Found problems: 3632
1975 AMC 12/AHSME, 28
In triangle $ABC$ shown in the adjoining figure, $M$ is the midpoint of side $BC$, $AB=12$ and $AC=16$. Points $E$ and $F$ are taken on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE=2AF$ then $\frac{EG}{GF}$ equals
[asy]
draw((0,0)--(12,0)--(14,7.75)--(0,0));
draw((0,0)--(13,3.875));
draw((5,0)--(8.75,4.84));
label("A", (0,0), S);
label("B", (12,0), S);
label("C", (14,7.75), E);
label("E", (8.75,4.84), N);
label("F", (5,0), S);
label("M", (13,3.875), E);
label("G", (7,1));
[/asy]
$ \textbf{(A)}\ \frac{3}{2} \qquad\textbf{(B)}\ \frac{4}{3} \qquad\textbf{(C)}\ \frac{5}{4} \qquad\textbf{(D)}\ \frac{6}{5} \\ \qquad\textbf{(E)}\ \text{not enough information to solve the problem} $
2008 AIME Problems, 3
A block of cheese in the shape of a rectangular solid measures $ 10$ cm by $ 13$ cm by $ 14$ cm. Ten slices are cut from the cheese. Each slice has a width of $ 1$ cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?
1988 AMC 12/AHSME, 16
$ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center, as in the figure. The distance between side $BC$ and side $B'C'$ is $\frac{1}{6}$ the altitude of $\triangle ABC$. The ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$ is
[asy]
size(170);
defaultpen(linewidth(0.7)+fontsize(10));
pair H=origin, B=(1,-(1/sqrt(3))), C=(-1,-(1/sqrt(3))), A=(0,(2/sqrt(3))), E=(2,-(2/sqrt(3))), F=(-2,-(2/sqrt(3))), D=(0,(4/sqrt(3)));
draw(A--B--C--A^^D--E--F--D);
label("$A'$", A, dir(90));
label("$B'$", B, SE);
label("$C'$", C, SW);
label("$A$", D, dir(90));
label("$B$", E, dir(0));
label("$C$", F, W);
[/asy]
$ \textbf{(A)}\ \frac{1}{36}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{\sqrt{3}}{4}\qquad\textbf{(E)}\ \frac{9+8\sqrt{3}}{36} $
1961 AMC 12/AHSME, 35
The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\times2!+a_3\times3!+ . . . a_n \times n!$ where $a_1, a_2, a_3 ... a_n$ are integers such that $0 \le a_k \le k$, and $n!$ means $n(n-1)(n-2)...2 \times 1$. Find $a_4$
${{ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3}\qquad\textbf{(E)}\ 4} $
2020 AMC 10, 3
Assuming $a\neq3$, $b\neq4$, and $c\neq5$, what is the value in simplest form of the following expression?
$$\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}$$
$\textbf{(A) } -1 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } \frac{abc}{60} \qquad \textbf{(D) } \frac{1}{abc} - \frac{1}{60} \qquad \textbf{(E) } \frac{1}{60} - \frac{1}{abc}$
1976 AMC 12/AHSME, 7
If $x$ is a real number, then the quantity $(1-|x|)(1+x)$ is positive if and only if
$\textbf{(A) }|x|<1\qquad\textbf{(B) }|x|>1\qquad\textbf{(C) }x<-1\text{ or }-1<x<1\qquad$
$\textbf{(D) }x<1\qquad \textbf{(E) }x<-1$
2024 AMC 12/AHSME, 4
What is the least value of $n$ such that $n!$ is a multiple of $2024$?
$
\textbf{(A) }11 \qquad
\textbf{(B) }21 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }23 \qquad
\textbf{(E) }253 \qquad
$
1985 USAMO, 3
Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$. Determine the maximum value of the sum of the six distances.
1999 AIME Problems, 4
The two squares shown share the same center $O$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
[asy]
real alpha = 25;
pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin;
pair w=dir(alpha)*W, x=dir(alpha)*X, y=dir(alpha)*Y, z=dir(alpha)*Z;
draw(W--X--Y--Z--cycle^^w--x--y--z--cycle);
pair A=intersectionpoint(Y--Z, y--z),
C=intersectionpoint(Y--X, y--x),
E=intersectionpoint(W--X, w--x),
G=intersectionpoint(W--Z, w--z),
B=intersectionpoint(Y--Z, y--x),
D=intersectionpoint(Y--X, w--x),
F=intersectionpoint(W--X, w--z),
H=intersectionpoint(W--Z, y--z);
dot(O);
label("$O$", O, SE);
label("$A$", A, dir(O--A));
label("$B$", B, dir(O--B));
label("$C$", C, dir(O--C));
label("$D$", D, dir(O--D));
label("$E$", E, dir(O--E));
label("$F$", F, dir(O--F));
label("$G$", G, dir(O--G));
label("$H$", H, dir(O--H));[/asy]
1986 AMC 12/AHSME, 24
Let $p(x) = x^{2} + bx + c$, where $b$ and $c$ are integers. If $p(x)$ is a factor of both \[x^{4} + 6x^{2} + 25\quad\text{and}\quad 3x^{4} + 4x^{2} + 28x + 5,\] what is $p(1)$?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 8 $
1960 AMC 12/AHSME, 7
Circle I passes through the center of, and is tangent to, circle II. The area of circle I is 4 square inches. Then the area of circle II, in square inches, is:
$ \textbf{(A) }8\qquad\textbf{(B) }8\sqrt{2}\qquad\textbf{(C) }8\sqrt{\pi}\qquad\textbf{(D) }16\qquad\textbf{(E) }16\sqrt{2} $
1979 AMC 12/AHSME, 4
For all real numbers $x$, $x[x\{x(2-x)-4\}+10]+1=$
$\textbf{(A) }-x^4+2x^3+4x^2+10x+1$
$\textbf{(B) }-x^4-2x^3+4x^2+10x+1$
$\textbf{(C) }-x^4-2x^3-4x^2+10x+1$
$\textbf{(D) }-x^4-2x^3-4x^2-10x+1$
$\textbf{(E) }-x^4+2x^3-4x^2+10x+1$
2023 AMC 10, 25
A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$
2009 AMC 10, 1
One can holds $ 12$ ounces of soda. What is the minimum number of cans to provide a gallon ($ 128$ ounces) of soda?
$ \textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 11$
1993 AMC 8, 11
Consider this histogram of the scores for $81$ students taking a test:
[asy]
unitsize(12);
draw((0,0)--(26,0));
draw((1,1)--(25,1));
draw((3,2)--(25,2));
draw((5,3)--(23,3));
draw((5,4)--(21,4));
draw((7,5)--(21,5));
draw((9,6)--(21,6));
draw((11,7)--(19,7));
draw((11,8)--(19,8));
draw((11,9)--(19,9));
draw((11,10)--(19,10));
draw((13,11)--(19,11));
draw((13,12)--(19,12));
draw((13,13)--(17,13));
draw((13,14)--(17,14));
draw((15,15)--(17,15));
draw((15,16)--(17,16));
draw((1,0)--(1,1));
draw((3,0)--(3,2));
draw((5,0)--(5,4));
draw((7,0)--(7,5));
draw((9,0)--(9,6));
draw((11,0)--(11,10));
draw((13,0)--(13,14));
draw((15,0)--(15,16));
draw((17,0)--(17,16));
draw((19,0)--(19,12));
draw((21,0)--(21,6));
draw((23,0)--(23,3));
draw((25,0)--(25,2));
for (int a = 1; a < 13; ++a)
{
draw((2*a,-.25)--(2*a,.25));
}
label("$40$",(2,-.25),S);
label("$45$",(4,-.25),S);
label("$50$",(6,-.25),S);
label("$55$",(8,-.25),S);
label("$60$",(10,-.25),S);
label("$65$",(12,-.25),S);
label("$70$",(14,-.25),S);
label("$75$",(16,-.25),S);
label("$80$",(18,-.25),S);
label("$85$",(20,-.25),S);
label("$90$",(22,-.25),S);
label("$95$",(24,-.25),S);
label("$1$",(2,1),N);
label("$2$",(4,2),N);
label("$4$",(6,4),N);
label("$5$",(8,5),N);
label("$6$",(10,6),N);
label("$10$",(12,10),N);
label("$14$",(14,14),N);
label("$16$",(16,16),N);
label("$12$",(18,12),N);
label("$6$",(20,6),N);
label("$3$",(22,3),N);
label("$2$",(24,2),N);
label("Number",(4,8),N);
label("of Students",(4,7),N);
label("$\textbf{STUDENT TEST SCORES}$",(14,18),N);
[/asy]
The median is in the interval labeled
$\text{(A)}\ 60 \qquad \text{(B)}\ 65 \qquad \text{(C)}\ 70 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 80$
1970 AMC 12/AHSME, 14
Consider $x^2+px+q=0$ where $p$ and $q$ are positive numbers. If the roots of this equation differ by $1$, then $p$ equals
$\textbf{(A) }\sqrt{4q+1}\qquad\textbf{(B) }q-1\qquad\textbf{(C) }-\sqrt{4q+1}\qquad\textbf{(D) }q+1\qquad \textbf{(E) }\sqrt{4q-1}$
2012 AIME Problems, 8
The complex numbers $z$ and $w$ satisfy the system
\begin{align*}z+\frac{20i}{w}&=5+i,\\w+\frac{12i}{z}&=-4+10i.\end{align*}
Find the smallest possible value of $|zw|^2$.
2024 AMC 10, 20
Let $S$ be a subset of $\{1, 2, 3, \dots, 2024\}$ such that the following two conditions hold:
- If $x$ and $y$ are distinct elements of $S$, then $|x-y| > 2$
- If $x$ and $y$ are distinct odd elements of $S$, then $|x-y| > 6$.
What is the maximum possible number of elements in $S$?
$
\textbf{(A) }436 \qquad
\textbf{(B) }506 \qquad
\textbf{(C) }608 \qquad
\textbf{(D) }654 \qquad
\textbf{(E) }675 \qquad
$
2023 AIME, 9
Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q$, and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B$, respectively. The line parallel to line $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Suppose $PX = 10, PY = 14,$ and $PQ = 5$. Then the area of trapezoid $XABY$ is $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.
2006 AMC 12/AHSME, 9
How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?
$ \textbf{(A) } 21 \qquad \textbf{(B) } 34 \qquad \textbf{(C) } 51 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 150$
2024 AMC 10, 9
Real numbers $a,b$ and $c$ have arithmetic mean $0$. The arithmetic mean of $a^2, b^2$ and $c^2$ is $10$. What is the arithmetic mean of $ab, ac$ and $bc$?
$
\textbf{(A) }-5 \qquad
\textbf{(B) }-\frac{10}{3} \qquad
\textbf{(C) }-\frac{10}{9} \qquad
\textbf{(D) }0 \qquad
\textbf{(E) }\frac{10}{9} \qquad
$
2021 AIME Problems, 2
Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$.
[asy]
pair A,B,C,D,E,F,G;
B=origin;
A=5*dir(60);
C=(5,0);
E=0.6*A+0.4*B;
F=0.6*A+0.4*C;
G=rotate(240,F)*A;
D=extension(E,F,B,dir(90));
draw(D--G--A,grey);
draw(B--0.5*A+rotate(60,B)*A*0.5,grey);
draw(A--B--C--cycle,linewidth(1.5));
dot(A^^B^^C^^D^^E^^F^^G);
label("$A$",A,dir(90));
label("$B$",B,dir(225));
label("$C$",C,dir(-45));
label("$D$",D,dir(180));
label("$E$",E,dir(-45));
label("$F$",F,dir(225));
label("$G$",G,dir(0));
label("$\ell$",midpoint(E--F),dir(90));
[/asy]
2007 AIME Problems, 15
Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA=5$ and $CD=2$. Point $E$ lies on side $CA$ such that $\angle DEF = 60^\circ$. The area of triangle $DEF$ is $14\sqrt{3}$. The two possible values of the length of side $AB$ are $p \pm q\sqrt{r}$, where $p$ and $q$ are rational, and $r$ is an integer not divisible by the square of a prime. Find $r$.
1968 AMC 12/AHSME, 33
A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is:
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$
1998 AMC 8, 2
If $ \begin{tabular}{r|l}a&b\\ \hline c&d\end{tabular}=\text{a}\cdot\text{d}-\text{b}\cdot\text{c} $, what is the value of $ \begin{tabular}{r|l}3&4\\ \hline 1&2\end{tabular} $
$ \text{(A)}\ -2\qquad\text{(B)}\ -1\qquad\text{(C)}\ 0\qquad\text{(D)}\ 1\qquad\text{(E)}\ 2 $