Subcontests
(30)External Diagonals
Which of the following sets could NOT be the lengths of the external diagonals of a right rectangular prism [a "box"]? (An external diagonal is a diagonal of one of the rectangular faces of the box.)<spanclass=′latex−bold′>(A)</span> {4,5,6}<spanclass=′latex−bold′>(B)</span> {4,5,7}<spanclass=′latex−bold′>(C)</span> {4,6,7}<spanclass=′latex−bold′>(D)</span> {5,6,7}<spanclass=′latex−bold′>(E)</span> {5,7,8} 1993 AMC 12 #21 - Arithmetic Sequence
Let a1,a2,...,ak be a finite arithmetic sequence with
a4+a7+a10=17 and a4+a5+a6+a7+a8+a9+a10+a11+a12+a13+a14=77 If ak=13, then k=<spanclass=′latex−bold′>(A)</span> 16<spanclass=′latex−bold′>(B)</span> 18<spanclass=′latex−bold′>(C)</span> 20<spanclass=′latex−bold′>(D)</span> 22<spanclass=′latex−bold′>(E)</span> 24 1993 AMC 12 #20 - Complex Equation
Consider the equation 10z2−3iz−k=0, where z is a complex variable and i2=−1. Which of the following statements is true?<spanclass=′latex−bold′>(A)</span> For all positive real numbers k, both roots are pure imaginary.<spanclass=′latex−bold′>(B)</span> For all negative real numbers k, both roots are pure imaginary.<spanclass=′latex−bold′>(C)</span> For all pure imaginary numbers k, both roots are real and rational.<spanclass=′latex−bold′>(D)</span> For all pure imaginary numbers k, both roots are real and irrational.<spanclass=′latex−bold′>(E)</span> For all complex numbers k, neither root is real. 1993 AMC 12 #16 - Sequence of Integers
Consider the non-decreasing sequence of positive integers
1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,... in which the nth positive integer appears n times. The remainder when the 1993rd term is divided by 5 is<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 1993 AMC 12 #1 - Special Operation
For integers a,b and c, define a,b,c to mean ab−bc+ca. Then 1,−1,2 equals<spanclass=′latex−bold′>(A)</span> −4<spanclass=′latex−bold′>(B)</span> −2<spanclass=′latex−bold′>(C)</span> 0<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> 4 1993 AMC 12 #22 - Stack of Blocks
Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of the numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.
[asy]
size((400));
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0), linewidth(1));
draw((5,0)--(10,0)--(15,0)--(20,0)--(20,5)--(15,5)--(10,5)--(5,5)--(6,7)--(11,7)--(16,7)--(21,7)--(21,2)--(20,0), linewidth(1));
draw((10,0)--(10,5)--(11,7), linewidth(1));
draw((15,0)--(15,5)--(16,7), linewidth(1));
draw((20,0)--(20,5)--(21,7), linewidth(1));
draw((0,5)--(1,7)--(6,7), linewidth(1));
draw((3.5,7)--(4.5,9)--(9.5,9)--(14.5,9)--(19.5,9)--(18.5,7)--(19.5,9)--(19.5,7), linewidth(1));
draw((8.5,7)--(9.5,9), linewidth(1));
draw((13.5,7)--(14.5,9), linewidth(1));
draw((7,9)--(8,11)--(13,11)--(18,11)--(17,9)--(18,11)--(18,9), linewidth(1));
draw((12,9)--(13,11), linewidth(1));
draw((10.5,11)--(11.5,13)--(16.5,13)--(16.5,11)--(16.5,13)--(15.5,11), linewidth(1));
draw((25,0)--(30,0)--(30,5)--(25,5)--(25,0), dashed);
draw((30,0)--(35,0)--(40,0)--(45,0)--(45,5)--(40,5)--(35,5)--(30,5)--(31,7)--(36,7)--(41,7)--(46,7)--(46,2)--(45,0), dashed);
draw((35,0)--(35,5)--(36,7), dashed);
draw((40,0)--(40,5)--(41,7), dashed);
draw((45,0)--(45,5)--(46,7), dashed);
draw((25,5)--(26,7)--(31,7), dashed);
draw((28.5,7)--(29.5,9)--(34.5,9)--(39.5,9)--(44.5,9)--(43.5,7)--(44.5,9)--(44.5,7), dashed);
draw((33.5,7)--(34.5,9), dashed);
draw((38.5,7)--(39.5,9), dashed);
draw((32,9)--(33,11)--(38,11)--(43,11)--(42,9)--(43,11)--(43,9), dashed);
draw((37,9)--(38,11), dashed);
draw((35.5,11)--(36.5,13)--(41.5,13)--(41.5,11)--(41.5,13)--(40.5,11), dashed);
draw((50,0)--(55,0)--(55,5)--(50,5)--(50,0), dashed);
draw((55,0)--(60,0)--(65,0)--(70,0)--(70,5)--(65,5)--(60,5)--(55,5)--(56,7)--(61,7)--(66,7)--(71,7)--(71,2)--(70,0), dashed);
draw((60,0)--(60,5)--(61,7), dashed);
draw((65,0)--(65,5)--(66,7), dashed);
draw((70,0)--(70,5)--(71,7), dashed);
draw((50,5)--(51,7)--(56,7), dashed);
draw((53.5,7)--(54.5,9)--(59.5,9)--(64.5,9)--(69.5,9)--(68.5,7)--(69.5,9)--(69.5,7), dashed);
draw((58.5,7)--(59.5,9), dashed);
draw((63.5,7)--(64.5,9), dashed);
draw((57,9)--(58,11)--(63,11)--(68,11)--(67,9)--(68,11)--(68,9), dashed);
draw((62,9)--(63,11), dashed);
draw((60.5,11)--(61.5,13)--(66.5,13)--(66.5,11)--(66.5,13)--(65.5,11), dashed);
draw((75,0)--(80,0)--(80,5)--(75,5)--(75,0), dashed);
draw((80,0)--(85,0)--(90,0)--(95,0)--(95,5)--(90,5)--(85,5)--(80,5)--(81,7)--(86,7)--(91,7)--(96,7)--(96,2)--(95,0), dashed);
draw((85,0)--(85,5)--(86,7), dashed);
draw((90,0)--(90,5)--(91,7), dashed);
draw((95,0)--(95,5)--(96,7), dashed);
draw((75,5)--(76,7)--(81,7), dashed);
draw((78.5,7)--(79.5,9)--(84.5,9)--(89.5,9)--(94.5,9)--(93.5,7)--(94.5,9)--(94.5,7), dashed);
draw((83.5,7)--(84.5,9), dashed);
draw((88.5,7)--(89.5,9), dashed);
draw((82,9)--(83,11)--(88,11)--(93,11)--(92,9)--(93,11)--(93,9), dashed);
draw((87,9)--(88,11), dashed);
draw((85.5,11)--(86.5,13)--(91.5,13)--(91.5,11)--(91.5,13)--(90.5,11), dashed);
draw((28,6)--(33,6)--(38,6)--(43,6)--(43,11)--(38,11)--(33,11)--(28,11)--(28,6), linewidth(1));
draw((28,11)--(29,13)--(34,13)--(39,13)--(44,13)--(43,11)--(44,13)--(44,8)--(43,6), linewidth(1));
draw((33,6)--(33,11)--(34,13)--(39,13)--(38,11)--(38,6), linewidth(1));
draw((31,13)--(32,15)--(37,15)--(36,13)--(37,15)--(42,15)--(41,13)--(42,15)--(42,13), linewidth(1));
draw((34.5,15)--(35.5,17)--(40.5,17)--(39.5,15)--(40.5,17)--(40.5,15), linewidth(1));
draw((53,6)--(58,6)--(63,6)--(68,6)--(68,11)--(63,11)--(58,11)--(53,11)--(53,6), dashed);
draw((53,11)--(54,13)--(59,13)--(64,13)--(69,13)--(68,11)--(69,13)--(69,8)--(68,6), dashed);
draw((58,6)--(58,11)--(59,13)--(64,13)--(63,11)--(63,6), dashed);
draw((56,13)--(57,15)--(62,15)--(61,13)--(62,15)--(67,15)--(66,13)--(67,15)--(67,13), dashed);
draw((59.5,15)--(60.5,17)--(65.5,17)--(64.5,15)--(65.5,17)--(65.5,15), dashed);
draw((78,6)--(83,6)--(88,6)--(93,6)--(93,11)--(88,11)--(83,11)--(78,11)--(78,6), dashed);
draw((78,11)--(79,13)--(84,13)--(89,13)--(94,13)--(93,11)--(94,13)--(94,8)--(93,6), dashed);
draw((83,6)--(83,11)--(84,13)--(89,13)--(88,11)--(88,6), dashed);
draw((81,13)--(82,15)--(87,15)--(86,13)--(87,15)--(92,15)--(91,13)--(92,15)--(92,13), dashed);
draw((84.5,15)--(85.5,17)--(90.5,17)--(89.5,15)--(90.5,17)--(90.5,15), dashed);
draw((56,12)--(61,12)--(66,12)--(66,17)--(61,17)--(56,17)--(56,12), linewidth(1));
draw((61,12)--(61,17)--(62,19)--(57,19)--(56,17)--(57,19)--(67,19)--(66,17)--(67,19)--(67,14)--(66,12), linewidth(1));
draw((59.5,19)--(60.5,21)--(65.5,21)--(64.5,19)--(65.5,21)--(65.5,19), linewidth(1));
draw((81,12)--(86,12)--(91,12)--(91,17)--(86,17)--(81,17)--(81,12), dashed);
draw((86,12)--(86,17)--(87,19)--(82,19)--(81,17)--(82,19)--(92,19)--(91,17)--(92,19)--(92,14)--(91,12), dashed);
draw((84.5,19)--(85.5,21)--(90.5,21)--(89.5,19)--(90.5,21)--(90.5,19), dashed);
draw((84,18)--(89,18)--(89,23)--(84,23)--(84,18)--(84,23)--(85,25)--(90,25)--(89,23)--(90,25)--(90,20)--(89,18), linewidth(1));[/asy]
<spanclass=′latex−bold′>(A)</span> 55<spanclass=′latex−bold′>(B)</span> 83<spanclass=′latex−bold′>(C)</span> 114<spanclass=′latex−bold′>(D)</span> 137<spanclass=′latex−bold′>(E)</span> 144 1993 AMC 12 #27 - Rolling Circle inside Triangle
The sides of △ABC have lengths 6,8 and 10. A circle with center P and radius 1 rolls around the inside of △ABC, always remaining tangent to at least one side of the triangle. When P first returns to its original position, through what distance has P traveled?
[asy]
draw((0,0)--(8,0)--(8,6)--(0,0));
draw(Circle((4.5,1),1));
draw((4.5,2.5)..(5.55,2.05)..(6,1), EndArrow);
dot((0,0));
dot((8,0));
dot((8,6));
dot((4.5,1));
label("A", (0,0), SW);
label("B", (8,0), SE);
label("C", (8,6), NE);
label("8", (4,0), S);
label("6", (8,3), E);
label("10", (4,3), NW);
label("P", (4.5,1), NW);
[/asy]
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 12<spanclass=′latex−bold′>(C)</span> 14<spanclass=′latex−bold′>(D)</span> 15<spanclass=′latex−bold′>(E)</span> 17 1993 AMC 12 #25 - Rays and a Point
Let S be the set of points on the rays forming the sides of a 120∘ angle, and let P be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles PQR with Q and R in S. (Points Q and R may be on the same ray, and switching the names of Q and R does not create a distinct triangle.) There are
[asy]
draw((0,0)--(6,10.2), EndArrow);
draw((0,0)--(6,-10.2), EndArrow);
draw((0,0)--(6,0), dotted);
dot((6,0));
label("P", (6,0), S);
[/asy]
<spanclass=′latex−bold′>(A)</span> exactly 2 such triangles<spanclass=′latex−bold′>(B)</span> exactly 3 such triangles<spanclass=′latex−bold′>(C)</span> exactly 7 such triangles<spanclass=′latex−bold′>(D)</span> exactly 15 such triangles<spanclass=′latex−bold′>(E)</span> more than 15 such triangles 1993 AMC 12 #23 - Angles in a Circle
Points A,B,C and D are on a circle of diameter 1, and X is on diameter AD. If BX=CX and 3∠BAC=∠BXC=36∘, then AX=
[asy]
draw(Circle((0,0),10));
draw((-10,0)--(8,6)--(2,0)--(8,-6)--cycle);
draw((-10,0)--(10,0));
dot((-10,0));
dot((2,0));
dot((10,0));
dot((8,6));
dot((8,-6));
label("A", (-10,0), W);
label("B", (8,6), NE);
label("C", (8,-6), SE);
label("D", (10,0), E);
label("X", (2,0), NW);
[/asy]
<spanclass=′latex−bold′>(A)</span> cos6∘cos12∘sec18∘<spanclass=′latex−bold′>(B)</span> cos6∘sin12∘csc18∘<spanclass=′latex−bold′>(C)</span> cos6∘sin12∘sec18∘<spanclass=′latex−bold′>(D)</span> sin6∘sin12∘csc18∘<spanclass=′latex−bold′>(E)</span> sin6∘sin12∘sec18∘ 1993 AMC 12 #17 - Painted Dartboard
Amy painted a dart board over a square clock face using the "hour positions" as boundaries. [See figure.] If t is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and q is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then tq=
[asy]
size((80));
draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--(.9,0)--(3.1,4)--(.9,4)--(3.1,0)--(2,0)--(2,4));
draw((0,3.1)--(4,.9)--(4,3.1)--(0,.9)--(0,2)--(4,2));
[/asy]
<spanclass=′latex−bold′>(A)</span> 23−2<spanclass=′latex−bold′>(B)</span> 23<spanclass=′latex−bold′>(C)</span> 25+1<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 2 1993 AMC 12 #14 - Convex Pentagon
The convex pentagon ABCDE has ∠A=∠B=120∘, EA=AB=BC=2 and CD=DE=4. What is the area of ABCDE?
[asy]
draw((0,0)--(1,0)--(1.5,sqrt(3)/2)--(0.5,3sqrt(3)/2)--(-0.5,sqrt(3)/2)--cycle);
dot((0,0));
dot((1,0));
dot((1.5,sqrt(3)/2));
dot((0.5,3sqrt(3)/2));
dot((-0.5,sqrt(3)/2));
label("A", (0,0), SW);
label("B", (1,0), SE);
label("C", (1.5,sqrt(3)/2), E);
label("D", (0.5,3sqrt(3)/2), N);
label("E", (-.5, sqrt(3)/2), W);
[/asy]
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 73<spanclass=′latex−bold′>(C)</span> 15<spanclass=′latex−bold′>(D)</span> 93<spanclass=′latex−bold′>(E)</span> 125 1993 AMC 12 #2 - Triangle
In △ABC, ∠A=55∘, ∠C=75∘, D is on side AB and E is on side BC. If DB=BE, then ∠BED=
[asy]
size((100));
draw((0,0)--(10,0)--(8,10)--cycle);
draw((4,5)--(9.2,4));
dot((0,0));
dot((10,0));
dot((8,10));
dot((4,5));
dot((9.2,4));
label("A", (0,0), SW);
label("B", (8,10), N);
label("C", (10,0), SE);
label("D", (4,5), NW);
label("E", (9.2,4), E);
label("55∘", (.5,0), NE);
label("75∘", (9.8,0), NW);
[/asy]
<spanclass=′latex−bold′>(A)</span> 50∘<spanclass=′latex−bold′>(B)</span> 55∘<spanclass=′latex−bold′>(C)</span> 60∘<spanclass=′latex−bold′>(D)</span> 65∘<spanclass=′latex−bold′>(E)</span> 70∘