Found problems: 3632
2006 AMC 12/AHSME, 21
Rectangle $ ABCD$ has area 2006. An ellipse with area $ 2006\pi$ passes through $ A$ and $ C$ and has foci at $ B$ and $ D$. What is the perimeter of the rectangle? (The area of an ellipse is $ \pi ab$, where $ 2a$ and $ 2b$ are the lengths of its axes.)
$ \textbf{(A) } \frac {16\sqrt {2006}}{\pi} \qquad \textbf{(B) } \frac {1003}4 \qquad \textbf{(C) } 8\sqrt {1003} \qquad \textbf{(D) } 6\sqrt {2006} \qquad \textbf{(E) } \frac {32\sqrt {1003}}\pi$
2012 AMC 10, 15
In a round-robin tournament with $6$ teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $
1967 AMC 12/AHSME, 21
In right triangle $ABC$ the hypotenuse $\overline{AB}=5$ and leg $\overline{AC}=3$. The bisector of angle $A$ meets the opposite side in $A_1$. A second right triangle $PQR$ is then constructed with hypotenuse $\overline{PQ}=A_1B$ and leg $\overline{PR}=A_1C$. If the bisector of angle $P$ meets the opposite side in $P_1$, the length of $PP_1$ is:
$\textbf{(A)}\ \frac{3\sqrt{6}}{4}\qquad
\textbf{(B)}\ \frac{3\sqrt{5}}{4}\qquad
\textbf{(C)}\ \frac{3\sqrt{3}}{4}\qquad
\textbf{(D)}\ \frac{3\sqrt{2}}{2}\qquad
\textbf{(E)}\ \frac{15\sqrt{2}}{16}$
2020 AMC 12/AHSME, 16
An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
$\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$
2019 AMC 12/AHSME, 17
Let $s_k$ denote the sum of the $\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$?
$\textbf{(A)} \; -6 \qquad \textbf{(B)} \; 0 \qquad \textbf{(C)} \; 6 \qquad \textbf{(D)} \; 10 \qquad \textbf{(E)} \; 26$
1993 AMC 12/AHSME, 25
Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not create a distinct triangle.) There are
[asy]
draw((0,0)--(6,10.2), EndArrow);
draw((0,0)--(6,-10.2), EndArrow);
draw((0,0)--(6,0), dotted);
dot((6,0));
label("P", (6,0), S);
[/asy]
$ \textbf{(A)}\ \text{exactly 2 such triangles} \\ \qquad\textbf{(B)}\ \text{exactly 3 such triangles} \\ \qquad\textbf{(C)}\ \text{exactly 7 such triangles} \\ \qquad\textbf{(D)}\ \text{exactly 15 such triangles} \\ \qquad\textbf{(E)}\ \text{more than 15 such triangles} $
2008 USAMO, 4
Let $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n \minus{} 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a [i]triangulation[/i] of $ \mathcal{P}$ into $ n \minus{} 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.
1969 AMC 12/AHSME, 16
When $(a-b)^n$, $n\geq 2$, $ab\neq 0$, is expanded by the binomial theorem, it is found that , when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals:
$\textbf{(A) }\tfrac12k(k-1)\qquad
\textbf{(B) }\tfrac12k(k+1)\qquad
\textbf{(C) }2k-1\qquad
\textbf{(D) }2k\qquad
\textbf{(E) }2k+1$
1989 AMC 12/AHSME, 21
A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up $36\%$ of the area of the flag, what percent of the area of the flag is blue?
[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--cycle);
draw((1,0)--(5,4));
draw((0,1)--(4,5));
draw((0,4)--(4,0));
draw((1,5)--(5,1));
label("RED", (1.2,3.7));
label("RED", (3.8,3.7));
label("RED", (1.2,1.3));
label("RED", (3.8,1.3));
label("BLUE", (2.5,2.5));
[/asy]
$ \textbf{(A)}\ 0.5 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6 $
2007 Romania Team Selection Test, 2
Let $ ABC$ be a triangle, let $ E, F$ be the tangency points of the incircle $ \Gamma(I)$ to the sides $ AC$, respectively $ AB$, and let $ M$ be the midpoint of the side $ BC$. Let $ N \equal{} AM \cap EF$, let $ \gamma(M)$ be the circle of diameter $ BC$, and let $ X, Y$ be the other (than $ B, C$) intersection points of $ BI$, respectively $ CI$, with $ \gamma$. Prove that
\[ \frac {NX} {NY} \equal{} \frac {AC} {AB}.
\]
[i]Cosmin Pohoata[/i]
1964 AMC 12/AHSME, 29
In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\dfrac{1}{2}$ inches. The length of $RS$, in inches, is:
[asy]
import olympiad;
pair F,R,S,D;
F=origin;
R=5*dir(aCos(9/16));
S=(7.5,0);
D=4*dir(aCos(9/16)+aCos(1/8));
label("$F$",F,SW);label("$R$",R,N); label("$S$",S,SE); label("$D$",D,W);
label("$7\frac{1}{2}$",(F+S)/2.5,SE);
label("$4$",midpoint(F--D),SW);
label("$5$",midpoint(F--R),W);
label("$6$",midpoint(D--R),N);
draw(F--D--R--F--S--R);
markscalefactor=0.1;
draw(anglemark(S,F,R)); draw(anglemark(F,D,R));
//Credit to throwaway1489 for the diagram[/asy]
$\textbf{(A)}\ \text{undetermined} \qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\dfrac{1}{2} \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 6\dfrac{1}{4}$
2018 AMC 12/AHSME, 5
What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common?
$
\textbf{(A) }3 \qquad
\textbf{(B) }4 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }10 \qquad
$
2019 AMC 12/AHSME, 15
Positive real numbers $a$ and $b$ have the property that
\[
\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100
\] and all four terms on the left are positive integers, where $\text{log}$ denotes the base 10 logarithm. What is $ab$?
$\textbf{(A) } 10^{52} \qquad \textbf{(B) } 10^{100} \qquad \textbf{(C) } 10^{144} \qquad \textbf{(D) } 10^{164} \qquad \textbf{(E) } 10^{200} $
1991 AMC 12/AHSME, 9
From time $t = 0$ to time $t = 1$ a population increased by $i\%$, and from time $t = 1$ to time $t = 2$ the population increased by $j\%$. Therefore, from time $t = 0$ to time $t = 2$ the population increased by
$ \textbf{(A)}\ (i + j)\%\qquad\textbf{(B)}\ ij\%\qquad\textbf{(C)}\ (i+ij)\%\qquad\textbf{(D)}\ \left(i + j + \frac{ij}{100}\right)\%\qquad\textbf{(E)}\left( i + j + \frac{i + j}{100}\right)\% $
2009 AMC 12/AHSME, 5
Kiana has two older twin brothers. The product of their ages is $ 128$. What is the sum of their three ages?
$ \textbf{(A)}\ 10\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 18\qquad
\textbf{(E)}\ 24$
1993 AMC 12/AHSME, 2
In $\triangle ABC$, $\angle A=55^{\circ}$, $\angle C=75^{\circ}$, $D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}$. If $DB=BE$, then $\angle BED=$
[asy]
size((100));
draw((0,0)--(10,0)--(8,10)--cycle);
draw((4,5)--(9.2,4));
dot((0,0));
dot((10,0));
dot((8,10));
dot((4,5));
dot((9.2,4));
label("A", (0,0), SW);
label("B", (8,10), N);
label("C", (10,0), SE);
label("D", (4,5), NW);
label("E", (9.2,4), E);
label("$55^{\circ}$", (.5,0), NE);
label("$75^{\circ}$", (9.8,0), NW);
[/asy]
$ \textbf{(A)}\ 50^{\circ} \qquad\textbf{(B)}\ 55^{\circ} \qquad\textbf{(C)}\ 60^{\circ} \qquad\textbf{(D)}\ 65^{\circ} \qquad\textbf{(E)}\ 70^{\circ} $
1980 USAMO, 3
Let $F_r=x^r\sin{rA}+y^r\sin{rB}+z^r\sin{rC}$, where $x,y,z,A,B,C$ are real and $A+B+C$ is an integral multiple of $\pi$. Prove that if $F_1=F_2=0$, then $F_r=0$ for all positive integral $r$.
2003 AIME Problems, 12
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of the committee?
2024 AMC 10, 5
In the following expression, Melanie changed some of the plus signs to minus signs: $$ 1 + 3+5+7+\cdots+97+99$$
When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
$
\textbf{(A) }14 \qquad
\textbf{(B) }15 \qquad
\textbf{(C) }16 \qquad
\textbf{(D) }17 \qquad
\textbf{(E) }18 \qquad
$
2020 AIME Problems, 10
Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\cdots+n^3$ is divided by $n+5$, the remainder is $17.$
2008 AMC 12/AHSME, 21
A permutation $ (a_1,a_2,a_3,a_4,a_5)$ of $ (1,2,3,4,5)$ is heavy-tailed if $ a_1 \plus{} a_2 < a_4 \plus{} a_5$. What is the number of heavy-tailed permutations?
$ \textbf{(A)}\ 36 \qquad
\textbf{(B)}\ 40 \qquad
\textbf{(C)}\ 44 \qquad
\textbf{(D)}\ 48 \qquad
\textbf{(E)}\ 52$
2011 AIME Problems, 10
A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder where $m+n$ is divided by 1000.
2014 AIME Problems, 12
Suppose that the angles of $\triangle ABC$ satisfy $\cos(3A) + \cos(3B) + \cos(3C) = 1$. Two sides of the triangle have lengths $10$ and $13$. There is a positive integer $m$ so that the maximum possible length for the remaining side of $\triangle ABC$ is $\sqrt{m}$. Find $m$.
2018 AMC 12/AHSME, 22
The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt 3i,$ where $i=\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q-r\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p+q+r+s?$
$\textbf{(A) } 20 \qquad
\textbf{(B) } 21 \qquad
\textbf{(C) } 22 \qquad
\textbf{(D) } 23 \qquad
\textbf{(E) } 24 $
2021 AMC 12/AHSME Fall, 10
What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40 ^{\circ}, \sin 40 ^{\circ}), (\cos 60 ^{\circ}, \sin 60 ^{\circ}),$ and $(\cos t ^{\circ}, \sin t ^{\circ})$ is isosceles?
$\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 150 \qquad\textbf{(C)}\ 330 \qquad\textbf{(D)}\
360 \qquad\textbf{(E)}\ 380$