Subcontests
(30)Part of a City
The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with probability 21 whether to go east or south. Find the probability that through any given morning, he goes through C. [asy]
defaultpen(linewidth(0.7)+fontsize(8));
size(250);
path p=origin--(5,0)--(5,3)--(0,3)--cycle;
path q=(5,19)--(6,19)--(6,20)--(5,20)--cycle;
int i,j;
for(i=0; i<5; i=i+1) {
for(j=0; j<6; j=j+1) {
draw(shift(6*i, 4*j)*p);
}}
clip((4,2)--(25,2)--(25,21)--(4,21)--cycle);
fill(q^^shift(18,-16)*q^^shift(18,-12)*q, black);
label("A", (6,19), SE);
label("B", (23,4), NW);
label("C", (23,8), NW);
draw((26,11.5)--(30,11.5), Arrows(5));
draw((28,9.5)--(28,13.5), Arrows(5));
label("N", (28,13.5), N);
label("W", (26,11.5), W);
label("E", (30,11.5), E);
label("S", (28,9.5), S);[/asy](A) 3211(B) 21(C) 74(D) 3221(E) 43 Circle and Triangle
In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If AG=2, GF=13, FC=1, and HJ=7, then DE equals [asy]
size(200);
defaultpen(fontsize(10));
real r=sqrt(22);
pair B=origin, A=16*dir(60), C=(16,0), D=(10-r,0), E=(10+r,0), F=C+1*dir(120), G=C+14*dir(120), H=13*dir(60), J=6*dir(60), O=circumcenter(G,H,J);
dot(A^^B^^C^^D^^E^^F^^G^^H^^J);
draw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7));
label("A", A, N);
label("B", B, dir(210));
label("C", C, dir(330));
label("D", D, SW);
label("E", E, SE);
label("F", F, dir(170));
label("G", G, dir(250));
label("H", H, SE);
label("J", J, dir(0));
label("2", A--G, dir(30));
label("13", F--G, dir(180+30));
label("1", F--C, dir(30));
label("7", H--J, dir(-30));[/asy](A) 222(B) 73(C) 9(D) 10(E) 13 Median Length
In the adjoining figure, the triangle ABC is a right triangle with ∠BCA=90∘. Median CM is perpendicular to median BN, and side BC=s. The length of BN is[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));real r=54.72;
pair B=origin, C=dir(r), A=intersectionpoint(B--(9,0), C--C+4*dir(r-90)), M=midpoint(B--A), N=midpoint(A--C), P=intersectionpoint(B--N, C--M);
draw(M--C--A--B--C^^B--N);
pair point=P;
markscalefactor=0.005;
draw(rightanglemark(C,P,B));
label("A", A, dir(point--A));
label("B", B, dir(point--B));
label("C", C, dir(point--C));
label("M", M, S);
label("N", N, dir(C--A)*dir(90));
label("s", B--C, NW);[/asy](A) s2(B) 23s2(C) 2s2(D) 21s5(E) 21s6 Cosine of Angle C
In the adjoining figure of a rectangular solid, ∠DHG=45∘ and ∠FHB=60∘. Find the cosine of ∠BHD.[asy]
size(200);
import three;defaultpen(linewidth(0.7)+fontsize(10));
currentprojection=orthographic(1/3+1/10,1-1/10,1/3);
real r=sqrt(3);
triple A=(0,0,r), B=(0,r,r), C=(1,r,r), D=(1,0,r), E=O, F=(0,r,0), G=(1,0,0), H=(1,r,0);
draw(D--G--H--D--A--B--C--D--B--F--H--B^^C--H);
draw(A--E^^G--E^^F--E, linetype("4 4"));
label("A", A, N);
label("B", B, dir(0));
label("C", C, N);
label("D", D, W);
label("E", E, NW);
label("F", F, S);
label("G", G, W);
label("H", H, S);
triple H45=(1,r-0.15,0.1), H60=(1-0.05, r, 0.07);
label("45∘", H45, dir(125), fontsize(8));
label("60∘", H60, dir(25), fontsize(8));[/asy](A) 63(B) 62(C) 36(D) 46(E) 46−2 Tangent Circles
In the adjoining figure, points B and C lie on line segment AD, and AB, BC, and CD are diameters of circle O, N, and P, respectively. Circles O, N, and P all have radius 15 and the line AG is tangent to circle P at G. If AG intersects circle N at points E and F, then chord EF has length[asy]
size(250);
defaultpen(fontsize(10));
pair A=origin, O=(1,0), B=(2,0), N=(3,0), C=(4,0), P=(5,0), D=(6,0), G=tangent(A,P,1,2), E=intersectionpoints(A--G, Circle(N,1))[0], F=intersectionpoints(A--G, Circle(N,1))[1];
draw(Circle(O,1)^^Circle(N,1)^^Circle(P,1)^^G--A--D, linewidth(0.7));
dot(A^^B^^C^^D^^E^^F^^G^^O^^N^^P);
label("A", A, W);
label("B", B, SE);
label("C", C, NE);
label("D", D, dir(0));
label("P", P, S);
label("N", N, S);
label("O", O, S);
label("E", E, dir(120));
label("F", F, NE);
label("G", G, dir(100));[/asy](A) 20(B) 152(C) 24<spanclass=′latex−bold′>(D)</span>25(E) none of these Perimeter of a triangle
In the adjoining diagram, BO bisects ∠CBA, CO bisects ∠ACB, and MN is parallel to BC. If AB=12, BC=24, and AC=18, then the perimeter of △AMN is[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair B=origin, C=(24,0), A=intersectionpoints(Circle(B,12), Circle(C,18))[0], O=incenter(A,B,C), M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0));
draw(B--M--O--B--C--O--N--C^^N--A--M);
label("A", A, dir(90));
label("B", B, dir(O--B));
label("C", C, dir(O--C));
label("M", M, dir(90)*dir(B--A));
label("N", N, dir(90)*dir(A--C));
label("O", O, dir(90));[/asy](A) 30(B) 33(C) 36(D) 39(E) 42 Dividing Area of a Triangle
A vertical line divides the triangle with vertices (0,0), (1,1), and (9,1) in the xy-plane into two regions of equal area. The equation of the line is x=(A) 2.5(B) 3.0(C) 3.5(D) 4.0(E) 4.5 Operation problem
If the operation x∗y is defined by x∗y=(x+1)(y+1)−1, then which one of the following is FALSE?<spanclass=′latex−bold′>(A)</span> x∗y=y∗x for all real x and y.<spanclass=′latex−bold′>(B)</span> x∗(y+z)=(x∗y)+(x∗z) for all real x,y, and z<spanclass=′latex−bold′>(C)</span> (x−1)∗(x+1)=(x∗x)−1 for all real x.<spanclass=′latex−bold′>(D)</span> x∗0=x for all real x.<spanclass=′latex−bold′>(E)</span> x∗(y∗z)=(x∗y)∗zfor all real x,y, and z. Combinatorics
By definition, r! \equal{} r(r \minus{} 1) \cdots 1 and \binom{j}{k} \equal{} \frac {j!}{k!(j \minus{} k)!}, where r,j,k are positive integers and k<j. If (1n),(2n),(3n) form an arithmetic progression with n>3, then n equals
<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 7<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 11<spanclass=′latex−bold′>(E)</span> 12