Subcontests
(30)1994 AMC 12 #29
Points A,B and C on a circle of radius r are situated so that AB=AC,AB>r, and the length of minor arc BC is r. If angles are measured in radians, then AB/BC=
[asy]
draw(Circle((0,0), 13));
draw((-13,0)--(12,5)--(12,-5)--cycle);
dot((-13,0));
dot((12,5));
dot((12,-5));
label("A", (-13,0), W);
label("B", (12,5), NE);
label("C", (12,-5), SE);
[/asy]
<spanclass=′latex−bold′>(A)</span> 21csc41<spanclass=′latex−bold′>(B)</span> 2cos21<spanclass=′latex−bold′>(C)</span> 4sin21<spanclass=′latex−bold′>(D)</span> csc21<spanclass=′latex−bold′>(E)</span> 2sec21 1994 AMC 12 #26
A regular polygon of m sides is exactly enclosed (no overlaps, no gaps) by m regular polygons of n sides each. (Shown here for m=4,n=8.) If m=10, what is the value of n?
[asy]
size(200);
defaultpen(linewidth(0.8));
draw(unitsquare);
path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle;
draw(p);
draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p);
draw(shift((0,-2-sqrt(2)))*p);
draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy]
<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 14<spanclass=′latex−bold′>(D)</span> 20<spanclass=′latex−bold′>(E)</span> 26 1994 AMC 12 #23
In the xy-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at (0,0),(0,3),(3,3),(3,1),(5,1) and (5,0). The slope of the line through the origin that divides the area of this region exactly in half is
[asy]
size(200);
Label l;
l.p=fontsize(6);
xaxis("x",0,6,Ticks(l,1.0,0.5),EndArrow);
yaxis("y",0,4,Ticks(l,1.0,0.5),EndArrow);
draw((0,3)--(3,3)--(3,1)--(5,1)--(5,0)--(0,0)--cycle,black+linewidth(2));[/asy]
<spanclass=′latex−bold′>(A)</span> 72<spanclass=′latex−bold′>(B)</span> 31<spanclass=′latex−bold′>(C)</span> 32<spanclass=′latex−bold′>(D)</span> 43<spanclass=′latex−bold′>(E)</span> 97 1994 AMC 12 #20
Suppose x,y,z is a geometric sequence with common ratio r and x=y. If x,2y,3z is an arithmetic sequence, then r is<spanclass=′latex−bold′>(A)</span> 41<spanclass=′latex−bold′>(B)</span> 31<spanclass=′latex−bold′>(C)</span> 21<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> 4 1994 AMC 12 #18
Triangle ABC is inscribed in a circle, and ∠B=∠C=4∠A. If B and C are adjacent vertices of a regular polygon of n sides inscribed in this circle, then n=
[asy]
draw(Circle((0,0), 5));
draw((0,5)--(3,-4)--(-3,-4)--cycle);
label("A", (0,5), N);
label("B", (-3,-4), SW);
label("C", (3,-4), SE);
dot((0,5));
dot((3,-4));
dot((-3,-4));
[/asy]
<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 7<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 15<spanclass=′latex−bold′>(E)</span> 18 1994 AMC 12 #15
For how many n in {1,2,3,...,100} is the tens digit of n2 odd?<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 20<spanclass=′latex−bold′>(C)</span> 30<spanclass=′latex−bold′>(D)</span> 40<spanclass=′latex−bold′>(E)</span> 50 1994 AMC 12 #13
In triangle ABC, AB=AC. If there is a point P strictly between A and B such that AP=PC=CB, then ∠A=
[asy]
draw((0,0)--(8,0)--(4,12)--cycle);
draw((8,0)--(1.6,4.8));
label("A", (4,12), N);
label("B", (0,0), W);
label("C", (8,0), E);
label("P", (1.6,4.8), NW);
dot((0,0));
dot((4,12));
dot((8,0));
dot((1.6,4.8));
[/asy]
<spanclass=′latex−bold′>(A)</span> 30∘<spanclass=′latex−bold′>(B)</span> 36∘<spanclass=′latex−bold′>(C)</span> 48∘<spanclass=′latex−bold′>(D)</span> 60∘<spanclass=′latex−bold′>(E)</span> 72∘ 1994 AMC 12 #10
For distinct real numbers x and y, let M(x,y) be the larger of x and y and let m(x,y) be the smaller of x and y. If a<b<c<d<e, then
M(M(a,m(b,c)),m(d,m(a,e)))=
<spanclass=′latex−bold′>(A)</span> a<spanclass=′latex−bold′>(B)</span> b<spanclass=′latex−bold′>(C)</span> c<spanclass=′latex−bold′>(D)</span> d<spanclass=′latex−bold′>(E)</span> e 1994 AMC 12 #8
In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is 56. The area of the region bounded by the polygon is
[asy]
draw((0,0)--(1,0)--(1,-1)--(2,-1)--(2,-2)--(3,-2)--(3,-3)--(4,-3)--(4,-2)--(5,-2)--(5,-1)--(6,-1)--(6,0)--(7,0)--(7,1)--(6,1)--(6,2)--(5,2)--(5,3)--(4,3)--(4,4)--(3,4)--(3,3)--(2,3)--(2,2)--(1,2)--(1,1)--(0,1)--cycle);
[/asy]
<spanclass=′latex−bold′>(A)</span> 84<spanclass=′latex−bold′>(B)</span> 96<spanclass=′latex−bold′>(C)</span> 100<spanclass=′latex−bold′>(D)</span> 112<spanclass=′latex−bold′>(E)</span> 196 1994 AMC 12 #7
Squares ABCD and EFGH are congruent, AB=10, and G is the center of square ABCD. The area of the region in the plane covered by these squares is
[asy]
draw((0,0)--(10,0)--(10,10)--(0,10)--cycle);
draw((5,5)--(12,-2)--(5,-9)--(-2,-2)--cycle);
label("A", (0,0), W);
label("B", (10,0), E);
label("C", (10,10), NE);
label("D", (0,10), NW);
label("G", (5,5), N);
label("F", (12,-2), E);
label("E", (5,-9), S);
label("H", (-2,-2), W);
dot((-2,-2));
dot((5,-9));
dot((12,-2));
dot((0,0));
dot((10,0));
dot((10,10));
dot((0,10));
dot((5,5));
[/asy]
<spanclass=′latex−bold′>(A)</span> 75<spanclass=′latex−bold′>(B)</span> 100<spanclass=′latex−bold′>(C)</span> 125<spanclass=′latex−bold′>(D)</span> 150<spanclass=′latex−bold′>(E)</span> 175 1994 AMC 12 #6
In the sequence
...,a,b,c,d,0,1,1,2,3,5,8,...
each term is the sum of the two terms to its left. Find a.<spanclass=′latex−bold′>(A)</span> −3<spanclass=′latex−bold′>(B)</span> −1<spanclass=′latex−bold′>(C)</span> 0<spanclass=′latex−bold′>(D)</span> 1<spanclass=′latex−bold′>(E)</span> 3 1994 AMC 12 #4
In the xy-plane, the segment with endpoints (−5,0) and (25,0) is the diameter of a circle. If the point (x,15) is on the circle, then x=<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 12.5<spanclass=′latex−bold′>(C)</span> 15<spanclass=′latex−bold′>(D)</span> 17.5<spanclass=′latex−bold′>(E)</span> 20 1994 AMC 12 #2
A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle?
[asy]
draw((0,0)--(10,0)--(10,7)--(0,7)--cycle);
draw((0,5)--(10,5));
draw((3,0)--(3,7));
label("6", (1.5,6));
label("?", (1.5,2.5));
label("14", (6.5,6));
label("35", (6.5,2.5));
[/asy]<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 15<spanclass=′latex−bold′>(C)</span> 20<spanclass=′latex−bold′>(D)</span> 21<spanclass=′latex−bold′>(E)</span> 25