2021 GMC 10B
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The radius of a circle that has an area of is . Find
What is the sum of the digits of the largest prime that divides ?
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
An equiangular 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
6. How many possible ordered pairs of nonnegative integers are there such that ?
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
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.
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 .
What is the remainder when is divided by ?
Two real numbers such that are chosen at random. What is the probability that ?
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 .
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 .
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 ?
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: .
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.
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 .
Let be the largest possible power of that divides . Find .
Find the remainder when is divided by .
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 ?
Find the remainder when is divided by .
James wrote all the positive divisors of on pieces of paper and randomly choose pieces with replacement. Find the probability that .
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.
can be expressed as such that are not necessarily distinct positive integers, and are maximized, and and and are minimized. Find
Let be an equilateral triangle with side length 2. Draw a circle such that arc inside triangle is a 120 degrees arc. Let be the center of the circle, and extend side and . Construct new lines and such that and are tangent to circle at point and , respectively, and they intersects line and at points and , respectively. and are perpendicular and is a square. Find the area of kite .