Difference between revisions of "2021 GMC 12B"
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==Problem 21== | ==Problem 21== | ||
− | Let <math>x^2=\sqrt{x^4+\frac{1}{x^2}}-\sqrt{x^4-\frac{1}{x^2}}</math> and <math>y=\sqrt{y^2+\frac{1}{y^4}}-\sqrt{y^2-\frac{1}{y^4}}</math>. The value of <math>xy</math> can be expressed as <math>\frac{a^{\frac{b}{c}}}{d^{-\frac{e}{f}}} such that < | + | Let <math>x^2=\sqrt{x^4+\frac{1}{x^2}}-\sqrt{x^4-\frac{1}{x^2}}</math> and <math>y=\sqrt{y^2+\frac{1}{y^4}}-\sqrt{y^2-\frac{1}{y^4}}</math>. The value of <math>xy</math> can be expressed as <math>\frac{a^{\frac{b}{c}}}{d^{-\frac{e}{f}}}</math> such that <math>a,b,c,d,e,f</math> are positive integers, <math>gcd(a,b,d,e)=1</math>. Find <math>a+b+c+d+e+f</math>. |
− | < | + | <math>\textbf{(A)} ~23 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~39 \qquad\textbf{(D)} ~41 \qquad\textbf{(E)} ~47</math> |
==Problem 22== | ==Problem 22== |
Revision as of 12:50, 13 May 2021
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
When of is a positive perfect square integer, what is such that is also an integer?
Problem 2
In the Great MICWELL civilization, each number digits of a number will be replaced by two times of the digit. For example: in MICWELL civilization is . Find the number that is equal to in MICWELL civilization.
Problem 3
The expression can be written as which and are natural numbers and they are relatively prime. Find .
Problem 4
How many possible ordered pairs of nonnegative integers are there such that ?
Problem 5
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 6
If and where . The value of can be expressed as . Find .
Problem 7
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 8
Bob is standing on the point on the Cartesian coordinate plane and he will move to the points or . Find the number of ways he can move such that he eventually reaches .
Problem 9
What is the remainder when is divided by ?
Problem 10
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 11
How many of the following statement are true for all parallelogram?
Statement 1: All parallelograms are cyclic quadrilaterals.
Statement 2: All cyclic quadrilaterals are parallelograms.
Statement 3: When all of the midpoint are chosen, the resulting figure is a parallelogram.
Statement 4: The length of a diagonal is the product of two adjacent sides.
Problem 12
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 13
Let and be two side lengthes of a right triangle with hypotenuse . Find the greatest possible value of
Problem 14
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 15
Let be the sum of base logarithms of the sum of all divisors of . Find the last two digits of .
Problem 16
Find the remainder when is divided by .
Problem 17
Let , find the remainder when is divided by .
Problem 18
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 19
What range does lies if ?
Problem 20
Find the remainder when is divided by .
Problem 21
Let and . The value of can be expressed as such that are positive integers, . Find .
Problem 22
How many ways are there to choose balls out of yellow balls, black balls and white balls? (Assume that the balls with same color are indistinguishable.)
Problem 23
Let . What is
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
In an unfair math competition audition, the coach, Mr.George, needs to choose people outside from participants, and he will partition the people into two different groups with each people and then choose people to become an temporary coach within the team. Then, the remaining teammates would either join one of the two teams, or left out and give up. Find the remainder when the total ways of arrangements is divided by .