Found problems: 3632
1976 AMC 12/AHSME, 21
What is the smallest positive odd integer $n$ such that the product \[2^{1/7}2^{3/7}\cdots2^{(2n+1)/7}\] is greater than $1000$? (In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.)
$\textbf{(A) }7\qquad\textbf{(B) }9\qquad\textbf{(C) }11\qquad\textbf{(D) }17\qquad \textbf{(E) }19$
1976 AMC 12/AHSME, 6
If $c$ is a real number and the negative of one of the solutions of $x^2-3x+c=0$ is a solution of $x^2+3x-c=0$, then the solutions of $x^2-3x+c=0$ are
$\textbf{(A) }1,~2\qquad\textbf{(B) }-1,~-2\qquad\textbf{(C) }0,~3\qquad\textbf{(D) }0,~-3\qquad \textbf{(E) }\frac{3}{2},~\frac{3}{2}$
1978 AMC 12/AHSME, 11
If $r$ is positive and the line whose equation is $x + y = r$ is tangen to the circle whose equation is $x^2 + y ^2 = r$, then $r$ equals
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{2}\qquad \textbf{(E) }2\sqrt{2}$
1986 AMC 12/AHSME, 26
It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The number of different constants $m$ for which such a triangle exists is
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \text{more than 3} $
2024 AMC 12/AHSME, 13
The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $(-1,\tfrac{1}{2})$ over this axis?
$\textbf{(A) }\left(-1,-\frac{3}{2}\right)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\tfrac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\frac{1}{2}\right)$
2015 USAJMO, 4
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that\[f(x)+f(t)=f(y)+f(z)\]for all rational numbers $x<y<z<t$ that form an arithmetic progression. ($\mathbb{Q}$ is the set of all rational numbers.)
2017 AIME Problems, 12
Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,...$ and a sequence of points on the circles $A_1,A_2,A_3,...$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r=\frac{11}{60}$, the distance from the center of $C_0$ to $B$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(6cm);
real r = 0.8;
pair nthCircCent(int n){
pair ans = (0, 0);
for(int i = 1; i <= n; ++i)
ans += rotate(90 * i - 90) * (r^(i - 1) - r^i, 0);
return ans;
}
void dNthCirc(int n){
draw(circle(nthCircCent(n), r^n));
}
dNthCirc(0);
dNthCirc(1);
dNthCirc(2);
dNthCirc(3);
dot("$A_0$", (1, 0), dir(0));
dot("$A_1$", nthCircCent(1) + (0, r), dir(135));
dot("$A_2$", nthCircCent(2) + (-r^2, 0), dir(0));
[/asy]
2010 AMC 12/AHSME, 2
A big $ L$ is formed as shown. What is its area?
[asy]unitsize(4mm);
defaultpen(linewidth(.8pt)+fontsize(12pt));
draw((0,0)--(5,0)--(5,2)--(2,2)--(2,8)--(0,8)--cycle);
label("5",(2.5,0),S);
label("2",(5,1),E);
label("2",(1,8),N);
label("8",(0,4),W);[/asy]$ \textbf{(A)}\ 22 \qquad
\textbf{(B)}\ 24 \qquad
\textbf{(C)}\ 26 \qquad
\textbf{(D)}\ 28 \qquad
\textbf{(E)}\ 30$
1989 AMC 12/AHSME, 22
A child has a set of $96$ distinct blocks. Each block is one of $2$ materials ([i]plastic, wood[/i]), $3$ sizes ([i]small, medium, large[/i]), $4$ colors ([i]blue, green, red, yellow[/i]), and $4$ shapes ([i]circle, hexagon, square, triangle[/i]). How many blocks in the set are different from the "[i]plastic medium red circle[/i]" in exactly two ways? (The "[i]wood medium red square[/i]" is such a block.)
$ \textbf{(A)}\ 29 \qquad\textbf{(B)}\ 39 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ 62 $
1969 AMC 12/AHSME, 28
Let $n$ be the number of points $P$ interior to the region bounded by a circle with radius $1$, such that the sum of the squares of the distances from $P$ to the endpoints of a given diameter is $3$. Then $n$ is:
$\textbf{(A) }0\qquad
\textbf{(B) }1\qquad
\textbf{(C) }2\qquad
\textbf{(D) }4\qquad
\textbf{(E) }\text{infinite}$
2017 AMC 12/AHSME, 18
The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$?
$\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$
1964 AMC 12/AHSME, 20
The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is:
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ -1\qquad\textbf{(E)}\ -19 $
2020 AMC 10, 15
A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 23$
1991 AMC 12/AHSME, 16
One hundred students at Century High School participated in the AHSME last year, and their mean score was $100$. The number of non-seniors taking the AHSME was $50\%$ more than the number of seniors, and the mean score of the seniors was $50\%$ higher than that of the non-seniors. What was the mean score of the seniors?
$ \textbf{(A)}\ 100\qquad\textbf{(B)}\ 112.5\qquad\textbf{(C)}\ 120\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 150 $
2016 AMC 10, 7
The ratio of the measures of two acute angles is $5:4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
$\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$
2012 AMC 12/AHSME, 5
A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $280$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?
$ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 96 $
2012 AMC 10, 11
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
$ \textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\ 2304 $
2018 AMC 10, 23
How many ordered pairs $(a, b)$ of positive integers satisfy the equation
$$a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),$$
where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$
2015 AMC 12/AHSME, 25
A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$?
$ \textbf{(A)}\ 2016 \qquad\textbf{(B)}\ 2024 \qquad\textbf{(C)}\ 2032 \qquad\textbf{(D)}\ 2040 \qquad\textbf{(E)}\ 2048$
1990 AMC 12/AHSME, 30
If $R_n=\frac{1}{2}(a^n+b^n)$ where $a=3+2\sqrt{2}$, $b=3-2\sqrt{2}$, and $n=0,1,2, ...,$ then $R_{12345}$ is an integer. Its units digit is
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 9 $
2024 AMC 12/AHSME, 15
A triangle in the coordinate plane has vertices $A(\log_21,\log_22)$, $B(\log_23,\log_24)$, and $C(\log_27,\log_28)$. What is the area of $\triangle ABC$?
$
\textbf{(A) }\log_2\frac{\sqrt3}7\qquad
\textbf{(B) }\log_2\frac3{\sqrt7}\qquad
\textbf{(C) }\log_2\frac7{\sqrt3}\qquad
\textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad
\textbf{(E) }\log_2\frac{11}{\sqrt3}\qquad
$
2013 AIME Problems, 7
A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$.
2025 AIME, 13
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and
\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]
$x_{2025}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$.
1968 AMC 12/AHSME, 29
Given the three numbers $x, y=x^x, z=x^{(x^x)}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are:
$\textbf{(A)}\ x, z, y \qquad\textbf{(B)}\ x, y, z \qquad\textbf{(C)}\ y, x, z \qquad\textbf{(D)}\ y, z, x \qquad\textbf{(E)}\ z, x, y$
1993 AMC 12/AHSME, 28
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \le x \le 4$ and $1 \le y \le 4$?
$ \textbf{(A)}\ 496 \qquad\textbf{(B)}\ 500 \qquad\textbf{(C)}\ 512 \qquad\textbf{(D)}\ 516 \qquad\textbf{(E)}\ 560 $