Found problems: 3632
1989 AMC 12/AHSME, 29
Find $\displaystyle \sum_{k=0}^{49}(-1)^k\binom{99}{2k}$, where $\binom{n}{j}=\frac{n!}{j!(n-j)!}$.
$ \textbf{(A)}\ -2^{50} \qquad\textbf{(B)}\ -2^{49} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2^{49} \qquad\textbf{(E)}\ 2^{50} $
2009 AMC 10, 8
In a certain year the price of gasoline rose by $ 20\%$ during January, fell by $ 20\%$ during February, rose by $ 25\%$ during March, and fell by $ x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $ x$?
$ \textbf{(A)}\ 12\qquad
\textbf{(B)}\ 17\qquad
\textbf{(C)}\ 20\qquad
\textbf{(D)}\ 25\qquad
\textbf{(E)}\ 35$
2011 AMC 10, 22
Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
$ \textbf{(A)}\ 2520\qquad\textbf{(B)}\ 2880\qquad\textbf{(C)}\ 3120\qquad\textbf{(D)}\ 3250\qquad\textbf{(E)}\ 3750 $
2017 AMC 12/AHSME, 23
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$?
$\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$
2017 AMC 10, 8
Points $A(11,9)$ and $B(2,-3)$ are vertices of $\triangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$?
$\textbf{(A) } (-8, 9)\qquad \textbf{(B) } (-4, 8)\qquad \textbf{(C) } (-4,9)\qquad \textbf{(D) } (-2, 3)\qquad \textbf{(E) } (-1, 0)$
2017 AMC 12/AHSME, 21
Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?
$\textbf{(A)} \text{ 92} \qquad \textbf{(B)} \text{ 94} \qquad \textbf{(C)} \text{ 96} \qquad \textbf{(D)} \text{ 98} \qquad \textbf{(E)} \text{ 100}$
1978 AMC 12/AHSME, 22
The following four statements, and only these are found on a card:
[asy]
pair A,B,C,D,E,F,G;
A=(0,1);
B=(0,5);
C=(11,5);
D=(11,1);
E=(0,4);
F=(0,3);
G=(0,2);
draw(A--B--C--D--cycle);
label("On this card exactly one statement is false.", B, SE);
label("On this card exactly two statements are false.", E, SE);
label("On this card exactly three statements are false.", F, SE);
label("On this card exactly four statements are false.", G, SE);
[/asy]
(Assume each statement is either true or false.) Among them the number of false statements is exactly
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
2008 AMC 12/AHSME, 17
Let $ a_1,a_2,\dots$ be a sequence of integers determined by the rule $ a_n\equal{}a_{n\minus{}1}/2$ if $ a_{n\minus{}1}$ is even and $ a_n\equal{}3a_{n\minus{}1}\plus{}1$ if $ a_{n\minus{}1}$ is odd. For how many positive integers $ a_1 \le 2008$ is it true that $ a_1$ is less than each of $ a_2$, $ a_3$, and $ a_4$?
$ \textbf{(A)}\ 250 \qquad
\textbf{(B)}\ 251 \qquad
\textbf{(C)}\ 501 \qquad
\textbf{(D)}\ 502 \qquad
\textbf{(E)}\ 1004$
2014 AMC 10, 19
Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
$\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{2-\sqrt{2}}{2}\qquad\textbf{(D) }\frac{1}{3}\qquad\textbf{(E) }\frac{1}{2}\qquad$
2024 USAMO, 6
Let $n > 2$ be an integer and let $\ell \in \{1, 2,\dots, n\}$. A collection $A_1,\dots,A_k$ of (not necessarily distinct) subsets of $\{1, 2,\dots, n\}$ is called $\ell$-large if $|A_i| \ge \ell$ for all $1 \le i \le k$. Find, in terms of $n$ and $\ell$, the largest real number $c$ such that the inequality
\[ \sum_{i=1}^k\sum_{j=1}^k x_ix_j\frac{|A_i\cap A_j|^2}{|A_i|\cdot|A_j|}\ge c\left(\sum_{i=1}^k x_i\right)^2 \]
holds for all positive integer $k$, all nonnegative real numbers $x_1,x_2,\dots,x_k$, and all $\ell$-large collections $A_1,A_2,\dots,A_k$ of subsets of $\{1,2,\dots,n\}$.
[i]Proposed by Titu Andreescu and Gabriel Dospinescu[/i]
2016 AMC 12/AHSME, 9
Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?
$\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$
2010 AIME Problems, 11
Define a [i]T-grid[/i] to be a $ 3\times3$ matrix which satisfies the following two properties:
(1) Exactly five of the entries are $ 1$'s, and the remaining four entries are $ 0$'s.
(2) Among the eight rows, columns, and long diagonals (the long diagonals are $ \{a_{13},a_{22},a_{31}\}$ and $ \{a_{11},a_{22},a_{33}\}$, no more than one of the eight has all three entries equal.
Find the number of distinct T-grids.
1977 AMC 12/AHSME, 20
\[\begin{tabular}{ccccccccccccc}
& & & & & & C & & & & & & \\
& & & & & C & O & C & & & & & \\
& & & & C & O & N & O & C & & & & \\
& & & C & O & N & T & N & O & C & & & \\
& & C & O & N & T & E & T & N & O & C & & \\
& C & O & N & T & E & S & E & T & N & O & C & \\
C & O & N & T & E & S & T & S & E & T & N & O & C
\end{tabular}\]
For how many paths consisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram above, is the word CONTEST spelled out as the path is traversed from beginning to end?
$\textbf{(A) }63\qquad\textbf{(B) }128\qquad\textbf{(C) }129\qquad\textbf{(D) }255\qquad \textbf{(E) }\text{none of these}$
2010 AIME Problems, 4
Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $ 100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $ 400$ feet or less to the new gate be a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
2019 AMC 10, 3
Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$
$\textbf{(A) } 3 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 12 \qquad\textbf{(E) } 15$
2008 AMC 10, 9
Suppose that
\[ \frac {2x}{3} \minus{} \frac {x}{6}
\]is an integer. Which of the following statements must be true about $ x$?
$ \textbf{(A)}\ \text{It is negative.} \qquad \textbf{(B)}\ \text{It is even, but not necessarily a multiple of }3\text{.}$
$ \textbf{(C)}\ \text{It is a multiple of }3\text{, but not necessarily even.}$
$ \textbf{(D)}\ \text{It is a multiple of }6\text{, but not necessarily a multiple of }12\text{.}$
$ \textbf{(E)}\ \text{It is a multiple of }12\text{.}$
1997 AMC 8, 23
There are positive integers that have these properties:
* the sum of the squares of their digits is 50, and
* each digit is larger than the one to its left.
The product of the digits of the largest integer with both properties is
$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60$
2011 AMC 12/AHSME, 21
Let $f_1(x)=\sqrt{1-x}$, and for integers $n \ge 2$, let $f_n(x)=f_{n-1}(\sqrt{n^2-x})$. If $N$ is the largest value of $n$ for which the domain of $f_n$ is nonempty, the domain of $f_N$ is ${c}$. What is $N+c$?
$ \textbf{(A)}\ -226 \qquad
\textbf{(B)}\ -144 \qquad
\textbf{(C)}\ -20 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 144$
2009 AIME Problems, 14
For $ t \equal{} 1, 2, 3, 4$, define $ \displaystyle S_t \equal{} \sum_{i \equal{} 1}^{350}a_i^t$, where $ a_i \in \{1,2,3,4\}$. If $ S_1 \equal{} 513$ and $ S_4 \equal{} 4745$, find the minimum possible value for $ S_2$.
2014 AMC 12/AHSME, 24
Let $f_0(x)=x+|x-100|-|x+100|$, and for $n\geq 1$, let $f_n(x)=|f_{n-1}(x)|-1$. For how many values of $x$ is $f_{100}(x)=0$?
$\textbf{(A) }299\qquad
\textbf{(B) }300\qquad
\textbf{(C) }301\qquad
\textbf{(D) }302\qquad
\textbf{(E) }303\qquad$
2021 AMC 12/AHSME Fall, 6
The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor of $16383$?
$\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }10\qquad\textbf{(D) }16\qquad\textbf{(E) }22$
1977 AMC 12/AHSME, 19
Let $E$ be the point of intersection of the diagonals of convex quadrilateral $ABCD$, and let $P,Q,R,$ and $S$ be the centers of the circles circumscribing triangles $ABE,$ $BCE$, $CDE$, and $ADE$, respectively. Then
$\textbf{(A) }PQRS\text{ is a parallelogram}$
$\textbf{(B) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rhombus}$
$\textbf{(C) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rectangle}$
$\textbf{(D) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a parallelogram}$
$\textbf{(E) }\text{none of the above are true}$
1960 AMC 12/AHSME, 35
From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, then $t$ may have the following number of values:
$ \textbf{(A)}\ \text{zero} \qquad\textbf{(B)}\ \text{one} \qquad\textbf{(C)}\ \text{two} \qquad\textbf{(D)}\ \text{three} \qquad\textbf{(E)}\ \text{infinitely many} $
1986 AMC 12/AHSME, 19
A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?
$ \textbf{(A)}\ \sqrt{13}\qquad\textbf{(B)}\ \sqrt{14}\qquad\textbf{(C)}\ \sqrt{15}\qquad\textbf{(D)}\ \sqrt{16}\qquad\textbf{(E)}\ \sqrt{17}$
2006 AMC 12/AHSME, 8
How many sets of two or more consecutive positive integers have a sum of 15?
$ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$