Difference between revisions of "User:Rowechen"
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Here's the AIME compilation I will be doing: | Here's the AIME compilation I will be doing: | ||
− | ==Problem | + | ==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>. | |
− | + | [[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? | |
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+ | [[2007 AIME I Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
+ | 5. 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> | |
− | < | + | 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>. |
− | ==Problem | + | ==Problem 9== |
− | + | 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>. | |
− | [[ | + | [[2017 AIME I Problems/Problem 9 | Solution]] |
− | == Problem | + | ==Problem 8== |
− | Let <math> | + | 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]] |
− | ==Problem | + | ==Problem 7== |
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− | + | 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. | |
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− | [[ | + | [[2018 AIME II Problems/Problem 7 | Solution]] |
+ | ==Problem 10== | ||
− | + | 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>. | |
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− | [[ | + | [[2018 AIME II Problems/Problem 10 | Solution]] |
==Problem 11== | ==Problem 11== | ||
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− | + | 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>. | |
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− | + | [[2018 AIME II Problems/Problem 11 | Solution]] | |
− | + | ==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>. | |
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− | + | [[2018 AIME II Problems/Problem 14 | Solution]] | |
− | + | == Problem 10 == | |
− | + | 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>. | |
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− | + | [[2009 AIME II Problems/Problem 10|Solution]] | |
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==Problem 11== | ==Problem 11== | ||
− | + | <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== | ||
− | + | 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 | |
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− | + | <cmath>|x - y| \leq |f(x) - f(y)| \leq 3|x - y|</cmath> | |
− | + | for all <math>x</math> and <math>y</math> in <math>\{0, 1, 2, 3, 4, 5, 6\}</math>. | |
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− | + | [[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 | + | == Problem 15 == |
− | Let <math>\ | + | 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== | |
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− | ==Problem | ||
− | + | 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>. | |
− | [[ | + | [[2015 AIME II Problems/Problem 14 | Solution]] |
Revision as of 15:02, 31 May 2020
Here's the AIME compilation I will be doing:
Contents
Problem 3
A triangle has vertices ,
, and
. The probability that a randomly chosen point inside the triangle is closer to vertex
than to either vertex
or vertex
can be written as
, where
and
are relatively prime positive integers. Find
.
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 ,
, and
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?
Problem 5
5. If
is written as a common fraction reduced to lowest terms, the result is . Compute the sum of the prime divisors of
plus the sum of the prime divisors of
.
Problem 9
Let , and for each integer
let
. Find the least
such that
is a multiple of
.
Problem 8
Two real numbers and
are chosen independently and uniformly at random from the interval
. Let
and
be two points on the plane with
. Let
and
be on the same side of line
such that the degree measures of
and
are
and
respectively, and
and
are both right angles. The probability that
is equal to
, where
and
are relatively prime positive integers. Find
.
Problem 7
Triangle has side lengths
,
, and
. Points
are on segment
with
between
and
for
, and points
are on segment
with
between
and
for
. Furthermore, each segment
,
, is parallel to
. The segments cut the triangle into
regions, consisting of
trapezoids and
triangle. Each of the
regions has the same area. Find the number of segments
,
, that have rational length.
Problem 10
Find the number of functions from
to
that satisfy
for all
in
.
Problem 11
Find the number of permutations of such that for each
with
, at least one of the first
terms of the permutation is greater than
.
Problem 14
The incircle of triangle
is tangent to
at
. Let
be the other intersection of
with
. Points
and
lie on
and
, respectively, so that
is tangent to
at
. Assume that
,
,
, and
, where
and
are relatively prime positive integers. Find
.
Problem 10
Four lighthouses are located at points ,
,
, and
. The lighthouse at
is
kilometers from the lighthouse at
, the lighthouse at
is
kilometers from the lighthouse at
, and the lighthouse at
is
kilometers from the lighthouse at
. To an observer at
, the angle determined by the lights at
and
and the angle determined by the lights at
and
are equal. To an observer at
, the angle determined by the lights at
and
and the angle determined by the lights at
and
are equal. The number of kilometers from
to
is given by
, where
,
, and
are relatively prime positive integers, and
is not divisible by the square of any prime. Find
.
Problem 11
lines and
circles divide the plane into at most
disjoint regions. Compute
.
Problem 15
Find the number of functions from
to the integers such that
,
, and
for all and
in
.
Problem 14
The sequence satisfies
and
for
. Find the greatest integer less than or equal to
.
Problem 15
Let be a diameter of a circle with diameter
. Let
and
be points on one of the semicircular arcs determined by
such that
is the midpoint of the semicircle and
. Point
lies on the other semicircular arc. Let
be the length of the line segment whose endpoints are the intersections of diameter
with the chords
and
. The largest possible value of
can be written in the form
, where
,
, and
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 14
Let and
be real numbers satisfying
and
. Evaluate
.