Subcontests
(25)1997 AJHSME #24 - Ratio of Regions
Diameter ACE is divided at C in the ratio 2:3. The two semicircles, ABC and CDE, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is[asy]pair A,B,C,D,EE;
A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0);
fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray);
draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW));
draw(circle((5,0),5));dot(A); dot(B); dot(C); dot(D); dot(EE);
label("A",A,W);
label("B",B,N);
label("C",C,E);
label("D",D,N);
label("E",EE,W);
[/asy]<spanclass=′latex−bold′>(A)</span> 2:3<spanclass=′latex−bold′>(B)</span> 1:1<spanclass=′latex−bold′>(C)</span> 3:2<spanclass=′latex−bold′>(D)</span> 9:4<spanclass=′latex−bold′>(E)</span> 5:2 1997 AJHSME #21 - Surface Area
Each corner cube is removed from this 3 cm×3 cm×3 cm cube. The surface area of the remaining figure is[asy]draw((2.7,3.99)--(0,3)--(0,0));
draw((3.7,3.99)--(1,3)--(1,0));
draw((4.7,3.99)--(2,3)--(2,0));
draw((5.7,3.99)--(3,3)--(3,0));draw((0,0)--(3,0)--(5.7,0.99));
draw((0,1)--(3,1)--(5.7,1.99));
draw((0,2)--(3,2)--(5.7,2.99));
draw((0,3)--(3,3)--(5.7,3.99));draw((0,3)--(3,3)--(3,0));
draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33));
draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66));
draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99));
[/asy]<spanclass=′latex−bold′>(A)</span> 19 sq.cm<spanclass=′latex−bold′>(B)</span> 24 sq.cm<spanclass=′latex−bold′>(C)</span> 30 sq.cm<spanclass=′latex−bold′>(D)</span> 54 sq.cm<spanclass=′latex−bold′>(E)</span> 72 sq.cm 1997 AJHSME #17 - Cube Diagonals
A cube has eight vertices (corners) and twelve edges. A segment, such as x, which joins two vertices not joined by an edge is called a diagonal. Segment y is also a diagonal. How many diagonals does a cube have?[asy]draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));
draw((3,0)--(3,3));
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);
draw((2.5,4)--(2.5,1),dashed);
label("x",(2.75,3.5),NNE);
label("y",(4.125,1.5),NNE);
[/asy]<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 8<spanclass=′latex−bold′>(C)</span> 12<spanclass=′latex−bold′>(D)</span> 14<spanclass=′latex−bold′>(E)</span> 16 1997 AJHSME #16 - Relative Values of Stocks
Penni Precisely buys $100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then<spanclass=′latex−bold′>(A)</span> A=B=C<spanclass=′latex−bold′>(B)</span> A=B<C<spanclass=′latex−bold′>(C)</span> C<B=A<spanclass=′latex−bold′>(D)</span> A<B<C<spanclass=′latex−bold′>(E)</span> B<A<C 1997 AJHSME #15 - Ratio of Areas
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2));
draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2));
draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8));
draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2));
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);
[/asy]<spanclass=′latex−bold′>(A)</span> 33<spanclass=′latex−bold′>(B)</span> 95<spanclass=′latex−bold′>(C)</span> 32<spanclass=′latex−bold′>(D)</span> 35<spanclass=′latex−bold′>(E)</span> 97 1997 AJHSME #12 - Finding an Angle
∠1+∠2=180∘∠3=∠4Find ∠4.[asy]pair H,I,J,K,L;
H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0);
draw(H--I--J--cycle);
draw(K--L--J);
draw(arc((0,0),dir(70),(1,0),CW)); label("70∘",dir(35),NE);
draw(arc(I,I+dir(250),I+dir(290),CCW)); label("40∘",I+1.25*dir(270),S);
label("1",J+0.25*dir(162.5),NW); label("2",J+0.25*dir(17.5),NE);
label("3",L+dir(162.5),WNW); label("4",K+dir(-52.5),SE);
[/asy]<spanclass=′latex−bold′>(A)</span> 20∘<spanclass=′latex−bold′>(B)</span> 25∘<spanclass=′latex−bold′>(C)</span> 30∘<spanclass=′latex−bold′>(D)</span> 35∘<spanclass=′latex−bold′>(E)</span> 40∘ 1997 AJHSME #10 - Shaded Region
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.[asy]
unitsize(8);
fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black);
fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white);
fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black);
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white);
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black);
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white);
draw((0,6)--(0,0)--(6,0));
[/asy]<spanclass=′latex−bold′>(A)</span> 125<spanclass=′latex−bold′>(B)</span> 21<spanclass=′latex−bold′>(C)</span> 127<spanclass=′latex−bold′>(D)</span> 32<spanclass=′latex−bold′>(E)</span> 65 1997 AJHSME #6 - Place Value
In the number 74982.1035 the value of the ''place'' occupied by the digit 9 is how many times as great as the value of the ''place'' occupied by the digit 3?<spanclass=′latex−bold′>(A)</span> 1,000<spanclass=′latex−bold′>(B)</span> 10,000<spanclass=′latex−bold′>(C)</span> 100,000<spanclass=′latex−bold′>(D)</span> 1,000,000<spanclass=′latex−bold′>(E)</span> 10,000,000