Found problems: 3632
1994 AIME Problems, 2
A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\overline{AB}$ can be written in the form $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
1968 AMC 12/AHSME, 2
The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is:
$\textbf{(A)}\ -\dfrac{2}{3} \qquad
\textbf{(B)}\ -\dfrac{1}{3} \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ \dfrac{1}{4} \qquad
\textbf{(E)}\ \dfrac{3}{8} $
2002 AMC 8, 23
A portion of a corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles?
[asy]/* AMC8 2002 #23 Problem */
fill((0,2)--(1,3)--(2,3)--(2,4)--(3,5)--(4,4)--(4,3)--(5,3)--(6,2)--(5,1)--(4,1)--(4,0)--(2,0)--(2,1)--(1,1)--cycle, mediumgrey);
fill((7,1)--(6,2)--(7,3)--(8,3)--(8,4)--(9,5)--(10,4)--(7,0)--cycle, mediumgrey);
fill((3,5)--(2,6)--(2,7)--(1,7)--(0,8)--(1,9)--(2,9)--(2,10)--(3,11)--(4,10)--(4,9)--(5,9)--(6,8)--(5,7)--(4,7)--(4,6)--cycle, mediumgrey);
fill((6,8)--(7,9)--(8,9)--(8,10)--(9,11)--(10,10)--(10,9)--(11,9)--(11,7)--(10,7)--(10,6)--(9,5)--(8,6)--(8,7)--(7,7)--cycle, mediumgrey);
draw((0,0)--(0,11)--(11,11));
for ( int x = 1; x < 11; ++x )
{
draw((x,11)--(x,0), linetype("4 4"));
}
for ( int y = 1; y < 11; ++y )
{
draw((0,y)--(11,y), linetype("4 4"));
}
clip((0,0)--(0,11)--(11,11)--(11,5)--(4,1)--cycle);[/asy]
$ \textbf{(A)}\ \frac13\qquad\textbf{(B)}\ \frac49\qquad\textbf{(C)}\ \frac12\qquad\textbf{(D)}\ \frac59\qquad\textbf{(E)}\ \frac58$
2004 Germany Team Selection Test, 2
Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.
2019 USAJMO, 2
Let $\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \colon \mathbb{Z}\rightarrow \mathbb{Z}$ and $g \colon \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying
\[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \]
for all integers $x$.
[i]Proposed by Ankan Bhattacharya[/i]
1968 AMC 12/AHSME, 22
A segment of length $1$ is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is:
$\textbf{(A)}\ \text{equal to}\ \frac{1}{4} \\ \qquad\textbf{(B)}\ \text{equal to or greater than}\ \frac{1}{8}\ \text{and less than}\ \frac{1}{2} \\ \qquad\textbf{(C)}\ \text{greater than}\ \frac{1}{8}\ \text{and less than}\ \frac{1}{2} \\ \qquad\textbf{(D)}\ \text{greater than}\ \frac{1}{8}\ \text{and less than}\ \frac{1}{4} \\ \qquad\textbf{(E)}\ \text{less than}\ \frac{1}{2}$
2015 USAJMO, 1
Given a sequence of real numbers, a move consists of choosing two terms and replacing each with their arithmetic mean. Show that there exists a sequence of 2015 distinct real numbers such that after one initial move is applied to the sequence -- no matter what move -- there is always a way to continue with a finite sequence of moves so as to obtain in the end a constant sequence.
2009 AIME Problems, 10
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $ 1$ to $ 15$ in clockwise order. Committee rules state that a Martian must occupy chair $ 1$ and an Earthling must occupy chair $ 15$. Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $ N\cdot (5!)^3$. Find $ N$.
2000 AMC 8, 5
Each principal of Lincoln High School serves exactly one $3$-year term. What is the maximum number of principals this school could have during an $8$-year period?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 8$
2015 AMC 12/AHSME, 2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
$ \textbf{(A) }\text{3:10 PM}\qquad\textbf{(B) }\text{3:30 PM}\qquad\textbf{(C) }\text{4:00 PM}\qquad\textbf{(D) }\text{4:10 PM}\qquad\textbf{(E) }\text{4:30 PM} $
2020 AMC 10, 9
A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$
$\textbf{(A) } 9 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 27 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 77$
2021 AMC 10 Fall, 5
The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$
$(\textbf{A})\: 1\qquad(\textbf{B}) \: 3\qquad(\textbf{C}) \: 5 \qquad(\textbf{D}) \: 7\qquad(\textbf{E}) \: 9$
1969 AMC 12/AHSME, 26
[asy]
size(180);
defaultpen(linewidth(0.8));
real r=4/5;
draw((-1,0)..(-6/7,r/3)..(0,r)..(6/7,r/3)..(1,0),linetype("4 4"));
draw((-1,0)--(1,0)^^origin--(0,r));
label("$A$",(-1,0),W);
label("$B$",(1,0),E);
label("$M$",origin,S);
label("$C$",(0,r),N);
[/asy]
A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at a point $5$ inches from the center of $M$ is:
$\textbf{(A) }1\qquad
\textbf{(B) }15\qquad
\textbf{(C) }15\tfrac13\qquad
\textbf{(D) }15\tfrac12\qquad
\textbf{(E) }15\tfrac34$
2001 AMC 12/AHSME, 23
A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial?
$ \textbf{(A)} \ \frac {1 \plus{} i \sqrt {11}}{2} \qquad \textbf{(B)} \ \frac {1 \plus{} i}{2} \qquad \textbf{(C)} \ \frac {1}{2} \plus{} i \qquad \textbf{(D)} \ 1 \plus{} \frac {i}{2} \qquad \textbf{(E)} \ \frac {1 \plus{} i \sqrt {13}}{2}$
2020 CHMMC Winter (2020-21), 2
Find the sum of all positive integers $x < 241$ such that both $x^{24} + x^{18} + x^{12} + x^6 + 1$ and $x^{20} + x^{10} + 1$ are multiples of $241$.
2020 AMC 12/AHSME, 13
Which of the following is the value of $\sqrt{\log_2{6}+\log_3{6}}?$
$\textbf{(A) } 1 \qquad\textbf{(B) } \sqrt{\log_5{6}} \qquad\textbf{(C) } 2 \qquad\textbf{(D) } \sqrt{\log_2{3}}+\sqrt{\log_3{2}} \qquad\textbf{(E) } \sqrt{\log_2{6}}+\sqrt{\log_3{6}}$
2024 AIME, 13
Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p^{2}$. Find the least positive integer $m$ such that $m^{4}+1$ is divisible by $p^{2}$.
2017 AIME Problems, 7
Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\log(kx)=2\log(x+2)$ has exactly one real solution.
1996 AMC 8, 20
Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $\frac{1}{1-x}$. For example, if the calculator is displaying $2$ and the special key is pressed, then the calculator will display $-1$ since $\frac{1}{1-2}=-1$. Now suppose that the calculator is displaying $5$. After the special key is pressed 100 times in a row, the calculator will display
$\text{(A)}\ -0.25 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 0.8 \qquad \text{(D)}\ 1.25 \qquad \text{(E)}\ 5$
2000 AMC 8, 9
Three-digit powers of 2 and 5 are used in this ''cross-number'' puzzle. What is the only possible digit for the outlined square?
\begin{tabular}{lcl}
\textbf{ACROSS} & & \textbf{DOWN} \\
\textbf{2}. $2^m$ & & \textbf{1}. $5^n$
\end{tabular}
[asy]
size(120);
draw((0,-1)--(1,-1)--(1,2)--(0,2)--cycle);
draw((0,1)--(3,1)--(3,0)--(0,0));
draw((3,0)--(2,0)--(2,1)--(3,1)--cycle,linewidth(1.3));
label("$1$",(0,2),SE);
label("$2$",(0,1),SE);
[/asy]
$\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$
2000 AMC 8, 3
How many whole numbers lie in the interval between $\frac{5}{3}$ and $2\pi$?
$\textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ \text{infinitely many}$
2005 Korea - Final Round, 2
Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the
arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ ,
\[\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}. \]
1959 AMC 12/AHSME, 47
Assume that the following three statements are true:
$I$. All freshmen are human. $II$. All students are human. $III$. Some students think.
Given the following four statements:
$ \textbf{(1)}\ \text{All freshmen are students.}\qquad$
$\textbf{(2)}\ \text{Some humans think.}\qquad$
$\textbf{(3)}\ \text{No freshmen think.}\qquad$
$\textbf{(4)}\ \text{Some humans who think are not students.}$
Those which are logical consequences of I,II, and III are:
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 2,3\qquad\textbf{(D)}\ 2,4\qquad\textbf{(E)}\ 1,2 $
2007 AMC 10, 1
One ticket to a show costs $ \$20$ at full price. Susan buys 4 tickets using a coupon that gives her a $25\%$ discount. Pam buys 5 tickets using a coupon that gives her a $30\%$ discount. How many more dollars does Pam pay than Susan?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 20$
2019 AMC 10, 13
Let $\Delta ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^{\circ}$. Contruct the circle with diameter $\overline{BC}$, and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$, respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$. What is the degree measure of $\angle BFC ?$
$\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$