Found problems: 3632
2022 USAMO, 6
There are $2022$ users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.)
Starting now, Mathbook will only allow a new friendship to be formed between two users if they have [i]at least two[/i] friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?
2015 AMC 12/AHSME, 4
David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place?
$\textbf{(A) } \text{David}
\qquad\textbf{(B) } \text{Hikmet}
\qquad\textbf{(C) } \text{Jack}
\qquad\textbf{(D) } \text{Rand}
\qquad\textbf{(E) } \text{Todd}
$
2014 AMC 8, 17
George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad \textbf{(E) }12$
1971 AMC 12/AHSME, 29
Given the progression $10^{\dfrac{1}{11}}, 10^{\dfrac{2}{11}}, 10^{\dfrac{3}{11}}, 10^{\dfrac{4}{11}},\dots , 10^{\dfrac{n}{11}}$. The least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $100,000$ is
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad \textbf{(E) }11$
1997 AMC 8, 17
A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have?
[asy]draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));
draw((3,0)--(3,3));
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);
draw((2.5,4)--(2.5,1),dashed);
label("$x$",(2.75,3.5),NNE);
label("$y$",(4.125,1.5),NNE);
[/asy]
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$
2016 AMC 12/AHSME, 8
What is the area of the shaded region of the given $8 \times 5$ rectangle?
[asy]
size(6cm);
defaultpen(fontsize(9pt));
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));
label("$1$",(1/2,5),dir(90));
label("$7$",(9/2,5),dir(90));
label("$1$",(8,1/2),dir(0));
label("$4$",(8,3),dir(0));
label("$1$",(15/2,0),dir(270));
label("$7$",(7/2,0),dir(270));
label("$1$",(0,9/2),dir(180));
label("$4$",(0,2),dir(180));
[/asy]
$\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf{(D)}\ 6\dfrac{1}{2} \qquad \textbf{(E)}\ 8$
1991 AIME Problems, 1
Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[xy+x+y = 71\qquad\text{and}\qquad x^2y+xy^2 = 880.\]
2016 AMC 12/AHSME, 15
Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$?
$\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}$
2011 AMC 10, 5
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of $12$, $15$, and $10$ minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
$ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ \frac{37}{3} \qquad\textbf{(C)}\ \frac{88}{7} \qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 14 $
2024 AMC 10, 10
Consider the following operation. Given a positive integer $n$, if $n$ is a multiple of $3$, then you replace $n$ by $\dfrac{n}3$. If $n$ is not a multiple of $3$, then you replace $n$ by $n + 10$. Then continue this process. For example, beginning with $n = 4$, this procedure gives $4 \to 14 \to 24 \to 8 \to 18 \to 6 \to 2 \to 12 \to \cdots$. Suppose you start with $n = 100$. What value results if you perform this operation exactly $100$ times?
$\textbf{(A) }10\qquad\textbf{(B) }20\qquad\textbf{(C) }30\qquad\textbf{(D) }40\qquad\textbf{(E) }50$
1961 AMC 12/AHSME, 36
In triangle $ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.
${{ \textbf{(A)}\ 4\qquad\textbf{(B)}\ \sqrt{17} \qquad\textbf{(C)}\ 4.25\qquad\textbf{(D)}\ 2\sqrt{5} }\qquad\textbf{(E)}\ 4.5} $
2020 AMC 12/AHSME, 25
For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that
$$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1.$$
What is the maximum value of $P(a)?$
$\textbf{(A)}\ \frac{7}{12} \qquad\textbf{(B)}\ 2 - \sqrt{2} \qquad\textbf{(C)}\ \frac{1+\sqrt{2}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(E)}\ \frac{5}{8}$
1993 AMC 12/AHSME, 20
Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true?
$ \textbf{(A)}\ \text{For all positive real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(B)}\ \text{For all negative real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(C)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and rational.} \\ \qquad\textbf{(D)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and irrational.} \\ \qquad\textbf{(E)}\ \text{For all complex numbers}\ k,\ \text{neither root is real.} $
2013 AMC 12/AHSME, 19
In $ \bigtriangleup ABC $, $ AB = 86 $, and $ AC = 97 $. A circle with center $ A $ and radius $ AB $ intersects $ \overline{BC} $ at points $ B $ and $ X $. Moreover $ \overline{BX} $ and $ \overline{CX} $ have integer lengths. What is $ BC $?
$ \textbf{(A)} \ 11 \qquad \textbf{(B)} \ 28 \qquad \textbf{(C)} \ 33 \qquad \textbf{(D)} \ 61 \qquad \textbf{(E)} \ 72 $
2013 AMC 10, 1
A taxi ride costs $\$1.50$ plus $\$0.25$ per mile traveled. How much does a $5$-mile taxi ride cost?
$ \textbf{(A)}\ \$2.25 \qquad\textbf{(B)}\ \$2.50 \qquad\textbf{(C)}\ \$2.75 \qquad\textbf{(D)}\ \$3.00 \qquad\textbf{(E)}\ \$ 3.25$
2025 USAJMO, 1
Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective.
[i]Note: [/i] A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.
2006 AIME Problems, 4
Let $(a_1,a_2,a_3,...,a_{12})$ be a permutation of $(1,2,3,...,12)$ for which \[ a_1>a_2>a_3>a_4>a_5>a_6 \text{ and } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}. \]
An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12)$. Find the number of such permutations.
2012 AMC 10, 22
Let $(a_1,a_2, \dots ,a_{10})$ be a list of the first $10$ positive integers such that for each $2 \le i \le 10$ either $a_i+1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?
$ \textbf{(A)}\ 120\qquad\textbf{(B)}\ 512\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 181,440\qquad\textbf{(E)}\ 362,880 $
2023 AMC 12/AHSME, 5
You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle?
$\textbf{(A)}~3\qquad\textbf{(B)}~5\qquad\textbf{(C)}~4\qquad\textbf{(D)}~8\qquad\textbf{(E)}~6$
2023 AMC 12/AHSME, 3
A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?
$\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}$
1993 AMC 12/AHSME, 3
$\frac{15^{30}}{45^{15}}=$
$ \textbf{(A)}\ \left(\frac{1}{3}\right)^{15} \qquad\textbf{(B)}\ \left(\frac{1}{3}\right)^2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 3^{15} \qquad\textbf{(E)}\ 5^{15}$
2014 USAJMO, 3
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
2019 AMC 10, 9
What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?
$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$
2017 AMC 10, 13
Define a sequence recursively by $F_0 = 0$, $F_1 = 1$, and $F_n = $ the remainder when $F_{n-1} + F_{n-2}$ is divided by $3$, for all $n \ge 2$. Thus the sequence starts $0,1,1,2,0,2 \ldots$. What is $F_{2017} + F_{2018} + F_{2019} + F_{2020} + F_{2021} + F_{2022} + F_{2023} + F_{2024}$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$
2008 AIME Problems, 9
A particle is located on the coordinate plane at $ (5,0)$. Define a [i]move[/i] for the particle as a counterclockwise rotation of $ \pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$-direction. Given that the particle's position after $ 150$ moves is $ (p,q)$, find the greatest integer less than or equal to $ |p|\plus{}|q|$.