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Difference between revisions of "User:Rowechen"

(→‎Problem 8)
(→‎Problem 13)
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<cmath>P(k) = \frac{1}{k(k+1)}</cmath>
 
<cmath>P(k) = \frac{1}{k(k+1)}</cmath>
  
−
for <math>k = 1, 2, 3, . . . , 10</math>. The value of <math>P(11)</math> can be written as <math>−\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively
+
for <math>k = 1, 2, 3, . . . , 10</math>. The value of <math>P(11)</math> can be written as <math>-\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively
 
prime positive integers. Determine <math>m + n</math>.
 
prime positive integers. Determine <math>m + n</math>.
  

Revision as of 10:12, 1 June 2020

Here's the AIME compilation I will be doing:

Problem 3

$x$, $y$, and $z$ are positive integers. Let $N$ denote the number of solutions of $2x + y + z = 2004$. Determine the remainder obtained when $N$ is divided by $1000$.


Problem 8

Find the number of sets $\{a,b,c\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,$ and $61$.

Solution

Problem 9

A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and $\textit{still}$ have at least one card of each color and at least one card with each number is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution

Problem 7

Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ = w$, the area of $PQRS$ can be expressed as the quadratic polynomial

\[\text{Area}(PQRS) = \alpha w - \beta \cdot w^2.\]

Then the coefficient $\beta = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 7

For integers $a$ and $b$ consider the complex number \[\frac{\sqrt{ab+2016}}{ab+100}-\left(\frac{\sqrt{|a+b|}}{ab+100}\right)i.\] Find the number of ordered pairs of integers $(a,b)$ such that this complex number is a real number.

Solution

Problem 8

A single atom of Uranium-238 rests at the origin. Each second, the particle has a $1/4$ chance of moving one unit in the negative x-direction and a $1/2$ chance of moving in the positive x-direction. If the particle reaches $(-3,0)$, it ignites fission that will consume the earth. If it reaches $(7, 0)$, it is harmlessly diffused. The probability that, eventually, the particle is safely contained can be expressed as $\frac{m}{n}$ for some relatively prime positive integers $m$ and $n$. Determine the remainder obtained when $m + n$ is divided by $1000$.

Problem 10

$ABCDE$ is a cyclic pentagon with $BC = CD = DE$. The diagonals $AC$ and $BE$ intersect at $M$. $N$ is the foot of the altitude from $M$ to $AB$. We have $MA = 25$, $MD = 113$, and $MN = 15$. The area of triangle $ABE$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the remainder obtained when $m + n$ is divided by $1000$.

Problem 12

$ABCD$ is a cyclic quadrilateral with $AB = 8$, $BC = 4$, $CD = 1$, and $DA = 7$. Let $O$ and $P$ denote the circumcenter and intersection of $AC$ and $BD$ respectively. The value of $OP^2$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime, positive integers. Determine the remainder obtained when $m + n$ is divided by $1000$.

Problem 11

For integers $a,b,c$ and $d,$ let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d.$ Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2))=g(f(4))=0.$

Solution

Problem 10

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumulated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B.$ The probability that team $A$ finishes with more points than team $B$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$


Solution

Problem 14

Centered at each lattice point in the coordinate plane are a circle radius $\frac{1}{10}$ and a square with sides of length $\frac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0,0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of the circles. Find $m + n$.

Solution

Problem 15

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$. The three points $C$, $Y$, $D$ are collinear, $XC = 67$, $XY = 47$, and $XD = 37$. Find $AB^2$.

Solution

Problem 15

$ABCD$ is a convex quadrilateral in which $AB \parallel CD$. Let $U$ denote the intersection of the extensions of $AD$ and $BC$. $\Omega_1$ is the circle tangent to line segment $BC$ which also passes through $A$ and $D$, and $\Omega_2$ is the circle tangent to $AD$ which passes through $B$ and $C$. Call the points of tangency $M$ and $S$. Let $O$ and $P$ be the points of intersection between $\Omega_1$ and $\Omega_2$. Finally, $MS$ intersects $OP$ at $V$. If $AB = 2$, $BC = 2005$, $CD = 4$, and $DA = 2004$, then the value of $UV^2$ is some integer $n$. Determine the remainder obtained when $n$ is divided by $1000$.

Problem 13

$P(x)$ is the polynomial of minimal degree that satisfies \[P(k) = \frac{1}{k(k+1)}\]

for $k = 1, 2, 3, . . . , 10$. The value of $P(11)$ can be written as $-\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $m + n$.

Problem 14

$3$ Elm trees, $4$ Dogwood trees, and $5$ Oak trees are to be planted in a line in front of a library such that i) No two Elm trees are next to each other. ii) No Dogwood tree is adjacent to an Oak tree. iii) All of the trees are planted. How many ways can the trees be situated in this manner?

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