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Difference between revisions of "2005 AMC 8 Problems"

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These are equivalent problems, but modified in form but preserving original math idea.
+
{{AMC8 Problems|year=2005|}}
−
 
+
==Problem 1==
−
1. Shannon multiplies a number by 2 and gets 64 as her answer. However, she should
+
Connie multiplies a number by 2 and gets 60 as her answer. However, she should
 
have divided the number by 2 to get the correct answer. What is the correct
 
have divided the number by 2 to get the correct answer. What is the correct
 
answer?
 
answer?
  
−
(A) 8 (B) 16 (C) 32 (D) 128 (E) 256
+
<math> \textbf{(A)}\ 7.5\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 240 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 1|Solution]]
 +
 
 +
==Problem 2==
 +
Karl bought five folders from Pay-A-Lot at a cost of <math> \textdollar 2.50 </math> each.
 +
Pay-A-Lot had a 20%-off sale the following day. How much could
 +
Karl have saved on the purchase by waiting a day?
 +
 
 +
<math> \textbf{(A)}\ \textdollar 1.00  \qquad\textbf{(B)}\ \textdollar 2.00 \qquad\textbf{(C)}\ \textdollar 2.50\qquad\textbf{(D)}\ \textdollar 2.75 \qquad\textbf{(E)}\ \textdollar 5.00 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 2|Solution]]
 +
 
 +
==Problem 3==
 +
What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal <math> \overline{BD}</math> of square <math> ABCD</math>?
 +
<asy>defaultpen(linewidth(1));
 +
for ( int x = 0; x < 5; ++x )
 +
{
 +
    draw((0,x)--(4,x));
 +
    draw((x,0)--(x,4));
 +
}
 +
 
 +
fill((1,0)--(2,0)--(2,1)--(1,1)--cycle);
 +
fill((0,3)--(1,3)--(1,4)--(0,4)--cycle);
 +
fill((2,3)--(4,3)--(4,4)--(2,4)--cycle);
 +
fill((3,1)--(4,1)--(4,2)--(3,2)--cycle);
 +
label("$A$", (0, 4), NW);
 +
label("$B$", (4, 4), NE);
 +
label("$C$", (4, 0), SE);
 +
label("$D$", (0, 0), SW);</asy>
 +
 
 +
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 3|Solution]]
 +
 
 +
==Problem 4==
 +
A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.1 cm, 8.2 cm and 9.7 cm. What is the area of the square in square centimeters?
 +
 
 +
<math> \textbf{(A)}\ 24\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 64 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 4|Solution]]
 +
 
 +
==Problem 5==
 +
Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda?
 +
 
 +
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 15 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 5|Solution]]
 +
 
 +
==Problem 6==
 +
Suppose <math>d</math> is a digit. For how many values of <math>d</math> is <math>2.00d5 > 2.005</math>?
 +
 
 +
<math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 10 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 6|Solution]]
 +
 
 +
==Problem 7==
 +
Bill walks <math>\tfrac12</math> mile south, then <math>\tfrac34</math> mile east, and finally <math>\tfrac12</math> mile south. How many miles is he, in a direct line, from his starting point?
 +
 
 +
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 1\tfrac14\qquad\textbf{(C)}\ 1\tfrac12\qquad\textbf{(D)}\ 1\tfrac34\qquad\textbf{(E)}\ 2 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 7|Solution]]
 +
 
 +
==Problem 8==
 +
Suppose m and n are positive odd integers. Which of the following must also be an odd integer?
 +
 
 +
<math> \textbf{(A)}\ m+3n\qquad\textbf{(B)}\ 3m-n\qquad\textbf{(C)}\ 3m^2 + 3n^2\qquad\textbf{(D)}\ (nm + 3)^2\qquad\textbf{(E)}\ 3mn </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 8|Solution]]
 +
 
 +
==Problem 9==
 +
In quadrilateral <math> ABCD</math>, sides <math> \overline{AB}</math> and <math> \overline{BC}</math> both have length 10, sides <math> \overline{CD}</math> and <math> \overline{DA}</math> both have length 17, and the measure of angle <math> ADC</math> is <math> 60^\circ</math>. What is the length of diagonal <math> \overline{AC}</math>?
 +
<asy>draw((0,0)--(17,0));
 +
draw(rotate(301, (17,0))*(0,0)--(17,0));
 +
picture p;
 +
draw(p, (0,0)--(0,10));
 +
draw(p, rotate(115, (0,10))*(0,0)--(0,10));
 +
add(rotate(3)*p);
 +
 
 +
draw((0,0)--(8.25,14.5), linetype("8 8"));
 +
 
 +
label("$A$", (8.25, 14.5), N);
 +
label("$B$", (-0.25, 10), W);
 +
label("$C$", (0,0), SW);
 +
label("$D$", (17, 0), E);</asy>
 +
 
 +
<math> \textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 15.5\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 18.5 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 9|Solution]]
 +
 
 +
==Problem 10==
 +
Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school?
 +
 
 +
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 7.3\qquad\textbf{(C)}\ 7.7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 8.3 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 10|Solution]]
 +
 
 +
==Problem 11==
 +
The sales tax rate in Bergville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its &#36;90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack brings up &#36;90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill brings up &#36;90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total minus Jill's total?
 +
 
 +
<math> \textbf{(A)}\ - \textdollar1.06\qquad\textbf{(B)}\  - \textdollar 0.53\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \textdollar 0.53\qquad\textbf{(E)}\ \textdollar 1.06 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 11|Solution]]
 +
 
 +
==Problem 12==
 +
Big Al the ape ate 100 delicious yellow bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many delicious bananas did Big Al eat on May 5?
 +
 
 +
<math> \textbf{(A)}\ 20\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 12|Solution]]
 +
 
 +
==Problem 13==
 +
The area of polygon <math> ABCDEF</math> is 52 with <math> AB=8</math>, <math> BC=9</math> and <math> FA=5</math>. What is <math> DE+EF</math>?
 +
<asy>pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4);
 +
draw(a--b--c--d--e--f--cycle);
 +
draw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a);
 +
draw(shift(-.25,0)*b--shift(-.25,-.25)*b--shift(0,-.25)*b);
 +
draw(shift(-.25,0)*c--shift(-.25,.25)*c--shift(0,.25)*c);
 +
draw(shift(.25,0)*d--shift(.25,.25)*d--shift(0,.25)*d);
 +
draw(shift(.25,0)*f--shift(.25,.25)*f--shift(0,.25)*f);
 +
label("$A$", a, NW);
 +
label("$B$", b, NE);
 +
label("$C$", c, SE);
 +
label("$D$", d, SW);
 +
label("$E$", e, SW);
 +
label("$F$", f, SW);
 +
label("5", (0,6.5), W);
 +
label("8", (4,9), N);
 +
label("9", (8, 4.5), E);</asy>
 +
 
 +
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 13|Solution]]
 +
 
 +
==Problem 14==
 +
The Little Twelve Basketball League has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many games are scheduled?
 +
 
 +
<math> \textbf{(A)}\ 80\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108\qquad\textbf{(E)}\ 192 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 14|Solution]]
 +
 
 +
==Problem 15==
 +
How many different isosceles triangles have integer side lengths and perimeter 23?
 +
 
 +
<math> \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 11</math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 15|Solution]]
 +
 
 +
==Problem 16==
 +
A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove from the drawer to be certain there will be 5 socks of the same color?
 +
 
 +
<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 16|Solution]]
 +
 
 +
==Problem 17==
 +
The results of a cross-country team's training run are graphed below. Which student has the greatest average speed?
 +
<asy>
 +
for ( int i = 1; i <= 7; ++i )
 +
{
 +
    draw((i,0)--(i,6));
 +
}
 +
 
 +
for ( int i = 1; i <= 5; ++i )
 +
{
 +
    draw((0,i)--(8,i));
 +
}
 +
draw((-0.5,0)--(8,0), linewidth(1));
 +
draw((0,-0.5)--(0,6), linewidth(1));
 +
label("$O$", (0,0), SW);
 +
label(scale(.85)*rotate(90)*"distance", (0, 3), W);
 +
label(scale(.85)*"time", (4, 0), S);
 +
dot((1.25, 4.5));
 +
label(scale(.85)*"Evelyn", (1.25, 4.8), N);
 +
dot((2.5, 2.2));
 +
label(scale(.85)*"Briana", (2.5, 2.2), S);
 +
dot((4.25,5.2));
 +
label(scale(.85)*"Carla", (4.25, 5.2), SE);
 +
dot((5.6, 2.8));
 +
label(scale(.85)*"Debra", (5.6, 2.8), N);
 +
dot((6.8, 1.4));
 +
label(scale(.85)*"Angela", (6.8, 1.4), E);
 +
</asy>
 +
 
 +
<math> \textbf{(A)}\ \text{Angela}\qquad\textbf{(B)}\ \text{Briana}\qquad\textbf{(C)}\ \text{Carla}\qquad\textbf{(D)}\ \text{Debra}\qquad\textbf{(E)}\ \text{Evelyn} </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 17|Solution]]
 +
 
 +
==Problem 18==
 +
How many three-digit numbers are divisible by 13?
 +
 
 +
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 77</math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 18|Solution]]
 +
 
 +
==Problem 19==
 +
What is the perimeter of trapezoid <math> ABCD</math>?
 +
 
 +
<asy>size(3inch, 1.5inch);
 +
pair a=(0,0), b=(18,24), c=(68,24), d=(75,0), f=(68,0), e=(18,0);
 +
draw(a--b--c--d--cycle);
 +
draw(b--e);
 +
draw(shift(0,2)*e--shift(2,2)*e--shift(2,0)*e);
 +
label("30", (9,12), W);
 +
label("50", (43,24), N);
 +
label("25", (71.5, 12), E);
 +
label("24", (18, 12), E);
 +
label("$A$", a, SW);
 +
label("$B$", b, N);
 +
label("$C$", c, N);
 +
label("$D$", d, SE);
 +
label("$E$", e, S);</asy>
 +
 
 +
<math> \textbf{(A)}\ 180\qquad\textbf{(B)}\ 188\qquad\textbf{(C)}\ 196\qquad\textbf{(D)}\ 200\qquad\textbf{(E)}\ 204 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 19|Solution]]
 +
 
 +
==Problem 20==
 +
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise.
 +
In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
 +
 
 +
<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 24 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 20|Solution]]
 +
 
 +
==Problem 21==
 +
How many distinct triangles can be drawn using three of the dots below as vertices?
 +
 
 +
<asy>dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));</asy>
 +
 
 +
<math> \textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24 </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 21|Solution]]
 +
 
 +
==Problem 22==
 +
A company sells detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs 50% more than the small size and contains 20% less detergent than the large size. The large size contains twice as much detergent as the small size and costs 30% more than the medium size. Rank the three sizes from best to worst buy.
 +
 
 +
<math> \textbf{(A)}\ \text{SML}\qquad\textbf{(B)}\ \text{LMS}\qquad\textbf{(C)}\ \text{MSL}\qquad\textbf{(D)}\ \text{LSM}\qquad\textbf{(E)}\ \text{MLS} </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 22|Solution]]
 +
 
 +
==Problem 23==
 +
Isosceles right triangle <math> ABC</math> encloses a semicircle of area <math> 2\pi</math>. The circle has its center <math> O</math> on hypotenuse <math> \overline{AB}</math> and is tangent to sides <math> \overline{AC}</math> and <math> \overline{BC}</math>. What is the area of triangle <math> ABC</math>?
 +
 
 +
<asy>pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2);
 +
draw(circle(o, 2));
 +
clip(a--b--c--cycle);
 +
draw(a--b--c--cycle);
 +
dot(o);
 +
label("$C$", c, NW);
 +
label("$A$", a, NE);
 +
label("$B$", b, SW);</asy>
 +
 
 +
<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 3\pi\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 4\pi </math>
 +
 
 +
[[2005 AMC 8 Problems/Problem 23|Solution]]
 +
 
 +
==Problem 24==
 +
A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically  displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"?
 +
 
 +
<math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12</math>
  
−
2. Kent bought five notebooks from Pay-More at a cost of 4 dollars each.
+
[[2005 AMC 8 Problems/Problem 24|Solution]]
−
Pay-More had a 25%-off sale a week later. How much could
 
−
Kent have saved on the purchase by waiting for the sale?
 
  
−
All numbers are in dollars:
+
==Problem 25==
−
(A) <math>1.00 (B) </math>2.50 (C) <math>5.00 (D) </math>7.50 (E) $10.00
+
A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?
  
−
3.
+
<asy>pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2);
 +
draw(a--d--b--c--cycle);
 +
draw(circle(o, 2.25));</asy>
 +
<math> \textbf{(A)}\ \frac{2}{\sqrt{\pi}} \qquad \textbf{(B)}\ \frac{1+\sqrt{2}}{2} \qquad \textbf{(C)}\ \frac{3}{2} \qquad \textbf{(D)}\ \sqrt{3} \qquad \textbf{(E)}\ \sqrt{\pi}</math>
  
−
4.
+
[[2005 AMC 8 Problems/Problem 25|Solution]]
  
−
5. Matha Cola bottles are sold in six-packs, 12-packs, and 20-packs.  What is the fewest
+
==See Also==
−
number of packs you must buy to get exactly 78 bottles of Matha Cola?
+
{{AMC8 box|year=2005|before=[[2004 AMC 8 Problems|2004 AMC 8]]|after=[[2006 AMC 8 Problems|2006 AMC 8]]}}
 +
* [[AMC 8]]
 +
* [[AMC 8 Problems and Solutions]]
 +
* [[Mathematics competition resources]]
  
−
(A) 4 (B) 5 (C) 6 (D) 8 (E) 13
 
  
−
Answers:
+
{{MAA Notice}}
−
1.B  2.C 3.B
 

Latest revision as of 22:18, 5 December 2023

2005 AMC 8 (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 1 point for each correct answer. There is no penalty for wrong answers.
  3. No aids are permitted other than plain scratch paper, writing utensils, ruler, and erasers. In particular, graph paper, compass, protractor, calculators, computers, smartwatches, and smartphones are not permitted. Rules
  4. Figures are not necessarily drawn to scale.
  5. You will have 40 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

Connie multiplies a number by 2 and gets 60 as her answer. However, she should have divided the number by 2 to get the correct answer. What is the correct answer?

$\textbf{(A)}\ 7.5\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 240$

Solution

Problem 2

Karl bought five folders from Pay-A-Lot at a cost of $\textdollar 2.50$ each. Pay-A-Lot had a 20%-off sale the following day. How much could Karl have saved on the purchase by waiting a day?

$\textbf{(A)}\ \textdollar 1.00 \qquad\textbf{(B)}\ \textdollar 2.00 \qquad\textbf{(C)}\ \textdollar 2.50\qquad\textbf{(D)}\ \textdollar 2.75 \qquad\textbf{(E)}\ \textdollar 5.00$

Solution

Problem 3

What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $\overline{BD}$ of square $ABCD$? [asy]defaultpen(linewidth(1)); for ( int x = 0; x < 5; ++x ) { draw((0,x)--(4,x)); draw((x,0)--(x,4)); } fill((1,0)--(2,0)--(2,1)--(1,1)--cycle); fill((0,3)--(1,3)--(1,4)--(0,4)--cycle); fill((2,3)--(4,3)--(4,4)--(2,4)--cycle); fill((3,1)--(4,1)--(4,2)--(3,2)--cycle); label("$A$", (0, 4), NW); label("$B$", (4, 4), NE); label("$C$", (4, 0), SE); label("$D$", (0, 0), SW);[/asy]

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution

Problem 4

A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.1 cm, 8.2 cm and 9.7 cm. What is the area of the square in square centimeters?

$\textbf{(A)}\ 24\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 64$

Solution

Problem 5

Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 15$

Solution

Problem 6

Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 10$

Solution

Problem 7

Bill walks $\tfrac12$ mile south, then $\tfrac34$ mile east, and finally $\tfrac12$ mile south. How many miles is he, in a direct line, from his starting point?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 1\tfrac14\qquad\textbf{(C)}\ 1\tfrac12\qquad\textbf{(D)}\ 1\tfrac34\qquad\textbf{(E)}\ 2$

Solution

Problem 8

Suppose m and n are positive odd integers. Which of the following must also be an odd integer?

$\textbf{(A)}\ m+3n\qquad\textbf{(B)}\ 3m-n\qquad\textbf{(C)}\ 3m^2 + 3n^2\qquad\textbf{(D)}\ (nm + 3)^2\qquad\textbf{(E)}\ 3mn$

Solution

Problem 9

In quadrilateral $ABCD$, sides $\overline{AB}$ and $\overline{BC}$ both have length 10, sides $\overline{CD}$ and $\overline{DA}$ both have length 17, and the measure of angle $ADC$ is $60^\circ$. What is the length of diagonal $\overline{AC}$? [asy]draw((0,0)--(17,0)); draw(rotate(301, (17,0))*(0,0)--(17,0)); picture p; draw(p, (0,0)--(0,10)); draw(p, rotate(115, (0,10))*(0,0)--(0,10)); add(rotate(3)*p); draw((0,0)--(8.25,14.5), linetype("8 8")); label("$A$", (8.25, 14.5), N); label("$B$", (-0.25, 10), W); label("$C$", (0,0), SW); label("$D$", (17, 0), E);[/asy]

$\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 15.5\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 18.5$

Solution

Problem 10

Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school?

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 7.3\qquad\textbf{(C)}\ 7.7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 8.3$

Solution

Problem 11

The sales tax rate in Bergville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack brings up $90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill brings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total minus Jill's total?

$\textbf{(A)}\ - \textdollar1.06\qquad\textbf{(B)}\ - \textdollar 0.53\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \textdollar 0.53\qquad\textbf{(E)}\ \textdollar 1.06$

Solution

Problem 12

Big Al the ape ate 100 delicious yellow bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many delicious bananas did Big Al eat on May 5?

$\textbf{(A)}\ 20\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34$

Solution

Problem 13

The area of polygon $ABCDEF$ is 52 with $AB=8$, $BC=9$ and $FA=5$. What is $DE+EF$? [asy]pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4); draw(a--b--c--d--e--f--cycle); draw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a); draw(shift(-.25,0)*b--shift(-.25,-.25)*b--shift(0,-.25)*b); draw(shift(-.25,0)*c--shift(-.25,.25)*c--shift(0,.25)*c); draw(shift(.25,0)*d--shift(.25,.25)*d--shift(0,.25)*d); draw(shift(.25,0)*f--shift(.25,.25)*f--shift(0,.25)*f); label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, SE); label("$D$", d, SW); label("$E$", e, SW); label("$F$", f, SW); label("5", (0,6.5), W); label("8", (4,9), N); label("9", (8, 4.5), E);[/asy]

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$

Solution

Problem 14

The Little Twelve Basketball League has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many games are scheduled?

$\textbf{(A)}\ 80\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108\qquad\textbf{(E)}\ 192$

Solution

Problem 15

How many different isosceles triangles have integer side lengths and perimeter 23?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 11$

Solution

Problem 16

A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove from the drawer to be certain there will be 5 socks of the same color?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15$

Solution

Problem 17

The results of a cross-country team's training run are graphed below. Which student has the greatest average speed? [asy] for ( int i = 1; i <= 7; ++i ) { draw((i,0)--(i,6)); } for ( int i = 1; i <= 5; ++i ) { draw((0,i)--(8,i)); } draw((-0.5,0)--(8,0), linewidth(1)); draw((0,-0.5)--(0,6), linewidth(1)); label("$O$", (0,0), SW); label(scale(.85)*rotate(90)*"distance", (0, 3), W); label(scale(.85)*"time", (4, 0), S); dot((1.25, 4.5)); label(scale(.85)*"Evelyn", (1.25, 4.8), N); dot((2.5, 2.2)); label(scale(.85)*"Briana", (2.5, 2.2), S); dot((4.25,5.2)); label(scale(.85)*"Carla", (4.25, 5.2), SE); dot((5.6, 2.8)); label(scale(.85)*"Debra", (5.6, 2.8), N); dot((6.8, 1.4)); label(scale(.85)*"Angela", (6.8, 1.4), E); [/asy]

$\textbf{(A)}\ \text{Angela}\qquad\textbf{(B)}\ \text{Briana}\qquad\textbf{(C)}\ \text{Carla}\qquad\textbf{(D)}\ \text{Debra}\qquad\textbf{(E)}\ \text{Evelyn}$

Solution

Problem 18

How many three-digit numbers are divisible by 13?

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 77$

Solution

Problem 19

What is the perimeter of trapezoid $ABCD$?

[asy]size(3inch, 1.5inch); pair a=(0,0), b=(18,24), c=(68,24), d=(75,0), f=(68,0), e=(18,0); draw(a--b--c--d--cycle); draw(b--e); draw(shift(0,2)*e--shift(2,2)*e--shift(2,0)*e); label("30", (9,12), W); label("50", (43,24), N); label("25", (71.5, 12), E); label("24", (18, 12), E); label("$A$", a, SW); label("$B$", b, N); label("$C$", c, N); label("$D$", d, SE); label("$E$", e, S);[/asy]

$\textbf{(A)}\ 180\qquad\textbf{(B)}\ 188\qquad\textbf{(C)}\ 196\qquad\textbf{(D)}\ 200\qquad\textbf{(E)}\ 204$

Solution

Problem 20

Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 24$

Solution

Problem 21

How many distinct triangles can be drawn using three of the dots below as vertices?

[asy]dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));[/asy]

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$

Solution

Problem 22

A company sells detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs 50% more than the small size and contains 20% less detergent than the large size. The large size contains twice as much detergent as the small size and costs 30% more than the medium size. Rank the three sizes from best to worst buy.

$\textbf{(A)}\ \text{SML}\qquad\textbf{(B)}\ \text{LMS}\qquad\textbf{(C)}\ \text{MSL}\qquad\textbf{(D)}\ \text{LSM}\qquad\textbf{(E)}\ \text{MLS}$

Solution

Problem 23

Isosceles right triangle $ABC$ encloses a semicircle of area $2\pi$. The circle has its center $O$ on hypotenuse $\overline{AB}$ and is tangent to sides $\overline{AC}$ and $\overline{BC}$. What is the area of triangle $ABC$?

[asy]pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2); draw(circle(o, 2)); clip(a--b--c--cycle); draw(a--b--c--cycle); dot(o); label("$C$", c, NW); label("$A$", a, NE); label("$B$", b, SW);[/asy]

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 3\pi\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 4\pi$

Solution

Problem 24

A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"?

$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$

Solution

Problem 25

A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

[asy]pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2); draw(a--d--b--c--cycle); draw(circle(o, 2.25));[/asy] $\textbf{(A)}\ \frac{2}{\sqrt{\pi}} \qquad \textbf{(B)}\ \frac{1+\sqrt{2}}{2} \qquad \textbf{(C)}\ \frac{3}{2} \qquad \textbf{(D)}\ \sqrt{3} \qquad \textbf{(E)}\ \sqrt{\pi}$

Solution

See Also

2005 AMC 8 (Problems • Answer Key • Resources)
Preceded by
2004 AMC 8 		Followed by
2006 AMC 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions


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