Subcontests
(25)Values of Absolute Values
Suppose a, b, and c are nonzero real numbers, and a+b+c=0. What are the possible value(s) for ∣a∣a+∣b∣b+∣c∣c+∣abc∣abc?<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1 and −1<spanclass=′latex−bold′>(C)</span>2 and −2<spanclass=′latex−bold′>(D)</span>0,2, and −2<spanclass=′latex−bold′>(E)</span>0,1, and −1 Easier than 12 Other Problems
In the figure shown, US and UT are line segments each of length 2, and m∠TUS=60∘. Arcs \overarc{TR} and \overarc{SR} are each one-sixth of a circle with radius 2. What is the area of the region shown?
[asy]draw((1,1.732)--(2,3.464)--(3,1.732));
draw(arc((0,0),(2,0),(1,1.732)));
draw(arc((4,0),(3,1.732),(2,0)));
label("U", (2,3.464), N);
label("S", (1,1.732), W);
label("T", (3,1.732), E);
label("R", (2,0), S);[/asy]
<spanclass=′latex−bold′>(A)</span>33−π<spanclass=′latex−bold′>(B)</span>43−34π<spanclass=′latex−bold′>(C)</span>23<spanclass=′latex−bold′>(D)</span>43−32π<spanclass=′latex−bold′>(E)</span>4+34π Subtracting Triangles
In the non-convex quadrilateral ABCD shown below, ∠BCD is a right angle, AB=12, BC=4, CD=3, and AD=13.
[asy]draw((0,0)--(2.4,3.6)--(0,5)--(12,0)--(0,0));
label("B", (0, 0), SW);
label("A", (12, 0), ESE);
label("C", (2.4, 3.6), SE);
label("D", (0, 5), N);[/asy]
What is the area of quadrilateral ABCD?
<spanclass=′latex−bold′>(A)</span>12<spanclass=′latex−bold′>(B)</span>24<spanclass=′latex−bold′>(C)</span>26<spanclass=′latex−bold′>(D)</span>30<spanclass=′latex−bold′>(E)</span>36 Finding the AMC 8 Path
In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path allows only moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.
[asy]
fill((0.5, 4.5)--(1.5,4.5)--(1.5,2.5)--(0.5,2.5)--cycle,lightgray);
fill((1.5,3.5)--(2.5,3.5)--(2.5,1.5)--(1.5,1.5)--cycle,lightgray);
label("8", (1, 0));
label("C", (2, 0));
label("8", (3, 0));
label("8", (0, 1));
label("C", (1, 1));
label("M", (2, 1));
label("C", (3, 1));
label("8", (4, 1));
label("C", (0, 2));
label("M", (1, 2));
label("A", (2, 2));
label("M", (3, 2));
label("C", (4, 2));
label("8", (0, 3));
label("C", (1, 3));
label("M", (2, 3));
label("C", (3, 3));
label("8", (4, 3));
label("8", (1, 4));
label("C", (2, 4));
label("8", (3, 4));[/asy]<spanclass=′latex−bold′>(A)</span>8<spanclass=′latex−bold′>(B)</span>9<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>24<spanclass=′latex−bold′>(E)</span>36 Equal Perimeters
In the figure below, choose point D on BC so that △ACD and △ABD have equal perimeters. What is the area of △ABD?
[asy]draw((0,0)--(4,0)--(0,3)--(0,0));
label("A", (0,0), SW);
label("B", (4,0), ESE);
label("C", (0, 3), N);
label("3", (0, 1.5), W);
label("4", (2, 0), S);
label("5", (2, 1.5), NE);[/asy]
<spanclass=′latex−bold′>(A)</span>43<spanclass=′latex−bold′>(B)</span>23<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>512<spanclass=′latex−bold′>(E)</span>25 The Election
Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received 36 votes, then how many votes were cast all together?[asy]
draw((-1,0)--(0,0)--(0,1));
draw((0,0)--(0.309, -0.951));
filldraw(arc((0,0), (0,1), (-1,0))--(0,0)--cycle, lightgray);
filldraw(arc((0,0), (0.309, -0.951), (0,1))--(0,0)--cycle, gray);
draw(arc((0,0), (-1,0), (0.309, -0.951)));
label("Colby", (-0.5, 0.5));
label("25\%", (-0.5, 0.3));
label("Alicia", (0.7, 0.2));
label("45\%", (0.7, 0));
label("Brenda", (-0.5, -0.4));
label("30\%", (-0.5, -0.6));[/asy]<spanclass=′latex−bold′>(A)</span>70<spanclass=′latex−bold′>(B)</span>84<spanclass=′latex−bold′>(C)</span>100<spanclass=′latex−bold′>(D)</span>106<spanclass=′latex−bold′>(E)</span>120 Semicircle in Triangle
In the right triangle ABC, AC=12, BC=5, and angle C is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
[asy] draw((0,0)--(12,0)--(12,5)--(0,0)); draw(arc((8.67,0),(12,0),(5.33,0))); label("A", (0,0), W); label("C", (12,0), E); label("B", (12,5), NE); label("12", (6, 0), S); label("5", (12, 2.5), E);[/asy]
<spanclass=′latex−bold′>(A)</span>67<spanclass=′latex−bold′>(B)</span>513<spanclass=′latex−bold′>(C)</span>1859<spanclass=′latex−bold′>(D)</span>310<spanclass=′latex−bold′>(E)</span>1360