Found problems: 3632
2022 AMC 12/AHSME, 4
The least common multiple of a positive integer $n$ and 18 is 180, and the greatest common divisor of $n$ and 45 is 15. What is the sum of the digits of $n$?
$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }12$
1968 AMC 12/AHSME, 16
If $x$ is such that $\dfrac{1}{x}<2$ and $\dfrac{1}{x}>-3$, then:
$\textbf{(A)}\ -\dfrac{1}{3}<x<\dfrac{1}{2} \qquad
\textbf{(B)}\ -\dfrac{1}{2}<x<3 \qquad
\textbf{(C)}\ x>\dfrac{1}{2} \qquad\\
\textbf{(D)}\ x>\dfrac{1}{2}\text{ or }-\dfrac{1}{3}<x<0 \qquad
\textbf{(E)}\ x>\dfrac{1}{2}\text{ or }x<-\dfrac{1}{3}$
2024 AMC 10, 18
How many different remainders can result when the $100$th power of an integer is divided by $125$?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }25 \qquad
\textbf{(E) }125 \qquad
$
1969 AMC 12/AHSME, 18
The number of points common to the graphs of \[(x-y+2)(3x+y-4)=0\text{ and }(x+y-2)(2x-5y+7)=0\] is:
$\textbf{(A) }2\qquad
\textbf{(B) }4\qquad
\textbf{(C) }6\qquad
\textbf{(D) }16\qquad
\textbf{(E) }\text{infinite}$
1971 AMC 12/AHSME, 1
The number of digits in the number $N=2^{12}\times 5^8$ is
$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad \textbf{(E) }20$
1959 AMC 12/AHSME, 35
The symbol $\ge$ means "greater than or equal to"; the symbol $\le$ means "less than or equal to".
In the equation $(x-m)^2-(x-n)^2=(m-n)^2$; m is a fixed positive number, and $n$ is a fixed negative number. The set of values $x$ satisfying the equation is:
$ \textbf{(A)}\ x\ge 0 \qquad\textbf{(B)}\ x\le n\qquad\textbf{(C)}\ x=0\qquad\textbf{(D)}\ \text{the set of all real numbers}\qquad\textbf{(E)}\ \text{none of these} $
2013 AMC 8, 4
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra \$2.50 to cover her portion of the total bill. What was the total bill?
$\textbf{(A)}\ \$120 \qquad \textbf{(B)}\ \$128 \qquad \textbf{(C)}\ \$140 \qquad \textbf{(D)}\ \$144 \qquad \textbf{(E)}\ \$160$
2018 AMC 10, 7
In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the semicircles. The ratio $A:B$ is $1:18$. What is $N$?
[asy] draw((0,0)--(18,0)); draw(arc((9,0),9,0,180));
filldraw(arc((1,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((3,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((5,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((7,0),1,0,180)--cycle,gray(0.8)); label("...",(9,0.5)); filldraw(arc((11,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((13,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((15,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((17,0),1,0,180)--cycle,gray(0.8));
[/asy]
$\textbf{(A) } 16 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 36$
2010 AMC 12/AHSME, 21
The graph of $ y \equal{} x^6 \minus{} 10x^5 \plus{} 29x^4 \minus{} 4x^3 \plus{} ax^2$ lies above the line $ y \equal{} bx \plus{} c$ except at three values of $ x$, where the graph and the line intersect. What is the largest of those values?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
2021 AMC 12/AHSME Spring, 11
Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$
$\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18$
2009 AMC 12/AHSME, 1
Each morning of her five-day workweek, Jane bought either a $ 50$-cent muffin or a $ 75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
1988 AMC 12/AHSME, 22
For how many integers $x$ does a triangle with side lengths $10$, $24$ and $x$ have all its angles acute?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{more than } 7 $
1988 AMC 12/AHSME, 10
In an experiment, a scientific constant $C$ is determined to be $2.43865$ with an error of at most $\pm 0.00312$. The experimenter wishes to announce a value for $C$ in which every digit is significant. That is, whatever $C$ is, the announced value must be the correct result when C is rounded to that number of digits. The most accurate value the experimenter can announce for $C$ is
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 2.4\qquad\textbf{(C)}\ 2.43\qquad\textbf{(D)}\ 2.44\qquad\textbf{(E)}\ 2.439 $
2020 USOJMO, 3
An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
[list=]
[*]The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.)
[*]No two beams have intersecting interiors.
[*]The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.
[/list]
What is the smallest positive number of beams that can be placed to satisfy these conditions?
[i]Proposed by Alex Zhai[/i]
2014 AMC 8, 15
The circumference of the circle with center $O$ is divided into 12 equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?
[asy]
size(230);
defaultpen(linewidth(0.65));
pair O=origin;
pair[] circum = new pair[12];
string[] let = {"$A$","$B$","$C$","$D$","$E$","$F$","$G$","$H$","$I$","$J$","$K$","$L$"};
draw(unitcircle);
for(int i=0;i<=11;i=i+1)
{
circum[i]=dir(120-30*i);
dot(circum[i],linewidth(2.5));
label(let[i],circum[i],2*dir(circum[i]));
}
draw(O--circum[4]--circum[0]--circum[6]--circum[8]--cycle);
label("$x$",circum[0],2.75*(dir(circum[0]--circum[4])+dir(circum[0]--circum[6])));
label("$y$",circum[6],1.75*(dir(circum[6]--circum[0])+dir(circum[6]--circum[8])));
label("$O$",O,dir(60));
[/asy]
$\textbf{(A) }75\qquad\textbf{(B) }80\qquad\textbf{(C) }90\qquad\textbf{(D) }120\qquad \textbf{(E) }150$
2011 AMC 12/AHSME, 18
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
$ \textbf{(A)}\ 5\sqrt{2}-7 \qquad
\textbf{(B)}\ 7-4\sqrt{3} \qquad
\textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad
\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad
\textbf{(E)}\ \frac{\sqrt{3}}{9} $
2024 AMC 8 -, 18
Three concentric circles centered at $O$ have radii of $1$, $2$, and $3$. Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle $BOC$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle{BOC}$ in degrees?
[asy]
size(100);
import graph;
draw(circle((0,0),3));
real radius = 3;
real angleStart = -54; // starting angle of the sector
real angleEnd = 54; // ending angle of the sector
label("$O$",(0,0),W);
pair O = (0, 0);
filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, lightgray);
filldraw(circle((0,0),2),lightgray);
filldraw(circle((0,0),1),white);
draw((1.763,2.427)--(0,0)--(1.763,-2.427));
label("$B$",(1.763,2.427),NE);
label("$C$",(1.763,-2.427),SE);
[/asy]
$\textbf{(A)}\ 108 \qquad \textbf{(B)}\ 120 \qquad \textbf{(C)}\ 135 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$
2011 AMC 10, 18
Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?
[asy]
pathpen = linewidth(.7); pointpen = black;
pair A=(-1,0), B=-A, C=(0,1); fill(arc(C,1,0,180)--arc(A,1,90,0)--arc(B,1,180,90)--cycle, gray(0.5)); D(CR(D("A",A,SW),1)); D(CR(D("B",B,SE),1)); D(CR(D("C",C,N),1));[/asy]
${
\textbf{(A)}\ 3 - \frac{\pi}{2} \qquad
\textbf{(B)}\ \frac{\pi}{2} \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ \frac{3\pi}{4} \qquad
\textbf{(E)}\ 1+\frac{\pi}{2}} $
1959 AMC 12/AHSME, 18
The arithmetic mean (average) of the first $n$ positive integers is:
$ \textbf{(A)}\ \frac{n}{2} \qquad\textbf{(B)}\ \frac{n^2}{2}\qquad\textbf{(C)}\ n\qquad\textbf{(D)}\ \frac{n-1}{2}\qquad\textbf{(E)}\ \frac{n+1}{2} $
1971 AMC 12/AHSME, 9
An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. If the distance between the points of contact of the belt with the pulleys is $24$ inches, then the distance between the centers of the pulleys in inches is
$\textbf{(A) }24\qquad\textbf{(B) }2\sqrt{119}\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad \textbf{(E) }4\sqrt{35}$
2019 USAMO, 4
Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that:
[list]
[*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and
[*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$.
[/list]
[i]Proposed by Ricky Liu[/i]
2022 AMC 10, 4
A donkey suffers an attack of hiccups and the first hiccup happens at $\text{4:00}$ one afternoon. Suppose that the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $\text{700th}$ hiccup occur?
$\textbf{(A) }$ $15$ seconds after $\text{4:58}$
$\textbf{(B) }$ $20$ seconds after $\text{4:58}$
$\textbf{(C)}$ $25$ seconds after $\text{4:58}$
$\textbf{(D) }$ $30$ seconds after $\text{4:58}$
$\textbf{(E) }$ $35$ seconds after $\text{4:58}$
2002 AMC 8, 9
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)
[asy]
/* AMC8 2002 #8, 9, 10 Problem */
size(3inch, 1.5inch);
for ( int y = 0; y <= 5; ++y )
{
draw((0,y)--(18,y));
}
draw((0,0)--(0,5));
draw((6,0)--(6,5));
draw((9,0)--(9,5));
draw((12,0)--(12,5));
draw((15,0)--(15,5));
draw((18,0)--(18,5));
draw(scale(0.8)*"50s", (7.5,4.5));
draw(scale(0.8)*"4", (7.5,3.5));
draw(scale(0.8)*"8", (7.5,2.5));
draw(scale(0.8)*"6", (7.5,1.5));
draw(scale(0.8)*"3", (7.5,0.5));
draw(scale(0.8)*"60s", (10.5,4.5));
draw(scale(0.8)*"7", (10.5,3.5));
draw(scale(0.8)*"4", (10.5,2.5));
draw(scale(0.8)*"4", (10.5,1.5));
draw(scale(0.8)*"9", (10.5,0.5));
draw(scale(0.8)*"70s", (13.5,4.5));
draw(scale(0.8)*"12", (13.5,3.5));
draw(scale(0.8)*"12", (13.5,2.5));
draw(scale(0.8)*"6", (13.5,1.5));
draw(scale(0.8)*"13", (13.5,0.5));
draw(scale(0.8)*"80s", (16.5,4.5));
draw(scale(0.8)*"8", (16.5,3.5));
draw(scale(0.8)*"15", (16.5,2.5));
draw(scale(0.8)*"10", (16.5,1.5));
draw(scale(0.8)*"9", (16.5,0.5));
label(scale(0.8)*"Country", (3,4.5));
label(scale(0.8)*"Brazil", (3,3.5));
label(scale(0.8)*"France", (3,2.5));
label(scale(0.8)*"Peru", (3,1.5));
label(scale(0.8)*"Spain", (3,0.5));
label(scale(0.9)*"Juan's Stamp Collection", (9,0), S);
label(scale(0.9)*"Number of Stamps by Decade", (9,5), N);
[/asy]
In dollars and cents, how much did his South American stampes issued before the '70s cost him?
$ \text{(A)}\ \textdollar 0.40\qquad\text{(B)}\ \textdollar 1.06\qquad\text{(C)}\ \textdollar 1.80\qquad\text{(D)}\ \textdollar 2.38\qquad\text{(E)}\ \textdollar 2.64 $
2012 AMC 12/AHSME, 19
A unit cube has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3'$, and $P_4'$. Vertices $P_2, P_3$, and $P_4$ are adjacent to $P_1$, and for $1\leq i\leq 4$, vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'$, and $P_1'P_4'$. What is the octahedron's side length?
[asy]
import three;
size(7.5cm);
triple eye = (-4, -8, 3);
currentprojection = perspective(eye);
triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience
triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]};
// draw octahedron
triple pt(int k){ return (3*P[k] + P[1])/4; }
triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; }
draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6));
draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6));
draw(ptp(2)--pt(4), gray(0.6));
draw(pt(2)--ptp(4), gray(0.6));
draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype("4 4"));
draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype("4 4"));
// draw cube
for(int i = 0; i < 4; ++i){
draw(P[1]--P[i]); draw(Pp[1]--Pp[i]);
for(int j = 0; j < 4; ++j){
if(i == 1 || j == 1 || i == j) continue;
draw(P[i]--Pp[j]); draw(Pp[i]--P[j]);
}
dot(P[i]); dot(Pp[i]);
dot(pt(i)); dot(ptp(i));
}
label("$P_1$", P[1], dir(P[1]));
label("$P_2$", P[2], dir(P[2]));
label("$P_3$", P[3], dir(-45));
label("$P_4$", P[4], dir(P[4]));
label("$P'_1$", Pp[1], dir(Pp[1]));
label("$P'_2$", Pp[2], dir(Pp[2]));
label("$P'_3$", Pp[3], dir(-100));
label("$P'_4$", Pp[4], dir(Pp[4]));
[/asy]
$ \textbf{(A)}\ \frac{3\sqrt{2}}{4}\qquad\textbf{(B)}\ \frac{7\sqrt{6}}{16}\qquad\textbf{(C)}\ \frac{\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \frac{\sqrt{6}}{2} $
2012 AIME Problems, 3
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible comittees that can be formed subject to these requirements.