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Difference between revisions of "User:Rowechen"

(→‎Problem 5)
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==Problem 3==
 
==Problem 3==
−
A triangle has vertices <math>A(0,0)</math>, <math>B(12,0)</math>, and <math>C(8,10)</math>. The probability that a randomly chosen point inside the triangle is closer to vertex <math>B</math> than to either vertex <math>A</math> or vertex <math>C</math> can be written as <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
+
<math>x</math>, <math>y</math>, and <math>z</math> are positive integers. Let <math>N</math> denote the number of solutions of <math>2x + y + z = 2004</math>. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>.
  
−
[[2017 AIME II Problems/Problem 3 | Solution]]
 
−
== Problem 4 ==
 
−
Three planets orbit a star circularly in the same plane.  Each moves in the same direction and moves at constant speed.  Their periods are <math>60</math>, <math>84</math>, and <math>140</math> years.  The three planets and the star are currently collinear.  What is the fewest number of years from now that they will all be collinear again?
 
  
−
[[2007 AIME I Problems/Problem 4|Solution]]
+
==Problem 8==
−
==Problem 5==
+
Find the number of sets <math>\{a,b,c\}</math> of three distinct positive integers with the property that the product of <math>a,b,</math> and <math>c</math> is equal to the product of <math>11,21,31,41,51,</math> and <math>61</math>.
−
If
 
  
−
<cmath>\frac{1}{0!10!}+\frac{1}{1!9!}+\frac{1}{2!8!}+\frac{1}{3!7!}+\frac{1}{4!6!}+\frac{1}{5!5!}</cmath>
+
[[2016 AIME II Problems/Problem 8 | Solution]]
 +
==Problem 9==
 +
A special deck of cards contains <math>49</math> cards, each labeled with a number from <math>1</math> to <math>7</math> 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 <math>\textit{still}</math> have at least one card of each color and at least one card with each number is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
  
−
is written as a common fraction reduced to lowest terms, the result is <math>\frac{m}{n}</math>. Compute the sum of the prime divisors of <math>m</math> plus the sum of the prime divisors of <math>n</math>.
+
[[2017 AIME II Problems/Problem 9 | Solution]]
 +
==Problem 7==
  
−
==Problem 9==
+
Triangle <math>ABC</math> has side lengths <math>AB = 12</math>, <math>BC = 25</math>, and <math>CA = 17</math>. Rectangle <math>PQRS</math> has vertex <math>P</math> on <math>\overline{AB}</math>, vertex <math>Q</math> on <math>\overline{AC}</math>, and vertices <math>R</math> and <math>S</math> on <math>\overline{BC}</math>. In terms of the side length <math>PQ = w</math>, the area of <math>PQRS</math> can be expressed as the quadratic polynomial
−
Let <math>a_{10} = 10</math>, and for each integer <math>n >10</math> let <math>a_n = 100a_{n - 1} + n</math>. Find the least <math>n > 10</math> such that <math>a_n</math> is a multiple of <math>99</math>.
+
 
 +
<cmath>\text{Area}(PQRS) = \alpha w - \beta \cdot w^2.</cmath>
  
−
[[2017 AIME I Problems/Problem 9 | Solution]]
+
Then the coefficient <math>\beta = \frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
−
==Problem 8==
 
−
Two real numbers <math>a</math> and <math>b</math> are chosen independently and uniformly at random from the interval <math>(0, 75)</math>. Let <math>O</math> and <math>P</math> be two points on the plane with <math>OP = 200</math>. Let <math>Q</math> and <math>R</math> be on the same side of line <math>OP</math> such that the degree measures of <math>\angle POQ</math> and <math>\angle POR</math> are <math>a</math> and <math>b</math> respectively, and <math>\angle OQP</math> and <math>\angle ORP</math> are both right angles. The probability that <math>QR \leq 100</math> is equal to <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
 
  
−
[[2017 AIME I Problems/Problem 8 | Solution]]
+
[[2015 AIME II Problems/Problem 7 | Solution]]
 
==Problem 7==
 
==Problem 7==
 +
For integers <math>a</math> and <math>b</math> consider the complex number <cmath>\frac{\sqrt{ab+2016}}{ab+100}-\left(\frac{\sqrt{|a+b|}}{ab+100}\right)i.</cmath> Find the number of ordered pairs of integers <math>(a,b)</math> such that this complex number is a real number.
  
−
Triangle <math>ABC</math> has side lengths <math>AB = 9</math>, <math>BC =</math> <math>5\sqrt{3}</math>, and <math>AC = 12</math>. Points <math>A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B</math> are on segment <math>\overline{AB}</math> with <math>P_{k}</math> between <math>P_{k-1}</math> and <math>P_{k+1}</math> for <math>k = 1, 2, ..., 2449</math>, and points <math>A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C</math> are on segment <math>\overline{AC}</math> with <math>Q_{k}</math> between <math>Q_{k-1}</math> and <math>Q_{k+1}</math> for <math>k = 1, 2, ..., 2449</math>. Furthermore, each segment <math>\overline{P_{k}Q_{k}}</math>, <math>k = 1, 2, ..., 2449</math>, is parallel to <math>\overline{BC}</math>. The segments cut the triangle into <math>2450</math> regions, consisting of <math>2449</math> trapezoids and <math>1</math> triangle. Each of the <math>2450</math> regions has the same area. Find the number of segments <math>\overline{P_{k}Q_{k}}</math>, <math>k = 1, 2, ..., 2450</math>, that have rational length.
+
[[2016 AIME I Problems/Problem 7 | Solution]]
 +
 
 +
==Problem 8==
 +
A single atom of Uranium-238 rests at the origin. Each second, the particle has a <math>1/4</math> chance of moving one unit in the negative x-direction and a <math>1/2</math> chance of moving in the positive x-direction. If the particle reaches <math>(−3, 0)</math>, it ignites fission that will consume the earth. If it reaches <math>(7, 0)</math>, it is harmlessly diffused. The probability that, eventually, the particle is safely contained can be expressed as <math>\frac{m}{n}</math> for some relatively prime positive integers <math>m</math> and <math>n</math>.
 +
Determine the remainder obtained when <math>m + n</math> is divided by <math>1000</math>.
  
−
[[2018 AIME II Problems/Problem 7 | Solution]]
 
 
==Problem 10==
 
==Problem 10==
 +
<math>ABCDE</math> is a cyclic pentagon with <math>BC = CD = DE</math>. The diagonals <math>AC</math> and <math>BE</math> intersect at <math>M</math>. <math>N</math> is the foot of the altitude from <math>M</math> to <math>AB</math>. We have <math>MA = 25</math>, <math>MD = 113</math>, and <math>MN = 15</math>. The area of triangle <math>ABE</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine the remainder obtained when <math>m + n</math> is divided by <math>1000</math>.
  
−
Find the number of functions <math>f(x)</math> from <math>\{1, 2, 3, 4, 5\}</math> to <math>\{1, 2, 3, 4, 5\}</math> that satisfy <math>f(f(x)) = f(f(f(x)))</math> for all <math>x</math> in <math>\{1, 2, 3, 4, 5\}</math>.
+
==Problem 12==
  
−
[[2018 AIME II Problems/Problem 10 | Solution]]
+
<math>ABCD</math> is a cyclic quadrilateral with <math>AB = 8</math>, <math>BC = 4</math>, <math>CD = 1</math>, and <math>DA = 7</math>. Let <math>O</math> and <math>P</math> denote the circumcenter and intersection of <math>AC</math> and <math>BD</math> respectively. The value of <math>OP^2</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime, positive integers. Determine the remainder obtained when <math>m + n</math> is divided by <math>1000</math>.
 
==Problem 11==
 
==Problem 11==
 +
For integers <math>a,b,c</math> and <math>d,</math> let <math>f(x)=x^2+ax+b</math> and <math>g(x)=x^2+cx+d.</math> Find the number of ordered triples <math>(a,b,c)</math> of integers with absolute values not exceeding <math>10</math> for which there is an integer <math>d</math> such that <math>g(f(2))=g(f(4))=0.</math>
  
−
Find the number of permutations of <math>1, 2, 3, 4, 5, 6</math> such that for each <math>k</math> with <math>1</math> <math>\leq</math> <math>k</math> <math>\leq</math> <math>5</math>, at least one of the first <math>k</math> terms of the permutation is greater than <math>k</math>.
+
[[2020 AIME I Problems/Problem 11 | 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 <math> 50\% </math> 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 <math> A </math> beats team <math> B. </math> The probability that team <math> A </math> finishes with more points than team <math> B </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers. Find <math> m+n. </math>
  
−
[[2018 AIME II Problems/Problem 11 | Solution]]
+
 
 +
[[2006 AIME II Problems/Problem 10|Solution]]
 
==Problem 14==
 
==Problem 14==
  
−
The incircle <math>\omega</math> of triangle <math>ABC</math> is tangent to <math>\overline{BC}</math> at <math>X</math>. Let <math>Y \neq X</math> be the other intersection of <math>\overline{AX}</math> with <math>\omega</math>. Points <math>P</math> and <math>Q</math> lie on <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, so that <math>\overline{PQ}</math> is tangent to <math>\omega</math> at <math>Y</math>. Assume that <math>AP = 3</math>, <math>PB = 4</math>, <math>AC = 8</math>, and <math>AQ = \dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
+
Centered at each lattice point in the coordinate plane are a circle radius <math>\frac{1}{10}</math> and a square with sides of length <math>\frac{1}{5}</math> whose sides are parallel to the coordinate axes. The line segment from <math>(0,0)</math> to <math>(1001, 429)</math> intersects <math>m</math> of the squares and <math>n</math> of the circles. Find <math>m + n</math>.
  
−
[[2018 AIME II Problems/Problem 14 | Solution]]
+
[[2016 AIME I Problems/Problem 14 | Solution]]
−
== Problem 10 ==
+
==Problem 15==
−
Four lighthouses are located at points <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>. The lighthouse at <math>A</math> is <math>5</math> kilometers from the lighthouse at <math>B</math>, the lighthouse at <math>B</math> is <math>12</math> kilometers from the lighthouse at <math>C</math>, and the lighthouse at <math>A</math> is <math>13</math> kilometers from the lighthouse at <math>C</math>. To an observer at <math>A</math>, the angle determined by the lights at <math>B</math> and <math>D</math> and the angle determined by the lights at <math>C</math> and <math>D</math> are equal. To an observer at <math>C</math>, the angle determined by the lights at <math>A</math> and <math>B</math> and the angle determined by the lights at <math>D</math> and <math>B</math> are equal. The number of kilometers from <math>A</math> to <math>D</math> is given by <math>\frac{p\sqrt{r}}{q}</math>, where <math>p</math>, <math>q</math>, and <math>r</math> are relatively prime positive integers, and <math>r</math> is not divisible by the square of any prime. Find <math>p+q+r</math>.
 
  
−
[[2009 AIME II Problems/Problem 10|Solution]]
+
Circles <math>\omega_1</math> and <math>\omega_2</math> intersect at points <math>X</math> and <math>Y</math>. Line <math>\ell</math> is tangent to <math>\omega_1</math> and <math>\omega_2</math> at <math>A</math> and <math>B</math>, respectively, with line <math>AB</math> closer to point <math>X</math> than to <math>Y</math>. Circle <math>\omega</math> passes through <math>A</math> and <math>B</math> intersecting <math>\omega_1</math> again at <math>D \neq A</math> and intersecting <math>\omega_2</math> again at <math>C \neq B</math>. The three points <math>C</math>, <math>Y</math>, <math>D</math> are collinear, <math>XC = 67</math>, <math>XY = 47</math>, and <math>XD = 37</math>. Find <math>AB^2</math>.
  
−
==Problem 11==
+
[[2016 AIME I Problems/Problem 15 | Solution]]
−
<math>10</math> lines and <math>10</math> circles divide the plane into at most <math>n</math> disjoint regions. Compute <math>n</math>.
 
  
 
==Problem 15==
 
==Problem 15==
 +
<math>ABCD</math> is a convex quadrilateral in which <math>AB \parallel CD</math>. Let <math>U</math> denote the intersection of the extensions of <math>AD</math> and <math>BC</math>. <math>\Omega_1</math> is the circle tangent to line segment <math>BC</math> which also passes through <math>A</math> and <math>D</math>, and <math>\Omega_2</math> is the circle tangent to <math>AD</math> which passes through <math>B</math> and <math>C</math>. Call the points of tangency <math>M</math> and <math>S</math>. Let <math>O</math> and <math>P</math> be the points of intersection between <math>\Omega_1</math> and <math>\Omega_2</math>.
 +
Finally, <math>MS</math> intersects <math>OP</math> at <math>V</math>. If <math>AB = 2</math>, <math>BC = 2005</math>, <math>CD = 4</math>, and <math>DA = 2004</math>, then the value of <math>UV^2</math> is some integer <math>n</math>. Determine the remainder obtained when <math>n</math> is divided by <math>1000</math>.
  
−
Find the number of functions <math>f</math> from <math>\{0, 1, 2, 3, 4, 5, 6\}</math> to the integers such that <math>f(0) = 0</math>, <math>f(6) = 12</math>, and
+
==Problem 13==
 +
<math>P(x)</math> is the polynomial of minimal degree that satisfies
 +
<cmath>P(k) = \frac{1}{k(k+1)}</cmath>
  
−
<cmath>|x - y|  \leq  |f(x) - f(y)|  \leq  3|x - y|</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
 +
prime positive integers. Determine <math>m + n</math>.
  
−
for all <math>x</math> and <math>y</math> in <math>\{0, 1, 2, 3, 4, 5, 6\}</math>.
 
−
 
−
[[2018 AIME II Problems/Problem 15 | Solution]]
 
−
 
−
== Problem 14 ==
 
−
The sequence <math>(a_n)</math> satisfies <math>a_0=0</math> and <math>a_{n + 1} = \frac{8}{5}a_n + \frac{6}{5}\sqrt{4^n - a_n^2}</math> for <math>n \geq 0</math>. Find the greatest integer less than or equal to <math>a_{10}</math>.
 
−
 
−
[[2009 AIME II Problems/Problem 14|Solution]]
 
−
== Problem 15 ==
 
−
Let <math>\overline{MN}</math> be a diameter of a circle with diameter <math>1</math>. Let <math>A</math> and <math>B</math> be points on one of the semicircular arcs determined by <math>\overline{MN}</math> such that <math>A</math> is the midpoint of the semicircle and <math>MB=\dfrac 35</math>. Point <math>C</math> lies on the other semicircular arc. Let <math>d</math> be the length of the line segment whose endpoints are the intersections of diameter <math>\overline{MN}</math> with the chords <math>\overline{AC}</math> and <math>\overline{BC}</math>. The largest possible value of <math>d</math> can be written in the form <math>r-s\sqrt t</math>, where <math>r</math>, <math>s</math>, and <math>t</math> are positive integers and <math>t</math> is not divisible by the square of any prime. Find <math>r+s+t</math>.
 
−
 
−
[[2009 AIME II Problems/Problem 15|Solution]]
 
 
==Problem 14==
 
==Problem 14==
−
 
+
<math>3</math> Elm trees, <math>4</math> Dogwood trees, and <math>5</math> Oak trees are to be planted in a line in front of a library such that
−
Let <math>x</math> and <math>y</math> be real numbers satisfying <math>x^4y^5+y^4x^5=810</math> and <math>x^3y^6+y^3x^6=945</math>. Evaluate <math>2x^3+(xy)^3+2y^3</math>.
+
i) No two Elm trees are next to each other.
−
 
+
ii) No Dogwood tree is adjacent to an Oak tree.
−
[[2015 AIME II Problems/Problem 14 | Solution]]
+
iii) All of the trees are planted.
 +
How many ways can the trees be situated in this manner?

Revision as of 11:03, 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 $(−3, 0)$ (Error compiling LaTeX. Unknown error_msg), 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 $−\frac{m}{n}$ (Error compiling LaTeX. Unknown error_msg), 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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