2021 GMC 12
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
Compute the number of ways to arrange distinguishable apples and indistinguishable books such that all five books must be adjacent.
Problem 2
In square with side length , point and are on side and respectively, such that is perpendicular to and . Find the area enclosed by the quadrilateral .
Problem 3
Lucas and Michael will arbitrarily choose seats to sit in a row of ten seats marked , respectively. Find the probability that Lucas would not sit next to Michael595 AND Michael595 chose an even seat.
Problem 4
If , find (Note that means logarithmic function that has a base of , and is the natural logarithm.).
Problem 5
Let be a sequence of positive integers with and and for all integers such that . Find .
Problem 6
Compute the remainder when the summation
is divided by .
Problem 7
The answer of this problem can be expressed as which are not necessarily distinct positive integers, and all of are not divisible by any square number. Find .
Problem 8
Let be a term sequence of positive integers such that ,, , , . Find the number of such sequences such that all of .
Problem 9
Find the largest possible such that is divisible by
Problem 10
Let be the sequence of positive integers such that all of the elements in the sequence only includes either a combination of digits of and , or only . For example: and are examples, but not . The first several terms of the sequence is . The th term of the sequence is . What is ?
Problem 11
Given that the two roots of polynomial are and which represents an angle. Find
Problem 12
Find the number of nonempty subsets of such that the product of all the numbers in the subset is NOT divisible by .
Problem 13
If , find the maximum possible value of such that both of and are integers and
Problem 14
A square with side length is rotated about its center. The square would externally swept out identical small regions as it rotates. Find the area of one of the small region.
Problem 15
In a circle with a radius of , four arcs are drawn inside the circle. Smaller circles are inscribed inside the eye-shape diagrams. Let denote the area of the greatest circle that can be inscribed inside the unshaded region. and let denote the total area of unshaded region. Find
Problem 16
The number ways are there to permute such that all five letters of exactly one of the three letters in are all adjacent is . Find the remainder when is divided by . (Examples: and )
Problem 17
There exists a polynomial which and are both integers. How many of the following statements are true about all quadratics ?
1. For every possible , there are at least of them such that but two quadratic that if the such has all integer roots.
2. For all roots of any quadratic in , there exists infinite number of quadratic such that if and only if has all real solutions and all terms of are real numbers.
3. For any quadratics in , there exists at least one quadratics such that they shares exactly one of the roots of and all of the roots are positive integers.
4. Statement
5. Statement
6. Statement
Problem 18
Among the roots of the polynomial , there are m values of such that is a real number. Find .
Problem 19
There exists an increasing sequence of positive integers such that the value of can be expressed with a remainder of . which is a prime number and are integers as small as possible. Find the sum of .
Problem 20
Given that . Find such that all of are nonnegative integers. (*Note that is the largest possible product of , and is the smallest possible product of .)
Problem 21
The exact value of can be expressed as which the fraction is in the most simplified form, and , , , and are not necessary distinct positive integers. Find
Problem 22
Given that on a complex plane, there is a polar coordinate . The point is rotated clockwise to form the new point , to form , and to form degrees around the origin. Evaluate:
Problem 23
Find the sum of last five digits of .
Problem 24
The Terminator is playing a game. He has a deck of card numbered from and he also has two dices. There are three green cards, three blue cards, and six orange cards. Terminator already knew that the three blue cards are but not necessary in this order, and the three green cards are in some order. Terminator will play this game two rounds by choose one card without replacement and roll the two dice. However, at the beginning of the second round, he MUST choose either by reroll one die , re-choose one card WITH replacement or re-choose the card and choose one die to reroll. He wins if in the first rounds his sum of the card number is equal to the sum of the numbers on the dice and in the second round the sum of numbers on the dice is greater than the card number. Given that terminator is a perfectionist and he always optimize his chance of winning. Find the probability that Terminator will replace only his card(He will replace the card and choose one die to reroll IF the probability of replace one card and reroll one die is equal.
Problem 25
All the solution of are vertices of a polygon. The smallest solution that when express in polar coordinate, has a value greater than but smaller than can be expressed as which are all not necessarily distinct positive integers, in this case represents the imagenary number, , and the fraction is in the most simplified form. Find .