Subcontests
(30)100 Lines, all distinct
Lines L1,L2,…,L100 are distinct. All lines L4n, n a positive integer, are parallel to each other. All lines L4n−3, n a positive integer, pass through a given point A. The maximum number of points of intersection of pairs of lines from the complete set {L1,L2,…,L100} is<spanclass=′latex−bold′>(A)</span>4350<spanclass=′latex−bold′>(B)</span>4351<spanclass=′latex−bold′>(C)</span>4900<spanclass=′latex−bold′>(D)</span>4901<spanclass=′latex−bold′>(E)</span>9851 What is true about the length of PQ?
[asy]
size(150);
dotfactor=4;
draw(circle((0,0),4));
draw(circle((10,-6),3));
pair O,A,P,Q;
O = (0,0);
A = (10,-6);
P = (-.55, -4.12);
Q = (10.7, -2.86);
dot("O", O, NE);
dot("O′", A, SW);
dot("P", P, SW);
dot("Q", Q, NE);
draw((2*sqrt(2),2*sqrt(2))--(10 + 3*sqrt(2)/2, -6 + 3*sqrt(2)/2)--cycle);
draw((-1.68*sqrt(2),-2.302*sqrt(2))--(10 - 2.6*sqrt(2)/2, -6 - 3.4*sqrt(2)/2)--cycle);
draw(P--Q--cycle);
//Credit to happiface for the diagram[/asy]In the adjoining figure, every point of circle O′ is exterior to circle O. Let P and Q be the points of intersection of an internal common tangent with the two external common tangents. Then the length of PQ is<spanclass=′latex−bold′>(A)</span>the average of the lengths of the internal and external common tangents<spanclass=′latex−bold′>(B)</span>equal to the length of an external common tangent if and only if circles O and O′ have equal radii<spanclass=′latex−bold′>(C)</span>always equal to the length of an external common tangent<spanclass=′latex−bold′>(D)</span>greater than the length of an external common tangent<spanclass=′latex−bold′>(E)</span>the geometric mean of the lengths of the internal and external common tangents Sequence u built upon a function
For a sequence u1,u2…, define Δ1(un)=un+1−un and, for all integer k>1, Δk(un)=Δ1(Δk−1(un)). If un=n3+n, then Δk(un)=0 for all n<spanclass=′latex−bold′>(A)</span>if k=1<spanclass=′latex−bold′>(B)</span>if k=2, but not if k=1<spanclass=′latex−bold′>(C)</span>if k=3, but not if k=2<spanclass=′latex−bold′>(D)</span>if k=4, but not if k=3<spanclass=′latex−bold′>(E)</span>for no value of k Find [K]:[M]
[asy]
size(150);
pair A=(0,0),B=(1,0),C=(0,1),D=(-1,0),E=(0,.5),F=(sqrt(2)/2,.25);
draw(circle(A,1)^^D--B);
draw(circle(E,.5)^^circle( F ,.25));
label("A", D, W);
label("K", A, S);
label("B", B, dir(0));
label("L", E, N);
label("M",shift(-.05,.05)*F);
//Credit to Klaus-Anton for the diagram[/asy]In the adjoining figure, circle K has diameter AB; cirlce L is tangent to circle K and to AB at the center of circle K; and circle M tangent to circle K, to circle L and AB. The ratio of the area of circle K to the area of circle M is<spanclass=′latex−bold′>(A)</span>12<spanclass=′latex−bold′>(B)</span>14<spanclass=′latex−bold′>(C)</span>16<spanclass=′latex−bold′>(D)</span>18<spanclass=′latex−bold′>(E)</span>not an integer Integer values of combinatoric identity
For integers k and n such that 1≤k<n, let Ckn=k!(n−k)!n!. Then (k+1n−2k−1)Ckn is an integer<spanclass=′latex−bold′>(A)</span>for all k and n<spanclass=′latex−bold′>(B)</span>for all even values of k and n, but not for all k and n<spanclass=′latex−bold′>(C)</span>for all odd values of k and n, but not for all k and n<spanclass=′latex−bold′>(D)</span>if k=1 or n−1, but not for all odd values k and n<spanclass=′latex−bold′>(E)</span>if n is divisible by k, but not for all even values k and n a,b,x satisfy logarithmic equation
Let a, b, and x be positive real numbers distinct from one. Then 4(logax)2+3(logbx)2=8(logax)(logbx)<spanclass=′latex−bold′>(A)</span>for all values of a, b, and x<spanclass=′latex−bold′>(B)</span>if and only if a=b2<spanclass=′latex−bold′>(C)</span>if and only if b=a2<spanclass=′latex−bold′>(D)</span>if and only if x=ab<spanclass=′latex−bold′>(E)</span>for none of these Find the radius of circle O
[asy]
//size(100);//local
size(200);
real r1=2;
pair
O=(0,0),
D=(.5,.5*sqrt(3)),
C=(D.x+.5*3,D.y),
B,
B_prime=endpoint(arc(D, 3, 0,-2));
B=B_prime;
path
c1=circle(O, r1);
pair C=midpoint(D--B_prime);
path arc2=arc(B_prime, 6/2, 158.25,250);
draw(c1);
draw(O--D);
draw(D--C);
draw(C--B_prime);
pair A=beginpoint(arc2);
draw(B_prime--A);
//dot(O^^D^^C^^A);
//dot(B_prime);
label("\scriptsize{O}",O,.6dir(D--O));
label("\scriptsize{C}",C,.5dir(-55));
label("\scriptsize{D}", D,.2NW);
//label("\scriptsize{B}",B,S);
label("\scriptsize{B}", B_prime, .5*dir(D--B_prime));
label("\scriptsize{A}",A,.5dir(NE));
label("\tiny{2}", O--D, .45*LeftSide);
label("\tiny{3}", D--C, .45*LeftSide);
label("\tiny{6}", B_prime--A, .45*RightSide);
label("\tiny{3}", waypoint(C--B_prime,.1), .45*N);
//Credit to Klaus-Anton for the diagram[/asy]In the adjoining figure, AB is tangent at A to the circle with center O; point D is interior to the circle; and DB intersects the circle at C. If BC=DC=3, OD=2, and AB=6, then the radius of the circle is<spanclass=′latex−bold′>(A)</span>3+3<spanclass=′latex−bold′>(B)</span>15/π<spanclass=′latex−bold′>(C)</span>9/2<spanclass=′latex−bold′>(D)</span>26<spanclass=′latex−bold′>(E)</span>22 Find the true statements
In triangles ABC and DEF, lengths AC, BC, DF, and EF are all equal. Length AB is twice the length of the altitude of △DEF from F to DE. Which of the following statements is (are) true?<spanclass=′latex−bold′>I.</span>∠ACB and ∠DFE must be complementary.<spanclass=′latex−bold′>II.</span>∠ACB and ∠DFE must be supplementary.<spanclass=′latex−bold′>III.</span>The area of △ABC must equal the area of △DEF.<spanclass=′latex−bold′>IV.</span>The area of △ABC must equal twice the area of △DEF.<spanclass=′latex−bold′>(A)</span><spanclass=′latex−bold′>II.</span>only<spanclass=′latex−bold′>(B)</span><spanclass=′latex−bold′>III.</span>only<spanclass=′latex−bold′>(C)</span><spanclass=′latex−bold′>IV.</span>only<spanclass=′latex−bold′>(D)</span>I. and <spanclass=′latex−bold′>III.</span>only<spanclass=′latex−bold′>(E)</span><spanclass=′latex−bold′>II.</span>and <spanclass=′latex−bold′>III.</span>only Real number m,n,p,q such that f(g(x))=g(f(x)) has a solution
If m, n, p, and q are real numbers and f(x)=mx+n and g(x)=px+q, then the equation f(g(x))=g(f(x)) has a solution<spanclass=′latex−bold′>(A)</span>for all choices of m, n, p, and q<spanclass=′latex−bold′>(B)</span>if and only if m=p and n=q<spanclass=′latex−bold′>(C)</span>if and only if mq−np=0<spanclass=′latex−bold′>(D)</span>if and only if n(1−p)−q(1−m)=0<spanclass=′latex−bold′>(E)</span>if and only if (1−n)(1−p)−(1−q)(1−m)=0 Solve x^2-3x+c=0
If c is a real number and the negative of one of the solutions of x2−3x+c=0 is a solution of x2+3x−c=0, then the solutions of x2−3x+c=0 are<spanclass=′latex−bold′>(A)</span>1, 2<spanclass=′latex−bold′>(B)</span>−1, −2<spanclass=′latex−bold′>(C)</span>0, 3<spanclass=′latex−bold′>(D)</span>0, −3<spanclass=′latex−bold′>(E)</span>23, 23