Found problems: 3632
2000 AMC 8, 16
In order for Mateen to walk a kilometer ($1000$m) in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters?
$\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$
2025 AIME, 10
The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle.
[asy]
unitsize(20);
add(grid(9,3));
draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2));
draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2));
real a = 0.5;
label("5",(a,a));
label("6",(1+a,a));
label("1",(2+a,a));
label("8",(3+a,a));
label("4",(4+a,a));
label("7",(5+a,a));
label("9",(6+a,a));
label("2",(7+a,a));
label("3",(8+a,a));
label("3",(a,1+a));
label("7",(1+a,1+a));
label("9",(2+a,1+a));
label("5",(3+a,1+a));
label("2",(4+a,1+a));
label("1",(5+a,1+a));
label("6",(6+a,1+a));
label("8",(7+a,1+a));
label("4",(8+a,1+a));
label("4",(a,2+a));
label("2",(1+a,2+a));
label("8",(2+a,2+a));
label("9",(3+a,2+a));
label("6",(4+a,2+a));
label("3",(5+a,2+a));
label("1",(6+a,2+a));
label("7",(7+a,2+a));
label("5",(8+a,2+a));
[/asy]
The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$, $q$, $r$, and $s$ are distinct prime numbers and $a$, $b$, $c$, $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$.
1968 AMC 12/AHSME, 32
$A$ and $B$ move uniformly along two straight paths intersecting at right angles in point $O$. When $A$ is at $O$, $B$ is $500$ yards short of $O$. In $2$ minutes, they are equidistant from $O$, and in $8$ minutes more they are again equidistant from $O$. Then the ratio of $A'$s speed to $B'$s speed is:
$\textbf{(A)}\ 4:5 \qquad\textbf{(B)}\ 5:6 \qquad\textbf{(C)}\ 2:3 \qquad\textbf{(D)}\ 5:8 \qquad\textbf{(E)}\ 1:2$
1978 AMC 12/AHSME, 4
If $a = 1,~ b = 10, ~c = 100$, and $d = 1000$, then \[(a+ b+ c-d) + (a + b- c+ d) +(a-b+ c+d)+ (-a+ b+c+d) \] is equal to
$\textbf{(A) }1111\qquad\textbf{(B) }2222\qquad\textbf{(C) }3333\qquad\textbf{(D) }1212\qquad \textbf{(E) }4242$
2022 AMC 12/AHSME, 11
Let $ f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n $, where $i = \sqrt{-1}$. What is $f(2022)$
$ \textbf{(A)}\ -2 \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ \sqrt{3} \qquad
\textbf{(E)}\ 2$
2016 AMC 10, 15
All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$
1978 AMC 12/AHSME, 20
If $a,b,c$ are non-zero real numbers such that \[\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a},\] and \[x=\frac{(a+b)(b+c)(c+a)}{abc},\] and $x<0$, then $x$ equals
$\textbf{(A) }-1\qquad\textbf{(B) }-2\qquad\textbf{(C) }-4\qquad\textbf{(D) }-6\qquad \textbf{(E) }-8$
2005 AIME Problems, 9
Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The $27$ cubes are randomly arranged to form a $3\times 3 \times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\frac{p^a}{q^br^c},$ where $p$,$q$, and $r$ are distinct primes and $a$,$b$, and $c$ are positive integers, find $a+b+c+p+q+r$.
2021 AIME Problems, 13
Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$. Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat{PQ}$ is $120^{\circ}$. Find the distance between the centers of $\omega_1$ and $\omega_2$.
1986 AMC 12/AHSME, 28
$ABCDE$ is a regular pentagon. $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively. Let $O$ be the center of the pentagon. If $OP = 1$, then $AO + AQ + AR$ equals
[asy]
size(200);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair O=origin, A=2*dir(90), B=2*dir(18), C=2*dir(306), D=2*dir(234), E=2*dir(162), P=(C+D)/2, Q=C+3.10*dir(C--B), R=D+3.10*dir(D--E), S=C+4.0*dir(C--B), T=D+4.0*dir(D--E);
draw(A--B--C--D--E--A^^E--T^^B--S^^R--A--Q^^A--P^^rightanglemark(A,Q,S,7)^^rightanglemark(A,R,T,7));
dot(O);
label("$O$",O,dir(B));
label("$1$",(O+P)/2,W);
label("$A$",A,dir(A));
label("$B$",B,dir(B));
label("$C$",C,dir(C));
label("$D$",D,dir(D));
label("$E$",E,dir(E));
label("$P$",P,dir(P));
label("$Q$",Q,dir(Q-A));
label("$R$",R,dir(R-A));
[/asy]
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 1 + \sqrt{5}\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 2 + \sqrt{5}\qquad\textbf{(E)}\ 5 $
2011 AMC 10, 17
In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4:5$. What is the degree measure of angle $BCD$?
[asy]
unitsize(7mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
real r=3;
pair A=(-3cos(80),-3sin(80));
pair D=(3cos(80),3sin(80)), C=(-3cos(80),3sin(80));
pair O=(0,0), E=(-3,0), B=(3,0);
path outer=Circle(O,r);
draw(outer);
draw(E--B);
draw(E--A);
draw(B--A);
draw(E--D);
draw(C--D);
draw(B--C);
pair[] ps={A,B,C,D,E,O};
dot(ps);
label("$A$",A,N);
label("$B$",B,NE);
label("$C$",C,S);
label("$D$",D,S);
label("$E$",E,NW);
label("$$",O,N);[/asy]
$ \textbf{(A)}\ 120 \qquad
\textbf{(B)}\ 125 \qquad
\textbf{(C)}\ 130 \qquad
\textbf{(D)}\ 135 \qquad
\textbf{(E)}\ 140 $
1959 AMC 12/AHSME, 37
When simplified the product $\left(1-\frac13\right)\left(1-\frac14\right)\left(1-\frac15\right)\cdots\left(1-\frac1n\right)$ becomes:
$ \textbf{(A)}\ \frac1n \qquad\textbf{(B)}\ \frac2n\qquad\textbf{(C)}\ \frac{2(n-1)}{n}\qquad\textbf{(D)}\ \frac{2}{n(n+1)}\qquad\textbf{(E)}\ \frac{3}{n(n+1)} $
2023 AMC 8, 3
[i]Wind chill[/i] is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation: $$(\text{wind chill}) = (\text{air temperature}) - 0.7 \times (\text{wind speed}),$$ where temperature is measured in degrees Fahrenheit $(^{\circ}\text{F})$ and and the wind speed is measured in miles per hour (mph). Suppose the air temperature is $36^{\circ}\text{F} $ and the wind speed is $18$ mph. Which of the following is closest to the approximate wind chill?
$\textbf{(A)}~18\qquad\textbf{(B)}~23\qquad\textbf{(C)}~28\qquad\textbf{(D)}~32\qquad\textbf{(E)}~35$
2014 AMC 8, 10
The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born?
$\textbf{(A) }1979\qquad\textbf{(B) }1980\qquad\textbf{(C) }1981\qquad\textbf{(D) }1982\qquad \textbf{(E) }1983$
2009 AMC 12/AHSME, 24
For how many values of $ x$ in $ [0,\pi]$ is $ \sin^{\minus{}1}(\sin 6x)\equal{}\cos^{\minus{}1}(\cos x)$?
Note: The functions $ \sin^{\minus{}1}\equal{}\arcsin$ and $ \cos^{\minus{}1}\equal{}\arccos$ denote inverse trigonometric functions.
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 7$
2021 AMC 12/AHSME Spring, 1
What is the value of $$2^{1+2+3}-(2^1+2^2+2^3)?$$
$\textbf{(A) }0 \qquad \textbf{(B) }50 \qquad \textbf{(C) }52 \qquad \textbf{(D) }54 \qquad \textbf{(E) }57$
Proposed by [b]djmathman[/b]
1987 AMC 12/AHSME, 17
In a mathematics competition, the sum of the scores of Bill and Dick equalled the sum of the scores of Ann and Carol. If the scores of Bill and Carol had been interchanged, then the sum of the scores of Ann and Carol would have exceeded the sum of the scores of the other two. Also, Dick's score exceeded the sum of the scores of Bill and Carol. Determine the order in which the four contestants finished, from highest to lowest. Assume all scores were nonnegative.
$ \textbf{(A)}\ \text{Dick, Ann, Carol, Bill} \qquad\textbf{(B)}\ \text{Dick, Ann, Bill, Carol} \qquad\textbf{(C)}\ \text{Dick, Carol, Bill, Ann} \\ \qquad\textbf{(D)}\ \text{Ann, Dick, Carol, Bill} \qquad\textbf{(E)}\ \text{Ann, Dick, Bill, Carol} $
2008 AMC 12/AHSME, 11
A cone-shaped mountain has its base on the ocean floor and has a height of $ 8000$ feet. The top $ \frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet?
$ \textbf{(A)}\ 4000 \qquad
\textbf{(B)}\ 2000(4\minus{}\sqrt{2}) \qquad
\textbf{(C)}\ 6000 \qquad
\textbf{(D)}\ 6400 \qquad
\textbf{(E)}\ 7000$
1979 AMC 12/AHSME, 30
[asy]
/*Using regular asymptote, this diagram would take 30 min to make. Using cse5, this takes 5 minutes. Conclusion? CSE5 IS THE BEST PACKAGE EVER CREATED!!!!*/
size(100);
import cse5;
pathpen=black;
anglefontpen=black;
pointpen=black;
anglepen=black;
dotfactor=3;
pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE;
EE=(B+C)/2;
D(MP("$A$",A,W)--MP("$B$",B,N)--MP("$C$",C,E)--cycle);
D(MP("$E$",EE,N)--MP("$D$",D,S));
D(D);D(EE);
MA("80^\circ",8,D,EE,C,0.1);
MA("20^\circ",8,EE,C,D,0.3,2,shift(1,3)*C);
draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow());
MA("100^\circ",8,A,B,C,0.1,0);
MA("60^\circ",8,C,A,B,0.1,0);
//Credit to TheMaskedMagician for the diagram
[/asy]
In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ$, $\measuredangle ABC = 100^\circ$, $\measuredangle ACB = 20^\circ$ and $\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equals
$\textbf{(A) }\frac{1}{4}\cos 10^\circ\qquad\textbf{(B) }\frac{\sqrt{3}}{8}\qquad\textbf{(C) }\frac{1}{4}\cos 40^\circ\qquad\textbf{(D) }\frac{1}{4}\cos 50^\circ\qquad\textbf{(E) }\frac{1}{8}$
2025 USAJMO, 5
Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.
2017 AMC 12/AHSME, 10
Chloé chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0,4034]$. What is the probability that Laurent's number is greater than Chloé's number?
$\textbf{(A)}~\frac12 \qquad
\textbf{(B)}~\frac23 \qquad
\textbf{(C)}~\frac34 \qquad
\textbf{(D)}~\frac56\qquad
\textbf{(E)}~\frac78$
1967 AMC 12/AHSME, 18
If $x^2-5x+6<0$ and $P=x^2+5x+6$ then
$\textbf{(A)}\ P \; \text{can take any real value} \qquad
\textbf{(B)}\ 20<P<30\\
\textbf{(C)}\ 0<P<20 \qquad
\textbf{(D)}\ P<0 \qquad
\textbf{(E)}\ P>30$
2015 AMC 10, 8
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2:1$?
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
2009 AMC 12/AHSME, 17
Let $ a\plus{}ar_1\plus{}ar_1^2\plus{}ar_1^3\plus{}\cdots$ and $ a\plus{}ar_2\plus{}ar_2^2\plus{}ar_2^3\plus{}\cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $ r_1$, and the sum of the second series is $ r_2$. What is $ r_1\plus{}r_2$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac{1\plus{}\sqrt{5}}{2}\qquad \textbf{(E)}\ 2$
2016 AMC 10, 19
Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$. and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?
[asy] pair A1=(2,0),A2=(4,4);
pair B1=(0,4),B2=(5,1);
pair C1=(5,0),C2=(0,4);
draw(A1--A2);
draw(B1--B2);
draw(C1--C2);
draw((0,0)--B1--(5,4)--C1--cycle);
dot((20/7,12/7));
dot((3.07692307692,2.15384615384));
label("$Q$",(3.07692307692,2.15384615384),N);
label("$P$",(20/7,12/7),W);
label("$A$",(0,4), NW);
label("$B$",(5,4), NE);
label("$C$",(5,0),SE);
label("$D$",(0,0),SW);
label("$F$",(2,0),S); label("$G$",(5,1),E);
label("$E$",(4,4),N);
dot(A1); dot(A2);
dot(B1); dot(B2);
dot(C1); dot(C2);
dot((0,0)); dot((5,4));[/asy]
$\textbf{(A)}~\frac{\sqrt{13}}{16} \qquad
\textbf{(B)}~\frac{\sqrt{2}}{13} \qquad
\textbf{(C)}~\frac{9}{82} \qquad
\textbf{(D)}~\frac{10}{91}\qquad
\textbf{(E)}~\frac19$