Found problems: 3632
2015 AMC 12/AHSME, 24
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$?
$ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $
1992 AMC 12/AHSME, 19
For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a [i]cuboctahedron[/i]. The ratio of the volume of the cuboctahedron to the volume of the original cube is closest to which of these?
$ \textbf{(A)}\ 75\%\qquad\textbf{(B)}\ 78\%\qquad\textbf{(C)}\ 81\%\qquad\textbf{(D)}\ 84\%\qquad\textbf{(E)}\ 87\% $
1996 AIME Problems, 13
In triangle $ABC, AB=\sqrt{30}, AC=\sqrt{6},$ and $BC=\sqrt{15}.$ There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$ and $\angle ADB$ is a right angle. The ratio \[ \frac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)} \] can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2021 AIME Problems, 12
A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1991 AMC 12/AHSME, 4
Which of the following triangles cannot exist?
$\textbf{(A)}\ \text{An acute isosceles triangle}$
$\textbf{(B)}\ \text{An isosceles right triangle}$
$\textbf{(C)}\ \text{An obtuse right triangle}$
$\textbf{(D)}\ \text{A scalene right triangle}$
$\textbf{(E)}\ \text{A scalene obtuse triangle}$
2019 AIME Problems, 13
Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1$. Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1}$, $\overline{PA_2}$, and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac17$, while the region bounded by $\overline{PA_3}$, $\overline{PA_4}$, and the minor arc $\widehat{A_3A_4}$ of the circle has area $\tfrac 19$. There is a positive integer $n$ such that the area of the region bounded by $\overline{PA_6}$, $\overline{PA_7}$, and the minor arc $\widehat{A_6A_7}$ is equal to $\tfrac18 - \tfrac{\sqrt 2}n$. Find $n$.
2010 AIME Problems, 12
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $ 8: 7$. Find the minimum possible value of their common perimeter.
2022 AMC 10, 16
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?
$\textbf{(A) }\frac{24}{5}\qquad\textbf{(B) }\frac{42}{5}\qquad\textbf{(C) }\frac{81}{5}\qquad\textbf{(D) }30\qquad\textbf{(E) }48$
1985 USAMO, 1
Determine whether or not there are any positive integral solutions of the simultaneous equations
\begin{align*}x_1^2+x_2^2+\cdots+x_{1985}^2&=y^3,\\
x_1^3+x_2^3+\cdots+x_{1985}^3&=z^2\end{align*}
with distinct integers $x_1$, $x_2$, $\ldots$, $x_{1985}$.
1989 AMC 12/AHSME, 23
A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly $1989$ minutes?
[asy]
draw((0,0)--(20,0), EndArrow);
draw((0,0)--(0,25), EndArrow);
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,10)--(10,10)--(10,0)--(15,0)--(15,15)--(0,15)--(0,20)--(10,20),linewidth(2));
draw((0,20)--(10,20), EndArrow);
draw((3.5,.5)--(4,.5)--(4,2), EndArrow);
draw((4,3.5)--(4,4)--(2.5,4), EndArrow);
draw((2,5.5)--(1,5.5)--(1,7), EndArrow);
draw((1,8)--(1,9)--(2.5,9), EndArrow);
draw((8,9.5)--(9,9.5)--(9,8), EndArrow);
draw((10.5,2)--(10.5,1)--(12,1), EndArrow);
draw((13,.5)--(14,.5)--(14,2), EndArrow);
draw((14.5,13)--(14.5,14)--(13,14), EndArrow);
draw((2,15.5)--(1,15.5)--(1,17), EndArrow);
draw((.5,18)--(.5,19)--(2,19), EndArrow);
label("x", (21,0), E);
label("y", (0,26), N);
label("4", (0,20), W);
label("3", (0,15), W);
label("2", (0,10), W);
label("1", (0,5), W);
label("0", (0,0), SW);
label("1", (5,0), S);
label("2", (10,0), S);
label("3", (15,0), S);
[/asy]
$\textbf{(A)}\ (35,44) \qquad\textbf{(B)}\ (36,45) \qquad\textbf{(C)}\ (37,45) \qquad\textbf{(D)}\ (44,35) \qquad\textbf{(E)}\ (45,36)$
1964 AMC 12/AHSME, 18
Let $n$ be the number of pairs of values of $b$ and $c$ such that $3x+by+c=0$ and $cx-2y+12=0$ have the same graph. Then $n$ is:
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ \text{finite but more than 2}\qquad\textbf{(E)}\ \text{greater than any finite number} $
1959 AMC 12/AHSME, 10
In triangle $ABC$ with $\overline{AB}=\overline{AC}=3.6$, a point $D$ is taken on $AB$ at a distance $1.2$ from $A$. Point $D$ is joined to $E$ in the prolongation of $AC$ so that triangle $AED$ is equal in area to $ABC$. Then $\overline{AE}$ is:
$ \textbf{(A)}\ 4.8 \qquad\textbf{(B)}\ 5.4\qquad\textbf{(C)}\ 7.2\qquad\textbf{(D)}\ 10.8\qquad\textbf{(E)}\ 12.6 $
1995 AIME Problems, 8
For how many ordered pairs of positive integers $(x,y),$ with $y<x\le 100,$ are both $\frac xy$ and $\frac{x+1}{y+1}$ integers?
1998 AMC 8, 5
Which of the following numbers is largest?
$ \text{(A)}\ 9.12344\qquad\text{(B)}\ 9.123\overline{4}\qquad\text{(C)}\ 9.12\overline{34}\qquad\text{(D)}\ 9.1\overline{234}\qquad\text{(E)}\ 9.\overline{1234} $
2001 AMC 12/AHSME, 20
Points $ A \equal{} (3,9), B \equal{} (1,1), C \equal{} (5,3),$ and $ D \equal{} (a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ ABCD$. The quadrilateral formed by joining the midpoints of $ \overline{AB}, \overline{BC}, \overline{CD},$ and $ \overline{DA}$ is a square. What is the sum of the coordinates of point $ D$?
$ \textbf{(A)} \ 7 \qquad \textbf{(B)} \ 9 \qquad \textbf{(C)} \ 10 \qquad \textbf{(D)} \ 12 \qquad \textbf{(E)} \ 16$
1991 AMC 12/AHSME, 24
The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?
$ \textbf{(A)}\ y = \log_{10}\left(\frac{x + 90}{9}\right)\qquad\textbf{(B)}\ y = \log_{x}10\qquad\textbf{(C)}\ y = \frac{1}{x + 1}\qquad\textbf{(D)}\ y = 10^{-x}\qquad\textbf{(E)}\ y = 10^{x} $
1961 AMC 12/AHSME, 31
In triangle $ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:
${{ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 3:4 \qquad\textbf{(C)}\ 4:3 \qquad\textbf{(D)}\ 3:1 }\qquad\textbf{(E)}\ 7:1 } $
1986 AMC 12/AHSME, 18
A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse of $50\%$ longer than the minor axis, the length of the major axis is
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ \frac{9}{4}\qquad\textbf{(E)}\ 3$
2024 AMC 12/AHSME, 5
A data set containing $20$ numbers, some of which are $6$, has mean $45$. When all the 6s are removed, the data set has mean $66$. How many 6s were in the original data set?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$
2023 AMC 10, 23
An arithmetic sequence has $n \geq 3$ terms, initial term $a$ and common difference $d > 1$. Carl wrote down all the terms in this sequence correctly except for one term which was off by $1$. The sum of the terms was $222$. What was $a + d + n$
$\textbf{(A) } 24 \qquad \textbf{(B) } 20 \qquad \textbf{(C) } 22 \qquad \textbf{(D) } 28 \qquad \textbf{(E) } 26$
2011 AMC 10, 21
Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?
$ \textbf{(A)}\ \frac{7}{11}\qquad\textbf{(B)}\ \frac{9}{13}\qquad\textbf{(C)}\ \frac{11}{15}\qquad\textbf{(D)}\ \frac{15}{19}\qquad\textbf{(E)}\ \frac{15}{16} $
2016 AMC 10, 24
A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?
$\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$
2011 AMC 12/AHSME, 9
At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her sibling and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and half the twins. How many handshakes took place?
$ \textbf{(A)}\ 324 \qquad
\textbf{(B)}\ 441 \qquad
\textbf{(C)}\ 630 \qquad
\textbf{(D)}\ 648 \qquad
\textbf{(E)}\ 882$
2019 USAJMO, 5
Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that:
[list]
[*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and
[*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$.
[/list]
[i]Proposed by Ricky Liu[/i]
2010 AMC 10, 11
A shopper plans to purchase an item that has a listed price greater than $ \$100$ and can use any one of the three coupons. Coupon A gives $ 15\%$ off the listed price, Coupon B gives $ \$30$ the listed price, and Coupon C gives $ 25\%$ off the amount by which the listed price exceeds $ \$100$.
Let $ x$ and $ y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $ y\minus{}x$?
$ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 80\qquad\textbf{(E)}\ 100$