Subcontests
(30)1987 AMC 12 #30 - Dividing a Triangle
In the figure, △ABC has ∠A=45∘ and ∠B=30∘. A line DE, with D on AB and ∠ADE=60∘, divides △ABC into two pieces of equal area. (Note: the figure may not be accurate; perhaps E is on CB instead of AC.) The ratio ABAD is
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label("B", (20,0), E);
label("C", (7,6), NE);
label("D", (9.5,-1), W);
label("E", (5.9, 6.1), SW);
label("45∘", (2.5,.5));
label("60∘", (7.8,.5));
label("30∘", (16.5,.5));
[/asy]
<spanclass=′latex−bold′>(A)</span> 21<spanclass=′latex−bold′>(B)</span> 2+22<spanclass=′latex−bold′>(C)</span> 31<spanclass=′latex−bold′>(D)</span> 361<spanclass=′latex−bold′>(E)</span> 4121 1987 AMC 12 #27 - Cuts in 3D Plane
A cube of cheese C={(x,y,z)∣0≤x,y,z≤1} is cut along the planes x=y, y=z and z=x. How many pieces are there? (No cheese is moved until all three cuts are made.)<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 7<spanclass=′latex−bold′>(D)</span> 8<spanclass=′latex−bold′>(E)</span> 9 1987 AMC 12 #21 - Square Inscribed in Triangle
There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is 441cm2. What is the area (in cm2) of the square inscribed in the same △ABC as shown in Figure 2 below?
[asy]
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draw((-25,0)--(-15,0)--(-25,10)--cycle);
draw((-20,0)--(-20,5)--(-25,5));
draw((6.5,3.25)--(3.25,0)--(0,3.25)--(3.25,6.5));
label("A", (-25,10), W);
label("B", (-25,0), W);
label("C", (-15,0), E);
label("Figure 1", (-20, -5));
label("Figure 2", (5, -5));
label("A", (0,10), W);
label("B", (0,0), W);
label("C", (10,0), E);
[/asy]
<spanclass=′latex−bold′>(A)</span> 378<spanclass=′latex−bold′>(B)</span> 392<spanclass=′latex−bold′>(C)</span> 400<spanclass=′latex−bold′>(D)</span> 441<spanclass=′latex−bold′>(E)</span> 484 1987 AMC 12 #18 - Math Books on a Shelf
It takes A algebra books (all the same thickness) and H geometry books (all the same thickness, which is greater than that of an algebra book) to completely fill a certain shelf. Also, S of the algebra books and M of the geometry books would fill the same shelf. Finally, E of the algebra books alone would fill this shelf. Given that A,H,S,M,E are distinct positive integers, it follows that E is<spanclass=′latex−bold′>(A)</span> M+HAM+SH<spanclass=′latex−bold′>(B)</span> M2+H2AM2+SH2<spanclass=′latex−bold′>(C)</span> M−HAH−SM<spanclass=′latex−bold′>(D)</span> M−HAM−SH<spanclass=′latex−bold′>(E)</span> M2−H2AM2−SH2 1987 AMC 12 #17 - Mathematics Competition
In a mathematics competition, the sum of the scores of Bill and Dick equalled the sum of the scores of Ann and Carol. If the scores of Bill and Carol had been interchanged, then the sum of the scores of Ann and Carol would have exceeded the sum of the scores of the other two. Also, Dick's score exceeded the sum of the scores of Bill and Carol. Determine the order in which the four contestants finished, from highest to lowest. Assume all scores were nonnegative.<spanclass=′latex−bold′>(A)</span> Dick, Ann, Carol, Bill<spanclass=′latex−bold′>(B)</span> Dick, Ann, Bill, Carol<spanclass=′latex−bold′>(C)</span> Dick, Carol, Bill, Ann<spanclass=′latex−bold′>(D)</span> Ann, Dick, Carol, Bill<spanclass=′latex−bold′>(E)</span> Ann, Dick, Bill, Carol 1987 AMC 12 #16 - Cryptography and Bases
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base 5. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base 5 and the elements of the set {V,W,X,Y,Z}. Using this correspondence, the cryptographer finds that three consecutive integers in increasing order are coded as VYZ, VYX, VVW, respectively. What is the base-10 expression for the integer coded as XYZ?<spanclass=′latex−bold′>(A)</span> 48<spanclass=′latex−bold′>(B)</span> 71<spanclass=′latex−bold′>(C)</span> 82<spanclass=′latex−bold′>(D)</span> 108<spanclass=′latex−bold′>(E)</span> 113 1987 AMC 12 #14 - Angle in a Square
ABCD is a square and M and N are the midpoints of BC and CD respectively. Then sinθ=
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label("B", (0,2), NW);
label("C", (2,2), NE);
label("D", (2,0), SE);
label("M", (1,2), N);
label("N", (2,1), E);
label("θ", (.5,.5), SW);
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<spanclass=′latex−bold′>(A)</span> 55<spanclass=′latex−bold′>(B)</span> 53<spanclass=′latex−bold′>(C)</span> 510<spanclass=′latex−bold′>(D)</span> 54<spanclass=′latex−bold′>(E)</span> none of these 1987 AMC 12 #8 - Sum of Distances
In the figure the sum of the distances AD and BD is
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label("B", (13,0), SE);
label("C", (13,4), NE);
label("D", (10,4), N);
label("13", (6.5,0), S);
label("4", (13,2), E);
label("3", (11.5,4), N);
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<spanclass=′latex−bold′>(A)</span> between 10 and 11<spanclass=′latex−bold′>(B)</span> 12<spanclass=′latex−bold′>(C)</span> between 15 and 16<spanclass=′latex−bold′>(D)</span> between 16 and 17<spanclass=′latex−bold′>(E)</span> 17 1987 AMC 12 #7 - Comparing Quantities
If a−1=b+2=c−3=d+4, which of the four quantities a,b,c,d is the largest?<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> no one is always largest 1987 AMC 12 #6 - Triangle Angle Chasing
In the △ABC shown, D is some interior point, and x,y,z,w are the measures of angles in degrees. Solve for x in terms of y,z and w.
[asy]
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label("D", (4,3), N);
label("x", (2.25,6));
label("y", (1.5,2), SW);
label("z", (7.88,1.5));
label("w", (4,2.85), S);
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<spanclass=′latex−bold′>(A)</span> w−y−z<spanclass=′latex−bold′>(B)</span> w−2y−2z<spanclass=′latex−bold′>(C)</span> 180−w−y−z<spanclass=′latex−bold′>(D)</span> 2w−y−z<spanclass=′latex−bold′>(E)</span> 180−w+y+z Basic Perimeter
A triangular corner with side lengths DB=EB=1 is cut from equilateral triangle ABC of side length 3. The perimeter of the remaining quadrilateral is[asy]
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label("B", (3,0), S);
label("C", (0,0), W);
label("D", (2.5,.87), NE);
label("E", (2,0), S);[/asy](A) 6(B) 621(C) 7(D) 721(E) 8