Found problems: 634
2021 AIME Problems, 1
Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or $383$.)
2018 AIME Problems, 3
Find the sum of all positive integers $b<1000$ such that the base-$b$ integer $36_b$ is a perfect square and the base-$b$ integer $27_b$ is a perfect cube.
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 $\tau (n) + \tau (n+1) = 7$.
2016 AIME Problems, 5
Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$.
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]
2013 AIME Problems, 7
A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$.
2025 AIME, 13
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and
\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]
$x_{2025}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$.
2013 AIME Problems, 14
For positive integers $n$ and $k$, let $f(n,k)$ be the remainder when $n$ is divided by $k$, and for $n>1$ let $F(n) = \displaystyle\max_{1 \le k \le \frac{n}{2}} f(n,k)$. Find the remainder when $\displaystyle\sum_{n=20}^{100} F(n)$ is divided by $1000$.
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]
2004 AIME Problems, 11
A right circular cone has a base with radius 600 and height $200\sqrt{7}$. A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is 125, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\sqrt{2}$. Find the least distance that the fly could have crawled.
2021 AIME Problems, 2
In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
pair A, B, C, D, E, F;
A = (0,3);
B=(0,0);
C=(11,0);
D=(11,3);
E=foot(C, A, (9/4,0));
F=foot(A, C, (35/4,3));
draw(A--B--C--D--cycle);
draw(A--E--C--F--cycle);
filldraw(A--(9/4,0)--C--(35/4,3)--cycle,gray*0.5+0.5*lightgray);
dot(A^^B^^C^^D^^E^^F);
label("$A$", A, W);
label("$B$", B, W);
label("$C$", C, (1,0));
label("$D$", D, (1,0));
label("$F$", F, N);
label("$E$", E, S);
[/asy]
1996 AIME Problems, 10
Find the smallest positive integer solution to $\tan 19x^\circ=\frac{\cos 96^\circ+\sin 96^\circ}{\cos 96^\circ-\sin 96^\circ}.$
2014 AIME Problems, 8
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.
2016 AIME Problems, 13
Beatrix is going to place six rooks on a $6\times6$ chessboard where both the rows and columns are labelled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The [i]value[/i] of a square is the sum of its row number and column number. The [i]score[/i] of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2005 AIME Problems, 13
Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.
2000 Manhattan Mathematical Olympiad, 4
An equilateral triangle $ABC$ is given, together with a point $P$ inside it.
[asy]
draw((0,0)--(4,0)--(2,3.464)--(0,0));
draw((1.3, 1.2)--(0,0));
draw((1.3, 1.2)--(2,3.464));
draw((1.3, 1.2)--(4,0));
label("$A$",(0,0),SW);
label("$B$",(4,0),SE);
label("$C$",(2,3.464),N);
label("$P$",(1.3,1.2),S);
[/asy]
Given that $PA = 3$ cm, $PB = 5$ cm, and $PC = 4$ cm, find the side of the equilateral triangle.
2016 AMC 10, 20
For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$?
$\textbf{(A) }9 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$
PEN O Problems, 54
Let $S$ be a subset of $\{1, 2, 3, \cdots, 1989 \}$ in which no two members differ by exactly $4$ or by exactly $7$. What is the largest number of elements $S$ can have?
2008 AMC 12/AHSME, 22
A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$?
[asy]unitsize(4mm);
defaultpen(linewidth(.8)+fontsize(8));
draw(Circle((0,0),4));
path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle;
draw(mat);
draw(rotate(60)*mat);
draw(rotate(120)*mat);
draw(rotate(180)*mat);
draw(rotate(240)*mat);
draw(rotate(300)*mat);
label("$x$",(-2.687,0),E);
label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$
2002 AIME Problems, 7
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $ x, y,$ and $ r$ with $ |x| > |y|,$
\[ (x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots
\]
What are the first three digits to the right of the decimal point in the decimal representation of $ \left(10^{2002} \plus{} 1\right)^{10/7}?$
2020 CHMMC Winter (2020-21), 8
$15$ ladies and $30$ gentlemen attend a luxurious party. At the start of the party, each one of the ladies shakes hands with a random gentleman. At the end of the party, each of the ladies shakes hands with another random gentleman. A lady may shake hands with the same gentleman twice (first at the start and then at the end of the party), and no two ladies shake hands with the same gentleman at the same time.
Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that the collection of ladies and gentlemen that shook hands at least once can be arranged in a single circle such that each lady is directly adjacent to someone if and only if she shook hands with that person. Find the remainder when $m$ is divided by $10000$.
1987 AIME Problems, 4
Find the area of the region enclosed by the graph of $|x-60|+|y|=|x/4|.$
2025 AIME, 7
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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]