Found problems: 162
2024 AIME, 6
Consider the paths of length $16$ that go from the lower left corner to the upper right corner of an $8\times 8$ grid. Find the number of such paths that change direction exactly $4$ times.
2014 AMC 12/AHSME, 19
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?
$\textbf{(A) }6\qquad
\textbf{(B) }12\qquad
\textbf{(C) }24\qquad
\textbf{(D) }48\qquad
\textbf{(E) }78\qquad$
2020 AIME Problems, 8
A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of 5 units due east. Each night the bug rotates $60 ^\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to point $P$. Then $OP^2 = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
CIME I 2018, 12
Define a permutation of the set $\{1,2,3,...,n\}$ to be $\textit{sortable}$ if upon cancelling an appropriate term of such permutation, the remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of sortable permutations of $\{1,2,3,...,n\}$, find the remainder when $$\sum\limits_{i=1}^{2018} (-1)^i \cdot f(i)$$ is divided by $1000$. Note that the empty set is considered sortable.
[i]Proposed by [b]FedeX333X[/b][/i]
2015 AIME Problems, 6
Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.[asy]
pair A,B,C,D,E,F,G,H,I,O;
O=(0,0);
C=dir(90);
B=dir(70);
A=dir(50);
D=dir(110);
E=dir(130);
draw(arc(O,1,50,130));
real x=2*sin(20*pi/180);
F=x*dir(228)+C;
G=x*dir(256)+C;
H=x*dir(284)+C;
I=x*dir(312)+C;
draw(arc(C,x,200,340));
label("$A$",A,dir(0));
label("$B$",B,dir(75));
label("$C$",C,dir(90));
label("$D$",D,dir(105));
label("$E$",E,dir(180));
label("$F$",F,dir(225));
label("$G$",G,dir(260));
label("$H$",H,dir(280));
label("$I$",I,dir(315));
[/asy]
CIME I 2018, 7
Let $A, B, C, D$ be points, in order, on a straight line such that $AB=BC=CD$. Let $E$ be a point closer to $B$ than $D$ such that $BE=EC=CD$ and let $F$ be the midpoint of $DE$. Let $AF$ intersect $EC$ at $G$ and let $BF$ intersect $EC$ at $H$. If $[BHC]+[GHF]=1$, then $AD^2 = \frac{a\sqrt{b}}{c}$ where $a,b,$ and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c$.
[i]Proposed by [b]AOPS12142015[/b][/i]
CIME I 2018, 3
Let $a, b, c >1$ be positive real numbers such that $a^{\log_b c}=27, b^{\log_c a}=81,$ and $c^{\log_a b}=243$. Then the value of $\log_3{abc}$ can be written as $\sqrt{x}+\sqrt{y}+\sqrt{z}$ for positive integers $x,y,$ and $z$. Find $x+y+z$.
[i]Proposed by [b]AOPS12142015[/b][/i]
2019 AIME Problems, 9
Let $\tau (n)$ denote the number of positive integer divisors of $n$. Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.
2023 AIME, 8
Rhombus $ABCD$ has $\angle BAD<90^{\circ}$. There is a point $P$ on the incircle of the rhombus such that the distances from $P$ to lines $DA$, $AB$, and $BC$ are $9$, $5$, and $16$, respectively. Find the perimeter of $ABCD$.
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]
2022 AIME Problems, 11
Let $ABCD$ be a parallelogram with $\angle BAD < 90^{\circ}$. A circle tangent to sides $\overline{DA}$, $\overline{AB}$, and $\overline{BC}$ intersects diagonal $\overline{AC}$ at points $P$ and $Q$ with $AP < AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$. Then the area of $ABCD$ can be expressed 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 $m+n$.
[asy]
defaultpen(linewidth(0.6)+fontsize(11));
size(8cm);
pair A,B,C,D,P,Q;
A=(0,0);
label("$A$", A, SW);
B=(6,15);
label("$B$", B, NW);
C=(30,15);
label("$C$", C, NE);
D=(24,0);
label("$D$", D, SE);
P=(5.2,2.6);
label("$P$", (5.8,2.6), N);
Q=(18.3,9.1);
label("$Q$", (18.1,9.7), W);
draw(A--B--C--D--cycle);
draw(C--A);
draw(Circle((10.95,7.45), 7.45));
dot(A^^B^^C^^D^^P^^Q);
[/asy]
CIME I 2018, 8
Mark has six boxes lined up in a straight line. Inside each of the first three boxes are a red ball, a blue ball, and a green ball. He randomly selects a ball from each of the three boxes and puts them into a fourth box. Then, he randomly selects a ball from each of the four boxes and puts them into a fifth box. Next, he randomly selects a ball from each of the five boxes and puts them into a sixth box, arriving at three boxes with $1, 3,$ and $5$ balls, respectively. The probability that the box with $3$ balls has each type of color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by [b]AOPS12142015[/b][/i]
2018 AIME Problems, 10
The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point \(A\). At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path \(AJABCHCHIJA\), which has \(10\) steps. Let \(n\) be the number of paths with \(15\) steps that begin and end at point \(A\). Find the remainder when \(n\) is divided by \(1000\).
[asy]
unitsize(32);
draw(unitcircle);
draw(scale(2) * unitcircle);
for(int d = 90; d < 360 + 90; d += 72){
draw(2 * dir(d) -- dir(d));
}
real s = 4;
dot(1 * dir( 90), linewidth(s));
dot(1 * dir(162), linewidth(s));
dot(1 * dir(234), linewidth(s));
dot(1 * dir(306), linewidth(s));
dot(1 * dir(378), linewidth(s));
dot(2 * dir(378), linewidth(s));
dot(2 * dir(306), linewidth(s));
dot(2 * dir(234), linewidth(s));
dot(2 * dir(162), linewidth(s));
dot(2 * dir( 90), linewidth(s));
defaultpen(fontsize(10pt));
real r = 0.05;
label("$A$", (1-r) * dir( 90), -dir( 90));
label("$B$", (1-r) * dir(162), -dir(162));
label("$C$", (1-r) * dir(234), -dir(234));
label("$D$", (1-r) * dir(306), -dir(306));
label("$E$", (1-r) * dir(378), -dir(378));
label("$F$", (2+r) * dir(378), dir(378));
label("$G$", (2+r) * dir(306), dir(306));
label("$H$", (2+r) * dir(234), dir(234));
label("$I$", (2+r) * dir(162), dir(162));
label("$J$", (2+r) * dir( 90), dir( 90));
[/asy]
CIME I 2018, 10
In circle $\Omega$, let $\overline{AB}=65$ be the diameter and let points $C$ and $D$ lie on the same side of arc $\overarc{AB}$ such that $CD=16$, with $C$ closer to $B$ and $D$ closer to $A$. Moreover, let $AD, BC, AC,$ and $BD$ all have integer lengths. Two other circles, circles $\omega_1$ and $\omega_2$, have $\overline{AC}$ and $\overline{BD}$ as their diameters, respectively. Let circle $\omega_1$ intersect $AB$ at a point $E \neq A$ and let circle $\omega_2$ intersect $AB$ at a point $F \neq B$. Then $EF=\frac{m}{n}$, for relatively prime integers $m$ and $n$. Find $m+n$.
[asy]
size(7cm);
pair A=(0,0), B=(65,0), C=(117/5,156/5), D=(125/13,300/13), E=(23.4,0), F=(9.615,0);
draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, NE);
dot("$D$", D, NW);
dot("$E$", E, S);
dot("$F$", F, S);
draw(circle((A + C)/2, abs(A - C)/2));
draw(circle((B + D)/2, abs(B - D)/2));
draw(circle((A + B)/2, abs(A - B)/2));
label("$\mathcal P$", (A + B)/2 + abs(A - B)/2 * dir(-45), dir(-45));
label("$\mathcal Q$", (A + C)/2 + abs(A - C)/2 * dir(-210), dir(-210));
label("$\mathcal R$", (B + D)/2 + abs(B - D)/2 * dir(70), dir(70));
[/asy]
[i]Proposed by [b]AOPS12142015[/b][/i]
2025 AIME, 12
The set of points in $3$-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities $$x-yz<y-zx<z-xy$$forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b},$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b.$
2010 AIME Problems, 7
Define an ordered triple $ (A, B, C)$ of sets to be minimally intersecting if $ |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1$ and $ A \cap B \cap C \equal{} \emptyset$. For example, $ (\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $ N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $ \{1,2,3,4,5,6,7\}$. Find the remainder when $ N$ is divided by $ 1000$.
[b]Note[/b]: $ |S|$ represents the number of elements in the set $ S$.
2019 AIME Problems, 1
Consider the integer $$N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.$$ Find the sum of the digits of $N$.
2020 AIME Problems, 1
In $\triangle ABC$ with $AB=AC$, point $D$ lies strictly between $A$ and $C$ on side $\overline{AC}$, and point $E$ lies strictly between $A$ and $B$ on side $\overline{AB}$ such that $AE=ED=DB=BC$. The degree measure of $\angle ABC$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2024 AIME, 12
Let $O(0,0)$, $A(\tfrac{1}{2},0)$, and $B(0, \tfrac{\sqrt{3}}{2})$ be points in the coordinate plane. Let $\mathcal{F}$ be the family of segments $\overline{PQ}$ of unit length lying in the first quadrant with $P$ on the $x$-axis and $Q$ on the $y$-axis. There is a unique point $C$ on $\overline{AB}$, distinct from $A$ and $B$, that does not belong to any segment from $\mathcal{F}$ other than $\overline{AB}$. Then $OC^2 = \tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
2007 AIME Problems, 10
In the $ 6\times4$ grid shown, $ 12$ of the $ 24$ squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $ N$ be the number of shadings with this property. Find the remainder when $ N$ is divided by $ 1000$.
[asy]size(100);
defaultpen(linewidth(0.7));
int i;
for(i=0; i<5; ++i) {
draw((i,0)--(i,6));
}
for(i=0; i<7; ++i) {
draw((0,i)--(4,i));
}[/asy]
2019 AIME Problems, 4
A soccer team has 22 available players. A fixed set of 11 players starts the game, while the other 11 are available as substitutes. During the game, the coach may make as many as 3 substitutions, where any one of the 11 players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let $n$ be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when $n$ is divided by 1000.
2019 AIME Problems, 6
In convex quadrilateral $KLMN$ side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$, side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$. Find $MO$.
2013 AIME Problems, 2
Find the number of five-digit positive integers, $n$, that satisfy the following conditions:
(a) the number $n$ is divisible by $5$,
(b) the first and last digits of $n$ are equal, and
(c) the sum of the digits of $n$ is divisible by $5$.
2015 AIME Problems, 13
With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.
2024 AIME, 12
Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$. Find the number of intersections of the graphs of $$y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).$$