# User:Lcz

Hi just getting some last minute A(O)IME geo prep in (my thing is alg>nt>combo>geo) from the Intermediate Geometry Problems page and then printing it out :)

(2015 II #9) https://artofproblemsolving.com/wiki/index.php/2015_AIME_II_Problems/Problem_9 A cylindrical barrel with radius feet and height feet is full of water. A solid cube with side length feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is cubic feet. Find .

[asy] import three; import solids; size(5cm); currentprojection=orthographic(1,-1/6,1/6); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,360)),white,nolight); triple A =(8*sqrt(6)/3,0,8*sqrt(3)/3), B = (-4*sqrt(6)/3,4*sqrt(2),8*sqrt(3)/3), C = (-4*sqrt(6)/3,-4*sqrt(2),8*sqrt(3)/3), X = (0,0,-2*sqrt(2)); draw(X--X+A--X+A+B--X+A+B+C); draw(X--X+B--X+A+B); draw(X--X+C--X+A+C--X+A+B+C); draw(X+A--X+A+C); draw(X+C--X+C+B--X+A+B+C,linetype("2 4")); draw(X+B--X+C+B,linetype("2 4")); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,240)),white,nolight); draw((-2,-2*sqrt(3),0)..(4,0,0)..(-2,2*sqrt(3),0)); draw((-4*cos(atan(5)),-4*sin(atan(5)),0)--(-4*cos(atan(5)),-4*sin(atan(5)),-10)..(4,0,-10)..(4*cos(atan(5)),4*sin(atan(5)),-10)--(4*cos(atan(5)),4*sin(atan(5)),0)); draw((-2,-2*sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype("2 4")); [/asy]

(2015 I #13) https://artofproblemsolving.com/wiki/index.php/2015_AIME_I_Problems/Problem_13 With all angles measured in degrees, the product , where and are integers greater than 1. Find .

(2015 I #11) https://artofproblemsolving.com/wiki/index.php/2015_AIME_I_Problems/Problem_11 Triangle has positive integer side lengths with . Let be the intersection of the bisectors of and . Suppose . Find the smallest possible perimeter of .

(2014 II #14) https://artofproblemsolving.com/wiki/index.php/2014_AIME_II_Problems/Problem_14 In , and . Let and be points on the line such that , , and . Point is the midpoint of the segment , and point is on ray such that . Then , where and are relatively prime positive integers. Find .

(2014 II #11) https://artofproblemsolving.com/wiki/index.php/2014_AIME_II_Problems/Problem_11 In , and . . Let be the midpoint of segment . Point lies on side such that . Extend segment through to point such that . Then , where and are relatively prime positive integers, and is a positive integer. Find .

(2011 II #13) https://artofproblemsolving.com/wiki/index.php/2011_AIME_II_Problems/Problem_13 Point lies on the diagonal of square with . Let and be the circumcenters of triangles and respectively. Given that and , then , where and are positive integers. Find .

(2011 II #10) https://artofproblemsolving.com/wiki/index.php/2011_AIME_II_Problems/Problem_10 A circle with center has radius 25. Chord of length 30 and chord of length 14 intersect at point . The distance between the midpoints of the two chords is 12. The quantity can be represented as , where and are relatively prime positive integers. Find the remainder when is divided by 1000.

(2011 I #13) https://artofproblemsolving.com/wiki/index.php/2011_AIME_I_Problems/Problem_13 A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled . The three vertices adjacent to vertex are at heights 10, 11, and 12 above the plane. The distance from vertex to the plane can be expressed as , where , , and are positive integers. Find .

(2011 II #9) https://artofproblemsolving.com/wiki/index.php/2010_AIME_II_Problems/Problem_9 Let be a regular hexagon. Let , , , , , and be the midpoints of sides , , , , , and , respectively. The segments , , , , , and bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of be expressed as a fraction where and are relatively prime positive integers. Find .