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1. In the figure below, a 3-inch by 3-inch square adjoins a 10-inch by 10-inch square. How many square inches are in the area of the shaded region? The number of square inches in the area of the shaded region can be represented as m/n, where m and n have no  common factors. What is m + n?

 

2. 

Steve has an isosceles triangle with base 8 inches and height 10 inches. He wants to cut it into eight pieces that
have equal areas, as shown below. The number of inches in the greatest perimeter among the eight pieces may
be written as a + √b + c√d, where a, b, c, d are positive integers and b and d have no perfect square factors. What is a + b + c + d?

3. Find the smallest positive integer n for which the sum 365 + 366 + · · · + n is equal to a power of 3.

 

4.The equations x3 + Ax + 10 = 0 and x3 + Bx2 + 50 = 0 have two roots in common for certain real numbers A and

B. Find the cube of the product of these common roots.

 

5. In how many ways is it possible to fill a 4 × 4 grid with X’es and O’s so that every pair of rows has matching
squares at exactly two of the four positions within the rows? One such configuration is shown here:

 

 Jan 30, 2021
edited by Guest  Jan 30, 2021
edited by Guest  Jan 30, 2021
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Please help with steps!!!

 Jan 30, 2021

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