±7, 0), four pairs.
17	E	Statistics & Data — The mode is 2 and the mean is (25+x)/7; split on whether the median is 2, x, or 4, then force an arithmetic progression.
18	D	Circles — Lines BC and FA meet at P in a 60-degree angle; both circles are inscribed in it, so each center is 2r from P.
19	B	Basic Counting — Label each song by the set of girls liking it (7 options, never all three); the three pair-labels must all appear, so patterns are 1-1-1-other or 2-1-1.
20	E	Circles — Tangency to both concentric circles forces radius 3 (center at distance 5) or radius 5 (center at distance 3); each case meets the third circle's tangency condition at four centers.