Found problems: 649
2017 AMC 12/AHSME, 21
A set $S$ is constructed as follows. To begin, $S=\{0,10\}$. Repeatedly, as long as possible, if $x$ is an integer root of some polynomial $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ for some $n\geq 1$, all of whose coefficients $a_i$ are elements of $S$, then $x$ is put into $S$. When no more elements can be added to $S$, how many elements does $S$ have?
$\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 11$
2024 AMC 12/AHSME, 13
There are real numbers $x,y,h$ and $k$ that satisfy the system of equations $$x^2 + y^2 - 6x - 8y = h$$ $$x^2 + y^2 - 10x + 4y = k$$
What is the minimum possible value of $h+k$?
$
\textbf{(A) }-54 \qquad
\textbf{(B) }-46 \qquad
\textbf{(C) }-34 \qquad
\textbf{(D) }-16 \qquad
\textbf{(E) }16 \qquad
$
2013 AMC 10, 21
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of $N$?
${ \textbf{(A)}\ 55\qquad\textbf{(B)}\ 89\qquad\textbf{(C)}\ 104\qquad\textbf{(D}}\ 144\qquad\textbf{(E)}\ 273 $
2019 AMC 12/AHSME, 7
Melanie computes the mean $\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1\text{s}$, $12$ $2\text{s}$, . . . , $12$ $28\text{s}$, $11$ $29\text{s}$, $11$ $30\text{s}$, and $7$ $31\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true?
$\textbf{(A) } \mu < d < M \qquad\textbf{(B) } M < d < \mu \qquad\textbf{(C) } d = M =\mu \qquad\textbf{(D) } d < M < \mu \qquad\textbf{(E) } d < \mu < M$
2020 AMC 10, 4
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
$\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26$
2012 AMC 12/AHSME, 21
Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square?
[asy]
size(200);
defaultpen(linewidth(1));
pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60);
pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A;
draw(A--B--C--D--E--F--cycle);
draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype("2 2"));
dot("$A$",A,W,linewidth(4));
dot("$B$",B,dir(0),linewidth(4));
dot("$C$",C,dir(0),linewidth(4));
dot("$D$",D,dir(20),linewidth(4));
dot("$E$",E,dir(100),linewidth(4));
dot("$F$",F,W,linewidth(4));
dot("$X$",X,dir(0),linewidth(4));
dot("$Y$",Y,N,linewidth(4));
dot("$Z$",Z,W,linewidth(4));
[/asy]
$ \textbf{(A)}\ 29\sqrt{3} \qquad\textbf{(B)}\ \frac{21}{2}\sqrt{2}+\frac{41}{2}\sqrt{3}\qquad\textbf{(C)}\ 20\sqrt{3}+16$
$\textbf{(D)}\ 20\sqrt{2}+13\sqrt{3}
\qquad\textbf{(E)}\ 21\sqrt{6}$
2018 AMC 12/AHSME, 1
A large urn contains $100$ balls, of which $36\%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72\%?$ (No red balls are to be removed.)
$
\textbf{(A) }28 \qquad
\textbf{(B) }32 \qquad
\textbf{(C) }36 \qquad
\textbf{(D) }50 \qquad
\textbf{(E) }64 \qquad
$
2020 AMC 12/AHSME, 2
What is the value of the following expression?
$$\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}$$
$\textbf{(A) } 1 \qquad \textbf{(B) } \frac{9951}{9950} \qquad \textbf{(C) } \frac{4780}{4779} \qquad \textbf{(D) } \frac{108}{107} \qquad \textbf{(E) } \frac{81}{80} $
2021 AMC 12/AHSME Fall, 18
Set $u_0 = \frac{1}{4},$ and for $k \geq 0$ let $u_{k+1}$ be determined by the recurrence $u_{k+1} = 2u_k - 2u_k^2.$ This sequence tends to a limit, call it $L.$ What is the least value of $k$ such that $$|u_k - L| \leq \frac{1}{2^{1000}}?$$
$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 97 \qquad\textbf{(C)}\ 253 \qquad\textbf{(D)}\
329 \qquad\textbf{(E)}\ 401$
2012 AMC 12/AHSME, 1
A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether?
$ \textbf{(A)}\ 9
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 13
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 15
$
2016 AMC 12/AHSME, 19
Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $a/b$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH$.)
$\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313$
2022 AMC 12/AHSME, 11
What is the product of all real numbers $x$ such that the distance on the number line between $\log_6x$ and $\log_69$ is twice the distance on the number line between $\log_610$ and $1$?
$\textbf{(A) }10\qquad\textbf{(B) }18\qquad\textbf{(C) }25\qquad\textbf{(D) }36\qquad\textbf{(E) }81$
2022 AMC 12/AHSME, 19
Don't have original wording:
In $\triangle{ABC}$ medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$ and $\triangle{AGE}$ is equilateral. Then $\cos(C)$ can be written as $\frac{m\sqrt p}n$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the square of any prime. What is $m+n+p?$
[asy]
import geometry;
unitsize(2cm);
real arg(pair p) {
return atan2(p.y, p.x) * 180/pi;
}
pair G=(0,0),E=(1,0),A=(1/2,sqrt(3)/2),D=1.5*G-0.5*A,C=2*E-A,B=2*D-C;
pair t(pair p) {
return rotate(-arg(dir(B--C)))*p;
}
path t(path p) {
return rotate(-arg(dir(B--C)))*p;
}
void d(path p, pen q = black+linewidth(1.5)) {
draw(t(p),q);
}
void o(pair p, pen q = 5+black) {
dot(t(p),q);
}
void l(string s, pair p, pair d) {
label(s, t(p),d);
}
d(A--B--C--cycle);
d(A--D);
d(B--E);
o(A);
o(B);
o(C);
o(D);
o(E);
o(G);
l("$A$",A,N);
l("$B$",B,SW);
l("$C$",C,SE);
l("$D$",D,S);
l("$E$",E,NE);
l("$G$",G,NW);
[/asy]
$\textbf{(A)}44~\textbf{(B)}48~\textbf{(C)}52~\textbf{(D)}56~\textbf{(E)}60$
2024 AMC 10, 7
The product of three integers is $60$. What is the least possible positive sum of the three integers?
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$
2020 AMC 12/AHSME, 10
In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$
$\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}$
2024 AMC 10, 3
For how many integer values of $x$ is $|2x|\leq 7\pi?$
$\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$
2023 AMC 12/AHSME, 6
When the roots of the polynomial \[P(x)=\prod_{i=1}^{10}(x-i)^{i}\] are removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of those intervals is $P(x)$ positive?
$\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$
2024 AMC 12/AHSME, 6
The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base $5$? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem.)
$
\textbf{(A) }18 \qquad
\textbf{(B) }20 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }24 \qquad
\textbf{(E) }26 \qquad
$
2016 AMC 10, 14
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$
$\textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 41 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 57$
2019 AMC 12/AHSME, 8
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?
$\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$
2022 AMC 10, 22
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]
$\textbf{(A) }4082\qquad\textbf{(B) }4095\qquad\textbf{(C) }4096\qquad\textbf{(D) }8178\qquad\textbf{(E) }8191$
2019 AMC 12/AHSME, 11
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$?
$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17$
2023 AMC 12/AHSME, 17
Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$?
$\textbf{(A)}\ 12\sqrt 3 \qquad\textbf{(B)}\ 8\sqrt 6 \qquad\textbf{(C)}\ 14\sqrt 2 \qquad\textbf{(D)}\ 20\sqrt 2 \qquad\textbf{(E)}\ 15\sqrt 3$
2023 AMC 12/AHSME, 15
Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true?
I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$.
II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.
III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.
$\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$
2024 AMC 10, 18
There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$?
$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21$