Subcontests
(30)Paper folding problem
Equilateral triangle ABC has been creased and folded so that vertex A now rests at A′ on BC as shown. If BA′=1 and A′C=2 then the length of crease PQ is
[asy]
size(170);
defaultpen(linewidth(0.7)+fontsize(10));
pair B=origin, A=(1.5,3*sqrt(3)/2), C=(3,0), D=(1,0), P=B+1.6*dir(B--A), Q=C+1.2*dir(C--A);
draw(B--P--D--B^^P--Q--D--C--Q);
draw(Q--A--P, linetype("4 4"));
label("A", A, N);
label("B", B, W);
label("C", C, E);
label("A′", D, S);
label("P", P, W);
label("Q", Q, E);
[/asy]
<spanclass=′latex−bold′>(A)</span> 58<spanclass=′latex−bold′>(B)</span> 20721<spanclass=′latex−bold′>(C)</span> 21+5<spanclass=′latex−bold′>(D)</span> 813<spanclass=′latex−bold′>(E)</span> 3 Number theory
An n-digit positive integer is cute if its n digits are an arrangement of the set {1,2,…,n} and its first k digits form an integer that is divisible by k, for k=1,2,…,n. For example 321 is a cute 3-digit integer because 1 divides 3, 2 divides 32, and 3 divides 321. How many cute 6-digit integers are there?<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 4 Quadrilateral area
If ABCD is a 2 X 2 square, E is the midpoint of AB, F is the midpoint of BC, AF and DE intersect at I, and BD and AF intersect at H, then the area of quadrilateral BEIH is[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair B=origin, A=(0,2), C=(2,0), D=(2,2), E=(0,1), F=(1,0);
draw(A--E--B--F--C--D--A--F^^E--D--B);
label("A", A, NW);
label("B", B, SW);
label("C", C, SE);
label("D", D, NE);
label("E", E, W);
label("F", F, S);
label("H", (.8,0.6));
label("I", (0.4,1.4));
[/asy]<spanclass=′latex−bold′>(A)</span> 31<spanclass=′latex−bold′>(B)</span> 52<spanclass=′latex−bold′>(C)</span> 157<spanclass=′latex−bold′>(D)</span> 158<spanclass=′latex−bold′>(E)</span> 53 Tangent circles
Two circles are externally tangent. Lines PAB and PA′B′ are common tangents with A and A′ on the smaller circle and B and B′ on the larger circle. If PA=AB=4, then the area of the smaller circle is
[asy]
size(250);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair O=origin, Q=(0,-3sqrt(2)), P=(0,-6sqrt(2)), A=(-4/3,3.77-6sqrt(2)), B=(-8/3,7.54-6sqrt(2)), C=(4/3,3.77-6sqrt(2)), D=(8/3,7.54-6sqrt(2));
draw(Arc(O,2sqrt(2),0,360));
draw(Arc(Q,sqrt(2),0,360));
dot(A);
dot(B);
dot(C);
dot(D);
dot(P);
draw(B--A--P--C--D);
label("A",A,dir(A));
label("B",B,dir(B));
label("A′",C,dir(C));
label("B′",D,dir(D));
label("P",P,S);[/asy]
<spanclass=′latex−bold′>(A)</span> 1.44π<spanclass=′latex−bold′>(B)</span> 2π<spanclass=′latex−bold′>(C)</span> 2.56π<spanclass=′latex−bold′>(D)</span> 8π<spanclass=′latex−bold′>(E)</span> 4π Solve the lengths
Triangle ABC has a right angle at C, AC=3 and BC=4. Triangle ABD has a right angle at A and AD=12. Points C and D are on opposite sides of AB. The line through D parallel to AC meets CB extended at E. If DBDE=nm, where m and n are relatively prime positive integers, then m+n=
[asy]
size(170);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair C=origin, A=(0,3), B=(4,0), D=(7.2,12.6), E=(7.2,0);
draw(A--C--B--A--D--B--E--D);
label("A",A,W);
label("B",B,S);
label("C",C,SW);
label("D",D,NE);
label("E",E,SE);
[/asy]
<spanclass=′latex−bold′>(A)</span> 25<spanclass=′latex−bold′>(B)</span> 128<spanclass=′latex−bold′>(C)</span> 153<spanclass=′latex−bold′>(D)</span> 243<spanclass=′latex−bold′>(E)</span> 256 Arrow area
In the arrow-shaped polygon [see figure], the angles at vertices A, C, D, E and F are right angles, BC=FG=5, CD=FE=20, DE=10, and AB=AG. The area of the polygon is closest to[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, B=(10,10), C=(10,5), D=(30,5), E=(30,-5), F=(10,-5), G=(10,-10);
draw(A--B--C--D--E--F--G--A);
label("A", A, W);
label("B", B, NE);
label("C", C, S);
label("D", D, NE);
label("E", E, SE);
label("F", F, N);
label("G", G, SE);
label("5", (11,7.5));
label("5", (11,-7.5));
label("20", (C+D)/2, N);
label("20", (F+E)/2, S);
label("10", (31,0));
[/asy]<spanclass=′latex−bold′>(A)</span> 288<spanclass=′latex−bold′>(B)</span> 291<spanclass=′latex−bold′>(C)</span> 294<spanclass=′latex−bold′>(D)</span> 297<spanclass=′latex−bold′>(E)</span> 300 Simple function
If for any three distinct numbers a, b and c we define a,b,c=c−bc+a, then 1,−2,−3=<spanclass=′latex−bold′>(A)</span> −2<spanclass=′latex−bold′>(B)</span> −52<spanclass=′latex−bold′>(C)</span> −41<spanclass=′latex−bold′>(D)</span> 52<spanclass=′latex−bold′>(E)</span> 2