Subcontests
(21)Meanest AMC 12 #13 Ever In The History of AMC 12s
Square ABCD has side length 30. Point P lies inside the square so that AP=12 and BP=26. The centroids of △ABP, △BCP, △CDP, and △DAP are the vertices of a convex quadrilateral. What is the area of that quadrilateral? [asy]
unitsize(120);
pair B = (0, 0), A = (0, 1), D = (1, 1), C = (1, 0), P = (1/4, 2/3);
draw(A--B--C--D--cycle);
dot(P);
defaultpen(fontsize(10pt));
draw(A--P--B);
draw(C--P--D);
label("A", A, W);
label("B", B, W);
label("C", C, E);
label("D", D, E);
label("P", P, N*1.5+E*0.5);
dot(A);
dot(B);
dot(C);
dot(D);
[/asy]
<spanclass=′latex−bold′>(A)</span>1002<spanclass=′latex−bold′>(B)</span>1003<spanclass=′latex−bold′>(C)</span>200<spanclass=′latex−bold′>(D)</span>2002<spanclass=′latex−bold′>(E)</span>2003 A Very Funky Equilateral Triangle
Circles ω1, ω2, and ω3 each have radius 4 and are placed in the plane so that each circle is externally tangent to the other two. Points P1, P2, and P3 lie on ω1, ω2, and ω3 respectively such that P1P2=P2P3=P3P1 and line PiPi+1 is tangent to ωi for each i=1,2,3, where P4=P1. See the figure below. The area of △P1P2P3 can be written in the form a+b for positive integers a and b. What is a+b?[asy]
unitsize(12);
pair A = (0, 8/sqrt(3)), B = rotate(-120)*A, C = rotate(120)*A;
real theta = 41.5;
pair P1 = rotate(theta)*(2+2*sqrt(7/3), 0), P2 = rotate(-120)*P1, P3 = rotate(120)*P1;
filldraw(P1--P2--P3--cycle, gray(0.9));
draw(Circle(A, 4));
draw(Circle(B, 4));
draw(Circle(C, 4));
dot(P1);
dot(P2);
dot(P3);
defaultpen(fontsize(10pt));
label("P1", P1, E*1.5);
label("P2", P2, SW*1.5);
label("P3", P3, N);
label("ω1", A, W*17);
label("ω2", B, E*17);
label("ω3", C, W*17);
[/asy]<spanclass=′latex−bold′>(A)</span>546<spanclass=′latex−bold′>(B)</span>548<spanclass=′latex−bold′>(C)</span>550<spanclass=′latex−bold′>(D)</span>552<spanclass=′latex−bold′>(E)</span>554 Mean and Median
For positive integers m and n such that m+10<n+1, both the mean and the median of the set {m,m+4,m+10,n+1,n+2,2n} are equal to n. What is m+n?<spanclass=′latex−bold′>(A)</span>20<spanclass=′latex−bold′>(B)</span>21<spanclass=′latex−bold′>(C)</span>22<spanclass=′latex−bold′>(D)</span>23<spanclass=′latex−bold′>(E)</span>24 Gauss Sum for High Schoolers
What is i=1∑100j=1∑100(i+j)?
<spanclass=′latex−bold′>(A)</span>100,100<spanclass=′latex−bold′>(B)</span>500,500<spanclass=′latex−bold′>(C)</span>505,000<spanclass=′latex−bold′>(D)</span>1,001,000<spanclass=′latex−bold′>(E)</span>1,010,000 Trigonometry
In △PAT, ∠P=36∘, ∠A=56∘, and PA=10. Points U and G lie on sides TP and TA, respectively, so that PU=AG=1. Let M and N be the midpoints of segments PA and UG, respectively. What is the degree measure of the acute angle formed by lines MN and PA?<spanclass=′latex−bold′>(A)</span>76<spanclass=′latex−bold′>(B)</span>77<spanclass=′latex−bold′>(C)</span>78<spanclass=′latex−bold′>(D)</span>79<spanclass=′latex−bold′>(E)</span>80 Complex Numbers
The solutions to the equations z2=4+415i and z2=2+23i, where i=−1, form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form pq−rs, where p, q, r, and s are positive integers and neither q nor s is divisible by the square of any prime number. What is p+q+r+s?<spanclass=′latex−bold′>(A)</span>20<spanclass=′latex−bold′>(B)</span>21<spanclass=′latex−bold′>(C)</span>22<spanclass=′latex−bold′>(D)</span>23<spanclass=′latex−bold′>(E)</span>24 Polynomial Counting
Consider polynomials P(x) of degree at most 3, each of whose coefficients is an element of {0,1,2,3,4,5,6,7,8,9}. How many such polynomials satisfy P(−1)=−9?<spanclass=′latex−bold′>(A)</span>110<spanclass=′latex−bold′>(B)</span>143<spanclass=′latex−bold′>(C)</span>165<spanclass=′latex−bold′>(D)</span>220<spanclass=′latex−bold′>(E)</span>286