Found problems: 3632
2008 AMC 12/AHSME, 2
What is the reciprocal of $ \frac{1}{2}\plus{}\frac{2}{3}$?
$ \textbf{(A)}\ \frac{6}{7} \qquad
\textbf{(B)}\ \frac{7}{6} \qquad
\textbf{(C)}\ \frac{5}{3} \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ \frac{7}{2}$
1967 AMC 12/AHSME, 38
Given a set $S$ consisting of two undefined elements "pib" and "maa", and the four postulates: $P_1$: Every pib is a collection of maas, $P_2$: Any two distinct pibs have one and only one maa in common, $P_3$: Every maa belongs to two and only two pibs, $P_4$: There are exactly four pibs. Consider the three theorems: $T_1$: There are exactly six maas, $T_2$: There are exactly three maas in each pib, $T_3$: For each maa there is exactly one other maa not in the same pid with it. The theorems which are deducible from the postulates are:
$\textbf{(A)}\ T_3 \; \text{only}\qquad
\textbf{(B)}\ T_2 \; \text{and} \; T_3 \; \text{only} \qquad
\textbf{(C)}\ T_1 \; \text{and} \; T_2 \; \text{only}\\
\textbf{(D)}\ T_1 \; \text{and} \; T_3 \; \text{only}\qquad
\textbf{(E)}\ \text{all}$
2022 AMC 12/AHSME, 9
The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that
\[2^{a_7}=2^{27} \cdot a_7.\]
What is the minimum possible value of $a_2$?
$\textbf{(A)}8~\textbf{(B)}12~\textbf{(C)}16~\textbf{(D)}17~\textbf{(E)}22$
1972 AMC 12/AHSME, 4
The number of solutions to $\{1,~2\}\subseteq~X~\subseteq~\{1,~2,~3,~4,~5\}$, where $X$ is a subset of $\{1,~2,~3,~4,~5\}$ is
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }\text{None of these}$
2022 AIME Problems, 5
A straight river that is $264$ meters wide flows from west to east at a rate of $14$ meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at $80$ meters per minute, and Sherry swims at $60$ meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find $D$.
2015 AMC 12/AHSME, 17
An unfair coin lands on heads with a probability of $\tfrac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of $n$?
$ \textbf {(A) } 5 \qquad \textbf {(B) } 8 \qquad \textbf {(C) } 10 \qquad \textbf {(D) } 11 \qquad \textbf {(E) } 13 $
1964 AMC 12/AHSME, 28
The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first interm is an integer, and $n>1$, then the number of possible values for $n$ is:
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $
2017 AMC 12/AHSME, 19
Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 44$
2001 AMC 10, 17
Which of the cones listed below can be formed from a $ 252^\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides?
[asy]import graph;unitsize(1.5cm);defaultpen(fontsize(8pt));draw(Arc((0,0),1,-72,180),linewidth(.8pt));draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));label("$10$",(-0.5,0),S);draw(Arc((0,0),0.1,-72,180));label("$252^{\circ}$",(0.05,0.05),NE);[/asy]
[asy]
import three;
picture mainframe;
defaultpen(fontsize(11pt));
picture conePic(picture pic, real r, real h, real sh)
{
size(pic, 3cm);
triple eye = (11, 0, 5);
currentprojection = perspective(eye);
real R = 1, y = 2;
triple center = (0, 0, 0);
triple radPt = (0, R, 0);
triple negRadPt = (0, -R, 0);
triple heightPt = (0, 0, y);
draw(pic, arc(center, radPt, negRadPt, heightPt, CW));
draw(pic, arc(center, radPt, negRadPt, heightPt, CCW), linetype("8 8"));
draw(pic, center--radPt, linetype("8 8"));
draw(pic, center--heightPt, linetype("8 8"));
draw(pic, negRadPt--heightPt--radPt);
label(pic, (string) r, center--radPt, dir(270));
if (h != 0)
{
label(pic, (string) h, heightPt--center, dir(0));
}
if (sh != 0)
{
label(pic, (string) sh, heightPt--radPt, dir(0));
}
return pic;
}
picture pic1;
pic1 = conePic(pic1, 6, 0, 10);
picture pic2;
pic2 = conePic(pic2, 6, 10, 0);
picture pic3;
pic3 = conePic(pic3, 7, 0, 10);
picture pic4;
pic4 = conePic(pic4, 7, 10, 0);
picture pic5;
pic5 = conePic(pic5, 8, 0, 10);
picture aux1; picture aux2; picture aux3;
add(aux1, pic1.fit(), (0,0), W);
label(aux1, "$\textbf{(A)}$", (0,0), 22W, linewidth(4));
label(aux1, "$\textbf{(B)}$", (0,0), 3E);
add(aux1, pic2.fit(), (0,0), 35E);
add(aux2, aux1.fit(), (0,0), W);
label(aux2, "$\textbf{(C)}$", (0,0), 3E);
add(aux2, pic3.fit(), (0,0), 35E);
add(aux3, aux2.fit(), (0,0), W);
label(aux3, "$\textbf{(D)}$", (0,0), 3E);
add(aux3, pic4.fit(), (0,0), 35E);
add(mainframe, aux3.fit(), (0,0), W);
label(mainframe, "$\textbf{(E)}$", (0,0), 3E);
add(mainframe, pic5.fit(), (0,0), 35E);
add(mainframe.fit(), (0,0), N);
[/asy]
2016 AMC 10, 17
All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
$\textbf{(A)}\ 312 \qquad
\textbf{(B)}\ 343 \qquad
\textbf{(C)}\ 625 \qquad
\textbf{(D)}\ 729 \qquad
\textbf{(E)}\ 1680$
2021 USAMO, 2
The Planar National Park is a subset of the Euclidean plane consisting of several trails which meet at junctions. Every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions. (An example of one possible layout of the park is shown to the left below, in which there are six junctions and nine trails.)
[center]
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZS9mLzc1YmNjN2YxMWZhZTNhMTVkZTQ4NWE1ZDIyMDNhN2I5NzY0NTBlLnBuZw==&rn=Z3JhcGguUE5H[/img]
[/center]
A visitor walks through the park as follows: she begins at a junction and starts walking along a trail. At the end of that first trail, she enters a junction and turns left. On the next junction she turns right, and so on, alternating left and right turns at each junction. She does this until she gets back to the junction where she started. What is the largest possible number of times she could have entered any junction during her walk, over all possible layouts of the park?
1974 AMC 12/AHSME, 21
In a geometric series of positive terms the difference between the fifth and fourth terms is $576$, and the difference between the second and first terms is $9$. What is the sum of the first five terms of this series?
$ \textbf{(A)}\ 1061 \qquad\textbf{(B)}\ 1023 \qquad\textbf{(C)}\ 1024 \qquad\textbf{(D)}\ 768 \qquad\textbf{(E)}\ \text{none of these} $
1998 AMC 12/AHSME, 22
What is the value of the expression
\[ \frac {1}{\log_2 100!} \plus{} \frac {1}{\log_3 100!} \plus{} \frac {1}{\log_4 100!} \plus{} \cdots \plus{} \frac {1}{\log_{100} 100!}?
\]$ \textbf{(A)}\ 0.01 \qquad \textbf{(B)}\ 0.1 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 10$
2024 AIME, 7
Find the largest possible real part of \[(75+117i)z+\frac{96+144i}{z}\] where $z$ is a complex number with $|z|=4$.
2018 AMC 10, 13
A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?
[asy]
draw((0,0)--(4,0)--(4,3)--(0,0));
label("$A$", (0,0), SW);
label("$B$", (4,3), NE);
label("$C$", (4,0), SE);
label("$4$", (2,0), S);
label("$3$", (4,1.5), E);
label("$5$", (2,1.5), NW);
fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray(0.9));
[/asy]
$\textbf{(A) } 1+\frac12 \sqrt2 \qquad \textbf{(B) } \sqrt3 \qquad \textbf{(C) } \frac74 \qquad \textbf{(D) } \frac{15}{8} \qquad \textbf{(E) } 2 $
2011 AMC 10, 5
In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His errorneous product was $161$. What is the correct value of the product of $a$ and $b$?
$ \textbf{(A)}\ 116 \qquad
\textbf{(B)}\ 161 \qquad
\textbf{(C)}\ 204 \qquad
\textbf{(D)}\ 214 \qquad
\textbf{(E)}\ 224 $
1993 AMC 8, 8
To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains $60$ pills, then the supply of medicine would last approximately
$\text{(A)}\ 1\text{ month} \qquad \text{(B)}\ 4\text{ months} \qquad \text{(C)}\ 6\text{ months} \qquad \text{(D)}\ 8\text{ months} \qquad \text{(E)}\ 1\text{ year}$
1983 USAMO, 5
Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\le q\le n$, contained in the given interval is at most $(n+1)/2$.
2007 AMC 10, 22
A finite sequence of three-digit integers has the property that the tens and units digits of each terms are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with terms $ 247$, $ 475$, and $ 756$ and end with the term $ 824$. Let $ S$ be the sum of all the terms in the sequence. What is the largest prime number that always divides $ S$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 43$
2016 AMC 12/AHSME, 13
Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice's position and $60^\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?
$\textbf{(A)}\ 3.5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 4.5 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 5.5$
2018 AMC 12/AHSME, 24
Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. How many real numbers $x$ satisfy the equation $x^2 + 10{,}000\lfloor x \rfloor = 10{,}000x$?
$\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201$
2017 AMC 10, 5
The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$
2010 AMC 10, 17
A solid cube has side length $ 3$ inches. A $ 2$-inch by $ 2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
$ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 15$
2014 AMC 8, 3
Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book?
$\textbf{(A) }240\qquad\textbf{(B) }250\qquad\textbf{(C) }260\qquad\textbf{(D) }270\qquad \textbf{(E) }280$
1967 AMC 12/AHSME, 10
If $\frac{a}{10^x-1}+\frac{b}{10^x+2}=\frac{2 \cdot 10^x+3}{(10^x-1)(10^x+2)}$ is an identity for positive rational values of $x$, then the value of $a-b$ is:
$\textbf{(A)}\ \frac{4}{3} \qquad
\textbf{(B)}\ \frac{5}{3} \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ \frac{11}{4} \qquad
\textbf{(E)}\ 3$