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Logic, Statistics & Miscellaneous set

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

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Logic, Statistics & Miscellaneous setAMC 10 Step by Step

Name: Date: September 13, 2026

20 problems · 60 minutes · 6 / 1.5 / 0

1.2002 AMC 10B #8
Suppose July of year $N$ has five Mondays. Which of the following must occur five times in the August of year $N$ ? (Note: Both months have $31$ days.)
1. A)
  $\text{Monday}$
2. B)
  $\text{Tuesday}$
3. C)
  $\text{Wednesday}$
4. D)
  $\text{Thursday}$
5. E)
  $\text{Friday}$
2.2007 AMC 10A #11
The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?
1. A)
  $14$
2. B)
  $16$
3. C)
  $18$
4. D)
  $20$
5. E)
  $24$
3.2008 AMC 10B #12
Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year?
1. A)
  $2500$
2. B)
  $3000$
3. C)
  $3500$
4. D)
  $4000$
5. E)
  $4500$
4.2010 AMC 10B #5
A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
1. A)
  $2$
2. B)
  $3$
3. C)
  $4$
4. D)
  $5$
5. E)
  $6$
5.2010 AMC 10A #10
Marvin had a birthday on Tuesday, May 27 in the leap year $2008$ . In what year will his birthday next fall on a Saturday?
1. A)
  $2011$
2. B)
  $2012$
3. C)
  $2013$
4. D)
  $2015$
5. E)
  $2017$
6.2011 AMC 10B #11
There are $52$ people in a room. What is the largest value of $n$ such that the statement "At least $n$ people in this room have birthdays falling in the same month" is always true?
1. A)
  $2$
2. B)
  $3$
3. C)
  $4$
4. D)
  $5$
5. E)
  $12$
7.2011 AMC 10B #8
At a certain beach if it is at least $80^{\circ} F$ and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?
1. A)
  $\text{The temperature was cooler than } 80^{\circ} \text{F and it was not sunny.}$
2. B)
  $\text{The temperature was cooler than } 80^{\circ} \text{F or it was not sunny.}$
3. C)
  $\text{If the temperature was at least } 80^{\circ} \text{F, then it was sunny.}$
4. D)
  $\text{If the temperature was cooler than } 80^{\circ} \text{F, then it was sunny.}$
5. E)
  $\text{If the temperature was cooler than } 80^{\circ} \text{F, then it was not sunny.}$
8.2015 AMC 10B #6
Marley practices exactly one sport each day of the week. She runs three days a week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the day after running or swimming. Which day of the week does Marley swim?
1. A)
  $\text{Sunday}$
2. B)
  $\text{Tuesday}$
3. C)
  $\text{Thursday}$
4. D)
  $\text{Friday}$
5. E)
  $\text{Saturday}$
9.2016 AMC 10A #7
The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$ . What is the value of $x$ ?
1. A)
  $50$
2. B)
  $60$
3. C)
  $75$
4. D)
  $90$
5. E)
  $100$
10.2016 AMC 10B #4
Zoey read $15$ books, one at a time. The first book took her $1$ day to read, the second book took her $2$ days to read, the third book took her $3$ days to read, and so on, with each book taking her $1$ more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day of the week did she finish her $15$ th book?
1. A)
  $\text{Sunday}$
2. B)
  $\text{Monday}$
3. C)
  $\text{Wednesday}$
4. D)
  $\text{Friday}$
5. E)
  $\text{Saturday}$
11.2021 AMC Fall 10A #10
A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$ . Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$ ?
1. A)
  ${-}18.5$
2. B)
  ${-}13.5$
3. C)
  $0$
4. D)
  $13.5$
5. E)
  $18.5$
12.2021 AMC 10A #7
Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that
- all of his happy snakes can add,
- none of his purple snakes can subtract, and
- all of his snakes that can't subtract also can't add.
Which of these conclusions can be drawn about Tom's snakes?
1. A)
  $Purple snakes can add.$
2. B)
  $Purple snakes are happy.$
3. C)
  $Snakes that can add are purple.$
4. D)
  $Happy snakes are not purple.$
5. E)
  $Happy snakes can't subtract.$
13.2021 AMC 10B #6
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$ , and the afternoon class's mean score is $70$ . The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{ 3}{4}$ . What is the mean of the scores of all the students?
1. A)
  $74$
2. B)
  $75$
3. C)
  $76$
4. D)
  $77$
5. E)
  $78$
14.2022 AMC 10B #11
All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?
1. A)
  $All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.$
2. B)
  $No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.$
3. C)
  $All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.$
4. D)
  $All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.$
5. E)
  $All schools smaller than Euclid HS sold more T-shirts than Euclid HS.$
15.2024 AMC 10A #12
Zelda played the Adventures of Math game on August 1 and scored $1,700$ points. She continued to play daily over the next $5$ days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was $1,700 + 80 = 1,780$ points.) What was Zelda's average score in points over the $6$ days?
1. A)
  $1700$
2. B)
  $1702$
3. C)
  $1703$
4. D)
  $1713$
5. E)
  $1715$
16.2002 AMC 10A #11
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
1. A)
  $12$
2. B)
  $13$
3. C)
  $14$
4. D)
  $15$
5. E)
  $16$
17.2010 AMC 10A #15
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs?
1. A)
  $0$
2. B)
  $1$
3. C)
  $2$
4. D)
  $3$
5. E)
  $4$
18.2012 AMC 10A #13
An iterative average of the numbers $1$ , $2$ , $3$ , $4$ , and $5$ is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
1. A)
  $\frac{31}{16}$
2. B)
  $2$
3. C)
  $\frac{17}{8}$
4. D)
  $3$
5. E)
  $\frac{65}{16}$
19.2022 AMC 10B #10
Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?
1. A)
  $5$
2. B)
  $7$
3. C)
  $9$
4. D)
  $11$
5. E)
  $13$
20.2000 AMC 10 #25
In year $N$ , the $300^{\text{th}}$ day of the year is a Tuesday. In year $N+1$ , the $200^{\text{th}}$ day is also a Tuesday. On what day of the week did the $100$ th day of year $N-1$ occur?
1. A)
  $\text{Thursday}$
2. B)
  $\text{Friday}$
3. C)
  $\text{Saturday}$
4. D)
  $\text{Sunday}$
5. E)
  $\text{Monday}$

Logic, Statistics & Miscellaneous 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	D	Clocks, Calendars & Time — In a 31-day month the weekdays of the 1st through 3rd occur five times; July 1 is Sat, Sun or Mon, so August 1 is Tue, Wed or Thu.
2	C	Sets, Estimation & Miscellaneous — Two opposite faces use every vertex exactly once, so two face sums equal 1 + 2 + ... + 8 = 36 and each face sums to 18.
3	A	Sets, Estimation & Miscellaneous — Each flip is 100,000 steps, so the year totals 44(100,000) + 50,000 = 4,450,000 steps; divided by 1800 that is about 2470 miles.
4	B	Clocks, Calendars & Time — 31 days is four weeks plus 3, so only the first three weekdays occur five times; Monday and Wednesday must both be in or both out of that block.
5	E	Clocks, Calendars & Time — May 27 shifts one weekday per year, two when a February 29 is crossed; tally the shifts from Tuesday until the total reaches 4 mod 7.
6	D	Sets, Estimation & Miscellaneous — Twelve months can hold at most 4 people each with 48 people total, so 52 people force some month to have at least 5, and 5 is sharp.
7	B	Logic Puzzles — Take the contrapositive: not crowded means the hypothesis failed, and the negation of 'hot AND sunny' is 'not hot OR not sunny'.
8	E	Logic Puzzles — Three non-consecutive running days among Tue, Thu, Fri, Sat, Sun force Tuesday plus two of Thu/Sat, Thu/Sun, Fri/Sun; only the first leaves a legal tennis day.
9	D	Statistics & Data — The mean condition alone gives (540 + x)/7 = x, so x = 90; then check that 90 is also the median and mode.
10	B	Clocks, Calendars & Time — Book 15 is finished on day 1+2+...+15 = 120; that is 119 = 7*17 days after the Monday of day 1, so also a Monday.
11	B	Statistics & Data — t is the plain average 20; s is size-weighted, (50^2 + 220^2 + 25^2)/100 = 33.5, so t - s = -13.5.
12	D	Logic Puzzles — Chain the implications: purple -> cannot subtract -> cannot add, while happy -> can add; so a happy snake cannot be purple.
13	C	Statistics & Data — Take 3 and 4 students: the total 3·84 + 4·70 = 532 over 7 students gives 76; the answer is a weighted mean.
14	B	Logic Puzzles — Rewrite the statement as 'if a school is bigger, then it did not sell more' and take the contrapositive: 'if a school sold more, then it is not bigger'.
15	E	Statistics & Data — Track deviations from 1700: the six scores are 1700 + (0, 80, −10, −20, 40, 0), and the deviations average 90/6 = 15, so the mean is 1715.
16	B	Sets, Estimation & Miscellaneous — Only 0.7+0.7, 0.8+0.4, 0.7+0.4 and 0.4+0.4+0.4 fit on a disk; the 15 large files force at least 9 disks with little room for the 15 small ones.
17	D	Logic Puzzles — Chris and LeRoy accuse each other, so exactly one is a toad; a toad Mike makes Brian's statement impossible, so Mike and Brian are frogs.
18	C	Statistics & Data — The result is a weighted average with weights 1/16, 1/16, 1/8, 1/4, 1/2 in entry order; put the largest number last to maximize and first to minimize.
19	D	Statistics & Data — Mode above median forces the top two values to equal the mode; the two smallest are then distinct with sum 2m − 10, so the mean m ≥ 7.
20	A	Clocks, Calendars & Time — The gap between the two dates is 265 or 266 days depending on a leap day; only the multiple of 7 fits, revealing which year is leap.