2021 JMC 10
Contents
[hide]- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
Problem 1
What is the value of
Problem 2
There exist irrational numbers and
How can
be expressed in terms of
and
Problem 3
A group of people are either honest or liars, where honest people always tell the truth and liars always lie. People
stand in a line, and person
calls
a liar where
Out of these eight people, how many liars are there?
Problem 4
A day in the format is called binary if all of the digits are either
s or
s with leading zeros allowed. How many days in a year are binary?
Problem 5
A mixture has grams of aluminum and
grams of barium. Nir the chemist uses magic to remove some aluminum. Now, exactly
of the mixture consists of aluminum. How many grams of the mixture now remain?
Problem 6
The sum of the ages of a family equals Fifteen years later, the sum of their ages is equal to
How many people are in this family?
Problem 7
For some real the area of a square equals
and the product of the lengths of its diagonals equals
What is the perimeter of this square?
Problem 8
A positive integer is pretentious if it has both even and odd digits. For example, and
are pretentious. How many pretentious three-digit numbers are odd?
Problem 9
In Malachar, the number system is identical to ours, but all real numbers are written with digits in reverse order. A citizen in Malachar writes What does this Malacharian write as the answer?
Problem 10
Let be a square with sides of length
Point
is on side
and point
is on side
such that
and angle
is right. What is
Problem 11
There exist positive integers that satisfy
What is the sum of all possible values of
Problem 12
Mihir draws line and Nathan draws line
for an integer
The two lines divide the region
into four regions, with regions possibly having infinite area. What is the sum of all possible values of
Problem 13
An angle chosen from and an angle chosen from
determine two angles of a triangle. What is the probability this triangle is obtuse?
Problem 14
For a certain the base
numbers
form an increasing arithmetic sequence in that specific order. Then, what is the value of
expressed in base
Problem 15
Let be a sequence such that
and
for positive integers
How many terms of this sequence are divisible by
Problem 16
If and
are randomly chosen numbers between
and
, what is the probability that
(Recall that
denotes the greatest integer less than or equal to
)
Problem 17
One lit lightbulb is units above the top of spherical ball with a radius of
The spherical ball, lying atop a flat floor, casts a shadow. What is the area of this shadow?
Problem 18
If and
are positive real numbers that satisfy the equation
what is the value of
?
Problem 19
Two distinct divisors of are mutual if their difference divides their product. For instance,
is mutual as
Suppose a mutual pair
exists where
for a positive integer
What is the sum of all possible
Problem 20
A particle is in a grid. Each second, it moves to an adjacent cell and when traveling from a cell to another cell, it takes one of the paths with shortest time. The particle starts at cell
and travels to cell
in
seconds, to cell
in
seconds, and finally back to cell
in
seconds. How many possible triples
exist?
Problem 21
Two identical circles and
with radius
have centers that are
units apart. Two externally tangent circles
and
of radius
and
respectively are each internally tangent to both
and
. If
, what is
?
Problem 22
Let be the roots of
Suppose
is the monic polynomial with all six roots in the form
for integers
What is the coefficient of the
term in the polynomial
Problem 23
An invisible ant and an anteater, at the same constant speed of edge length per second, start at (not necessarily distinct) randomly chosen vertices of a cube. Each second, the ant first pings its location to the anteater, then randomly chooses one of the
edges emerging from its vertex to traverse immediately. The anteater traverses the edge on the closest path to the ping at the same time the ant travels. If multiple optimal paths exist, one is randomly chosen. The anteater eats the ant if at some time they are both at the same point, not necessarily a vertex. What is the ant's expected lifespan in seconds?
Problem 24
In cyclic convex hexagon diagonals
,
, and
concur at the circumcenter of the hexagon, and quadrilateral
has area
If the sum of the areas of
and the original hexagon is equal to
what is the sum of the areas of quadrilaterals
and
Problem 25
How many ordered pairs of positive integers with
and
exist such that neither the numerator nor denominator of the below fraction, when completely simplified (i.e. numerator and denominator are relatively prime), are divisible by five?