Found problems: 3632
2006 AIME Problems, 13
How many integers $ N$ less than 1000 can be written as the sum of $ j$ consecutive positive odd integers from exactly 5 values of $ j\ge 1$?
2021 AMC 12/AHSME Spring, 8
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34.$ What is the distance between two adjacent parallel lines?
$\textbf{(A)}\ 5\frac{1}{2} \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 6\frac{1}{2} \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 7\frac{1}{2}$
1959 AMC 12/AHSME, 14
Given the set $S$ whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements: (1) addition (2) subtraction (3) multiplication (4) division (5) finding the arithmetic mean (average), those elements that only yield elements of $S$ are:
$ \textbf{(A)}\ \text{all} \qquad\textbf{(B)}\ 1,2,3,4\qquad\textbf{(C)}\ 1,2,3,5\qquad\textbf{(D)}\ 1,2,3\qquad\textbf{(E)}\ 1,3,5 $
1964 AMC 12/AHSME, 3
When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers. What is the remainder when $x+2uy$ is divided by $y$?
${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2u \qquad\textbf{(C)}\ 3u \qquad\textbf{(D)}\ v }\qquad\textbf{(E)}\ 2v } $
2008 AMC 10, 12
In a collection of red, blue, and green marbles, there are $ 25\%$ more red marbles than blue marbles, and there are $ 60\%$ more green marbles than red marbles. Suppose that there are $ r$ red marbles. What is the total number of marbles in that collection?
$ \textbf{(A)}\ 2.85r \qquad \textbf{(B)}\ 3r \qquad \textbf{(C)}\ 3.4r \qquad \textbf{(D)}\ 3.85r \qquad \textbf{(E)}\ 4.25r$
2011 AMC 12/AHSME, 1
A cell phone plan costs $\$20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?
$ \textbf{(A)}\ \$ 24.00 \qquad
\textbf{(B)}\ \$ 24.50\qquad
\textbf{(C)}\ \$ 25.50\qquad
\textbf{(D)}\ \$ 28.00\qquad
\textbf{(E)}\ \$ 30.00$
2015 Switzerland Team Selection Test, 12
Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.
2013 AMC 12/AHSME, 20
Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \leq 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$?
$ \textbf{(A)} \ 810 \qquad \textbf{(B)} \ 855 \qquad \textbf{(C)} \ 900 \qquad \textbf{(D)} \ 950 \qquad \textbf{(E)} \ 988$
2011 AMC 12/AHSME, 6
Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2:3$. What is the degree measure of $\angle BAC$?
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 36 \qquad
\textbf{(D)}\ 48 \qquad
\textbf{(E)}\ 60 $
1993 AMC 8, 18
The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. The area of quadrilateral $ABDF$ is
[asy]
pair A,B,C,D,EE,F;
A = (0,20); B = (16,20); C = (32,20); D = (32,0); EE = (0,0); F = (0,10);
draw(A--C--D--EE--cycle);
draw(B--D--F);
dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F);
label("$A$",A,NW);
label("$B$",B,N);
label("$C$",C,NE);
label("$D$",D,SE);
label("$E$",EE,SW);
label("$F$",F,W);
[/asy]
$\text{(A)}\ 320 \qquad \text{(B)}\ 325 \qquad \text{(C)}\ 330 \qquad \text{(D)}\ 335 \qquad \text{(E)}\ 340$
2009 AMC 12/AHSME, 4
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents?
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 25 \qquad
\textbf{(C)}\ 35 \qquad
\textbf{(D)}\ 45 \qquad
\textbf{(E)}\ 55$
2015 AMC 8, 25
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?
$ \textbf{(A) } 9\qquad \textbf{(B) } 12\frac{1}{2}\qquad \textbf{(C) } 15\qquad \textbf{(D) } 15\frac{1}{2}\qquad \textbf{(E) } 17$
[asy]
draw((0,0)--(0,5)--(5,5)--(5,0)--cycle);
filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray);
filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray);
filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray);
filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray);
[/asy]
2020 CHMMC Winter (2020-21), 5
Let $S$ be the sum of all positive integers $n$ such that $\frac{3}{5}$ of the positive divisors of $n$ are multiples of $6$ and $n$ has no prime divisors greater than $3$. Compute $\frac{S}{36}$.
1964 AMC 12/AHSME, 30
If $(7+4\sqrt{3})x^2+(2+\sqrt{3})x-2=0$, the larger root minus the smaller root is:
$ \textbf{(A)}\ -2+3\sqrt{3}\qquad\textbf{(B)}\ 2-\sqrt{3}\qquad\textbf{(C)}\ 6+3\sqrt{3}\qquad\textbf{(D)}\ 6-3\sqrt{3}\qquad\textbf{(E)}\ 3\sqrt{3}+2 $
2024 AMC 12/AHSME, 4
Balls numbered $1,2,3,\ldots$ are deposited in $5$ bins, labeled $A,B,C,D,$ and $E$, using the following procedure. Ball $1$ is deposited in bin $A$, and balls $2$ and $3$ are deposted in $B$. The next three balls are deposited in bin $C$, the next $4$ in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. (For example, $22,23,\ldots,28$ are despoited in bin $B$ at step 7 of this process.) In which bin is ball $2024$ deposited?
$\textbf{(A) }A\qquad\textbf{(B) }B\qquad\textbf{(C) }C\qquad\textbf{(D) }D\qquad\textbf{(E) }E$
1963 AMC 12/AHSME, 39
In triangle $ABC$ lines $CE$ and $AD$ are drawn so that
$\dfrac{CD}{DB}=\dfrac{3}{1}$ and $\dfrac{AE}{EB}=\dfrac{3}{2}$. Let $r=\dfrac{CP}{PE}$
where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals:
[asy]
size(8cm);
pair A = (0, 0), B = (9, 0), C = (3, 6);
pair D = (7.5, 1.5), E = (6.5, 0);
pair P = intersectionpoints(A--D, C--E)[0];
draw(A--B--C--cycle);
draw(A--D);
draw(C--E);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("$D$", D, NE);
label("$E$", E, S);
label("$P$", P, S);
//Credit to MSTang for the asymptote
[/asy]
$\textbf{(A)}\ 3 \qquad
\textbf{(B)}\ \dfrac{3}{2}\qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ \dfrac{5}{2}$
1974 AMC 12/AHSME, 24
A fair die is rolled six times. The probability of rolling at least a five at least five times is
$ \textbf{(A)}\ \frac{13}{729} \qquad\textbf{(B)}\ \frac{12}{729} \qquad\textbf{(C)}\ \frac{2}{729} \qquad\textbf{(D)}\ \frac{3}{729} \qquad\textbf{(E)}\ \text{none of these} $
1993 AMC 12/AHSME, 15
For how many values of $n$ will an $n$-sided regular polygon have interior angles with integral degree measures?
$ \textbf{(A)}\ 16 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ 24 $
2008 AIME Problems, 7
Let $ r$, $ s$, and $ t$ be the three roots of the equation
\[ 8x^3\plus{}1001x\plus{}2008\equal{}0.\]Find $ (r\plus{}s)^3\plus{}(s\plus{}t)^3\plus{}(t\plus{}r)^3$.
2021 AMC 12/AHSME Spring, 8
A sequence of numbers is defined by $D_0=0,D_1=0,D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021},D_{2022},D_{2023})$, where $E$ denotes even and $O$ denotes odd?
$\textbf{(A) }(O,E,O) \qquad \textbf{(B) }(E,E,O) \qquad \textbf{(C) }(E,O,E) \qquad \textbf{(D) }(O,O,E) \qquad \textbf{(E) }(O,O,O)$
2006 AMC 10, 21
How many four-digit positive integers have at least one digit that is a 2 or a 3?
$ \textbf{(A) } 2439 \qquad \textbf{(B) } 4096 \qquad \textbf{(C) } 4903 \qquad \textbf{(D) } 4904 \qquad \textbf{(E) } 5416$
2011 AMC 12/AHSME, 12
A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river $9$ hours after leaving dock $A.$ How many hours did it take the power boat to go from $A $ to $B$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 3.5 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 4.5 \qquad
\textbf{(E)}\ 5
$
2018 AMC 12/AHSME, 24
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between $\tfrac{1}{2}$ and $\tfrac{2}{3}.$ Armed with this information, what number should Carol choose to maximize her chance of winning?
$
\textbf{(A) }\frac{1}{2}\qquad
\textbf{(B) }\frac{13}{24} \qquad
\textbf{(C) }\frac{7}{12} \qquad
\textbf{(D) }\frac{5}{8} \qquad
\textbf{(E) }\frac{2}{3}\qquad
$
2022 AIME Problems, 6
Find the number of ordered pairs of integers $(a, b)$ such that the sequence $$3, 4, 5, a, b, 30, 40, 50$$ is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.
2012 AMC 8, 19
In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar?
$\textbf{(A)}\hspace{.05in}6 \qquad \textbf{(B)}\hspace{.05in}8 \qquad \textbf{(C)}\hspace{.05in}9 \qquad \textbf{(D)}\hspace{.05in}10 \qquad \textbf{(E)}\hspace{.05in}12$