Found problems: 162
2023 AIME, 15
Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying [list]
[*] the real and imaginary part of $z$ are both integers;
[*] $|z|=\sqrt{p}$, and
[*] there exists a triangle whose three side lengths are $p$, the real part of $z^{3}$, and the imaginary part of $z^{3}$.
[/list]
2023 AIME, 7
Call a positive integer $n$ [i]extra-distinct[/i] if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$.
2018 AIME Problems, 1
Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$.
2015 AIME Problems, 7
In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\overline{AD}$. Points $F$ and $G$ lie on $\overline{CE}$, and $H$ and $J$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\overline{GH}$, and $M$ and $N$ lie on $\overline{AD}$ and $\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is 99. Find the area of $FGHJ$.
[asy]
pair A,B,C,D,E,F,G,H,J,K,L,M,N;
B=(0,0);
real m=7*sqrt(55)/5;
J=(m,0);
C=(7*m/2,0);
A=(0,7*m/2);
D=(7*m/2,7*m/2);
E=(A+D)/2;
H=(0,2m);
N=(0,2m+3*sqrt(55)/2);
G=foot(H,E,C);
F=foot(J,E,C);
draw(A--B--C--D--cycle);
draw(C--E);
draw(G--H--J--F);
pair X=foot(N,E,C);
M=extension(N,X,A,D);
K=foot(N,H,G);
L=foot(M,H,G);
draw(K--N--M--L);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NE);
label("$E$",E,dir(90));
label("$F$",F,NE);
label("$G$",G,NE);
label("$H$",H,W);
label("$J$",J,S);
label("$K$",K,SE);
label("$L$",L,SE);
label("$M$",M,dir(90));
label("$N$",N,dir(180));
[/asy]
2022 AIME Problems, 9
Ellina has twelve blocks, two each of red $\left({\bf R}\right),$ blue $\left({\bf B}\right),$ yellow $\left({\bf Y}\right),$ green $\left({\bf G}\right),$ orange $\left({\bf O}\right),$ and purple $\left({\bf P}\right).$ Call an arrangement of blocks [i]even[/i] if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement
$$ {\text {\bf R B B Y G G Y R O P P O}}$$is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2023 AIME, 9
Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$, such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$.
2015 AIME Problems, 1
The expressions $A=1\times2+3\times4+5\times6+\cdots+37\times38+39$ and $B=1+2\times3+4\times5+\cdots+36\times37+38\times39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$.
2024 AIME, 5
Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16,$ $AB=107,$ $FG=17,$ and $EF=184,$ what is the length of $CE$?
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F, G, H;
A = (0,0);
B = (5,0);
C = (5,1.5);
D = (0,1.5);
E = (1,1.5);
F = (8,1.5);
G = (8,3.5);
H = (1,3.5);
draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
dot("A", A, SW);
dot("B", B, SE);
dot("C", C, SE);
dot("D", D, NW);
dot("E", E, NW);
dot("F", F, SE);
dot("G", G, NE);
dot("H", H, NW);
[/asy]
2019 AIME Problems, 2
Jenn randomly chooses a number $J$ from $1, 2, 3,\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2020 AIME Problems, 7
A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that can be formed. Find the sum of the prime numbers that divide $N$.
2015 AIME Problems, 3
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.
2023 AIME, 13
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below.
[asy]
unitsize(2cm);
pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70);
draw(o--u--(u+v));
draw(o--v--(u+v), dotted);
draw(shift(w)*(o--u--(u+v)--v--cycle));
draw(o--w);
draw(u--(u+w));
draw(v--(v+w), dotted);
draw((u+v)--(u+v+w));
[/asy]
2018 AIME Problems, 6
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.
2015 AIME Problems, 2
The nine delegates to the Economic Cooperation Conference include 2 officials from Mexico, 3 officials from Canada, and 4 officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2022 AIME Problems, 2
Find the three-digit positive integer $\underline{a} \ \underline{b} \ \underline{c}$ whose representation in base nine is $\underline{b} \ \underline{c} \ \underline{a}_{\hspace{.02in}\text{nine}}$, where $a$, $b$, and $c$ are (not necessarily distinct) digits.
2008 AIME Problems, 7
Let $ r$, $ s$, and $ t$ be the three roots of the equation
\[ 8x^3\plus{}1001x\plus{}2008\equal{}0.\]Find $ (r\plus{}s)^3\plus{}(s\plus{}t)^3\plus{}(t\plus{}r)^3$.
2022 AIME Problems, 6
Find the number of ordered pairs of integers $(a, b)$ such that the sequence $$3, 4, 5, a, b, 30, 40, 50$$ is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.
2019 AIME Problems, 13
Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
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 $b$.
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$
2024 AIME, 3
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes $1$ token or $4$ tokens from the stack. The player who removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ such that there is a strategy that guarantees that Bob wins, regardless of Alice’s moves.
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.
2002 AIME Problems, 14
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatet number of elements that $\mathcal{S}$ can have?
2025 AIME, 5
There are $8!= 40320$ eight-digit positive integers that use each of the digits 1, 2, 3, 4, 5, 6, 7, 8 exactly once. Let N be the number of these integers that are divisible by $22$. Find the difference between $N$ and 2025.
2018 AIME Problems, 6
A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial \[x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2\] are all real can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.