Found problems: 634
2000 AIME Problems, 6
One base of a trapezoid is 100 units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3.$ Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100.$
1952 AMC 12/AHSME, 30
When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is:
$ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 2: 1 \qquad\textbf{(C)}\ 1: 4 \qquad\textbf{(D)}\ 4: 1 \qquad\textbf{(E)}\ 1: 1$
2005 AIME Problems, 15
Triangle $ABC$ has $BC=20$. The incircle of the triangle evenly trisects the median $AD$. If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n$.
2008 AIME Problems, 11
Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?
1986 AIME Problems, 9
In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.
2006 AMC 10, 1
Sandwiches at Joe's Fast Food cost $ \$3$ each and sodas cost $ \$2$ each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?
$ \textbf{(A) } 31\qquad \textbf{(B) } 32\qquad \textbf{(C) } 33\qquad \textbf{(D) } 34\qquad \textbf{(E) } 35$
2004 AIME Problems, 8
How many positive integer divisors of $2004^{2004}$ are divisible by exactly $2004$ positive integers?
1989 AIME Problems, 8
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[ \begin{array}{r} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\,\,\,\,\,\,\,\, \\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\,\,\,\,\, \\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123. \\ \end{array} \] Find the value of \[16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7.\]
1986 AIME Problems, 3
If $\tan x+\tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$?
2015 AIME Problems, 9
Let $S$ be the set of all ordered triples of integers $(a_1,a_2,a_3)$ with $1 \le a_1,a_2,a_3 \le 10$. Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$ for all $n\ge 4$. Find the number of such sequences for which $a_n=0$ for some $n$.
2023 AIME, 3
Let $\triangle{ABC}$ be an isoceles triangle with $\angle A=90^{\circ}$. There exists a point $P$ inside $\triangle{ABC}$ such that $\angle PAB=\angle PBC=\angle PCA$ and $AP=10$. Find the area of $\triangle{ABC}$.
1985 ITAMO, 8
The sum of the following seven numbers is exactly 19:
\[a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71,\]
\[a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86.\]
It is desired to replace each $a_i$ by an integer approximation $A_i$, $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "errors" $|A_i - a_i|$, is as small as possible. For this minimum $M$, what is $100M$?
2011 AIME Problems, 7
Find the number of positive integers $m$ for which there exist nonnegative integers $x_0,x_1,\ldots,x_{2011}$ such that \[ m^{x_0}=\sum_{k=1}^{2011}m^{x_k}. \]
1999 AIME Problems, 4
The two squares shown share the same center $O$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
[asy]
real alpha = 25;
pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin;
pair w=dir(alpha)*W, x=dir(alpha)*X, y=dir(alpha)*Y, z=dir(alpha)*Z;
draw(W--X--Y--Z--cycle^^w--x--y--z--cycle);
pair A=intersectionpoint(Y--Z, y--z),
C=intersectionpoint(Y--X, y--x),
E=intersectionpoint(W--X, w--x),
G=intersectionpoint(W--Z, w--z),
B=intersectionpoint(Y--Z, y--x),
D=intersectionpoint(Y--X, w--x),
F=intersectionpoint(W--X, w--z),
H=intersectionpoint(W--Z, y--z);
dot(O);
label("$O$", O, SE);
label("$A$", A, dir(O--A));
label("$B$", B, dir(O--B));
label("$C$", C, dir(O--C));
label("$D$", D, dir(O--D));
label("$E$", E, dir(O--E));
label("$F$", F, dir(O--F));
label("$G$", G, dir(O--G));
label("$H$", H, dir(O--H));[/asy]
2023 AIME, 9
Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q$, and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B$, respectively. The line parallel to line $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Suppose $PX = 10, PY = 14,$ and $PQ = 5$. Then the area of trapezoid $XABY$ is $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.
2021 AIME Problems, 2
Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$.
[asy]
pair A,B,C,D,E,F,G;
B=origin;
A=5*dir(60);
C=(5,0);
E=0.6*A+0.4*B;
F=0.6*A+0.4*C;
G=rotate(240,F)*A;
D=extension(E,F,B,dir(90));
draw(D--G--A,grey);
draw(B--0.5*A+rotate(60,B)*A*0.5,grey);
draw(A--B--C--cycle,linewidth(1.5));
dot(A^^B^^C^^D^^E^^F^^G);
label("$A$",A,dir(90));
label("$B$",B,dir(225));
label("$C$",C,dir(-45));
label("$D$",D,dir(180));
label("$E$",E,dir(-45));
label("$F$",F,dir(225));
label("$G$",G,dir(0));
label("$\ell$",midpoint(E--F),dir(90));
[/asy]
2007 AIME Problems, 15
Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA=5$ and $CD=2$. Point $E$ lies on side $CA$ such that $\angle DEF = 60^\circ$. The area of triangle $DEF$ is $14\sqrt{3}$. The two possible values of the length of side $AB$ are $p \pm q\sqrt{r}$, where $p$ and $q$ are rational, and $r$ is an integer not divisible by the square of a prime. Find $r$.
2025 AIME, 1
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$
2016 AIME Problems, 6
In $\triangle ABC$ let $I$ be the center of the inscribed circle, and let the bisector of $\angle ACB$ intersect $AB$ at $L$. The line through $C$ and $L$ intersects the circumscribed circle of $\triangle ABC$ at the two points $C$ and $D$. If $LI = 2$ and $LD = 3$, then $IC = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
2019 AIME Problems, 8
Let $x$ be a real number such that $\sin^{10}x+\cos^{10} x = \tfrac{11}{36}$. Then $\sin^{12}x+\cos^{12} x = \tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2002 AIME Problems, 11
Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8,$ and the second term of both series can be written in the form $\frac{\sqrt{m}-n}{p},$ where $m,$ $n,$ and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m+10n+p.$
2009 AIME Problems, 12
From the set of integers $ \{1,2,3,\ldots,2009\}$, choose $ k$ pairs $ \{a_i,b_i\}$ with $ a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $ a_i\plus{}b_i$ are distinct and less than or equal to $ 2009$. Find the maximum possible value of $ k$.
1982 AMC 12/AHSME, 2
If a number eight times as large as $x$ is increased by two, then one fourth of the result equals
$\textbf{(A)} \ 2x + \frac{1}{2} \qquad \textbf{(B)} \ x + \frac{1}{2} \qquad \textbf{(C)} \ 2x+2 \qquad \textbf{(D)} \ 2x+4 \qquad \textbf{(E)} \ 2x+16$
1990 AIME Problems, 13
Let $T = \{9^k : k \ \text{is an integer}, 0 \le k \le 4000\}$. Given that $9^{4000}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T$ have 9 as their leftmost digit?
2014 AIME Problems, 15
In $ \triangle ABC $, $ AB = 3 $, $ BC = 4 $, and $ CA = 5 $. Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$, $\overline{BC}$ at $B$ and $D$, and $\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\tfrac{DG}{EG} = \tfrac{3}{4}$, length $DE=\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.