Found problems: 3632
1991 AMC 12/AHSME, 26
An $n$-digit positive integer is [i]cute[/i] if its $n$ digits are an arrangement of the set $\{1,2,\ldots,n\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,\ldots,n$. For example 321 is a cute 3-digit integer because 1 divides 3, 2 divides 32, and 3 divides 321. How many cute 6-digit integers are there?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4 $
2013 AMC 12/AHSME, 25
Let $G$ be the set of polynomials of the form
\[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\]
where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has $n$ distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?
${ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D}}\ 992\qquad\textbf{(E)}\ 1056 $
2024 AMC 12/AHSME, 16
A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$?
$
\textbf{(A) }5 \qquad
\textbf{(B) }6 \qquad
\textbf{(C) }7 \qquad
\textbf{(D) }8 \qquad
\textbf{(E) }9 \qquad
$
1992 AMC 12/AHSME, 24
Let $ABCD$ be a parallelogram of area $10$ with $AB = 3$ and $BC = 5$. Locate $E$, $F$ and $G$ on segments $\overline{AB}$, $\overline{BC}$ and $\overline{AD}$, respectively, with $AE = BF = AG = 2$. Let the line through $G$ parallel to $\overline{EF}$ intersect $\overline{CD}$ at $H$. The area of the quadrilateral $EFHG$ is
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 4.5\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 5.5\qquad\textbf{(E)}\ 6 $
1952 AMC 12/AHSME, 30
When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is:
$ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 2: 1 \qquad\textbf{(C)}\ 1: 4 \qquad\textbf{(D)}\ 4: 1 \qquad\textbf{(E)}\ 1: 1$
1979 AMC 12/AHSME, 7
The square of an integer is called a [i]perfect square[/i]. If $x$ is a perfect square, the next larger perfect square is
$\textbf{(A) }x+1\qquad\textbf{(B) }x^2+1\qquad\textbf{(C) }x^2+2x+1\qquad\textbf{(D) }x^2+x\qquad\textbf{(E) }x+2\sqrt{x}+1$
1996 AMC 8, 12
What number should be removed from the list
\[1,2,3,4,5,6,7,8,9,10,11\]
so that the average of the remaining numbers is $6.1$?
$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$
2001 AMC 12/AHSME, 24
In $ \triangle ABC$, $ \angle ABC \equal{} 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD \equal{} CD$ and $ \angle DAB \equal{} 15^\circ$. Find $ \angle ACB$.
[asy]
pair A, B, C, D;
A = origin;
real Bcoord = 3*sqrt(2) + sqrt(6);
B = Bcoord/2*dir(180);
C = sqrt(6)*dir(120);
draw(A--B--C--cycle);
D = (C-B)/2.4 + B;
draw(A--D);
label("$A$", A, dir(0));
label("$B$", B, dir(180));
label("$C$", C, dir(110));
label("$D$", D, dir(130));
[/asy]
$ \textbf{(A)} \ 54^\circ \qquad \textbf{(B)} \ 60^\circ \qquad \textbf{(C)} \ 72^\circ \qquad \textbf{(D)} \ 75^\circ \qquad \textbf{(E)} \ 90^\circ$
1961 AMC 12/AHSME, 40
Find the minimum value of $\sqrt{x^2+y^2}$ if $5x+12y=60$.
${{ \textbf{(A)}\ \frac{60}{13} \qquad\textbf{(B)}\ \frac{13}{5} \qquad\textbf{(C)}\ \frac{13}{12} \qquad\textbf{(D)}\ 1}\qquad\textbf{(E)}\ 0 } $
1979 AMC 12/AHSME, 5
Find the sum of the digits of the largest even three digit number (in base ten representation) which is not changed when its units and hundreds digits are interchanged.
$\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }25\qquad\textbf{(E) }26$
1963 AMC 12/AHSME, 38
Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 32$ and $GF = 24$, then $BE$ equals:
[asy]
size(7cm);
pair A = (0, 0), B = (7, 0), C = (10, 5), D = (3, 5), F = (5.7, 9.5);
pair G = intersectionpoints(B--F, D--C)[0];
pair E = intersectionpoints(A--C, B--F)[0];
draw(A--D--C--B--cycle);
draw(A--C);
draw(D--F--B);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, NE);
label("$D$", D, NW);
label("$F$", F, N);
label("$G$", G, NE);
label("$E$", E, SE);
//Credit to MSTang for the asymptote
[/asy]
$\textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 16$
2023 AMC 10, 3
A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?
$\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}$
2018 AMC 12/AHSME, 6
For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$?
$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$
1990 USAMO, 4
Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left. (Your answer should be an explicit function of $n$ in simplest form.)
2010 AMC 10, 10
Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?
$ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 30$
1972 AMC 12/AHSME, 27
If the area of $\triangle ABC$ is $64$ square units and the geometric mean (mean proportional) between sides $AB$ and $AC$ is $12$ inches, then $\sin A$ is equal to
$\textbf{(A) }\dfrac{\sqrt{3}}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{4}{5}\qquad\textbf{(D) }\frac{8}{9}\qquad \textbf{(E) }\frac{15}{17}$
1964 AMC 12/AHSME, 1
What is the value of $[\log_{10}(5\log_{10}100)]^2$?
${{ \textbf{(A)}\ \log_{10}50 \qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 2}\qquad\textbf{(E)}\ 1 } $
2015 AMC 10, 16
Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?
$\textbf{(A) } \dfrac{9}{1000}
\qquad\textbf{(B) } \dfrac{1}{90}
\qquad\textbf{(C) } \dfrac{1}{80}
\qquad\textbf{(D) } \dfrac{1}{72}
\qquad\textbf{(E) } \dfrac{2}{121}
$
2023 AMC 12/AHSME, 19
What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$
$\textbf{(A) }(\log_{2023}7 \cdot \log_{2023}289)^2 \qquad\textbf{(B) }\log_{2023}7 \cdot \log_{2023}289\qquad\textbf{(C) }1\qquad\textbf{(D) }\log_{7}2023 \cdot \log_{289}2023\qquad\textbf{(E) }(\log_{7}2023 \cdot \log_{289}2023)^2$
2011 AMC 12/AHSME, 14
Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$?
$ \textbf{(A)}\ \frac{11}{81} \qquad
\textbf{(B)}\ \frac{13}{81} \qquad
\textbf{(C)}\ \frac{5}{27} \qquad
\textbf{(D)}\ \frac{17}{81} \qquad
\textbf{(E)}\ \frac{19}{81}
$
2008 AIME Problems, 11
Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?
1986 AIME Problems, 9
In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.
2014 AMC 10, 9
The two legs of a right triangle, which are altitudes, have lengths $2\sqrt3$ and $6$. How long is the third altitude of the triangle?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $
2014 AMC 10, 10
Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?
${ \textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}}\ a+6\qquad\textbf{(E)}\ a+7$
2012 AMC 10, 14
Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
$ \textbf{(A)}\ 480
\qquad\textbf{(B)}\ 481
\qquad\textbf{(C)}\ 482
\qquad\textbf{(D)}\ 483
\qquad\textbf{(E)}\ 484
$