AMC 10A, 2021, Problem 10
Which of the following is equivalent to
AMC 10A, 2021, Problem 11
For which of the following integers is the base-
number
not divisible by
?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2021, Problem 16
In the following list of numbers, the integer appears
times in the list for
.
AMC 10A, 2021, Problem 19
The area of the region bounded by the graph of
AMC 10B, 2021, Problem 1
How many integer values of satisfy
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2021, Problem 13
Let be a positive integer and
be a digit such that the value of the numeral
in base
equals 263 , and the value of the numeral
in base
equals the value of the numeral
in base six. What is
?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2021, Problem 16
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, , and
are all uphill integers, but
, and
are not. How many uphill integers are divisible by
?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2021, Problem 19
Suppose that is a finite set of positive integers. If the greatest integer in
is removed from
, then the average value (arithmetic mean) of the integers remaining is 32 . If the least integer in
is also removed, then the average value of the integers remaining is 35 . If the greatest integer is then returned to the set, the average value of the integers rises to 40 . The greatest integer in the original set
is 72 greater than the least integer in
. What is the average value of all the integers in the set
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2020, Problem 4
A driver travels for hours at
miles per hour, during which her car gets
miles per gallon of gasoline. She is paid
per mile, and her only expense is gasoline at
per gallon. What is her net rate of pay, in dollars per hour, after this expense?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2020, Problem 6
How many -digit positive integers (that is, integers between
and
, inclusive) having only even digits are divisible by
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2020, Problem 8
What is the value of
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2020, Problem 9
A single bench section at a school event can hold either adults or
children. When
bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2020, Problem 17
Define .How many integers
are there such that
?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2020, Problem 21
There exists a unique strictly increasing sequence of nonnegative integers such that
.What is
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2020, Problem 22
For how many positive integers is
not divisible by
? (Recall that
is the greatest integer less than or equal to
.)
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2020, Problem 24
Let be the least positive integer greater than
for which
.What is the sum of the digits of
?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2020, Problem 24
How many positive integers satisfy
AMC 10B, 2020, Problem 25
Let denote the number of ways of writing the positive integer
as a product
where , the
are integers strictly greater than 1 , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number 6 can be written as
, and
, so
. What is
?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2019, Problem 5
What is the greatest number of consecutive integers whose sum is ?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2019, Problem 9
What is the greatest three-digit positive integer for which the sum of the first
positive integers is not a divisor of the product of the first
positive integers?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2019, Problem 15
A sequence of numbers is defined recursively by , and
AMC 10A, 2019, Problem 18
For some positive integer , the repeating base-
representation of the (base-ten) fraction
is
. What is
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2019, Problem 19
What is the least possible value of
AMC 10B, 2018, Problem 5
How many subsets of contain at least one prime number?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2018, Problem 11
Which of the following expressions is never a prime number when is a prime number?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2018, Problem 13
How many of the first 2018 numbers in the sequence are divisible by
?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2018, Problem 14
A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the list?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2018, Problem 16
Let be a strictly increasing sequence of positive integers such that
AMC 10B, 2018, Problem 20
A function is defined recursively by
and
AMC 10B, 2018, Problem 21
Mary chose an even 4 -digit number . She wrote down all the divisors of
in increasing order from left to right:
. At some moment Mary wrote 323 as a divisor of
. What is the smallest possible value of the next divisor written to the right of
?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2018, Problem 23
How many ordered pairs of positive integers satisfy the equation
AMC 10B, 2018, Problem 25
Let denote the greatest integer less than or equal to
. How many real numbers
satisfy the equation
?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2018, Problem 12
Let be a set of points
in the coordinate plane such that two of the three quantities
, and
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
?
(A) a single point
(B) two intersecting lines
(C) three lines whose pairwise intersections are three distinct points
(D) a triangle
(E) three rays with a common endpoint
AMC 10A, 2017, Problem 13
Define a sequence recursively by , and
the remainder when
is divided by 3, for all
. Thus the sequence starts
What is
?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2017, Problem 17
Distinct points lie on the circle
and have integer coordinates. The distances
and
are irrational numbers. What is the greatest possible value of the ratio
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2017, Problem 20
Let equal the sum of the digits of positive integer
. For example,
. For a particular positive integer
. Which of the following could be the value of
?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2017, Problem 23
How many triangles with positive area have all their vertices at points in the coordinate plane, where
and
are integers between 1 and 5, inclusive?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2017, Problem 25
How many integers between 100 and 999 , inclusive, have the property that some permutation of its digits is a multiple of 11 between 100 and 999? For example, both 121 and 211 have this property.
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2017, Problem 14
An integer is selected at random in the range
. What is the probability that the remainder when
is divided by 5 is 1 ?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2017, Problem 16
How many of the base-ten numerals for the positive integers less than or equal to 2017 contain the digit 0 ?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2017, Problem 23
Let be the 79 -digit number that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when
is divided by 45 ?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2017, Problem 25
Last year Isabella took math tests and received 7 different scores, each an integer between
and 100 , inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was
. What was her score on the sixth test?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2016, Problem 4
The remainder can be defined for all real numbers and
with
by
where denotes the greatest integer less than or equal to
. What is the value of
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2016, Problem 9
A triangular array of coins has
coin in the first row,
coins in the second row,
coins in the third row, and so on up to
coins in the
th row. What is the sum of the digits of
?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2016, Problem 17
Let be a positive multiple of 5 . One red ball and
green balls are arranged in a line in random order. Let
be the probability that at least
of the green balls are on the same side of the red ball. Observe that
and that
approaches
as
grows large. What is the sum of the digits of the least value of
such that
?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2016, Problem 20
For some particular value of , when
is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables
, and
, each to some positive power. What is
?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2016, Problem 22
For some positive integer , the number
has 110 positive integer divisors, including 1 and the number
. How many positive integer divisors does the number
have?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2016, Problem 25
How many ordered triples of positive integers satisfy
and
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2016, Problem 6
Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number . What is the smallest possible value for the sum of the digits of
?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2016, Problem 13
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 1000 of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these 1000 babies were in sets of quadruplets?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2016, Problem 24
How many four-digit integers , with
, have the property that the three two-digit integers
form an increasing arithmetic sequence? One such number is 4692 , where
, and
.
(A) 9
(B) 15
(C) 16
(D) 17
(E) 20
AMC 10B, 2016, Problem 25
Let , where
denotes the greatest integer less than or equal to
.
How many distinct values does assume for
(A) 32
(B) 36
(C) 45
(D) 46
(E) infinitely many
AMC 10A, 2015, Problem 18
Hexadecimal (base-16) numbers are written using numeric digits 0 through 9 as well as the letters through
to represent 10 through
Among the first 1000 positive integers, there are
whose hexadecimal representation contains only numeric digits. What is the sum of the digits of
?
(A) 17
(B) 18
(C) 19
(D) 20
(E) 21
AMC 10A, 2015, Problem 23
The zeroes of the function are integers. What is the sum of the possible values of
(A) 7
(B) 8
(C) 16
(D) 17
(E) 18
AMC 10A, 2015, Problem 25
Let be a square of side length 1 . Two points are chosen independently at random on the sides of
. The probability that the straight-line distance between the points is at least
is
, where
, and
are positive integers with
What is
?
(A) 59
(B) 60
(C) 61
(D) 62
(E) 63
AMC 10B, 2015, Problem 10
What are the sign and units digit of the product of all the odd negative integers strictly greater than ?
(A) It is a negative number ending with a 1.
(B) It is a positive number ending with a 1 .
(C) It is a negative number ending with a 5 .
(D) It is a positive number ending with a
(E) It is a negative number ending with a 0 .
AMC 10B, 2015, Problem 14
Let , and
be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation
?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2015, Problem 21
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of
?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2015, Problem 23
Let be a positive integer greater than 4 such that the decimal representation of
ends in
zeros and the decimal representation of
. ends in
zeros. Let
denote the sum of the four least possible values of
. What is the sum of the digits of
?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2014, Problem 20
The product , where the second factor has
digits, is an integer whose digits have a sum of 1000 . What is
?
(A) 901
(B) 911
(C) 919
(D) 991
(E) 999
AMC 10A, 2014, Problem 24
A sequence of natural numbers is constructed by listing the first 4 , then skipping one, listing the next 5 , skipping 2 , listing 6 , skipping 3 , and, on the th iteration, listing
and skipping
. The sequence begins
. What is the 500,000 th number in the sequence?
(A) 996,506
(B) 996,507
(C) 996,508
(D) 996,509
(E) 996,510
AMC 10A, 2014, Problem 25
The number is between
and
How many pairs of integers
are there such that
and
AMC 10B, 2014, Problem 12
The largest divisor of is itself. What is its fifth-largest divisor?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2014, Problem 14
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, abe miles was displayed on the odometer, where is a 3-digit number with
and
. At the end of the trip, the odometer showed
miles. What is
(A) 26
(B) 27
(C) 36
(D) 37
(E) 41
AMC 10B, 2014, Problem 17
What is the greatest power of 2 that is a factor of ?
(A)
(B)
(C)
(D)
(E)
AMC 10B, 2014, Problem 20
For how many integers is the number
negative?
(A) 8
(B) 10
(C) 12
(D) 14
(E) 16
AMC 10A, 2013, Problem 13
How many three-digit numbers are not divisible by 5 , have digits that sum to less than 20 , and have the first digit equal to the third digit?
(A) 52
(B) 60
(C) 66
(D) 68
(E) 70
AMC 10A, 2013, Problem 19
In base 10, the number 2013 ends in the digit 3 . In base 9 , on the other hand, the same number is written as and ends in the digit 6 . For how many positive integers
does the base-
-representation of 2013 end in the digit 3 ?
(A) 6
(B) 9
(C) 13
(D) 16
(E) 18
AMC 10B, 2013, Problem 4
When counting from 3 to 201,53 is the number counted. When counting backwards from 201 to 3,53 is the
number counted. What is
?
(A) 146
(B) 147
(C) 148
(D) 149
(E) 150
AMC 10B, 2013, Problem 5
Positive integers and
are each less than 6 . What is the smallest possible value for
?
(A)
(B)
(C)
(D) 0
(E) 2
AMC 10B, 2013, Problem 9
Three positive integers are each greater than 1 , have a product of 27000 , and are pairwise relatively prime. What is their sum?
(A) 100
(B) 137
(C) 156
(D) 160
(E) 165
AMC 10B, 2013, Problem 18
The number 2013 has the property that its units digit is the sum of its other digits, that is . How many integers less than 2013 but greater than 1000 have this property?
(A) 33
(B) 34
(C) 45
(D) 46
(E) 58
AMC 10B, 2013, Problem 20
The number 2013 is expressed in the form
AMC 10B, 2013, Problem 14
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is . What is the smallest possible value of
?
(A) 55
(B) 89
(C) 104
(D) 144
(E) 273
AMC 10B, 2013, Problem 18
The number 2013 has the property that its units digit is the sum of its other digits, that is . How many integers less than 2013 but greater than 1000 have this property?
(A) 33
(B) 34
(C) 45
(D) 46
(E) 58
AMC 10B, 2013, Problem 24
A positive integer is nice if there is a positive integer
with exactly four positive divisors (including 1 and
) such that the sum of the four divisors is equal to
. How many numbers in the set
are nice?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
AMC 10B, 2013, Problem 25
Bernardo chooses a three-digit positive integer and writes both its base- 5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer
. For example, if
, Bernardo writes the numbers 10,444 and 3,245 , and LeRoy obtains the sum
. For how many choices of
are the two rightmost digits of
, in order, the same as those of
?
(A) 5
(B) 10
(C) 15
(D) 20
(E) 25
AMC 10A, 2012, Problem 24
Let , and
be positive integers with
such that
and
.
What is
(A) 249
(B) 250
(C) 251
(D) 252
(E) 253
AMC 10B, 2012, Problem 8
What is the sum of all integer solutions to ?
(A) 10
(B) 12
(C) 15
(D) 19
(E) 25
AMC 10B, 2012, Problem 10
How many ordered pairs of positive integers satisfy the equation
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
AMC 10B, 2012, Problem 20
Bernardo and Silvia play the following game. An integer between 0 and 999 inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000 . Let be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of
?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11
AMC 10A, 2011, Problem 13
How many even integers are there between and
whose digits are all different and come from the set
(A) 12
(B) 20
(C) 72
(D) 120
(E) 200
AMC 10A, 2011, Problem 17
In the eight term sequence , the value of
is 5 and the sum of any three consecutive terms is 30 . What is
?
(A) 17
(B) 18
(C) 25
(D) 26
(E) 43
AMC 10A, 2011, Problem 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?
(A) 42
(B) 47
(C) 52
(D) 57
(E) 62
AMC 10A, 2011, Problem 23
Seven students count from 1 to 1000 as follows:
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
Finally, George says the only number that no one else says.
What number does George say?
(A) 37
(B) 242
(C) 365
(D) 728
(E) 998
AMC 10A, 2011, Problem 25
Let be a unit square region and
an integer. A point
in the interior of
is called
-ray partitional if there are
rays emanating from
that divide
into
triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
(A) 1500
(B) 1560
(C) 2320
(D) 2480
(E) 2500
AMC 10B, 2011, Problem 10
Consider the set of numbers . The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?
(A) 1
(B) 9
(C) 10
(D) 11
(E) 101
AMC 10B, 2011, Problem 21
Brian writes down four integers whose sum is 44 . The pairwise positive differences of these numbers are
, and 9 What is the sum of the possible values for
?
(A) 16
(B) 31
(C) 48
(D) 62
(E) 93
AMC 10B, 2011, Problem 23
What is the hundreds digit of ?
(A) 1
(B) 4
(C) 5
(D) 6
(E) 9
AMC 10B, 2011, Problem 24
A lattice point in an -coordinate system is any point
where both
and
are integers. The graph of
passes through no lattice point with
for all
such that
. What is the maximum possible value of
?
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2010, Problem 9
A palindrome, such as 83438 , is a number that remains the same when its digits are reversed. The numbers and
are three-digit and four-digit palindromes, respectively. What is the sum of the digits of
(A) 20
(B) 21
(C) 22
(D) 23
(E) 24
AMC 10A, 2010, Problem 25
Jim starts with a positive integer and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with
, then his sequence contains 5 numbers:
AMC 10A, 2009, Problem 5
What is the sum of the digits of the square of 1111111111 ?
(A) 18
(B) 27
(C) 45
(D) 63
(E) 81
AMC 10A, 2009, Problem 13
Suppose that and
. Which of the following is equal to
for every pair of integers
(A)
(B)
(C)
(D)
(E)
AMC 10A, 2009, Problem 25
For , let
, where there are
zeros between the 1 and the 6 . Let
) be the number of factors of 2 in the prime factorization of
. What is the maximum value of
?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
AMC 10A, 2009, Problem 21
What is the remainder when is divided by 8 ?
(A) 0
(B) 1
(C) 2
(D) 4
(E) 6
AMC 10A, 2008, Problem 24
Let What is the units digit of
?
(A) 0
(B) 2
(C) 4
(D) 6
(E) 8
AMC 10B, 2008, Problem 13
For each positive integer , the mean of the first
terms of a sequence is
. What is the
term of the sequence?
(A) 2008
(B) 4015
(C) 4016
(D)
(E)