What is the difference betweenand ?

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Al, Bob, Clayton, Derek, Ethan, and Frank are six Boy Scouts that will be split up into two groups of three Boy Scouts for a boating trip. How many ways are there to split up the six boys if the two groups are indistinguishable?

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Which of these numbers is a rational number?

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In the diagram below,is an isosceles right triangle with a right angle atand with a hypotenuse ofunits. Find the greatest integer less than or equal to the value of the radius of the quarter circle inscribed inside .

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The three medians of the unit equilateral triangleintersect at point . Find .

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Mark rolled two standard dice. Given that he rolled two distinct values, find the probability that he rolled two primes.

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What is the sum of the solutions to ?, whereis a positive integer?

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In the following diagram, Bob starts at the origin and makes a certain number of moves. A move is defined as him starting atand moves to , , , andwith equal probability. The probability that Bob will eventually reach the pointis . Find the number of distinct points, including , that satisfy that the probability that he will eventually reach that point is .

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Consider the line segment , which has endpointsand . Letbe a positive integer greater than .is constructed by rotatingabout the pointclockwisedegrees for all positive integerssuch that . Letbe the sum of the areas of the trianglesAsapproaches infinity,approaches a constant . Find .

Note: The original problem was set up as below but was unsolvable. This question has been rewritten for the sake of solvability.

Consider the line segment , which has two endpointsand .is constructed by rotatingabout the pointclockwisedegrees, whereis a positive integer greater than 2 and . After this operation, the line segments , , , , ,are drawn. Letbe the sum of the areas of the Triangles . Asapproaches infinity,approaches a constant . Find .

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A certain period of timestarts at exactly 6:09PM on a Tuesday and ends at exactly 6:09AM on a Thursday. Which of these numbers listed in the choices here is a possible length in days for ?

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Consider Square , a square with side length . Let Points , , ,be the midpoints of sides , , , and , respectively. Find the area of the square formed by the four line segments , , , and .

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In the figure shown here, the triangle has two legs of lengthand , and the semicircle has diameter . The area of Regioncan be expressed as , whereare positive integers,is square-free, and . Find .

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Kevin has a friend named Anna. The two of them are both in the same class, BC Calculus, which is a class that hasstudents. To split the class up into partners that work on a group project involving integrals, the teacher, Mrs. Jannesen, randomly partitions the class into groups of two. If he is assigned to be partners with his friend, he will be happy. What is the probability that Kevin is assigned to be with Anna?

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Letbe the number of distinct triangles that can be formed fromcoplanar points. Find the sum of all possible values of .

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In the figure below, a square of areais inscribed inside a square of area . There are two segments, labeledand . The value ofcan be expressed as , whereare positive integers andis square-free. Find .

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For a particular positive integer , the number of ordered sextuples of positive integersthat satisfyis exactly . Find .

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Letbe a regular octagon. How many distinct quadrilaterals can be formed from the vertices ofgiven that two quadrilaterals are not distinct if the latter can be obtained by a rotation of the former?

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Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. What is the volume of the set of points swept out by the larger log as it rolls over the smaller one?

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What is the largest power ofthat divides ?

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Define a permutationof the setto beiffor all . Find the number ofpermutations.

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There aredistinctarrays of integers that satisfy:1. Each integer in the array is aor .2. Every row and column contains all the integersand .3. No row or column contains two of the same number.Find .

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Letbe the set of all possible remainders whenis divided by , whereis a positive integer andis the number of elements in . The sumcan be expressed as whereare positive integers andandare as small as possible. Find .

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Four real numbersare randomly and independently selected from the range . Let the sets , , ,contain all of the real numbers in the rangeand , respectively. The probability that the four aforementioned sets are disjoint can be expressed as , whereandare relatively prime positive integers. Find .

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Four elementary schoolers, four middle schoolers, and four high schoolers sit around a round table withseats. There is a rule that no two people of the same school may sit adjacent to each other. Letbe the number of distinct seating arrangements following the rule. Find .

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Let . Find the remainder whenis divided by .

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