Found problems: 3632
2008 AMC 10, 10
Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$?
$ \textbf{(A)}\ \sqrt{10} \qquad
\textbf{(B)}\ \frac{7}{2} \qquad
\textbf{(C)}\ \sqrt{14} \qquad
\textbf{(D)}\ \sqrt{15} \qquad
\textbf{(E)}\ 4$
1987 AMC 12/AHSME, 30
In the figure, $\triangle ABC$ has $\angle A =45^{\circ}$ and $\angle B =30^{\circ}$. A line $DE$, with $D$ on $AB$ and $\angle ADE =60^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. (Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC$.) The ratio $\frac{AD}{AB}$ is
[asy]
size((220));
draw((0,0)--(20,0)--(7,6)--cycle);
draw((6,6)--(10,-1));
label("A", (0,0), W);
label("B", (20,0), E);
label("C", (7,6), NE);
label("D", (9.5,-1), W);
label("E", (5.9, 6.1), SW);
label("$45^{\circ}$", (2.5,.5));
label("$60^{\circ}$", (7.8,.5));
label("$30^{\circ}$", (16.5,.5));
[/asy]
$ \textbf{(A)}\ \frac{1}{\sqrt{2}} \qquad\textbf{(B)}\ \frac{2}{2+\sqrt{2}} \qquad\textbf{(C)}\ \frac{1}{\sqrt{3}} \qquad\textbf{(D)}\ \frac{1}{\sqrt[3]{6}} \qquad\textbf{(E)}\ \frac{1}{\sqrt[4]{12}} $
1996 AMC 8, 10
When Walter drove up to the gasoline pump, he noticed that his gasoline tank was $\frac{1}{8}$ full. He purchased $7.5$ gallons of gasoline for $ \$10$. With this additional gasoline, his gasoline tank was then $\frac{5}{8}$ full. The number of gallons of gasoline his tank holds when it is full is
$\text{(A)}\ 8.75 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11.5 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 22.5$
1969 AMC 12/AHSME, 8
Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$, and $CA$ are, respectively, $x+75^\circ$, $2x+25^\circ$, $3x-22^\circ$. Then one interior angle of the triangle, in degrees, is:
$\textbf{(A) }57\tfrac12\qquad
\textbf{(B) }59\qquad
\textbf{(C) }60\qquad
\textbf{(D) }61\qquad
\textbf{(E) }122$
2002 Moldova National Olympiad, 4
At least two of the nonnegative real numbers $ a_1,a_2,...,a_n$ aer nonzero. Decide whether $ a$ or $ b$ is larger if
$ a\equal{}\sqrt[2002]{a_1^{2002}\plus{}a_2^{2002}\plus{}\ldots\plus{}a_n^{2002}}$
and
$ b\equal{}\sqrt[2003]{a_1^{2003}\plus{}a_2^{2003}\plus{}\ldots\plus{}a_n^{2003} }$
2013 AMC 8, 9
The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?
$\textbf{(A)}\ 9^\text{th} \qquad \textbf{(B)}\ 10^\text{th} \qquad \textbf{(C)}\ 11^\text{th} \qquad \textbf{(D)}\ 12^\text{th} \qquad \textbf{(E)}\ 13^\text{th}$
2020 AIME Problems, 1
In $\triangle ABC$ with $AB=AC$, point $D$ lies strictly between $A$ and $C$ on side $\overline{AC}$, and point $E$ lies strictly between $A$ and $B$ on side $\overline{AB}$ such that $AE=ED=DB=BC$. The degree measure of $\angle ABC$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2024 AMC 12/AHSME, 20
Suppose $A$, $B$, and $C$ are points in the plane with $AB=40$ and $AC=42$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p,q)$, and the maximum value $r$ of $f(x)$ occurs at $x=s$. What is $p+q+r+s$?
$
\textbf{(A) }909\qquad
\textbf{(B) }910\qquad
\textbf{(C) }911\qquad
\textbf{(D) }912\qquad
\textbf{(E) }913\qquad
$
2001 AIME Problems, 2
Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$.
2011 AIME Problems, 5
The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits $1$ through $9$ in such a way that the sum of the numbers on every three consecutive vertices is a multiple of $3$. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.
1970 AMC 12/AHSME, 30
In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice the measure of angle $B$, and the measures of segments $AB$ and $CD$ are $a$ and $b$ respectively. Then the measure of $AB$ is equal to
$\textbf{(A) }\dfrac{1}{2}a+2b\qquad\textbf{(B) }\dfrac{3}{2}b+\dfrac{3}{4}a\qquad\textbf{(C) }2a-b\qquad\textbf{(D) }4b-\dfrac{1}{2}a\qquad \textbf{(E) }a+b$
[asy]
size(175);
defaultpen(linewidth(0.8));
real r=50, a=4,b=2.5,c=6.25;
pair A=origin,B=c*dir(r),D=(a,0),C=shift(b*dir(r))*D;
draw(A--B--C--D--cycle);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,E);
label("$D$",D,S);
label("$a$",D/2,N);
label("$b$",(C+D)/2,NW);
//Credit to djmathman for the diagram[/asy]
2006 AMC 12/AHSME, 24
The expression
\[ (x \plus{} y \plus{} z)^{2006} \plus{} (x \minus{} y \minus{} z)^{2006}
\]is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
$ \textbf{(A) } 6018 \qquad \textbf{(B) } 671,676 \qquad \textbf{(C) } 1,007,514 \qquad \textbf{(D) } 1,008,016 \qquad \textbf{(E) } 2,015,028$
2016 AMC 12/AHSME, 16
In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$
2000 AMC 8, 11
The number $64$ has the property that it is divisible by its units digit. How many whole numbers between $10$ and $50$ have this property?
$\text{(A)}\ 15 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 20$
2015 AMC 10, 14
The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?
[asy]
size(170);
defaultpen(linewidth(0.9)+fontsize(13pt));
draw(unitcircle^^circle((0,1.5),0.5));
path arrow = origin--(-0.13,-0.35)--(-0.06,-0.35)--(-0.06,-0.7)--(0.06,-0.7)--(0.06,-0.35)--(0.13,-0.35)--cycle;
for(int i=1;i<=12;i=i+1)
{
draw(0.9*dir(90-30*i)--dir(90-30*i));
label("$"+(string) i+"$",0.78*dir(90-30*i));
}
dot(origin);
draw(shift((0,1.87))*arrow);
draw(arc(origin,1.5,68,30),EndArrow(size=12));[/asy]
$ \textbf{(A) }\text{2 o'clock} \qquad\textbf{(B) }\text{3 o'clock} \qquad\textbf{(C) }\text{4 o'clock} \qquad\textbf{(D) }\text{6 o'clock} \qquad\textbf{(E) }\text{8 o'clock} $
2012 AMC 8, 17
A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?
$\textbf{(A)}\hspace{.05in}3 \qquad \textbf{(B)}\hspace{.05in}4 \qquad \textbf{(C)}\hspace{.05in}5 \qquad \textbf{(D)}\hspace{.05in}6 \qquad \textbf{(E)}\hspace{.05in}7 $
2024 AMC 8 -, 13
Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }12$
2007 AMC 10, 5
A school store sells 7 pencils and 8 notebooks for $ \$4.15$. It also sells 5 pencils and 3 notebooks for $ \$1.77$. How much do 16 pencils and 10 notebooks cost?
$ \textbf{(A)}\ \$1.76 \qquad \textbf{(B)}\ \$5.84 \qquad \textbf{(C)}\ \$6.00 \qquad \textbf{(D)}\ \$6.16 \qquad \textbf{(E)}\ \$6.32$
2009 AMC 10, 6
Kiana has two older twin brothers. The product of their ages is $ 128$. What is the sum of their three ages?
$ \textbf{(A)}\ 10\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 18\qquad
\textbf{(E)}\ 24$
1998 AMC 12/AHSME, 21
In an $ h$-meter race, Sunny is exactly $ d$ meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts $ d$ meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finishes the second race?
$ \textbf{(A)}\ \frac {d}{h} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ \frac {d^2}{h} \qquad \textbf{(D)}\ \frac {h^2}{d} \qquad \textbf{(E)}\ \frac {d^2}{h \minus{} d}$
2015 AMC 10, 9
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata?
[asy]
import cse5;pathpen=black;pointpen=black;
size(1.5inch);
D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W));
path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle;
draw(P,linewidth(2));
fill(P,gray);
[/asy]
$\textbf{(A) } \dfrac{4\pi}{5}
\qquad\textbf{(B) } \dfrac{9\pi}{8}
\qquad\textbf{(C) } \dfrac{4\pi}{3}
\qquad\textbf{(D) } \dfrac{7\pi}{5}
\qquad\textbf{(E) } \dfrac{3\pi}{2}
$
2021 AIME Problems, 6
For any finite set $S$, let $|S|$ denote the number of elements in $S$. FInd the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy
$$|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$$
1986 AMC 12/AHSME, 14
Suppose hops, skips and jumps are specific units of length. If $b$ hops equals $c$ skips, $d$ jumps equals $e$ hops, and $f$jumps equals $g$ meters, then one meter equals how many skips?
$ \textbf{(A)}\ \frac{bdg}{cef}\qquad\textbf{(B)}\ \frac{cdf}{beg}\qquad\textbf{(C)}\ \frac{cdg}{bef}\qquad\textbf{(D)}\ \frac{cef}{bdg}\qquad\textbf{(E)}\ \frac{ceg}{bdf} $
2010 AMC 10, 25
Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that
\[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\]
\[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\]
What is the smallest possible value of $ a$?
$ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$
2008 AMC 12/AHSME, 10
Doug can paint a room in $ 5$ hours. Dave can paint the same room in $ 7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $ t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $ t$?
$ \textbf{(A)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\plus{}1)\equal{}1 \qquad
\textbf{(B)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\plus{}1\equal{}1 \qquad
\textbf{(C)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\equal{}1 \\
\textbf{(D)}\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\minus{}1)\equal{}1 \qquad
\textbf{(E)}\ (5\plus{}7)t\equal{}1$