Found problems: 3632
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]
2010 Contests, 2
A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were $ 100$ tourists on the ferry boat, and that on each successive trip, the number of tourists was $ 1$ fewer than on the previous trip. How many tourists did the ferry take to the island that day?
$ \textbf{(A)}\ 585\qquad \textbf{(B)}\ 594\qquad \textbf{(C)}\ 672\qquad \textbf{(D)}\ 679\qquad \textbf{(E)}\ 694$
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.
1970 AMC 12/AHSME, 18
$\sqrt{3+2\sqrt{2}}-\sqrt{3-2\sqrt{2}}$ is equal to
$\textbf{(A) }2\qquad\textbf{(B) }2\sqrt{3}\qquad\textbf{(C) }4\sqrt{2}\qquad\textbf{(D) }\sqrt{6}\qquad \textbf{(E) }2\sqrt{2}$
2023 AMC 10, 20
Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curcve is $\pi \sqrt{n}$. What is $n$?
$\textbf{(A) } 32 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 48 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 27$
1968 AMC 12/AHSME, 8
A positive number is mistakenly divided by $6$ instead of being multiplied by $6$. Based on the correct answer, the error thus comitted, to the nearest percent, is:
$\textbf{(A)}\ 100 \qquad
\textbf{(B)}\ 97 \qquad
\textbf{(C)}\ 83 \qquad
\textbf{(D)}\ 17 \qquad
\textbf{(E)}\ 3 $
2012 AMC 8, 16
Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?
$\textbf{(A)}\hspace{.05in}76531 \qquad \textbf{(B)}\hspace{.05in}86724 \qquad \textbf{(C)}\hspace{.05in}87431 \qquad \textbf{(D)}\hspace{.05in}96240 \qquad \textbf{(E)}\hspace{.05in}97403 $
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]
1972 AMC 12/AHSME, 35
[asy]
draw(unitsquare);draw((0,0)--(.25,sqrt(3)/4)--(.5,0));
label("Z",(0,1),NW);label("Y",(1,1),NE);label("A",(0,0),SW);label("X",(1,0),SE);label("B",(.5,0),S);label("P",(.25,sqrt(3)/4),N);
//Credit to Zimbalono for the diagram[/asy]
Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The length of the path in inches traversed by vertex $P$ is equal to
$\textbf{(A) }20\pi/3\qquad\textbf{(B) }32\pi/3\qquad\textbf{(C) }12\pi\qquad\textbf{(D) }40\pi/3\qquad \textbf{(E) }15\pi$
1984 AMC 12/AHSME, 14
The product of all real roots of the equation $x^{\log_{10} x} = 10$ is
A. 1
B. -1
C. 10
D. $10^{-1}$
E. None of these
2017 AMC 12/AHSME, 25
A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are among those $9$ people is equal to the reciprocal of the average, over all subsets of size $8$ of the set of $n$ participants, of the number of complete teams whose members are among those $8$ people. How many values $n$, $9\leq n\leq 2017$, can be the number of participants?
$\textbf{(A) } 477 \qquad \textbf{(B) } 482 \qquad \textbf{(C) } 487 \qquad \textbf{(D) } 557 \qquad \textbf{(E) } 562$
1969 AMC 12/AHSME, 21
If the graph of $x^2+y^2=m$ is tangent to that of $x+y=\sqrt{2m}$, then:
$\textbf{(A) }m\text{ must equal }\tfrac12\qquad
\textbf{(B) }m\text{ must equal }\tfrac1{\sqrt2}\qquad$
$\textbf{(C) }m\text{ must equal }\sqrt2\qquad
\textbf{(D) }m\text{ must equal }2\qquad$
$\textbf{(E) }m\text{ may be any nonnegative real number}$
1996 AIME Problems, 10
Find the smallest positive integer solution to $\tan 19x^\circ=\frac{\cos 96^\circ+\sin 96^\circ}{\cos 96^\circ-\sin 96^\circ}.$
2010 AMC 12/AHSME, 23
Monic quadratic polynomials $ P(x)$ and $ Q(x)$ have the property that $ P(Q(x))$ has zeroes at $ x\equal{}\minus{}23,\minus{}21,\minus{}17, \text{and} \minus{}15$, and $ Q(P(x))$ has zeroes at $ x\equal{}\minus{}59, \minus{}57, \minus{}51, \text{and} \minus{}49$. What is the sum of the minimum values of $ P(x)$ and $ Q(x)$?
$ \textbf{(A)}\ \text{\minus{}100} \qquad \textbf{(B)}\ \text{\minus{}82} \qquad \textbf{(C)}\ \text{\minus{}73} \qquad \textbf{(D)}\ \text{\minus{}64} \qquad \textbf{(E)}\ 0$
2012 AMC 10, 11
Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$. What is $BC$?
$ \textbf{(A)}\ 4
\qquad\textbf{(B)}\ 4.8
\qquad\textbf{(C)}\ 10.2
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ 14.4
$
2024 AMC 8 -, 21
A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and yellow when in the sun. Initially the ratio of green to yellow frogs was 3:1. Then 3 green frogs moved to the sunny side and 5 yellow frogs moved to the shady side. Now the ratio is 4:1. What is the difference between the number of green frogs and yellow frogs now?
$\textbf{(A) } 10\qquad\textbf{(B) } 12\qquad\textbf{(C) } 16\qquad\textbf{(D) } 20\qquad\textbf{(E) } 24$
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 $abc$.
2013 AMC 12/AHSME, 23
$ ABCD$ is a square of side length $ \sqrt{3} + 1 $. Point $ P $ is on $ \overline{AC} $ such that $ AP = \sqrt{2} $. The square region bounded by $ ABCD $ is rotated $ 90^{\circ} $ counterclockwise with center $ P $, sweeping out a region whose area is $ \frac{1}{c} (a \pi + b) $, where $a $, $b$, and $ c $ are positive integers and $ \text{gcd}(a,b,c) = 1 $. What is $ a + b + c $?
$\textbf{(A)} \ 15 \qquad \textbf{(B)} \ 17 \qquad \textbf{(C)} \ 19 \qquad \textbf{(D)} \ 21 \qquad \textbf{(E)} \ 23 $
1977 AMC 12/AHSME, 26
Let $a,b,c,$ and $d$ be the lengths of sides $MN,NP,PQ,$ and $QM$, respectively, of quadrilateral $MNPQ$. If $A$ is the area of $MNPQ$, then
$\textbf{(A) }A=\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is convex}$
$\textbf{(B) }A=\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a rectangle}$
$\textbf{(C) }A\le\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a rectangle}$
$\textbf{(D) }A\le\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a parallelogram}$
$\textbf{(E) }A\ge\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a parallelogram}$
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 $p+q$.
2009 AMC 12/AHSME, 2
Which of the following is equal to $ 1\plus{}\frac{1}{1\plus{}\frac{1}{1\plus{}1}}$?
$ \textbf{(A)}\ \frac{5}{4} \qquad
\textbf{(B)}\ \frac{3}{2} \qquad
\textbf{(C)}\ \frac{5}{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 3$
2021 AMC 10 Spring, 3
In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. among the 28 students in the program, 25% of the juniors and 10% of the seniors are on the debate team. how many juniors are in the program?
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 20.$
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 $n_1\cdot n_2$.
2007 AIME Problems, 3
The complex number $z$ is equal to $9+bi$, where $b$ is a positive real number and $i^{2}=-1$. Given that the imaginary parts of $z^{2}$ and $z^{3}$ are equal, find $b$.