Found problems: 3632
1961 AMC 12/AHSME, 21
Medians $AD$ and and $CE$ of triangle $ABC$ intersect in $M$. The midpoint of $AE$ is $N$. Let the area of triangle $MNE$ be $k$ times the area of triangle $ABC$. Then $k$ equals:
${{ \textbf{(A)}\ \frac{1}{6} \qquad\textbf{(B)}\ \frac{1}{8} \qquad\textbf{(C)}\ \frac{1}{9} \qquad\textbf{(D)}\ \frac{1}{12} }\qquad\textbf{(E)}\ \frac{1}{16} } $
2023 AMC 10, 12
How many three-digit positive integers $N$ satisfy the following properties?
- The number $N$ is divisible by $7$.
- The number formed by reversing the digits of $N$ is divisible by $5$.
$\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }15\qquad\textbf{(D) }16\qquad\textbf{(E) }17$
2011 Math Prize For Girls Problems, 9
Let $ABC$ be a triangle. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AD}$, and let $F$ be the midpoint of $\overline{BE}$. Let $G$ be the point where the lines $AB$ and $CF$ intersect. What is the value of $\frac{AG}{AB}$?
1987 AMC 12/AHSME, 27
A cube of cheese $C=\{(x, y, z)| 0 \le x, y, z \le 1\}$ is cut along the planes $x=y$, $y=z$ and $z=x$. How many pieces are there? (No cheese is moved until all three cuts are made.)
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 9 $
2020 AMC 10, 25
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product$$n = f_1\cdot f_2\cdots f_k,$$where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$?
$\textbf{(A) } 112 \qquad\textbf{(B) } 128 \qquad\textbf{(C) } 144 \qquad\textbf{(D) } 172 \qquad\textbf{(E) } 184$
2000 AIME Problems, 14
In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP=PQ=QB=BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find the greatest integer that does not exceed $1000r.$
2020 AMC 12/AHSME, 3
The ratio of $w$ to $x$ is $4 : 3$, the ratio of $y$ to $z$ is $3 : 2$, and the ratio of $z$ to $x$ is $1 : 6$. What is the ratio of $w$ to $y$?
$\textbf{(A) }4:3 \qquad \textbf{(B) }3:2 \qquad \textbf{(C) } 8:3 \qquad \textbf{(D) } 4:1 \qquad \textbf{(E) } 16:3 $
2020 AMC 10, 9
How many ordered pairs of integers $(x, y)$ satisfy the equation$$x^{2020}+y^2=2y?$$
$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}$
2021 AMC 10 Spring, 19
The area of the region bounded by the graph of $$x^2 + y^2 = 3|x-y| + 3|x+y|$$ is $m + n \pi,$ where $m$ and $n$ are integers. What is $m+n$?
$\textbf{(A)} 18\qquad\textbf{(B)} 27\qquad\textbf{(C)} 36\qquad\textbf{(D)} 45\qquad\textbf{(E)} 54$
2023 AMC 12/AHSME, 10
In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?
$\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$
1988 AMC 12/AHSME, 13
If $\sin\ x\ =\ 3\ \cos\ x$ then what is $\sin\ x\ \cos\ x$?
$ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{2}{9}\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ \frac{3}{10} $
2005 District Olympiad, 1
Prove that for all $a\in\{0,1,2,\ldots,9\}$ the following sum is divisible by 10:
\[ S_a = \overline{a}^{2005} + \overline{1a}^{2005} + \overline{2a}^{2005} + \cdots + \overline{9a}^{2005}. \]
2022 AMC 12/AHSME, 8
What is the graph of $y^4+1=x^4+2y^2$ in the coordinate plane?
$ \textbf{(A)}\ \textbf{Two intersecting parabolas} \qquad
\textbf{(B)}\ \textbf{Two nonintersecting parabolas} \qquad
\textbf{(C)}\ \textbf{Two intersecting circles} \qquad
\textbf{(D)}\ \textbf{A circle and a hyperbola} \qquad
\textbf{(E)}\ \textbf{A circle and two parabolas}$
2011 AMC 12/AHSME, 16
Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
$ \textbf{(A)}\ 2520 \qquad
\textbf{(B)}\ 2880 \qquad
\textbf{(C)}\ 3120 \qquad
\textbf{(D)}\ 3250 \qquad
\textbf{(E)}\ 3750
$
2006 AIME Problems, 15
Given that $x$, $y$, and $z$ are real numbers that satisfy:
\[ x=\sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}} \]
\[ y=\sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}} \]
\[ z=\sqrt{x^2-\frac{1}{36}}+\sqrt{y^2-\frac{1}{36}} \]
and that $x+y+z=\frac{m}{\sqrt{n}}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime, find $m+n$.
2015 AMC 10, 9
Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?
$\textbf{(A) }\text{The second height is 10\% less than the first.}$
$\textbf{(B) }\text{The first height is 10\% more than the second.}$
$\textbf{(C) }\text{The second height is 21\% less than the first.}$
$\textbf{(D) }\text{The first height is 21\% more than the second.}$
$\textbf{(E) }\text{The second height is 80\% of the first.}$
2013 AMC 8, 11
Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
2013 AMC 12/AHSME, 1
On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature, and the average of the high and low temperatures was $3^{\circ}$. In degrees, what was the low temperature in Lincoln that day?
$\textbf{(A) }-13\qquad\textbf{(B) }-8\qquad\textbf{(C) }-5\qquad\textbf{(D) }3\qquad\textbf{(E) }11$
2022 AMC 10, 10
Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?
$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad\textbf{(E) }13$
2021 AMC 12/AHSME Fall, 1
What is the value of $\frac{(2112-2021)^2}{169}$?
$\textbf{(A) }7\qquad\textbf{(B) }21\qquad\textbf{(C) }49\qquad\textbf{(D) }64\qquad\textbf{(E) }91$
2016 Switzerland Team Selection Test, Problem 6
Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.
2007 AIME Problems, 5
The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C=\frac{5}{9}(F-32)$. An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer. For how many integer Fahrenheit temperatures $T$ with $32 \leq T \leq 1000$ does the original temperature equal the final temperature?
2007 AIME Problems, 4
The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In three hours, 50 workers can produce 150 widgets and m whoosits. Find m.
2016 USAJMO, 2
Prove that there exists a positive integer $n < 10^6$ such that $5^n$ has six consecutive zeros in its decimal representation.
[i]Proposed by Evan Chen[/i]
2015 USAJMO, 6
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform [i]stone moves[/i], defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i<j$ and $k<l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively, or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
How many different non-equivalent ways can Steve pile the stones on the grid?