Found problems: 730
2018 AMC 10, 13
How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \dots$ are divisible by $101$?
$
\textbf{(A) }253 \qquad
\textbf{(B) }504 \qquad
\textbf{(C) }505 \qquad
\textbf{(D) }506 \qquad
\textbf{(E) }1009 \qquad
$
2019 AMC 10, 1
What is the value of $$2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9\,?$$
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$
2002 AMC 10, -1
This test and the matching AMC 12P were developed for the use of a group of Taiwan schools, in early January of 2002. When Taiwan had taken the contests, the AMC released the questions here as a set of practice questions for the 2002 AMC 10 and AMC 12 contests.
2024 AMC 12/AHSME, 14
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
$
2020 AMC 10, 5
What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$
$\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 25$
2022 AMC 12/AHSME, 15
The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism. A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?
$\textbf{(A) }\frac{24}{5}\qquad\textbf{(B) }\frac{42}{5}\qquad\textbf{(C) }\frac{81}{5}\qquad\textbf{(D) }30\qquad\textbf{(E) }48$
2002 AMC 12/AHSME, 25
The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
[asy]//Choice A
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.101562 x^4+0.265625 x^3+0.0546875 x^2-0.109375 x+0.125;
}
real g(real x) {
return 0.0625 x^4+0.0520833 x^3-0.21875 x^2-0.145833 x-2.5;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(A)}$", (-5,4.5));
[/asy]
[asy]//Choice B
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.541667 x^4+0.458333 x^3-0.510417 x^2-0.927083 x-2;
}
real g(real x) {
return -0.791667 x^4-0.208333 x^3-0.177083 x^2-0.260417 x-1;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(B)}$", (-5,4.5));
[/asy]
[asy]//Choice C
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.21875 x^2+0.28125 x+0.5;
}
real g(real x) {
return -0.375 x^2-0.75 x+0.5;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(C)}$", (-5,4.5));
[/asy]
[asy]//Choice D
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.015625 x^5-0.244792 x^3+0.416667 x+0.6875;
}
real g(real x) {
return 0.0284722 x^6-0.340278 x^4+0.874306 x^2-1.5625;
}
real z=3.14;
draw(graph(f,-z, z), heavygray);
draw(graph(g,-z, z), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(D)}$", (-5,4.5));
[/asy]
[asy]//Choice E
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.026067 x^4-0.0136612 x^3-0.157131 x^2-0.00961796 x+1.21598;
}
real g(real x) {
return -0.166667 x^3+0.125 x^2+0.479167 x-0.375;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(E)}$", (-5,4.5));
[/asy]
2019 AMC 10, 2
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$
2022 AMC 12/AHSME, 1
What is the value of
$$3 + \frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$
$\textbf{(A) } \frac{31}{10} \qquad \textbf{(B) } \frac{49}{15} \qquad \textbf{(C) } \frac{33}{10} \qquad \textbf{(D) } \frac{109}{33} \qquad \textbf{(E) } \frac{15}{4}$
2017 AMC 12/AHSME, 19
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$?
$\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$
2010 AMC 12/AHSME, 11
The solution of the equation $ 7^{x\plus{}7}\equal{}8^x$ can be expressed in the form $ x\equal{}\log_b 7^7$. What is $ b$?
$ \textbf{(A)}\ \frac{7}{15} \qquad
\textbf{(B)}\ \frac{7}{8} \qquad
\textbf{(C)}\ \frac{8}{7} \qquad
\textbf{(D)}\ \frac{15}{8} \qquad
\textbf{(E)}\ \frac{15}{7}$
2016 AMC 10, 25
How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600$ and $\text{lcm}(y,z)=900$?
$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64$
2009 AMC 10, 15
The figures $ F_1$, $ F_2$, $ F_3$, and $ F_4$ shown are the first in a sequence of figures. For $ n\ge3$, $ F_n$ is constructed from $ F_{n \minus{} 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $ F_{n \minus{} 1}$ had on each side of its outside square. For example, figure $ F_3$ has $ 13$ diamonds. How many diamonds are there in figure $ F_{20}$?
[asy]unitsize(3mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle;
marker m=marker(scale(5)*d,Fill);
path f1=(0,0);
path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1);
path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1);
path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2);
path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2);
path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)--
(3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3);
path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^
(-2,2)--(-3,3);
draw(f1,m);
draw(shift(5,0)*f2,m);
draw(shift(5,0)*g2);
draw(shift(12,0)*f3,m);
draw(shift(12,0)*g3);
draw(shift(21,0)*f4,m);
draw(shift(21,0)*g4);
label("$F_1$",(0,-4));
label("$F_2$",(5,-4));
label("$F_3$",(12,-4));
label("$F_4$",(21,-4));[/asy]$ \textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761$
2016 AMC 10, 21
Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$?
$\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}$
2020 AMC 12/AHSME, 11
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length $2$ so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region—inside the hexagon but outside all of the semicircles?
[asy]
size(140);
fill((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--cycle,gray(0.4));
fill(arc((2,0),1,180,0)--(2,0)--cycle,white);
fill(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle,white);
fill(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle,white);
fill(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle,white);
fill(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle,white);
fill(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle,white);
draw((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--(1,0));
draw(arc((2,0),1,180,0)--(2,0)--cycle);
draw(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle);
draw(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle);
draw(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle);
draw(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle);
draw(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle);
label("$2$",(3.5,3sqrt(3)/2),NE);
[/asy]
$\textbf{(A)}\ 6\sqrt3-3\pi \qquad\textbf{(B)}\ \frac{9\sqrt3}{2}-2\pi \qquad\textbf{(C)}\ \frac{3\sqrt3}{2}-\frac{\pi}{3} \qquad\textbf{(D)}\ 3\sqrt3-\pi \\ \qquad\textbf{(E)}\ \frac{9\sqrt3}{2}-\pi$
2024 AMC 12/AHSME, 17
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$
\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad
$
2022 AMC 10, 15
Quadrilateral $ABCD$ with side lengths $AB=7, BC = 24, CD = 20, DA = 15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi - b}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c$?
$\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997$
2017 AMC 12/AHSME, 9
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2021 AMC 10 Fall, 19
A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a+\frac{b\pi}{c}$, where $a,b$, and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c$?
$\textbf{(A)} ~10\qquad\textbf{(B)} ~11\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~13\qquad\textbf{(E)} ~14$
2022 AMC 12/AHSME, 19
Suppose that 13 cards numbered $1, 2, 3, \dots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes?
[asy]
size(11cm);
draw((0,0)--(2,0)--(2,3)--(0,3)--cycle);
label("7", (1,1.5));
draw((3,0)--(5,0)--(5,3)--(3,3)--cycle);
label("11", (4,1.5));
draw((6,0)--(8,0)--(8,3)--(6,3)--cycle);
label("8", (7,1.5));
draw((9,0)--(11,0)--(11,3)--(9,3)--cycle);
label("6", (10,1.5));
draw((12,0)--(14,0)--(14,3)--(12,3)--cycle);
label("4", (13,1.5));
draw((15,0)--(17,0)--(17,3)--(15,3)--cycle);
label("5", (16,1.5));
draw((18,0)--(20,0)--(20,3)--(18,3)--cycle);
label("9", (19,1.5));
draw((21,0)--(23,0)--(23,3)--(21,3)--cycle);
label("12", (22,1.5));
draw((24,0)--(26,0)--(26,3)--(24,3)--cycle);
label("1", (25,1.5));
draw((27,0)--(29,0)--(29,3)--(27,3)--cycle);
label("13", (28,1.5));
draw((30,0)--(32,0)--(32,3)--(30,3)--cycle);
label("10", (31,1.5));
draw((33,0)--(35,0)--(35,3)--(33,3)--cycle);
label("2", (34,1.5));
draw((36,0)--(38,0)--(38,3)--(36,3)--cycle);
label("3", (37,1.5));
[/asy]
$\textbf{(A) }4082\qquad\textbf{(B) }4095\qquad\textbf{(C) }4096\qquad\textbf{(D) }8178\qquad\textbf{(E) }8191$
2021 AMC 10 Fall, 7
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form [i]special[/i] if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\
12 \qquad\textbf{(E)}\ 13$
2021 AMC 10 Fall, 15
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR=6$ and $PR=7$. What is the area of the square?
[asy]
size(170);
defaultpen(linewidth(0.6));
real r = 3.5;
pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0),
R = intersectionpoint(B--P,C--Q);
draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7));
dot("$A$",A,S);
dot("$B$",B,S);
dot("$C$",C,N);
dot("$D$",D,N);
dot("$Q$",Q,S);
dot("$P$",P,W);
dot("$R$",R,1.3*S);
label("$7$",(P+R)/2,NE);
label("$6$",(R+B)/2,NE);
[/asy]
$\textbf{(A) }85\qquad\textbf{(B) }93\qquad\textbf{(C) }100\qquad\textbf{(D) }117\qquad\textbf{(E) }125$
2006 AMC 10, 23
Circles with centers $ A$ and $ B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $ C$ and $ D$, respectively. Lines $ AB$ and $ CD$ intersect at $ E$, and $ AE \equal{} 5$. What is $ CD$?
[asy]unitsize(2.5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=3;
pair A=(0,0), Ep=(5,0), B=(5+40/3,0);
pair M=midpoint(A--Ep);
pair C=intersectionpoints(Circle(M,2.5),Circle(A,3))[1];
pair D=B+8*dir(180+degrees(C));
dot(A);
dot(C);
dot(B);
dot(D);
draw(C--D);
draw(A--B);
draw(Circle(A,3));
draw(Circle(B,8));
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,SE);
label("$E$",Ep,SSE);
label("$D$",D,NW);[/asy]$ \textbf{(A) } 13\qquad \textbf{(B) } \frac {44}{3}\qquad \textbf{(C) } \sqrt {221}\qquad \textbf{(D) } \sqrt {255}\qquad \textbf{(E) } \frac {55}{3}$
2020 AMC 10, 14
Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$. What is the value of \[x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?\]
$\textbf{(A)}\ 360\qquad\textbf{(B)}\ 400\qquad\textbf{(C)}\ 420\qquad\textbf{(D)}\ 440\qquad\textbf{(E)}\ 480$
2018 AMC 12/AHSME, 1
Kate bakes a $20$-inch by $18$-inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain?
$
\textbf{(A) }90 \qquad
\textbf{(B) }100 \qquad
\textbf{(C) }180 \qquad
\textbf{(D) }200 \qquad
\textbf{(E) }360 \qquad
$