Found problems: 3632
2021 AIME Problems, 10
Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[img]https://imgur.com/1mfBNNL.png[/img]
2023 AMC 12/AHSME, 23
When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$?
$\textbf{(A)}~6\qquad\textbf{(B)}~8\qquad\textbf{(C)}~9\qquad\textbf{(D)}~10\qquad\textbf{(E)}~11$
2023 AIME, 9
Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$, such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$.
2017 AMC 12/AHSME, 7
Define a function on the positive integers recursively by $f(1) = 2$, $f(n) = f(n-1) + 1$ if $n$ is even, and $f(n) = f(n-2) + 2$ if $n$ is odd and greater than $1$. What is $f(2017)$?
$\textbf{(A) } 2017 \qquad \textbf{(B) } 2018 \qquad \textbf{(C) } 4034 \qquad \textbf{(D) } 4035 \qquad \textbf{(E) } 4036$
2019 AMC 10, 5
Triangle $ABC$ lies in the first quadrant. Points $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is [u][i]not[/i][/u] always true?
$(A)$ Triangle $A'B'C'$ lies in the first quadrant.
$(B)$ Triangles $ABC$ and $A'B'C'$ have the same area.
$(C)$ The slope of line $AA'$ is $-1$.
$(D)$ The slopes of lines $AA'$ and $CC'$ are the same.
$(E)$ Lines $AB$ and $A'B'$ are perpendicular to each other.
2014 AMC 10, 4
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
${ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6$
2024 AMC 10, 23
The Fibonacci numbers are defined by $F_1=1,$ $F_2=1,$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3.$ What is $$\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?$$
$\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$
2008 ITest, 90
For $a,b,c$ positive reals, let \[N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}.\] Find the minimum value of $\lfloor 2008N\rfloor$.
2007 AIME Problems, 14
Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$
2023 AMC 10, 8
Barb the baker creates a new temperature system for baking bread, Breadus, which is linearly based on Fahrenheit. Bread rises at $110$ F$^\circ$, which is $0$ on the Breadus scale. Bread bakes at $350$ F$^\circ$, which is $100$ on the Breadus scale. Bread is done when it’s internal temperature is $200$ F$^\circ.$ What is this temperature on the Breadus scale?
$\textbf{(A) }33\qquad\textbf{(B) }34.5\qquad\textbf{(C) }36\qquad\textbf{(D) }37.5\qquad\textbf{(E) }39$
2015 AIME Problems, 1
The expressions $A=1\times2+3\times4+5\times6+\cdots+37\times38+39$ and $B=1+2\times3+4\times5+\cdots+36\times37+38\times39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$.
2009 AMC 10, 18
Rectangle $ ABCD$ has $ AB\equal{}8$ and $ BC\equal{}6$. Point $ M$ is the midpoint of diagonal $ \overline{AC}$, and E is on $ \overline{AB}$ with $ \overline{ME}\perp\overline{AC}$. What is the area of $ \triangle AME$?
$ \textbf{(A)}\ \frac{65}{8} \qquad
\textbf{(B)}\ \frac{25}{3} \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ \frac{75}{8} \qquad
\textbf{(E)}\ \frac{85}{8}$
2013 AMC 10, 14
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have?
$\textbf{(A) }36\qquad
\textbf{(B) }60\qquad
\textbf{(C) }72\qquad
\textbf{(D) }84\qquad
\textbf{(E) }108\qquad$
2006 AMC 12/AHSME, 4
Mary is about to pay for five items at the grocery store. The prices of the items are $ \$$7.99, $ \$$ 4.99, $ \$$2.99, $ \$$1.99, and $ \$$0.99. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the $ \$$20.00 that she will receive in change?
$ \textbf{(A) } 5 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 25$
2011 AMC 10, 20
Rhombus $ABCD$ has side length $2$ and $\angle B = 120 ^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?
$ \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad
\textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad
\textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad
\textbf{(D)}\ 1+\frac{\sqrt{3}}{3} \qquad
\textbf{(E)}\ 2 $
2014 AMC 10, 12
The largest divisor of $2,014,000,000$ is itself. What is its fifth largest divisor?
$\textbf{(A) }125,875,000\qquad\textbf{(B) }201,400,000\qquad\textbf{(C) }251,750,000\qquad\textbf{(D) }402,800,000\qquad\textbf{(E) }503,500,000\qquad$
2024 AMC 10, 13
Two transformations are said to [i]commute[/i] if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:
- A translation $2$ units to the right
- A $90^\circ$- rotation counterclockwise about the origin.
- A reflection across the $x$-axis, and
- A dilation centered at the origin with scale factor $2$.
Of the $6$ pairs of distinct transformations from this list, how many commute?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) }5 \qquad
$
2024 AIME, 5
Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16,$ $AB=107,$ $FG=17,$ and $EF=184,$ what is the length of $CE$?
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F, G, H;
A = (0,0);
B = (5,0);
C = (5,1.5);
D = (0,1.5);
E = (1,1.5);
F = (8,1.5);
G = (8,3.5);
H = (1,3.5);
draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
dot("A", A, SW);
dot("B", B, SE);
dot("C", C, SE);
dot("D", D, NW);
dot("E", E, NW);
dot("F", F, SE);
dot("G", G, NE);
dot("H", H, NW);
[/asy]
2010 AMC 10, 24
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half?
$ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$
2019 AIME Problems, 2
Jenn randomly chooses a number $J$ from $1, 2, 3,\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2011 AIME Problems, 15
Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.
2018 AMC 10, 11
Which of the following expressions is never a prime number when $p$ is a prime number?
$\textbf{(A) } p^2+16 \qquad \textbf{(B) } p^2+24 \qquad \textbf{(C) } p^2+26 \qquad \textbf{(D) } p^2+46 \qquad \textbf{(E) } p^2+96$
2010 AMC 10, 13
What is the sum of all the solutions of $ x \equal{} |2x \minus{} |60\minus{}2x\parallel{}$?
$ \textbf{(A)}\ 32\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 124$
1970 AMC 12/AHSME, 3
If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is:
$\textbf{(A) }\dfrac{x+1}{x-1}\qquad\textbf{(B) }\dfrac{x+2}{x-1}\qquad\textbf{(C) }\dfrac{x}{x-1}\qquad\textbf{(D) }2-x\qquad \textbf{(E) }\dfrac{x-1}{x}$
2007 AMC 10, 11
The numbers from $ 1$ to $ 8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?
$ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 24$