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Difference between revisions of "2005 AMC 10A Problems"

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[[2005 AMC 10A Problems/Problem 1|#1]]
+
{{AMC10 Problems|year=2005|ab=A}}
 +
== Problem 1 ==
 +
While eating out, Mike and Joe each tipped their server <math>\$2</math>. Mike tipped <math>10\%</math> of his bill and Joe tipped <math>20\%</math> of his bill. What was the difference, in dollars, between their bills?
  
−
While eating out, Mike and Joe each tipped their server <math>\$2</math>. Mike tipped <math>10%</math> of his bll and Joe tipped <math>20%</math> of his bill. What was the difference, in dollars between their bills?
+
<math>
 +
\textbf{(A) } 2\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 10\qquad \textbf{(E) } 20
 +
</math>
  
−
<math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 20 </math>
+
[[2005 AMC 10A Problems/Problem 1|Solution]]
  
−
----
+
== Problem 2 ==
 +
For each pair of real numbers <math>a \neq b</math>, define the operation <math>\star</math> as
  
−
[[2005 AMC 10A Problems/Problem 2|#2]]
+
<cmath>(a \star b) = \frac{a+b}{a-b}.</cmath>
  
−
For each pair of real numbers <math>a</math><math>\neq</math><math>b</math>, define the [[operation]] <math>\star</math> as
+
What is the value of <math>\left(\left(1 \star 2\right) \star 3\right)</math>?
  
−
<math> (a \star b) = \frac{a+b}{a-b} </math>.
+
<math>
 +
\textbf{(A) } -\frac{2}{3}\qquad \textbf{(B) } -\frac{1}{5}\qquad \textbf{(C) } 0\qquad \textbf{(D) } \frac{1}{2}\qquad \textbf{(E) } \text{This value is not defined.}
 +
</math>
  
−
What is the value of <math> ((1 \star 2) \star 3)</math>?
+
[[2005 AMC 10A Problems/Problem 2|Solution]]
  
−
<math> \mathrm{(A) \ } -\frac{2}{3}\qquad \mathrm{(B) \ } -\frac{1}{5}\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \textrm{This\, value\, is\, not\, defined.} </math>
+
== Problem 3 ==
 +
The equations <math>2x + 7 = 3</math> and <math>bx - 10 = -2</math> have the same solution <math>x</math>. What is the value of <math>b</math>?
  
−
----
+
<math>
 +
\textbf{(A) } -8\qquad \textbf{(B) } -4\qquad \textbf{(C) } -2\qquad \textbf{(D) } 4\qquad \textbf{(E) } 8
 +
</math>
  
−
[[2005 AMC 10A Problems/Problem 3|#3]]
+
[[2005 AMC 10A Problems/Problem 3|Solution]]
  
 +
== Problem 4 ==
 +
A rectangle with a diagonal of length <math>x</math> is twice as long as it is wide. What is the area of the rectangle?
  
−
The equations <math> 2x + 7 = 3 </math> and <math> bx - 10 = -2 </math> have the same solution <math>x</math>. What is the value of <math>b</math>?
+
<math>
 +
\textbf{(A) } \frac{1}{4}x^2\qquad \textbf{(B) } \frac{2}{5}x^2\qquad \textbf{(C) } \frac{1}{2}x^2\qquad \textbf{(D) } x^2\qquad \textbf{(E) } \frac{3}{2}x^2
 +
</math>
  
−
<math> \mathrm{(A) \ } -8\qquad \mathrm{(B) \ } -4\qquad \mathrm{(C) \ } -2\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 8 </math>
+
[[2005 AMC 10A Problems/Problem 4|Solution]]
  
−
----
+
== Problem 5 ==
 +
A store normally sells windows at <math>\$100</math> each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
  
−
[[2005 AMC 10A Problems/Problem 4|#4]]
+
<math>
 +
\textbf{(A) } 100\qquad \textbf{(B) } 200\qquad \textbf{(C) } 300\qquad \textbf{(D) } 400\qquad \textbf{(E) } 500
 +
</math>
  
 +
[[2005 AMC 10A Problems/Problem 5|Solution]]
  
−
A rectangle with a [[diagonal]] of length <math>x</math> is twice as long as it is wide. What is the area of the rectangle?  
+
== Problem 6 ==
 +
The average (mean) of <math>20</math> numbers is <math>30</math>, and the average of <math>30</math> other numbers is <math>20</math>. What is the average of all <math>50</math> numbers?
  
−
<math> \mathrm{(A) \ } \frac{1}{4}x^2\qquad \mathrm{(B) \ } \frac{2}{5}x^2\qquad \mathrm{(C) \ } \frac{1}{2}x^2\qquad \mathrm{(D) \ } x^2\qquad \mathrm{(E) \ } \frac{3}{2}x^2 </math>
+
<math>
 +
\textbf{(A) } 23\qquad \textbf{(B) } 24\qquad \textbf{(C) } 25\qquad \textbf{(D) } 26\qquad \textbf{(E) } 27
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 6|Solution]]
 +
 
 +
== Problem 7 ==
 +
Josh and Mike live <math>13</math> miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?
 +
 
 +
<math>
 +
\textbf{(A) } 4\qquad \textbf{(B) } 5\qquad \textbf{(C) } 6\qquad \textbf{(D) } 7\qquad \textbf{(E) } 8
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 7|Solution]]
 +
 
 +
== Problem 8 ==
 +
In the figure, the length of side <math>AB</math> of square <math>ABCD</math> is <math>\sqrt{50}</math>, <math>E</math> is between <math>B</math> and <math>H</math>, and <math>BE = 1</math>. What is the area of the inner square <math>EFGH</math>?
 +
 
 +
<asy>
 +
unitsize(4cm);
 +
defaultpen(linewidth(.8pt)+fontsize(10pt));
 +
 
 +
pair D=(0,0), C=(1,0), B=(1,1), A=(0,1);
 +
pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0];
 +
pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H);
 +
 
 +
draw(A--B--C--D--cycle);
 +
draw(D--F);
 +
draw(C--E);
 +
draw(B--H);
 +
draw(A--G);
 +
 
 +
label("$A$",A,NW);
 +
label("$B$",B,NE);
 +
label("$C$",C,SE);
 +
label("$D$",D,SW);
 +
label("$E$",E,NNW);
 +
label("$F$",F,ENE);
 +
label("$G$",G,SSE);
 +
label("$H$",H,WSW);
 +
</asy>
 +
 
 +
<math>
 +
\textbf{(A) } 25\qquad\textbf{(B) } 32\qquad\textbf{(C) } 36\qquad\textbf{(D) } 40\qquad\textbf{(E) } 42
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 8|Solution]]
 +
 
 +
== Problem 9 ==
 +
Three tiles are marked <math>X</math> and two other tiles are marked <math>O</math>. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads <math>XOXOX</math>?
 +
 
 +
<math>
 +
\textbf{(A) } \frac{1}{12}\qquad \textbf{(B) } \frac{1}{10}\qquad \textbf{(C) } \frac{1}{6}\qquad \textbf{(D) } \frac{1}{4}\qquad \textbf{(E) } \frac{1}{3}
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 9|Solution]]
 +
 
 +
== Problem 10 ==
 +
There are two values of <math>a</math> for which the equation <math>4x^2 + ax + 8x + 9 = 0</math> has only one solution for <math>x</math>. What is the sum of those values of <math>a</math>?
 +
 
 +
<math>
 +
\textbf{(A) } -16\qquad \textbf{(B) } -8\qquad \textbf{(C) } 0\qquad \textbf{(D) } 8\qquad \textbf{(E) } 20
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 10|Solution]]
 +
 
 +
== Problem 11 ==
 +
A wooden cube <math>n</math> units on a side is painted red on all six faces and then cut into <math>n^3</math> unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is <math>n</math>?
 +
 
 +
<math>
 +
\textbf{(A) } 3\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 6\qquad \textbf{(E) } 7
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 11|Solution]]
 +
 
 +
== Problem 12 ==
 +
The figure shown is called a ''trefoil'' and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length <math>2</math>?
 +
 
 +
<asy>
 +
unitsize(1.5cm);
 +
defaultpen(linewidth(.8pt)+fontsize(12pt));
 +
 
 +
pair O=(0,0), A=dir(0), B=dir(60), C=dir(120), D=dir(180);
 +
pair E=B+C;
 +
 
 +
draw(D--E--B--O--C--B--A,linetype("4 4"));
 +
draw(Arc(O,1,0,60),linewidth(1.2pt));
 +
draw(Arc(O,1,120,180),linewidth(1.2pt));
 +
draw(Arc(C,1,0,60),linewidth(1.2pt));
 +
draw(Arc(B,1,120,180),linewidth(1.2pt));
 +
draw(A--D,linewidth(1.2pt));
 +
draw(O--dir(40),EndArrow(HookHead,4));
 +
draw(O--dir(140),EndArrow(HookHead,4));
 +
draw(C--C+dir(40),EndArrow(HookHead,4));
 +
draw(B--B+dir(140),EndArrow(HookHead,4));
 +
 
 +
label("2",O,S);
 +
draw((0.1,-0.12)--(1,-0.12),EndArrow(HookHead,4),EndBar);
 +
draw((-0.1,-0.12)--(-1,-0.12),EndArrow(HookHead,4),EndBar);
 +
</asy>
 +
 
 +
<math>
 +
\textbf{(A) } \frac{1}{3}\pi+\frac{\sqrt{3}}{2}\qquad \textbf{(B) } \frac{2}{3}\pi\qquad \textbf{(C) } \frac{2}{3}\pi+\frac{\sqrt{3}}{4}\qquad \textbf{(D) } \frac{2}{3}\pi+\frac{\sqrt{3}}{3}\qquad \textbf{(E) } \frac{2}{3}\pi+\frac{\sqrt{3}}{2}
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 12|Solution]]
 +
 
 +
== Problem 13 ==
 +
How many positive integers <math>n</math> satisfy the following condition:
 +
 
 +
<cmath>\left(130n\right)^{50} > n^{100} > 2^{200} \ \text{?}</cmath>
 +
 
 +
<math>
 +
\textbf{(A) } 0\qquad \textbf{(B) } 7\qquad \textbf{(C) } 12\qquad \textbf{(D) } 65\qquad \textbf{(E) } 125
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 13|Solution]]
 +
 
 +
== Problem 14 ==
 +
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
 +
 
 +
<math>
 +
\textbf{(A) } 41\qquad \textbf{(B) } 42\qquad \textbf{(C) } 43\qquad \textbf{(D) } 44\qquad \textbf{(E) } 45
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 14|Solution]]
 +
 
 +
== Problem 15 ==
 +
How many positive cubes divide <math>3! \cdot 5! \cdot 7!</math>?
 +
 
 +
<math>
 +
\textbf{(A) } 2\qquad \textbf{(B) } 3\qquad \textbf{(C) } 4\qquad \textbf{(D) } 5\qquad \textbf{(E) } 6
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 15|Solution]]
 +
 
 +
== Problem 16 ==
 +
The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is <math>6</math>. How many two-digit numbers have this property?
 +
 
 +
<math>
 +
\textbf{(A) } 5\qquad \textbf{(B) } 7\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 19
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 16|Solution]]
 +
 
 +
== Problem 17 ==
 +
In the five-sided star shown, the letters <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math> are replaced by the numbers <math>3</math>, <math>5</math>, <math>6</math>, <math>7</math>, and <math>9</math>, although not necessarily in this order. The sums of the numbers at the ends of the line segments <math>\overline{AB}</math>, <math>\overline{BC}</math>, <math>\overline{CD}</math>, <math>\overline{DE}</math>, and <math>\overline{EA}</math> form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?
 +
 
 +
<asy>
 +
size(150);
 +
defaultpen(linewidth(0.8));
 +
string[] strng = {'A','D','B','E','C'};
 +
pair A=dir(90),B=dir(306),C=dir(162),D=dir(18),E=dir(234);
 +
draw(A--B--C--D--E--cycle);
 +
for(int i=0;i<=4;i=i+1)
 +
{
 +
path circ=circle(dir(90-72*i),0.125);
 +
unfill(circ);
 +
draw(circ);
 +
label("$"+strng[i]+"$",dir(90-72*i));
 +
}
 +
</asy>
 +
 
 +
<math>
 +
\textbf{(A) } 9\qquad \textbf{(B) } 10\qquad \textbf{(C) } 11\qquad \textbf{(D) } 12\qquad \textbf{(E) } 13
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 17|Solution]]
 +
 
 +
== Problem 18 ==
 +
Team <math>A</math> and team <math>B</math> play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team <math>B</math> wins the second game and team <math>A</math> wins the series, what is the probability that team <math>B</math> wins the first game?
 +
 
 +
<math>
 +
\textbf{(A) } \frac{1}{5}\qquad \textbf{(B) } \frac{1}{4}\qquad \textbf{(C) } \frac{1}{3}\qquad \textbf{(D) } \frac{1}{2}\qquad \textbf{(E) } \frac{2}{3}
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 18|Solution]]
 +
 
 +
== Problem 19 ==
 +
Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated <math>45^{\circ}</math>, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point <math>B</math> from the line on which the bases of the original squares were placed?
 +
 
 +
<asy>
 +
unitsize(1inch);
 +
defaultpen(linewidth(.8pt)+fontsize(8pt));
 +
draw((0,0)--((1/3) + 3*(1/2),0));
 +
fill(((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6) + 1,(1/2))--((1/6) + 1,0)--cycle, rgb(.7,.7,.7));
 +
draw(((1/6),0)--((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6),(1/2))--cycle);
 +
draw(((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6) + 1,(1/2))--((1/6) + 1,0)--cycle);
 +
draw(((1/6) + 1,0)--((1/6) + 1,(1/2))--((1/6) + (3/2),(1/2))--((1/6) + (3/2),0)--cycle);
 +
draw((2,0)--(2 + (1/3) + (3/2),0));
 +
draw(((2/3) + (3/2),0)--((2/3) + 2,0)--((2/3) + 2,(1/2))--((2/3) + (3/2),(1/2))--cycle);
 +
draw(((2/3) + (5/2),0)--((2/3) + (5/2),(1/2))--((2/3) + 3,(1/2))--((2/3) + 3,0)--cycle);
 +
label("$B$",((1/6) + (1/2),(1/2)),NW);
 +
label("$B$",((2/3) + 2 + (1/4),(29/30)),NNE);
 +
draw(((1/6) + (1/2),(1/2)+0.05)..(1,.8)..((2/3) + 2 + (1/4)-.05,(29/30)),EndArrow(HookHead,3));
 +
fill(((2/3) + 2 + (1/4),(1/4))--((2/3) + (5/2) + (1/10),(1/2) + (1/9))--((2/3) + 2 + (1/4),(29/30))--((2/3) + 2 - (1/10),(1/2) + (1/9))--cycle, rgb(.7,.7,.7));
 +
draw(((2/3) + 2 + (1/4),(1/4))--((2/3) + (5/2) + (1/10),(1/2) + (1/9))--((2/3) + 2 + (1/4),(29/30))--((2/3) + 2 - (1/10),(1/2) + (1/9))--cycle);</asy>
 +
 
 +
<math>
 +
\textbf{(A) } 1\qquad\textbf{(B) } \sqrt{2}\qquad\textbf{(C) } \frac{3}{2}\qquad\textbf{(D) } \sqrt{2}+\frac{1}{2}\qquad\textbf{(E) } 2
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 19|Solution]]
 +
 
 +
== Problem 20 ==
 +
An equiangular octagon has four sides of length <math>1</math> and four sides of length <math>\sqrt{2}/2</math>, arranged so that no two consecutive sides have the same length.  What is the area of the octagon?
 +
 
 +
<math>
 +
\textbf{(A) } \frac{7}{2}\qquad \textbf{(B) } \frac{7\sqrt{2}}{2}\qquad \textbf{(C) } \frac{5+4\sqrt{2}}{2}\qquad \textbf{(D) } \frac{4+5\sqrt{2}}{2}\qquad \textbf{(E) } 7
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 20|Solution]]
 +
 
 +
== Problem 21 ==
 +
For how many positive integers <math>n</math> does <math>1+2+\dotsb+n</math> evenly divide <math>6n</math>?
 +
 
 +
<math>
 +
\textbf{(A) } 3\qquad \textbf{(B) } 5\qquad \textbf{(C) } 7\qquad \textbf{(D) } 9\qquad \textbf{(E) } 11
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 21|Solution]]
 +
 
 +
== Problem 22 ==
 +
Let <math>S</math> be the set of the <math>2005</math> smallest positive multiples of <math>4</math>, and let <math>T</math> be the set of the <math>2005</math> smallest positive multiples of <math>6</math>. How many elements are common to <math>S</math> and <math>T</math>?
 +
 
 +
<math>
 +
\textbf{(A) } 166\qquad \textbf{(B) } 333\qquad \textbf{(C) } 500\qquad \textbf{(D) } 668\qquad \textbf{(E) } 1001
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 22|Solution]]
 +
 
 +
== Problem 23 ==
 +
Let <math>\overline{AB}</math> be a diameter of a circle and <math>C</math> be a point on <math>\overline{AB}</math> with <math>2 \cdot AC = BC</math>. Let <math>D</math> and <math>E</math> be points on the circle such that <math>\overline{DC} \perp \overline{AB}</math> and <math>\overline{DE}</math> is a second diameter. What is the ratio of the area of <math>\triangle DCE</math> to the area of <math>\triangle ABD</math>?
 +
 
 +
<asy>
 +
unitsize(2.5cm);
 +
defaultpen(fontsize(10pt)+linewidth(.8pt));
 +
dotfactor=3;
 +
pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);
 +
pair D=dir(aCos(C.x)), E=(-D.x,-D.y);
 +
draw(A--B--D--cycle);
 +
draw(D--E--C);
 +
draw(unitcircle,white);
 +
drawline(D,C);
 +
dot(O);
 +
clip(unitcircle);
 +
draw(unitcircle);
 +
label("$E$",E,SSE);
 +
label("$B$",B,E);
 +
label("$A$",A,W);
 +
label("$D$",D,NNW);
 +
label("$C$",C,SW);
 +
draw(rightanglemark(D,C,B,2));</asy>
 +
 
 +
<math>
 +
\textbf{(A) } \frac{1}{6} \qquad \textbf{(B) } \frac{1}{4} \qquad \textbf{(C) } \frac{1}{3} \qquad \textbf{(D) } \frac{1}{2} \qquad \textbf{(E) } \frac{2}{3}
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 23|Solution]]
 +
 
 +
== Problem 24 ==
 +
For each positive integer <math>m > 1</math>, let <math>P(m)</math> denote the greatest prime factor of <math>m</math>. For how many positive integers <math>n</math> is it true that both <math>P(n) = \sqrt{n}</math> and <math>P(n+48) = \sqrt{n+48}</math>?
 +
 
 +
<math>
 +
\textbf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 24|Solution]]
 +
 
 +
== Problem 25 ==
 +
In <math>\triangle ABC</math> we have <math>AB = 25</math>, <math>BC = 39</math>, and <math>AC = 42</math>. Points <math>D</math> and <math>E</math> are on <math>\overline{AB}</math> and <math>\overline{AC}</math> respectively, with <math>AD = 19</math> and <math>AE = 14</math>. What is the ratio of the area of triangle <math>ADE</math> to the area of the quadrilateral <math>BCED</math>?
 +
 
 +
<math>
 +
\textbf{(A) } \frac{266}{1521}\qquad \textbf{(B) } \frac{19}{75}\qquad \textbf{(C) } \frac{1}{3}\qquad \textbf{(D) } \frac{19}{56}\qquad \textbf{(E) } 1
 +
</math>
 +
 
 +
[[2005 AMC 10A Problems/Problem 25|Solution]]
 +
 
 +
== See also ==
 +
{{AMC10 box|year=2005|ab=A|before=[[2004 AMC 10B Problems]]|after=[[2005 AMC 10B Problems]]}}
 +
*[[AMC 10 Problems and Solutions]]
 +
* [[AMC Problems and Solutions]]
 +
{{MAA Notice}}

Latest revision as of 16:52, 1 July 2025

2005 AMC 10A (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 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 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

While eating out, Mike and Joe each tipped their server $$2$. Mike tipped $10\%$ of his bill and Joe tipped $20\%$ of his bill. What was the difference, in dollars, between their bills?

$\textbf{(A) } 2\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 10\qquad \textbf{(E) } 20$

Solution

Problem 2

For each pair of real numbers $a \neq b$, define the operation $\star$ as

\[(a \star b) = \frac{a+b}{a-b}.\]

What is the value of $\left(\left(1 \star 2\right) \star 3\right)$?

$\textbf{(A) } -\frac{2}{3}\qquad \textbf{(B) } -\frac{1}{5}\qquad \textbf{(C) } 0\qquad \textbf{(D) } \frac{1}{2}\qquad \textbf{(E) } \text{This value is not defined.}$

Solution

Problem 3

The equations $2x + 7 = 3$ and $bx - 10 = -2$ have the same solution $x$. What is the value of $b$?

$\textbf{(A) } -8\qquad \textbf{(B) } -4\qquad \textbf{(C) } -2\qquad \textbf{(D) } 4\qquad \textbf{(E) } 8$

Solution

Problem 4

A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?

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

Solution

Problem 5

A store normally sells windows at $$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?

$\textbf{(A) } 100\qquad \textbf{(B) } 200\qquad \textbf{(C) } 300\qquad \textbf{(D) } 400\qquad \textbf{(E) } 500$

Solution

Problem 6

The average (mean) of $20$ numbers is $30$, and the average of $30$ other numbers is $20$. What is the average of all $50$ numbers?

$\textbf{(A) } 23\qquad \textbf{(B) } 24\qquad \textbf{(C) } 25\qquad \textbf{(D) } 26\qquad \textbf{(E) } 27$

Solution

Problem 7

Josh and Mike live $13$ miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?

$\textbf{(A) } 4\qquad \textbf{(B) } 5\qquad \textbf{(C) } 6\qquad \textbf{(D) } 7\qquad \textbf{(E) } 8$

Solution

Problem 8

In the figure, the length of side $AB$ of square $ABCD$ is $\sqrt{50}$, $E$ is between $B$ and $H$, and $BE = 1$. What is the area of the inner square $EFGH$?

[asy] unitsize(4cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(1,0), B=(1,1), A=(0,1); pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0]; pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H); draw(A--B--C--D--cycle); draw(D--F); draw(C--E); draw(B--H); draw(A--G); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,NNW); label("$F$",F,ENE); label("$G$",G,SSE); label("$H$",H,WSW); [/asy]

$\textbf{(A) } 25\qquad\textbf{(B) } 32\qquad\textbf{(C) } 36\qquad\textbf{(D) } 40\qquad\textbf{(E) } 42$

Solution

Problem 9

Three tiles are marked $X$ and two other tiles are marked $O$. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads $XOXOX$?

$\textbf{(A) } \frac{1}{12}\qquad \textbf{(B) } \frac{1}{10}\qquad \textbf{(C) } \frac{1}{6}\qquad \textbf{(D) } \frac{1}{4}\qquad \textbf{(E) } \frac{1}{3}$

Solution

Problem 10

There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of those values of $a$?

$\textbf{(A) } -16\qquad \textbf{(B) } -8\qquad \textbf{(C) } 0\qquad \textbf{(D) } 8\qquad \textbf{(E) } 20$

Solution

Problem 11

A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$?

$\textbf{(A) } 3\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 6\qquad \textbf{(E) } 7$

Solution

Problem 12

The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length $2$?

[asy] unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(12pt)); pair O=(0,0), A=dir(0), B=dir(60), C=dir(120), D=dir(180); pair E=B+C; draw(D--E--B--O--C--B--A,linetype("4 4")); draw(Arc(O,1,0,60),linewidth(1.2pt)); draw(Arc(O,1,120,180),linewidth(1.2pt)); draw(Arc(C,1,0,60),linewidth(1.2pt)); draw(Arc(B,1,120,180),linewidth(1.2pt)); draw(A--D,linewidth(1.2pt)); draw(O--dir(40),EndArrow(HookHead,4)); draw(O--dir(140),EndArrow(HookHead,4)); draw(C--C+dir(40),EndArrow(HookHead,4)); draw(B--B+dir(140),EndArrow(HookHead,4)); label("2",O,S); draw((0.1,-0.12)--(1,-0.12),EndArrow(HookHead,4),EndBar); draw((-0.1,-0.12)--(-1,-0.12),EndArrow(HookHead,4),EndBar); [/asy]

$\textbf{(A) } \frac{1}{3}\pi+\frac{\sqrt{3}}{2}\qquad \textbf{(B) } \frac{2}{3}\pi\qquad \textbf{(C) } \frac{2}{3}\pi+\frac{\sqrt{3}}{4}\qquad \textbf{(D) } \frac{2}{3}\pi+\frac{\sqrt{3}}{3}\qquad \textbf{(E) } \frac{2}{3}\pi+\frac{\sqrt{3}}{2}$

Solution

Problem 13

How many positive integers $n$ satisfy the following condition:

\[\left(130n\right)^{50} > n^{100} > 2^{200} \ \text{?}\]

$\textbf{(A) } 0\qquad \textbf{(B) } 7\qquad \textbf{(C) } 12\qquad \textbf{(D) } 65\qquad \textbf{(E) } 125$

Solution

Problem 14

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

$\textbf{(A) } 41\qquad \textbf{(B) } 42\qquad \textbf{(C) } 43\qquad \textbf{(D) } 44\qquad \textbf{(E) } 45$

Solution

Problem 15

How many positive cubes divide $3! \cdot 5! \cdot 7!$?

$\textbf{(A) } 2\qquad \textbf{(B) } 3\qquad \textbf{(C) } 4\qquad \textbf{(D) } 5\qquad \textbf{(E) } 6$

Solution

Problem 16

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $6$. How many two-digit numbers have this property?

$\textbf{(A) } 5\qquad \textbf{(B) } 7\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 19$

Solution

Problem 17

In the five-sided star shown, the letters $A$, $B$, $C$, $D$, and $E$ are replaced by the numbers $3$, $5$, $6$, $7$, and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$, and $\overline{EA}$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?

[asy] size(150); defaultpen(linewidth(0.8)); string[] strng = {'A','D','B','E','C'}; pair A=dir(90),B=dir(306),C=dir(162),D=dir(18),E=dir(234); draw(A--B--C--D--E--cycle); for(int i=0;i<=4;i=i+1) { path circ=circle(dir(90-72*i),0.125); unfill(circ); draw(circ); label("$"+strng[i]+"$",dir(90-72*i)); } [/asy]

$\textbf{(A) } 9\qquad \textbf{(B) } 10\qquad \textbf{(C) } 11\qquad \textbf{(D) } 12\qquad \textbf{(E) } 13$

Solution

Problem 18

Team $A$ and team $B$ play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team $B$ wins the second game and team $A$ wins the series, what is the probability that team $B$ wins the first game?

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

Solution

Problem 19

Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated $45^{\circ}$, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original squares were placed?

[asy] unitsize(1inch); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((0,0)--((1/3) + 3*(1/2),0)); fill(((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6) + 1,(1/2))--((1/6) + 1,0)--cycle, rgb(.7,.7,.7)); draw(((1/6),0)--((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6),(1/2))--cycle); draw(((1/6) + (1/2),0)--((1/6) + (1/2),(1/2))--((1/6) + 1,(1/2))--((1/6) + 1,0)--cycle); draw(((1/6) + 1,0)--((1/6) + 1,(1/2))--((1/6) + (3/2),(1/2))--((1/6) + (3/2),0)--cycle); draw((2,0)--(2 + (1/3) + (3/2),0)); draw(((2/3) + (3/2),0)--((2/3) + 2,0)--((2/3) + 2,(1/2))--((2/3) + (3/2),(1/2))--cycle); draw(((2/3) + (5/2),0)--((2/3) + (5/2),(1/2))--((2/3) + 3,(1/2))--((2/3) + 3,0)--cycle); label("$B$",((1/6) + (1/2),(1/2)),NW); label("$B$",((2/3) + 2 + (1/4),(29/30)),NNE); draw(((1/6) + (1/2),(1/2)+0.05)..(1,.8)..((2/3) + 2 + (1/4)-.05,(29/30)),EndArrow(HookHead,3)); fill(((2/3) + 2 + (1/4),(1/4))--((2/3) + (5/2) + (1/10),(1/2) + (1/9))--((2/3) + 2 + (1/4),(29/30))--((2/3) + 2 - (1/10),(1/2) + (1/9))--cycle, rgb(.7,.7,.7)); draw(((2/3) + 2 + (1/4),(1/4))--((2/3) + (5/2) + (1/10),(1/2) + (1/9))--((2/3) + 2 + (1/4),(29/30))--((2/3) + 2 - (1/10),(1/2) + (1/9))--cycle);[/asy]

$\textbf{(A) } 1\qquad\textbf{(B) } \sqrt{2}\qquad\textbf{(C) } \frac{3}{2}\qquad\textbf{(D) } \sqrt{2}+\frac{1}{2}\qquad\textbf{(E) } 2$

Solution

Problem 20

An equiangular octagon has four sides of length $1$ and four sides of length $\sqrt{2}/2$, arranged so that no two consecutive sides have the same length. What is the area of the octagon?

$\textbf{(A) } \frac{7}{2}\qquad \textbf{(B) } \frac{7\sqrt{2}}{2}\qquad \textbf{(C) } \frac{5+4\sqrt{2}}{2}\qquad \textbf{(D) } \frac{4+5\sqrt{2}}{2}\qquad \textbf{(E) } 7$

Solution

Problem 21

For how many positive integers $n$ does $1+2+\dotsb+n$ evenly divide $6n$?

$\textbf{(A) } 3\qquad \textbf{(B) } 5\qquad \textbf{(C) } 7\qquad \textbf{(D) } 9\qquad \textbf{(E) } 11$

Solution

Problem 22

Let $S$ be the set of the $2005$ smallest positive multiples of $4$, and let $T$ be the set of the $2005$ smallest positive multiples of $6$. How many elements are common to $S$ and $T$?

$\textbf{(A) } 166\qquad \textbf{(B) } 333\qquad \textbf{(C) } 500\qquad \textbf{(D) } 668\qquad \textbf{(E) } 1001$

Solution

Problem 23

Let $\overline{AB}$ be a diameter of a circle and $C$ be a point on $\overline{AB}$ with $2 \cdot AC = BC$. Let $D$ and $E$ be points on the circle such that $\overline{DC} \perp \overline{AB}$ and $\overline{DE}$ is a second diameter. What is the ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$?

[asy] unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y); draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O); clip(unitcircle); draw(unitcircle); label("$E$",E,SSE); label("$B$",B,E); label("$A$",A,W); label("$D$",D,NNW); label("$C$",C,SW); draw(rightanglemark(D,C,B,2));[/asy]

$\textbf{(A) } \frac{1}{6} \qquad \textbf{(B) } \frac{1}{4} \qquad \textbf{(C) } \frac{1}{3} \qquad \textbf{(D) } \frac{1}{2} \qquad \textbf{(E) } \frac{2}{3}$

Solution

Problem 24

For each positive integer $m > 1$, let $P(m)$ denote the greatest prime factor of $m$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$?

$\textbf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Solution

Problem 25

In $\triangle ABC$ we have $AB = 25$, $BC = 39$, and $AC = 42$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{AC}$ respectively, with $AD = 19$ and $AE = 14$. What is the ratio of the area of triangle $ADE$ to the area of the quadrilateral $BCED$?

$\textbf{(A) } \frac{266}{1521}\qquad \textbf{(B) } \frac{19}{75}\qquad \textbf{(C) } \frac{1}{3}\qquad \textbf{(D) } \frac{19}{56}\qquad \textbf{(E) } 1$

Solution

See also

2005 AMC 10A (Problems • Answer Key • Resources)
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2004 AMC 10B Problems 		Followed by
2005 AMC 10B Problems
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All AMC 10 Problems and Solutions

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