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?