2021 Fall AMC 10B Problems
2021 Fall AMC 10B (Answer Key) Printable versions: • Fall AoPS Resources • Fall PDF | ||
Instructions
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Contents
Problem 1
What is the value of
Problem 2
Problem 3
The expression is equal to the fraction
in which
and
are positive integers whose greatest common divisor is
. What is
Problem 4
At noon on a certain day, Minneapolis is degrees warmer than St. Louis. At
the temperature in Minneapolis has fallen by
degrees while the temperature in St. Louis has risen by
degrees, at which time the temperatures in the two cities differ by
degrees. What is the product of all possible values of
Problem 5
Let . Which of the following is equal to
Problem 6
The least positive integer with exactly distinct positive divisors can be written in the form
, where
and
are integers and
is not a divisor of
. What is
Problem 7
Call a fraction , not necessarily in the simplest form special if
and
are positive integers whose sum is
. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
Problem 8
The largest prime factor of is
, because
. What is the sum of the digits of the largest prime factor of
?
Problem 9
The knights in a certain kingdom come in two colors. of them are red, and the rest are blue. Furthermore,
of the knights are magical, and the fraction of red knights who are magical is
times the fraction of blue knights who are magical. What fraction of red knights are magical?
Problem 10
Fourty slips of paper numbered to
are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by
and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?
Problem 11
A regular hexagon of side length is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these
reflected arcs?
Problem 19
Let be the positive integer
, a
-digit number where each digit is a
. Let
be the leading digit of the
th root of
. What is
Problem 22
For each integer , let
be the sum of all products
, where
and
are integers and
. What is the sum of the 10 least values of
such that
is divisible by
?
Problem 23
Each of the sides and the
diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?
Problem 16
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?
See also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2021 Fall AMC 10A |
Followed by 2022 AMC 10A | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.