Found problems: 3632
2000 AIME Problems, 6
For how many ordered pairs $(x,y)$ of integers is it true that $0<x<y<10^{6}$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y?$
1979 AMC 12/AHSME, 28
Circles with centers $A ,~ B,$ and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals
$\textbf{(A) }3r-2\qquad\textbf{(B) }r^2\qquad\textbf{(C) }r+\sqrt{3(r-1)}\qquad$
$\textbf{(D) }1+\sqrt{3(r^2-1)}\qquad\textbf{(E) }\text{none of these}$
[asy]
//Holy crap, CSE5 is freaking amazing!
import cse5;
pathpen=black;
pointpen=black;
dotfactor=3;
size(200);
pair A=(1,2),B=(2,0),C=(0,0);
D(CR(A,1.5));
D(CR(B,1.5));
D(CR(C,1.5));
D(MP("$A$",A));
D(MP("$B$",B));
D(MP("$C$",C));
pair[] BB,CC;
CC=IPs(CR(A,1.5),CR(B,1.5));
BB=IPs(CR(A,1.5),CR(C,1.5));
D(BB[0]--CC[1]);
MP("$B'$",BB[0],NW);MP("$C'$",CC[1],NE);
//Credit to TheMaskedMagician for the diagram
[/asy]
2023 AMC 10, 23
Positive integer divisors $a$ and $b$ of $n$ are called [i]complementary[/i] if $ab=n$. Given that $N$ has a pair of complementary divisors that differ by $20$ and a pair of complementary divisors that differ by $23$, find the sum of the digits of $N$.
$\textbf{(A) } 11 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$
2000 AMC 8, 10
Ara and Shea were once the same height. Since then Shea has grown $20\%$ while Ara has grow half as many inches as Shea. Shea is now $60$ inches tall. How tall, in inches, is Ara now?
$\text{(A)}\ 48 \qquad \text{(B)}\ 51 \qquad \text{(C)}\ 52 \qquad \text{(D)}\ 54 \qquad \text{(E)}\ 55$
1999 AMC 8, 17
Problems 17, 18, and 19 refer to the following:
At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups flour, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes.
Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.)
$ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 5\qquad\text{(D)}\ 7\qquad\text{(E)}\ 15 $
2020 AMC 12/AHSME, 25
The number $a = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying $$\lfloor x \rfloor \cdot \{x\} = a \cdot x^2$$ is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p + q?$
$\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332$
2006 AMC 10, 2
Define $ x\otimes y \equal{} x^3 \minus{} y$. What is $ h\otimes (h\otimes h)$?
$ \textbf{(A) } \minus{} h\qquad \textbf{(B) } 0\qquad \textbf{(C) } h\qquad \textbf{(D) } 2h\qquad \textbf{(E) } h^3$
2014 AMC 12/AHSME, 12
A set S consists of triangles whose sides have integer lengths less than $5$, and no two elements of S are congruent or similar. What is the largest number of elements that $S$ can have?
${\textbf{(A)}\ \ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}}\ 11\qquad\textbf{(E)}\ 12 $
2015 AMC 12/AHSME, 23
A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?
$ \textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26 $
2017 AMC 10, 6
What is the largest number of solid $2\text{-in}\times 2\text{-in}\times 1\text{-in}$ blocks that can fit in a $3\text{-in}\times 2\text{-in}\times 3\text{-in}$ box?
$\textbf{(A) } 3\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 6\qquad \textbf{(E) } 7$
2015 AMC 12/AHSME, 5
Amelia needs to estimate the quantity $\tfrac ab-c$, where $a$, $b$, and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\tfrac ab-c$?
$\textbf{(A) }\text{She rounds all three numbers up.}$
$\textbf{(B) }\text{She rounds }a\text{ and }b\text{ up, and she rounds }c\text{ down.}$
$\textbf{(C) }\text{She rounds }a\text{ and }c\text{ up, and she rounds }b\text{ down.}$
$\textbf{(D) }\text{She rounds }a\text{ up, and she rounds }b\text{ and }c\text{ down.}$
$\textbf{(E) }\text{She rounds }c\text{ up, and she rounds }a\text{ and }b\text{ down.}$
2019 AMC 10, 16
In $\triangle ABC$ with a right angle at $C,$ point $D$ lies in the interior of $\overline{AB}$ and point $E$ lies in the interior of $\overline{BC}$ so that $AC=CD,$ $DE=EB,$ and the ratio $AC:DE=4:3.$ What is the ratio $AD:DB?$
$\textbf{(A) } 2:3
\qquad\textbf{(B) } 2:\sqrt{5}
\qquad\textbf{(C) } 1:1
\qquad\textbf{(D) } 3:\sqrt{5}
\qquad\textbf{(E) } 3:2$
2010 AMC 10, 14
The average of the numbers $ 1,2,3,...,98,99$, and $ x$ is $ 100x$. What is $ x$?
$ \textbf{(A)}\ \frac{49}{101} \qquad\textbf{(B)}\ \frac{50}{101} \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ \frac{51}{101} \qquad\textbf{(E)}\ \frac{50}{99}$
1967 AMC 12/AHSME, 5
A triangle is circumscribed about a circle of radius $r$ inches. If the perimeter of the triangle is $P$ inches and the area is $K$ square inches, then $\frac{P}{K}$ is:
$ \text{(A)}\text{independent of the value of} \; r\qquad\text{(B)}\ \frac{\sqrt{2}}{r}\qquad\text{(C)}\ \frac{2}{\sqrt{r}}\qquad\text{(D)}\ \frac{2}{r}\qquad\text{(E)}\ \frac{r}{2} $
1976 AMC 12/AHSME, 17
If $\theta$ is an acute angle, and $\sin 2\theta=a$, then $\sin\theta+\cos\theta$ equals
$\textbf{(A) }\sqrt{a+1}\qquad\textbf{(B) }(\sqrt{2}-1)a+1\qquad\textbf{(C) }\sqrt{a+1}-\sqrt{a^2-a}\qquad$
$\textbf{(D) }\sqrt{a+1}+\sqrt{a^2-a}\qquad \textbf{(E) }\sqrt{a+1}+a^2-a$
1970 AMC 12/AHSME, 31
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to $43$, what is the probability that this number is divisible by $11$?
$\textbf{(A) }2/5\qquad\textbf{(B) }1/5\qquad\textbf{(C) }1/6\qquad\textbf{(D) }1/11\qquad \textbf{(E) }1/15$
1960 AMC 12/AHSME, 13
The polygon(s) formed by $y=3x+2$, $y=-3x+2$, and $y=-2$, is (are):
$ \textbf{(A) }\text{An equilateral triangle}\qquad\textbf{(B) }\text{an isosceles triangle} \qquad\textbf{(C) }\text{a right triangle} \qquad$
$\textbf{(D) }\text{a triangle and a trapezoid}\qquad\textbf{(E) }\text{a quadrilateral} $
1978 USAMO, 3
An integer $n$ will be called [i]good[/i] if we can write \[n=a_1+a_2+\cdots+a_k,\] where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying \[\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.\] Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.
2017 AIME Problems, 11
Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$.
2012 AMC 12/AHSME, 2
Cagney can frost a cupcake every $20$ seconds and Lacey can frost a cupcake every $30$ seconds. Working together, how many cupcakes can they frost in $5$ minutes?
$ \textbf{(A)}\ 10
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 25
\qquad\textbf{(E)}\ 30
$
1979 AMC 12/AHSME, 8
Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$.
$\textbf{(A) }\frac{\pi}{4}\qquad\textbf{(B) }\frac{3\pi}{4}\qquad\textbf{(C) }\pi\qquad\textbf{(D) }\frac{3\pi}{2}\qquad\textbf{(E) }2\pi$
2023 AMC 12/AHSME, 20
Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below:
[asy]
size(4.5cm);
label("$1$", (0,0));
label("$1$", (-0.5,-2/3));
label("$1$", (0.5,-2/3));
label("$1$", (-1,-4/3));
label("$3$", (0,-4/3));
label("$1$", (1,-4/3));
label("$1$", (-1.5,-2));
label("$5$", (-0.5,-2));
label("$5$", (0.5,-2));
label("$1$", (1.5,-2));
label("$1$", (-2,-8/3));
label("$7$", (-1,-8/3));
label("$11$", (0,-8/3));
label("$7$", (1,-8/3));
label("$1$", (2,-8/3));
[/asy]
Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the 2023 numbers in the 2023rd row?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }9$
2007 AMC 10, 12
Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?
$ \textbf{(A)}\ 56 \qquad \textbf{(B)}\ 58 \qquad \textbf{(C)}\ 60 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 64$
1963 AMC 12/AHSME, 19
In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red. Thereafter, $7$ out of every $8$ counted were red. If, in all, $90\%$ or more of the balls counted were red, the maximum value of $n$ is:
$\textbf{(A)}\ 225 \qquad
\textbf{(B)}\ 210 \qquad
\textbf{(C)}\ 200 \qquad
\textbf{(D)}\ 180 \qquad
\textbf{(E)}\ 175$
2015 AMC 10, 5
Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$. After he graded Payton's test, the class average became $81$. What was Payton's score on the test?
$\textbf{(A) }81\qquad\textbf{(B) }85\qquad\textbf{(C) }91\qquad\textbf{(D) }94\qquad\textbf{(E) }95$