Found problems: 3632
1960 AMC 12/AHSME, 3
Applied to a bill for $\$10,000$ the difference between a discount of $40\%$ and two successive discounts of $36\%$ and $4\%$, expressed in dollars, is:
$ \textbf{(A) }0\qquad\textbf{(B) }144\qquad\textbf{(C) }256\qquad\textbf{(D) }400\qquad\textbf{(E) }416 $
2017 AMC 12/AHSME, 11
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
$\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 117\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 163$
1970 AMC 12/AHSME, 29
It is now between $10:00$ and $11:00$ o'clock, and six minutes form now, the minute hand of the watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now?
$\textbf{(A) }10:05\dfrac{5}{11}\qquad\textbf{(B) }10:07\dfrac{1}{2}\qquad\textbf{(C) }10:10\qquad\textbf{(D) }10:15\qquad$
$\textbf{(E) }10:17\dfrac{1}{2}$
2007 AMC 10, 14
A triangle with side lengths in the ratio $ 3: 4: 5$ is inscribed in a circle of radius $ 3$. What is the area of the triangle?
$ \textbf{(A)}\ 8.64 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 5\pi \qquad \textbf{(D)}\ 17.28 \qquad \textbf{(E)}\ 18$
2014 AMC 8, 20
Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?
[asy]
draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0));
draw(Circle((0,0),1));
draw(Circle((0,3),2));
draw(Circle((5,3),3));
label("A",(0.2,0),W);
label("B",(0.2,2.8),NW);
label("C",(4.8,2.8),NE);
label("D",(5,0),SE);
label("5",(2.5,0),N);
label("3",(5,1.5),E);
[/asy]
$\textbf{(A) }3.5\qquad\textbf{(B) }4.0\qquad\textbf{(C) }4.5\qquad\textbf{(D) }5.0\qquad \textbf{(E) }5.5$
2023 AMC 10, 2
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew that he had to pay a 7.5% sales tax on the discounted price. He had 43 dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
A)$46$ B)$50$ C)$48$ D)$47$ E)$49$
2009 AIME Problems, 6
Let $ m$ be the number of five-element subsets that can be chosen from the set of the first $ 14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $ m$ is divided by $ 1000$.
2022 AMC 12/AHSME, 7
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy]
size(5.5cm);
draw((0,0)--(0,2)--(2,2)--(2,0)--cycle);
draw((2,0)--(8,0)--(8,2)--(2,2)--cycle);
draw((8,0)--(12,0)--(12,2)--(8,2)--cycle);
draw((0,2)--(6,2)--(6,4)--(0,4)--cycle);
draw((6,2)--(12,2)--(12,4)--(6,4)--cycle);
[/asy]
$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$
2016 AMC 10, 10
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?
$\textbf{(A)}\ 14.0\qquad\textbf{(B)}\ 16.0\qquad\textbf{(C)}\ 20.0\qquad\textbf{(D)}\ 33.3\qquad\textbf{(E)}\ 55.6$
2017 AMC 10, 22
Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle?
$ \textbf{(A) }\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) }\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}$
2025 AIME, 2
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. The area of quadrilateral $DEGF$ is $288$. Find the area of heptagon $AFNBCEM$, as shown in the figure below.
[asy]
unitsize(14);
pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G;
filldraw(A--F--N--B--C--E--M--cycle, lightgray);
draw(A--B--C--cycle);
draw(D--M);
draw(N--G);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(M);
dot(N);
label("$A$", A, dir(90));
label("$B$", B, dir(225));
label("$C$", C, dir(315));
label("$D$", D, dir(135));
label("$E$", E, dir(135));
label("$F$", F, dir(45));
label("$G$", G, dir(45));
label("$M$", M, dir(45));
label("$N$", N, dir(135));
[/asy]
2008 AMC 10, 3
For the positive integer $n$, let $\left< n \right>$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\left<4\right> = 1+2=3$ and $\left<12\right>=1+2+3+4+6=16$ What is $\left< \left< \left< 6 \right>\right>\right>$?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 36$
2009 AMC 10, 7
A carton contains milk that is $ 2\%$ fat, and amount that is $ 40\%$ less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
$ \textbf{(A)}\ \frac{12}{5} \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ \frac{10}{3} \qquad
\textbf{(D)}\ 38 \qquad
\textbf{(E)}\ 42$
2009 AMC 12/AHSME, 12
How many positive integers less than $ 1000$ are $ 6$ times the sum of their digits?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 12$
2007 Pre-Preparation Course Examination, 16
Prove that $2^{2^{n}}+2^{2^{{n-1}}}+1$ has at least $n$ distinct prime divisors.
2019 AMC 12/AHSME, 16
The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
$\textbf{(A) }1/21\qquad\textbf{(B) }1/14\qquad\textbf{(C) }5/63\qquad\textbf{(D) }2/21\qquad\textbf{(E) } 1/7$
2006 AMC 10, 17
Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process, the contents of the two bags are the same?
$ \textbf{(A) } \frac 1{10} \qquad \textbf{(B) } \frac 16 \qquad \textbf{(C) } \frac 15 \qquad \textbf{(D) } \frac 13 \qquad \textbf{(E) } \frac 12$
1991 AMC 12/AHSME, 14
If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be
$ \textbf{(A)}\ 200\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 202\qquad\textbf{(D)}\ 203\qquad\textbf{(E)}\ 204 $
2017 AIME Problems, 6
Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer.
2023 AIME, 10
There exists a unique positive integer $a$ for which the sum \[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\] is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$.
(Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)
2015 AMC 10, 10
What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$?
$\textbf{(A) } \text{It is a negative number ending with a 1.}$
$\textbf{(B) } \text{It is a positive number ending with a 1.}$
$\textbf{(C) } \text{It is a negative number ending with a 5.}$
$\textbf{(D) } \text{It is a positive number ending with a 5.}$
$\textbf{(E) } \text{It is a negative number ending with a 0.}$
2015 AIME Problems, 12
Consider all 1000-element subsets of the set $\{1,2,3,\dots,2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2017 AMC 10, 2
Sofia ran 5 laps around the 400-meter track at her school. For each lap, she ran the first 100 meters at an average speed of 4 meters per second and the remaining 300 meters at an average speed of 5 meters per second. How much time did Sofia take running the 5 laps?
$\textbf{(A) } \text{5 minutes and 35 seconds} $
$\textbf{(B) } \text{6 minutes and 40 seconds} $
$\textbf{(C) } \text{7 minutes and 5 seconds} $
$\textbf{(D) } \text{7 minutes and 25 seconds} $
$\textbf{(E) } \text{8 minutes and 10 seconds} $
2004 AIME Problems, 13
The polynomial \[P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}\] has 34 complex roots of the form $z_k=r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34$, with $0<a_1\le a_2\le a_3\le\cdots\le a_{34}<1$ and $r_k>0$. Given that $a_1+a_2+a_3+a_4+a_5=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
2014 AMC 10, 17
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?
$ \textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29 $