Found problems: 3632
1991 AMC 12/AHSME, 7
If $x = \frac{a}{b}$, $a \ne b$ and $b \ne 0$, then $\frac{a + b}{a - b} = $
$ \textbf{(A)}\ \frac{x}{x + 1}\qquad\textbf{(B)}\ \frac{x + 1}{x - 1}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ x - \frac{1}{x}\qquad\textbf{(E)}\ x + \frac{1}{x} $
1994 AMC 12/AHSME, 1
$4^4 \cdot 9^4 \cdot 4^9 \cdot 9^9=$
$ \textbf{(A)}\ 13^{13} \qquad\textbf{(B)}\ 13^{36} \qquad\textbf{(C)}\ 36^{13} \qquad\textbf{(D)}\ 36^{36} \qquad\textbf{(E)}\ 1296^{26} $
1987 AMC 12/AHSME, 14
$ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Then $\sin \theta=$
[asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((0,0)--(2,1));
draw((0,0)--(1,2));
label("A", (0,0), SW);
label("B", (0,2), NW);
label("C", (2,2), NE);
label("D", (2,0), SE);
label("M", (1,2), N);
label("N", (2,1), E);
label("$\theta$", (.5,.5), SW);
[/asy]
$ \textbf{(A)}\ \frac{\sqrt{5}}{5} \qquad\textbf{(B)}\ \frac{3}{5} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{5} \qquad\textbf{(D)}\ \frac{4}{5} \qquad\textbf{(E)}\ \text{none of these} $
2015 AMC 12/AHSME, 20
For every positive integer $n$, let $\operatorname{mod_5}(n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f : \{0, 1, 2, 3, \dots\} \times \{0, 1, 2, 3, 4\} \to \{0, 1, 2, 3, 4\}$ recursively as follows:
\[f(i, j) = \begin{cases}
\operatorname{mod_5}(j+1) & \text{if }i=0\text{ and }0\leq j\leq 4 \\
f(i-1, 1) & \text{if }i\geq 1\text{ and }j=0 \text{, and}\\
f(i-1, f(i, j-1)) & \text{if }i\geq 1\text{ and }1\leq j\leq 4
\end{cases}\]
What is $f(2015, 2)$?
$\textbf{(A) }0 \qquad\textbf{(B) }1 \qquad\textbf{(C) }2 \qquad\textbf{(D) }3 \qquad\textbf{(E) }4$
1979 AMC 12/AHSME, 12
[asy]
size(200);
pair A=(-2,0),B,C=(-1,0),D=(1,0),EE,O=(0,0);
draw(arc(O,1, 0, 180));
EE=midpoint(arc(O,1, 0, 90));
draw(A--EE);
draw(A--D);
B=intersectionpoint(arc(O,1, 180, 0),EE--A);
draw(O--EE);
label("$A$",A,W);
label("$B$",B,NW);
label("$C$",C,S);label("$D$",D,E);label("$E$",EE,NE);label("$O$",O,S);label("$45^\circ$",(0.25,0.1),fontsize(10pt));
//Credit to TheMaskedMagician for the diagram
[/asy]
In the adjoining figure, $CD$ is the diameter of a semi-circle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semi-circle, and $B$ is the point of intersection (distinct from $E$ ) of line segment $AE$ with the semi-circle. If length $AB$ equals length $OD$, and the measure of $\measuredangle EOD$ is $45^\circ$, then the
measure of $\measuredangle BAO$ is
$\textbf{(A) }10^\circ\qquad\textbf{(B) }15^\circ\qquad\textbf{(C) }20^\circ\qquad\textbf{(D) }25^\circ\qquad\textbf{(E) }30^\circ$
2012 AIME Problems, 3
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people such that exactly one person receives the type of meal ordered by that person.
2015 AIME Problems, 12
There are $2^{10}=1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical.
2025 AIME, 15
Let $N$ denote the numbers of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \le 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.
1970 AMC 12/AHSME, 24
An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is $2$, then the area of the hexagon is
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad \textbf{(E) }12$
2024 AMC 12/AHSME, 2
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T = aL + bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimate it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?
$\textbf{(A) } 240 \qquad \textbf{(B) } 246 \qquad \textbf{(C) } 252 \qquad \textbf{(D) } 258 \qquad \textbf{(E) } 264$
2019 AMC 12/AHSME, 23
Define binary operations $\diamondsuit$ and $\heartsuit$ by $$a \, \diamondsuit \, b = a^{\log_{7}(b)} \qquad \text{and} \qquad a \, \heartsuit \, b = a^{\frac{1}{\log_{7}(b)}}$$
for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3\, \heartsuit\, 2$ and $$a_n = (n\, \heartsuit\, (n-1)) \,\diamondsuit\, a_{n-1}$$
for all integers $n \geq 4$. To the nearest integer, what is $\log_{7}(a_{2019})$?
$\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$
2009 India IMO Training Camp, 3
Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following:
$ a_1 \equal{} a \\
a_2 \equal{} b \\
a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2})$.
Prove that there exists a natural number $ N$ such that $ a_n \equal{} gcd(a,b) \forall n\ge N$.
2022 AIME Problems, 13
Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\overline{abcd},$ where at least one of the digits $a, b, c, $ or $d$ is nonzero. Let $N$ be the number of distinct numerators when numbers in $S$ are written as fractions in lowest terms. For example, both $4$ and $410$ are counted among the distinct numerators for numbers in $S$ because $0.\overline{3636} = \frac{4}{11}$ and $0.\overline{1230} = \frac{410}{3333}.$ Find the remainder when $N$ is divided by $1000.$
2015 AMC 12/AHSME, 12
The parabolas $y=ax^2-2$ and $y=4-bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. What is $a+b$?
$\textbf{(A) }1\qquad\textbf{(B) }1.5\qquad\textbf{(C) }2\qquad\textbf{(D) }2.5\qquad\textbf{(E) }3$
1963 AMC 12/AHSME, 5
If $x$ and $\log_{10} x$ are real numbers and $\log_{10} x<0$, then:
$\textbf{(A)}\ x<0 \qquad
\textbf{(B)}\ -1<x<1 \qquad
\textbf{(C)}\ 0<x\le 1 $
$
\textbf{(D)}\ -1<x<0 \qquad
\textbf{(E)}\ 0<x<1$
2013 AMC 10, 25
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum $S=13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$?
${ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25 $
2006 AIME Problems, 7
Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.
2012 AMC 10, 6
The product of two positive numbers is $9$. The reciprocal of one of these numbers is $4$ times the reciprocal of the other number. What is the sum of the two numbers?
$ \textbf{(A)}\ \dfrac{10}{3}
\qquad\textbf{(B)}\ \dfrac{20}{3}
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ \dfrac{15}{2}
\qquad\textbf{(E)}\ 8
$
2020 CHMMC Winter (2020-21), 2
Find the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$.
2004 AIME Problems, 10
A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2016 AMC 12/AHSME, 23
Three numbers in the interval [0,1] are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?
$\textbf{(A) }\frac16\qquad\textbf{(B) }\frac13\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac23\qquad\textbf{(E) }\frac56$
2024 AMC 10, 17
In a race among 5 snails, there is at most one tie, but that tie can involve any number of snails. For example, the result of the race might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second, and Bruna is fifth. How many different results of the race are possible?
$
\textbf{(A) }180 \qquad
\textbf{(B) }361 \qquad
\textbf{(C) }420 \qquad
\textbf{(D) }431 \qquad
\textbf{(E) }720 \qquad
$
2020 AIME Problems, 9
Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m.$
1969 AMC 12/AHSME, 20
Let $P$ equal the product of $3,659,893,456,789,325,678$ and $342,973,489,379,256$. The number of digits in $P$ is:
$\textbf{(A) }36\qquad
\textbf{(B) }35\qquad
\textbf{(C) }34\qquad
\textbf{(D) }33\qquad
\textbf{(E) }32$
2017 AMC 12/AHSME, 6
The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point?
$\textbf{(A)}\ 4\sqrt2 \qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 5\sqrt2 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 6\sqrt2$