Subcontests
(35)1968 AMC 12 #35 - Ratio of Areas in Semicircle
In this diagram the center of the circle is O, the radius is a inches, chord EF is parallel to chord CD,O,G,H,J are collinear, and G is the midpoint of CD. Let K (sq. in.) represent the area of trapezoid CDFE and let R (sq. in.) represent the area of rectangle ELMF. Then, as CD and EF are translated upward so that OG increases toward the value a, while JH always equals HG, the ratio K:R become arbitrarily close to:
[asy]
size((270));
draw((0,0)--(10,0)..(5,5)..(0,0));
draw((5,0)--(5,5));
draw((9,3)--(1,3)--(1,1)--(9,1)--cycle);
draw((9.9,1)--(.1,1));
label("O", (5,0), S);
label("a", (7.5,0), S);
label("G", (5,1), SE);
label("J", (5,5), N);
label("H", (5,3), NE);
label("E", (1,3), NW);
label("L", (1,1), S);
label("C", (.1,1), W);
label("F", (9,3), NE);
label("M", (9,1), S);
label("D", (9.9,1), E);
[/asy]
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 21+21<spanclass=′latex−bold′>(E)</span> 21+1 1968 AMC 12 #31 - Three Figures
In this diagram, not drawn to scale, figures I and III are equilateral triangular regions with respective areas of 323 and 83 square inches. Figure II is a square region with area 32 sq. in. Let the length of segment AD be decreased by 1221% of itself, while the lengths of AB and CD remain unchanged. The percent decrease in the area of the square is:
[asy]
draw((0,0)--(22.6,0));
draw((0,0)--(5.66,9.8)--(11.3,0)--(11.3,5.66)--(16.96,5.66)--(16.96,0)--(19.45,4.9)--(22.6,0));
label("A", (0,0), S);
label("B", (11.3,0), S);
label("C", (16.96,0), S);
label("D", (22.6,0), S);
label("I", (5.66, 3.9));
label("II", (14.15,2.83));
label("III", (19.7,2));
[/asy]
<spanclass=′latex−bold′>(A)</span> 1221<spanclass=′latex−bold′>(B)</span> 25<spanclass=′latex−bold′>(C)</span> 50<spanclass=′latex−bold′>(D)</span> 75<spanclass=′latex−bold′>(E)</span> 8721 1968 AMC 12 #29 - Ordering Numbers
Given the three numbers x,y=xx,z=x(xx) with .9<x<1.0. Arranged in order of increasing magnitude, they are:<spanclass=′latex−bold′>(A)</span> x,z,y<spanclass=′latex−bold′>(B)</span> x,y,z<spanclass=′latex−bold′>(C)</span> y,x,z<spanclass=′latex−bold′>(D)</span> y,z,x<spanclass=′latex−bold′>(E)</span> z,x,y Set of Possible Values
Let f(n)=nx1+x2+...+xn, where n is a positive integer. If xk=(−1)k,k=1,2,...,n, the set of possible values of f(n) is:<spanclass=′latex−bold′>(A)</span> {0}<spanclass=′latex−bold′>(B)</span> {n1}<spanclass=′latex−bold′>(C)</span> {0,−n1}<spanclass=′latex−bold′>(D)</span> {0,n1}<spanclass=′latex−bold′>(E)</span> {1,n1} Find the True Statement
Let side AD of convex quadrilateral ABCD be extended through D, and let side BC be extended through C, to meet in point E. Let S represent the degree-sum of angles CDE and DCE, and let S′ represent the degree-sum of angles BAD and ABC. If r=S/S′, then:<spanclass=′latex−bold′>(A)</span> r=1 sometimes, r>1 sometimes<spanclass=′latex−bold′>(B)</span> r=1 sometimes, r<1 sometimes<spanclass=′latex−bold′>(C)</span> 0<r<1<spanclass=′latex−bold′>(D)</span> r>1<spanclass=′latex−bold′>(E)</span> r=1