2021 GMC 10B
Contents
- 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
Problem 2
The radius of a circle that has an area of is
. Find
Problem 3
What is the sum of the digits of the largest prime that divides ?
Problem 4
Ary wants to go to the park at afternoon. he walked to halfway, and he's pretty hungry. Therefore, he searched on his phone and found that exactly on the halfway between his house and the park there is a restaurant. After he eats, he continues to walk, however, he stops at the way between the restaurant and park to take a break. Let
be the length that he need to walk to reach the park, and
be the distance between his house and the park. Find
Problem 5
An octagon with diagonal length and other 4 length
has four given vertices
,
,and it partially covers all the four quadrants. Let
be the area of the portion of the octagon that lies in the
th quadrant. Find
Problem 6
6. How many possible ordered pairs of nonnegative integers are there such that
?
Problem 7
In the diagram below, 9 squares with side length grid has 16 circles with radius of
such that all circles have vertices of the square as center. Assume that the diagram continues on forever. Given that the area of the circle is
of the entire infinite diagram, find
Problem 8
A three digit natural number is if it has two even digits and one odd digit as its number digits. Find the number of alternative positive integers.
Problem 9
Given a natural number is has
divisors and its product of digits is divisible by
, find the number of
that are less than or equal to
.
Problem 10
What is the remainder when is divided by
?
Problem 11
Two real numbers such that
are chosen at random. What is the probability that
?
Problem 12
In square , let
be the midpoint of side
, and let
and
be reflections of the center of the square across side
and
, respectively. Let
be the reflection of
across side
. Find the ratio between the area of kite
and square
.
Problem 13
Let be the positive integer and
be the sum of digits when
is expressed in base
. Find
such that
has the greatest possible value and
.
Problem 14
Let polynomial such that
has three roots
. Let
be the polynomial with leading coefficient 1 and roots
.
can be expressed in the form of
. What is
?
Problem 15
Given that a number is if the last 2 digits are the last two digits of
and it is divisible by
. How many
are there below
? Example:
.
Problem 16
Keel is choosing classes. His chose Algebra 2, US History, Honor Geometry, English, Advanced Spanish, PE, Math olympiad prep, and Honor Science. He can arrange the eight classes in any order of 9 class periods, and the fifth period is always lunch. Find the number of ways Keel can arrange classes such that none of his math classes are the last period before lunch, nor first period after lunch and last period of the day.
Problem 17
Let be an equilateral triangle with side length
, and let
,
and
be the midpoints of side
,
, and
, respectively. Let
be the reflection of
across the point
and let
be the intersection of line segment
and
. A circle is constructed with radius
and center at
. Find the area of pentagon
that lines outside the circle
.
Problem 18
Let be the largest possible power of
that divides
. Find
.
Problem 19
Sigre won a national competition of Mathocontition in an infinite-populated world and each of the person in such world has exactly 5 best friends. He would disseminate such honor by first day telling all of his best friends, and at the same day each of the best friends would tell 1 of their best friends. Each of the five people would then tell one of their five best friends, and in the same day all of the five best friends of the five people would tell another one of their five best friends and so on. Let be the number of people that got informed on the end of the
th day, find the remainder when
is divided by
.
Problem 20
In the diagram below, let square with side length inscribed in the circle. Each new squares are constructed by connecting points that divide the side of the previous square into a ratio of
. The new square also forms four right triangular regions. Let
be the
th square inside the circle and let
be the sum of the four arcs that are included in the circle but excluded from
.
can be expressed as which
. What is
?
Problem 21
How many ordered pairs satisfies
?
Problem 22
James wrote all the positive divisors of out pieces of paper and randomly choose
pieces with replacement. Find the probability that
.
Problem 23
In the game of "Infinite war", James need to put 5 different portal: no 1, no 2, no 3, no 4 and no 5 into 3 different boxes such that no boxes can be empty, and then he would choose to transfigure himself temporarily into light or shadow to transfer through the portal into three different locations, and then transfigure back into his initial composition. The four locations that he's able to transfer to are Experiment room, Weapon house, Poison gas station and food house. Given that no 1 and no 3 can only go to food house, find the probability that he would go to experiment room by jumping into a portal inside box
and through transfiguration of light.
Problem 24
can be expressed as
such that
are not necessarily distinct positive integers,
and
are maximized, and
and
and
are minimized. Find
Problem 25
Given the three polynomials:
1.
2.
3.
For which has roots
,
has roots
, and
has roots
. Also,
and
are all real numbers. Find