Found problems: 3632
1964 AMC 12/AHSME, 12
Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?
$ \textbf{(A)}\ \text{For all x}, x^2 < 0\qquad$
$\textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad$
$\textbf{(C)}\ \text{For no x}, x^2>0\qquad$
${\textbf{(D)}\ \text{For some x}, x^2>0 }\qquad$
${{\textbf{(E)}\ \text{For some x}, x^2 \le 0}} $
2019 AMC 12/AHSME, 21
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c,a\neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.)
$\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } \text{infinitely many}$
1970 AMC 12/AHSME, 15
Lines in the xy-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1).$ One of these lines has the equation
$\textbf{(A) }3x-2y-1=0\qquad\textbf{(B) }4x-5y+8=0\qquad\textbf{(C) }5x+2y-23=0\qquad$
$\textbf{(D) }x+7y-31=0\qquad \textbf{(E) }x-4y+13=0$
1960 AMC 12/AHSME, 18
The pair of equations $3^{x+y}=81$ and $81^{x-y}=3$ has:
$ \textbf{(A)}\ \text{no common solution} \qquad\textbf{(B)}\ \text{the solution} \text{ } x=2, y=2\qquad$
$\textbf{(C)}\ \text{the solution} \text{ } x=2\frac{1}{2}, y=1\frac{1}{2} \qquad$
$\textbf{(D)}\text{ a common solution in positive and negative integers} \qquad$
$\textbf{(E)}\ \text{none of these} $
2015 AMC 10, 19
In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
$ \textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32 $
2025 AIME, 7
Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2008 AMC 10, 1
A basketball player made $ 5$ baskets during a game. Each basket was worth either $ 2$ or $ 3$ points. How many different numbers could represent the total points scored by the player?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 6$
1970 AMC 12/AHSME, 4
Let $S$ be the set of all numbers which are the sum of the squares of three consecutive integers. Then we can say that:
$\textbf{(A) }\text{No member of }S\text{ is divisible by }2\qquad$
$\textbf{(B) }\text{No member of }S\text{ is divisible by }3\text{ but some member is divisible by }11\qquad$
$\textbf{(C) }\text{No member of }S\text{ is divisible by }3\text{ or }5\qquad$
$\textbf{(D) }\text{No member of }S\text{ is divisible by }3\text{ or }7\qquad$
$\textbf{(E) }\text{None of these}$
2018 AIME Problems, 10
The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point \(A\). At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path \(AJABCHCHIJA\), which has \(10\) steps. Let \(n\) be the number of paths with \(15\) steps that begin and end at point \(A\). Find the remainder when \(n\) is divided by \(1000\).
[asy]
unitsize(32);
draw(unitcircle);
draw(scale(2) * unitcircle);
for(int d = 90; d < 360 + 90; d += 72){
draw(2 * dir(d) -- dir(d));
}
real s = 4;
dot(1 * dir( 90), linewidth(s));
dot(1 * dir(162), linewidth(s));
dot(1 * dir(234), linewidth(s));
dot(1 * dir(306), linewidth(s));
dot(1 * dir(378), linewidth(s));
dot(2 * dir(378), linewidth(s));
dot(2 * dir(306), linewidth(s));
dot(2 * dir(234), linewidth(s));
dot(2 * dir(162), linewidth(s));
dot(2 * dir( 90), linewidth(s));
defaultpen(fontsize(10pt));
real r = 0.05;
label("$A$", (1-r) * dir( 90), -dir( 90));
label("$B$", (1-r) * dir(162), -dir(162));
label("$C$", (1-r) * dir(234), -dir(234));
label("$D$", (1-r) * dir(306), -dir(306));
label("$E$", (1-r) * dir(378), -dir(378));
label("$F$", (2+r) * dir(378), dir(378));
label("$G$", (2+r) * dir(306), dir(306));
label("$H$", (2+r) * dir(234), dir(234));
label("$I$", (2+r) * dir(162), dir(162));
label("$J$", (2+r) * dir( 90), dir( 90));
[/asy]
2009 AMC 10, 16
Points $ A$ and $ C$ lie on a circle centered at $ O$, each of $ \overline{BA}$ and $ \overline{BC}$ are tangent to the circle, and $ \triangle ABC$ is equilateral. The circle intersects $ \overline{BO}$ at $ D$. What is $ \frac {BD}{BO}$?
$ \textbf{(A)}\ \frac {\sqrt2}{3} \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {\sqrt3}{3} \qquad \textbf{(D)}\ \frac {\sqrt2}{2} \qquad \textbf{(E)}\ \frac {\sqrt3}{2}$
2021 AMC 10 Spring, 19
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer is $S$ is [i]also[/i] removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?
$\textbf{(A)} ~36.2 \qquad\textbf{(B)} ~36.4 \qquad\textbf{(C)} ~36.6 \qquad\textbf{(D)} ~36.8 \qquad\textbf{(E)} ~37$
2011 Purple Comet Problems, 21
If a, b, and c are non-negative real numbers satisfying $a + b + c = 400$, fi nd the maximum possible value of $\sqrt{2a+b}+\sqrt{2b+c}+\sqrt{2c+a}$.
2020 AMC 12/AHSME, 3
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
$\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26$
2016 AMC 10, 3
Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$?
$\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$
2008 USAMO, 3
Let $n$ be a positive integer. Denote by $S_n$ the set of points $(x, y)$ with integer coordinates such that \[ \left\lvert x\right\rvert + \left\lvert y + \frac{1}{2} \right\rvert < n. \] A path is a sequence of distinct points $(x_1 , y_1), (x_2, y_2), \ldots, (x_\ell, y_\ell)$ in $S_n$ such that, for $i = 2, \ldots, \ell$, the distance between $(x_i , y_i)$ and $(x_{i-1} , y_{i-1} )$ is $1$ (in other words, the points $(x_i, y_i)$ and $(x_{i-1} , y_{i-1} )$ are neighbors in the lattice of points with integer coordinates). Prove that the points in $S_n$ cannot be partitioned into fewer than $n$ paths (a partition of $S_n$ into $m$ paths is a set $\mathcal{P}$ of $m$ nonempty paths such that each point in $S_n$ appears in exactly one of the $m$ paths in $\mathcal{P}$).
2018 AMC 10, 1
What is the value of \[\bigg(\Big((2+1)^{-1}+1\Big)^{-1}+1\bigg)^{-1}+1?\]
$\textbf{(A) } \frac{5}{8} \qquad\textbf{(B) } \frac{11}{7} \qquad\textbf{(C) } \frac{8}{5} \qquad\textbf{(D) } \frac{18}{11} \qquad\textbf{(E) } \frac{15}{8}$
2023 AMC 12/AHSME, 8
Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$. What is the mean of her quiz scores currently?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$
2010 AMC 12/AHSME, 7
Logan is constructing a scaled model of his town. The city's water tower stands $ 40$ meters high, and the top portion is a sphere that holds $ 100,000$ liters of water. Logan's miniature water tower holds $ 0.1$ liters. How tall, in meters, should Logan make his tower?
$ \textbf{(A)}\ 0.04\qquad \textbf{(B)}\ \frac{0.4}{\pi}\qquad \textbf{(C)}\ 0.4\qquad \textbf{(D)}\ \frac{4}{\pi}\qquad \textbf{(E)}\ 4$
2019 AMC 10, 23
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. What is the $2019$th number said by Tadd?
$ \textbf{(A)}\ 5743
\qquad\textbf{(B)}\ 5885
\qquad\textbf{(C)}\ 5979
\qquad\textbf{(D)}\ 6001
\qquad\textbf{(E)}\ 6011
$
2014 AMC 12/AHSME, 21
In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?
[asy]
unitsize(150);
pair A,B,C,D,E,F,G,H,J,K;
A=(1,0); B=(0,0); C=(0,1); D=(1,1);
draw(A--B--C--D--A);
E=(2-sqrt(3),0); F=(2-sqrt(3),1);
draw(E--F);
G=(1,sqrt(3)/2); H=(2.5-sqrt(3),1);
K=(2-sqrt(3),1-sqrt(3)/2); J=(0.5,0);
draw(G--H--K--J--G);
label("$A$",A,SE);
label("$B$",B,SW);
label("$C$",C,NW);
label("$D$",D,NE);
label("$E$",E,S);
label("$F$",F,N);
label("$G$",G,E);
label("$H$",H,N);
label("$K$",K,W);
label("$J$",J,S);
[/asy]
$ \textbf{(A) }\dfrac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\dfrac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\dfrac{\sqrt{3}}{6}\qquad\textbf{(E) }1-\dfrac{\sqrt{2}}{2} $
2006 AMC 10, 20
Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5?
$ \textbf{(A) } \frac 12 \qquad \textbf{(B) } \frac 35 \qquad \textbf{(C) } \frac 23 \qquad \textbf{(D) } \frac 45 \qquad \textbf{(E) } 1$
1984 AIME Problems, 6
Three circles, each of radius 3, are drawn with centers at $(14,92)$, $(17,76)$, and $(19,84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?
2010 AMC 12/AHSME, 16
Bernardo randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a $ 3$-digit number. Silvia randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a $ 3$-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
$ \textbf{(A)}\ \frac {47}{72}\qquad
\textbf{(B)}\ \frac {37}{56}\qquad
\textbf{(C)}\ \frac {2}{3}\qquad
\textbf{(D)}\ \frac {49}{72}\qquad
\textbf{(E)}\ \frac {39}{56}$
2025 USAJMO, 6
Let $S$ be a set of integers with the following properties:
[list]
[*] $\{ 1, 2, \dots, 2025 \} \subseteq S$.
[*] If $a, b \in S$ and $\gcd(a, b) = 1$, then $ab \in S$.
[*] If for some $s \in S$, $s + 1$ is composite, then all positive divisors of $s + 1$ are in $S$.
[/list]
Prove that $S$ contains all positive integers.
1995 AMC 12/AHSME, 7
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is:
$\textbf{(A)}\ 25 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 75 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 100$