Found problems: 634
2013 AIME Problems, 10
There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let $p_{a,b}$ be the sum of the zeroes of $P(x)$. Find the sum of the $p_{a,b}$'s for all possible combinations of $a$ and $b$.
2012 AIME Problems, 10
Let $\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$. Let $\mathcal{T}$ be the set of all numbers of the form $\frac{x-256}{1000}$, where $x$ is in $\mathcal{S}$. In other words, $\mathcal{T}$ is the set of numbers that result when the last three digits of each number in $\mathcal{S}$ are truncated. Find the remainder when the tenth smallest element of $\mathcal{T}$ is divided by $1000$.
2007 AIME Problems, 1
A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in $2007$. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in $2007$. A set of plates in which each possible sequence appears exactly once contains $N$ license plates. Find $\frac{N}{10}$.
2025 AIME, 2
Find the sum of all positive integers $n$ such that $n+2$ divides the product $3(n+3)(n^2+9)$.
2011 India IMO Training Camp, 1
Find all positive integer $n$ satisfying the conditions
$a)n^2=(a+1)^3-a^3$
$b)2n+119$ is a perfect square.
2004 AMC 12/AHSME, 5
The graph of the line $ y \equal{} mx \plus{} b$ is shown. Which of the following is true?
[asy]import math;
unitsize(8mm);
defaultpen(linewidth(1pt)+fontsize(6pt));
dashed=linetype("4 4")+linewidth(.8pt);
draw((-2,-2.5)--(-2,2.5)--(2.5,2.5)--(2.5,-2.5)--cycle,white);
label("$-1$",(-1,0),SW);
label("$1$",(1,0),SW);
label("$2$",(2,0),SW);
label("$1$",(0,1),NE);
label("$2$",(0,2),NE);
label("$-1$",(0,-1),SW);
label("$-2$",(0,-2),SW);
drawline((0,0),(1,0));
drawline((0,0),(0,1));
drawline((0,0.8),(1.8,0));
drawline((1,0),(1,1),dashed);
drawline((2,0),(2,1),dashed);
drawline((-1,0),(-1,1),dashed);
drawline((0,1),(1,1),dashed);
drawline((0,2),(1,2),dashed);
drawline((0,-1),(1,-1),dashed);
drawline((0,-2),(1,-2),dashed);[/asy]
$ \textbf{(A)}\ mb < \minus{} 1 \qquad \textbf{(B)}\ \minus{} 1 < mb < 0 \qquad \textbf{(C)}\ mb \equal{} 0$
$ \textbf{(D)}\ 0 < mb < 1\qquad \textbf{(E)}\ mb > 1$
2007 AMC 12/AHSME, 25
Points $ A$, $ B$, $ C$, $ D$, and $ E$ are located in 3-dimensional space with $ AB \equal{} BC \equal{} CD \equal{} DE \equal{} EA \equal{} 2$ and $ \angle ABC \equal{} \angle CDE \equal{} \angle DEA \equal{} 90^\circ.$ The plane of $ \triangle ABC$ is parallel to $ \overline{DE}$. What is the area of $ \triangle BDE$?
$ \textbf{(A)}\ \sqrt2 \qquad \textbf{(B)}\ \sqrt3 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt5 \qquad \textbf{(E)}\ \sqrt6$
2019 AIME Problems, 1
Points $C\neq D$ lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2005 AMC 10, 15
How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 6$
2009 AIME Problems, 15
In triangle $ ABC$, $ AB \equal{} 10$, $ BC \equal{} 14$, and $ CA \equal{} 16$. Let $ D$ be a point in the interior of $ \overline{BC}$. Let $ I_B$ and $ I_C$ denote the incenters of triangles $ ABD$ and $ ACD$, respectively. The circumcircles of triangles $ BI_BD$ and $ CI_CD$ meet at distinct points $ P$ and $ D$. The maximum possible area of $ \triangle BPC$ can be expressed in the form $ a\minus{}b\sqrt{c}$, where $ a$, $ b$, and $ c$ are positive integers and $ c$ is not divisible by the square of any prime. Find $ a\plus{}b\plus{}c$.
2018 AIME Problems, 10
Find the number of functions $f(x)$ from $\{1,2,3,4,5\}$ to $\{1,2,3,4,5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1,2,3,4,5\}$.
2025 AIME, 13
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these $27$ line segments divide the disk.
2008 AIME Problems, 5
A right circular cone has base radius $ r$ and height $ h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $ 17$ complete rotations. The value of $ h/r$ can be written in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
2025 AIME, 3
The 9 members of a baseball team went to an ice-cream parlor after their game. Each player had a singlescoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let $N$ be the number of different assignments of flavors to players that meet these conditions. Find the remainder when $N$ is divided by $1000.$
2010 AMC 12/AHSME, 19
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half?
$ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$
2004 AMC 12/AHSME, 3
For how many ordered pairs of positive integers $ (x,y)$ is $ x \plus{} 2y \equal{} 100$?
$ \textbf{(A)}\ 33 \qquad \textbf{(B)}\ 49 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 99 \qquad \textbf{(E)}\ 100$
2008 AIME Problems, 4
There exist $ r$ unique nonnegative integers $ n_1 > n_2 > \cdots > n_r$ and $ r$ unique integers $ a_k$ ($ 1\le k\le r$) with each $ a_k$ either $ 1$ or $ \minus{} 1$ such that
\[ a_13^{n_1} \plus{} a_23^{n_2} \plus{} \cdots \plus{} a_r3^{n_r} \equal{} 2008.
\]Find $ n_1 \plus{} n_2 \plus{} \cdots \plus{} n_r$.
2014 Moldova Team Selection Test, 4
Define $p(n)$ to be th product of all non-zero digits of $n$. For instance $p(5)=5$, $p(27)=14$, $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:
\[p(1)+p(2)+p(3)+...+p(999).\]
2019 AIME Problems, 7
Triangle $ABC$ has side lengths $AB=120$, $BC=220$, and $AC=180$. Lines $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ are drawn parallel to $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively, such that the intersection of $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ with the interior of $\triangle ABC$ are segments of length $55$, $45$, and $15$, respectively. Find the perimeter of the triangle whose sides lie on $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$.
1995 AMC 12/AHSME, 28
Two parallel chords in a circle have lengths $10$ and $14$, and the distance between them is $6$. The chord parallel to these chords and midway between them is of length $\sqrt{a}$ where $a$ is
[asy]
// note: diagram deliberately not to scale -- azjps
void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); }
size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3);
real min = -0.6, step = 0.5;
pair[] A, B; D(unitcircle);
for(int i = 0; i < 3; ++i) {
A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]);
D(D(A[i])--D(B[i]));
}
MP("10",(A[0]+B[0])/2,N);
MP("\sqrt{a}",(A[1]+B[1])/2,N);
MP("14",(A[2]+B[2])/2,N);
htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);[/asy]
$\textbf{(A)}\ 144 \qquad
\textbf{(B)}\ 156 \qquad
\textbf{(C)}\ 168 \qquad
\textbf{(D)}\ 176 \qquad
\textbf{(E)}\ 184$
1983 AIME Problems, 13
For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique [b]alternating sum[/b] is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $9 - 6 + 4 - 2 + 1 = 6$ and for $\{5\}$ it is simply 5.) Find the sum of all such alternating sums for $n = 7$.
2013 AIME Problems, 1
Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $9{:}99$ just before midnight, $0{:}00$ at midnight, $1{:}25$ at the former $3{:}00$ $\textsc{am}$, and $7{:}50$ at the former $6{:}00$ $\textsc{pm}$. After the conversion, a person who wanted to wake up at the equivalent of the former $6{:}36$ $\textsc{am}$ would have to set his new digital alarm clock for $\text{A:BC}$, where $\text{A}$, $\text{B}$, and $\text{C}$ are digits. Find $100\text{A} + 10\text{B} + \text{C}$.
2004 AMC 10, 20
In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$, respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT \equal{} 3$ and $ BT/ET \equal{} 4$, what is $ CD/BD$?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt));
pair A = (0,0);
pair C = (2,0);
pair B = dir(57.5)*2;
pair E = waypoint(C--A,0.25);
pair D = waypoint(C--B,0.25);
pair T = intersectionpoint(D--A,E--B);
label("$B$",B,NW);label("$A$",A,SW);label("$C$",C,SE);label("$D$",D,NE);label("$E$",E,S);label("$T$",T,2*W+N);
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);[/asy]$ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {2}{9}\qquad \textbf{(C)}\ \frac {3}{10}\qquad \textbf{(D)}\ \frac {4}{11}\qquad \textbf{(E)}\ \frac {5}{12}$
2018 AIME Problems, 4
In equiangular octagon $CAROLINE$, $CA = RO = LI = NE = \sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, that total area of the six triangular regions. Then $K=\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
2011 AIME Problems, 3
Let $L$ be the line with slope $\tfrac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis, and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis. The point $P$ with coordinates $(-14,27)$ in the original system has coordinates $(\alpha,\beta)$ in the new coordinate system. Find $\alpha+\beta$.