Found problems: 3632
2019 AMC 12/AHSME, 19
In $\triangle ABC$ with integer side lengths,
\[
\cos A=\frac{11}{16}, \qquad \cos B= \frac{7}{8}, \qquad \text{and} \qquad\cos C=-\frac{1}{4}.
\] What is the least possible perimeter for $\triangle ABC$?
$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 23 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 44$
1998 AMC 12/AHSME, 24
Call a $ 7$-digit telephone number $ d_1d_2d_3 \minus{} d_4d_5d_6d_7$ [i]memorable[/i] if the prefix sequence $ d_1d_2d_3$ is exactly the same as either of the sequences $ d_4d_5d_6$ or $ d_5d_6d_7$ (possibly both). Assuming that each $ d_i$ can be any of the ten decimal digits $ 0,1,2,\ldots9$, the number of different memorable telephone numbers is
$ \textbf{(A)}\ 19,\!810 \qquad \textbf{(B)}\ 19,\!910 \qquad \textbf{(C)}\ 19,\!990 \qquad \textbf{(D)}\ 20,\!000 \qquad \textbf{(E)}\ 20,\!100$
1997 AMC 8, 21
Each corner cube is removed from this $3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}$ cube. The surface area of the remaining figure is
[asy]draw((2.7,3.99)--(0,3)--(0,0));
draw((3.7,3.99)--(1,3)--(1,0));
draw((4.7,3.99)--(2,3)--(2,0));
draw((5.7,3.99)--(3,3)--(3,0));
draw((0,0)--(3,0)--(5.7,0.99));
draw((0,1)--(3,1)--(5.7,1.99));
draw((0,2)--(3,2)--(5.7,2.99));
draw((0,3)--(3,3)--(5.7,3.99));
draw((0,3)--(3,3)--(3,0));
draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33));
draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66));
draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99));
[/asy]
$\textbf{(A)}\ 19\text{ sq.cm} \qquad \textbf{(B)}\ 24\text{ sq.cm} \qquad \textbf{(C)}\ 30\text{ sq.cm} \qquad \textbf{(D)}\ 54\text{ sq.cm} \qquad \textbf{(E)}\ 72\text{ sq.cm}$
2006 AMC 10, 15
Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?
$ \textbf{(A) } 29 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 47 \qquad \textbf{(E) } 50$
2023 AMC 10, 20
Each square in a $3\times 3$ grid of squares is colored red, white, blue, or green so that every $2\times 2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?\\
[asy]
size(8cm);
pen grey1, grey2, grey3;
grey1 = RGB(211, 211, 211);
grey2 = RGB(173, 173, 173);
grey3 = RGB(138, 138, 138);
for(int i = 0; i < 4; ++i) {
draw((i, 0)--(i, 3));
draw((0, i)--(3, i));
}
filldraw((5, 3)--(6, 3)--(6, 2)--(5, 2)--cycle, grey1);
label('B', (5.5, 2.5));
filldraw((6, 3)--(7, 3)--(7, 2)--(6, 2)--cycle, grey2);
label('R', (6.5, 2.5));
filldraw((7, 3)--(8, 3)--(8, 2)--(7, 2)--cycle, grey1);
label('B', (7.5, 2.5));
filldraw((5, 2)--(6, 2)--(6, 1)--(5, 1)--cycle, grey3);
label('G', (5.5, 1.5));
filldraw((6, 2)--(7, 2)--(7, 1)--(6, 1)--cycle, white);
filldraw((7, 2)--(8, 2)--(8, 1)--(7, 1)--cycle, grey3);
label('G', (7.5, 1.5));
filldraw((5, 1)--(6, 1)--(6, 0)--(5, 0)--cycle, grey2);
label('R', (5.5, 0.5));
filldraw((6, 1)--(7, 1)--(7, 0)--(6, 0)--cycle, grey1);
label('B', (6.5, 0.5));
filldraw((7, 1)--(8, 1)--(8, 0)--(7, 0)--cycle, grey2);
label('R', (7.5, 0.5));
[/asy]
$\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }96$
2022 AMC 12/AHSME, 2
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$?
[asy]
import olympiad;
size(180);
real r = 3, s = 5, t = sqrt(r*r+s*s);
defaultpen(linewidth(0.6) + fontsize(10));
pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0);
draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D));
label("$A$",A,SW);
label("$B$", B, NW);
label("$C$",C,NE);
label("$D$",D,SE);
label("$P$",P,S);
[/asy]
$\textbf{(A) }3\sqrt 5 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }6\sqrt 5 \qquad
\textbf{(D) }20\qquad
\textbf{(E) }25$
1960 AMC 12/AHSME, 17
The formula $N=8 \times 10^{8} \times x^{-3/2}$ gives, for a certain group, the number of individuals whose income exceeds $x$ dollars. The lowest income, in dollars, of the wealthiest $800$ individuals is at least:
$ \textbf{(A)}\ 10^4\qquad\textbf{(B)}\ 10^6\qquad\textbf{(C)}\ 10^8\qquad\textbf{(D)}\ 10^{12} \qquad\textbf{(E)}\ 10^{16} $
2025 AIME, 15
Let
\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]
There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$.
1978 AMC 12/AHSME, 23
[asy]
size(100);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((0,1)--(1,0));
draw((0,0)--(.5,sqrt(3)/2)--(1,0));
label("$A$",(0,0),SW);
label("$B$",(1,0),SE);
label("$C$",(1,1),NE);
label("$D$",(0,1),NW);
label("$E$",(.5,sqrt(3)/2),E);
label("$F$",intersectionpoint((0,0)--(.5,sqrt(3)/2),(0,1)--(1,0)),2W);
//Credit to chezbgone2 for the diagram[/asy]
Vertex $E$ of equilateral triangle $ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\sqrt{1+\sqrt{3}}$ then the area of $\triangle ABF$ is
$\textbf{(A) }1\qquad\textbf{(B) }\frac{\sqrt{2}}{2}\qquad\textbf{(C) }\frac{\sqrt{3}}{2}$
$\qquad\textbf{(D) }4-2\sqrt{3}\qquad \textbf{(E) }\frac{1}{2}+\frac{\sqrt{3}}{4}$
1992 AMC 12/AHSME, 28
Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is
$ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $
1966 AMC 12/AHSME, 30
If three of the roots of $x^4+ax^2+bx+c=0$ are $1$, $2$, and $3$, then the value of $a+c$ is:
$\text{(A)}\ 35 \qquad
\text{(B)}\ 24\qquad
\text{(C)}\ -12\qquad
\text{(D)}\ -61 \qquad
\text{(E)}\ -63$
2024 AIME, 11
Each vertex of a regular octagon is coloured either red or blue with equal probability. The probability that the octagon can then be rotated in such a way that all of the blue vertices end up at points that were originally red is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
2009 AMC 10, 10
A flagpole is originally $ 5$ meters tall. A hurricane snaps the flagpole at a point $ x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $ 1$ meter away from the base. What is $ x$?
$ \textbf{(A)}\ 2.0 \qquad \textbf{(B)}\ 2.1 \qquad \textbf{(C)}\ 2.2 \qquad \textbf{(D)}\ 2.3 \qquad \textbf{(E)}\ 2.4$
2024 AMC 10, 19
In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the $12$ entries will be "Possible"?
\begin{tabular}{|c|c|c|c|c|} \cline{2-5}
\multicolumn{1}{c|}{} & \textbf{zero} & \textbf{exactly one} & \textbf{exactly two} & \textbf{more than two}\\ \hline
\textbf{zero slope} & ? & ? & ? & ?\\ \hline
\textbf{nonzero rational slope} & ? & ? & ? & ?\\ \hline
\textbf{irrational slope} & ? & ? & ? & ?\\ \hline
\end{tabular}
$
\textbf{(A) }4 \qquad
\textbf{(B) }5 \qquad
\textbf{(C) }6 \qquad
\textbf{(D) }7 \qquad
\textbf{(E) }9 \qquad
$
2015 AMC 12/AHSME, 21
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
$\textbf{(A) } 9
\qquad\textbf{(B) } 11
\qquad\textbf{(C) } 12
\qquad\textbf{(D) } 13
\qquad\textbf{(E) } 15
$
1970 AMC 12/AHSME, 7
Inside square $ABCD$ with side $s$, quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at point $X$ inside the square. How far is $X$ from side $CD$?
$\textbf{(A) }\frac{1}{2}s(\sqrt{3}+4)\qquad\textbf{(B) }\frac{1}{2}s\sqrt{3}\qquad\textbf{(C) }\frac{1}{2}s(1+\sqrt{3})\qquad$
$\textbf{(D) }\frac{1}{2}s(\sqrt{3}-1)\qquad \textbf{(E) }\frac{1}{2}s(2-\sqrt{3})$
2018 AMC 10, 4
4. A three-dimensional rectangular box with dimensions $X$, $Y$, and $Z$ has faces whose surface areas are $24$, $24$, $48$, $48$, $72$, and $72$ square units. What is $X + Y + Z$?
$\textbf{(A)} \text{ 18} \qquad \textbf{(B)} \text{ 22} \qquad \textbf{(C)} \text{ 24} \qquad \textbf{(D)} \text{ 30} \qquad \textbf{(E)} \text{ 36}$
2019 AMC 12/AHSME, 10
The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L,$ without traveling along any portion of a road more than once. (Paula [i]is[/i] allowed to visit a city more than once.) How many different routes can Paula take?
[asy]
import olympiad;
unitsize(50);
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 4; ++j) {
pair A = (j,i);
dot(A);
}
}
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 4; ++j) {
if (j != 3) {
draw((j,i)--(j+1,i));
}
if (i != 2) {
draw((j,i)--(j,i+1));
}
}
}
label("$A$", (0,2), W);
label("$L$", (3,0), E);
[/asy]
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$
1993 AMC 12/AHSME, 21
Let $a_1, a_2, ..., a_k$ be a finite arithmetic sequence with
\[ a_4+a_7+a_{10}=17 \] and \[ a_4+a_5+a_6+a_7+a_8+a_9+a_{10}+a_{11}+a_{12}+a_{13}+a_{14}=77 \] If $a_k=13$, then $k=$
$ \textbf{(A)}\ 16 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ 24 $
2012 AMC 10, 15
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?
[asy]
size(200);
defaultpen(linewidth(.6pt)+fontsize(12pt));
dotfactor=4;
draw((0,0)--(0,2));
draw((0,0)--(1,0));
draw((1,0)--(1,2));
draw((0,1)--(2,1));
draw((0,0)--(1,2));
draw((0,2)--(2,1));
draw((0,2)--(2,2));
draw((2,1)--(2,2));
label("$A$",(0,2),NW);
label("$B$",(1,2),N);
label("$C$",(4/5,1.55),W);
dot((0,2));
dot((1,2));
dot((4/5,1.6));
dot((2,1));
dot((0,0));
[/asy]
$ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{2}{9}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt2}{4} $
2016 AMC 10, 24
How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.
$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 20$
2006 AIME Problems, 12
Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200$.
1964 AMC 12/AHSME, 34
If $n$ is a multiple of $4$, the sum $s=1+2i+3i^2+ ... +(n+1)i^{n}$, where $i=\sqrt{-1}$, equals:
$ \textbf{(A)}\ 1+i\qquad\textbf{(B)}\ \frac{1}{2}(n+2) \qquad\textbf{(C)}\ \frac{1}{2}(n+2-ni) \qquad$
$ \textbf{(D)}\ \frac{1}{2}[(n+1)(1-i)+2]\qquad\textbf{(E)}\ \frac{1}{8}(n^2+8-4ni) $
2022 AMC 10, 21
A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?
[asy]
size(200);
defaultpen(linewidth(0.8));
draw((342,-662) -- (600, -727) -- (757,-619) -- (967,-400) -- (1016,-300) -- (912,-116) -- (651,-46) -- (238,-90) -- (82,-204) -- (184, -388) -- (447,-458) -- (859,-410) -- (1016,-300));
draw((82,-204) -- (133,-490) -- (342, -662));
draw((652,-626) -- (600,-727));
draw((447,-458) -- (652,-626) -- (859,-410));
draw((133,-490) -- (184, -388));
draw((967,-400) -- (912,-116)^^(342,-662) -- (496, -545) -- (757,-619)^^(496, -545) -- (446, -262) -- (238, -90)^^(446, -262) -- (651, -46),linewidth(0.6)+linetype("5 5")+gray(0.4));
[/asy]
$\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }5+2\sqrt{2}\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2025 AIME, 9
The parabola with equation $y = x^2 - 4$ is rotated $60^\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a - \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.