Difference between revisions of "2017 AIME II Problems"
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Find the number of subsets of <math>\{1, 2, 3, 4, 5, 6, 7, 8\}</math> that are subsets of neither <math>\{1, 2, 3, 4, 5\}</math> nor <math>\{4, 5, 6, 7, 8\}</math>. | Find the number of subsets of <math>\{1, 2, 3, 4, 5, 6, 7, 8\}</math> that are subsets of neither <math>\{1, 2, 3, 4, 5\}</math> nor <math>\{4, 5, 6, 7, 8\}</math>. | ||
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<math>\textbf{Problem 2}</math> | <math>\textbf{Problem 2}</math> | ||
Theams <math>T_1</math>, <math>T_2</math>, <math>T_3</math>, and <math>T_4</math> are in the playoffs. In the semifinal matches, <math>T_1</math> plays <math>T_4</math>, and <math>T_2</math> plays <math>T_3</math>. The winners of those two matches will play each other in the final match to determine the champion. When <math>T_i</math> plays <math>T_j</math>, the probability that <math>T_i</math> wins is <math>\frac{i}{i+j}</math>, and the outcomes of all the matches are independent. The probability that <math>T_4</math> will be the champion is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | Theams <math>T_1</math>, <math>T_2</math>, <math>T_3</math>, and <math>T_4</math> are in the playoffs. In the semifinal matches, <math>T_1</math> plays <math>T_4</math>, and <math>T_2</math> plays <math>T_3</math>. The winners of those two matches will play each other in the final match to determine the champion. When <math>T_i</math> plays <math>T_j</math>, the probability that <math>T_i</math> wins is <math>\frac{i}{i+j}</math>, and the outcomes of all the matches are independent. The probability that <math>T_4</math> will be the champion is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||
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<math>\textbf{Problem 3}</math> | <math>\textbf{Problem 3}</math> | ||
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>. | 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>. | ||
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<math>\textbf{Problem 4}</math> | <math>\textbf{Problem 4}</math> | ||
Find the number of positive integers less than or equal to <math>2017</math> whose base-three representation contains no digit equal to <math>0</math>. | Find the number of positive integers less than or equal to <math>2017</math> whose base-three representation contains no digit equal to <math>0</math>. | ||
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<math>\textbf{Problem 5}</math> | <math>\textbf{Problem 5}</math> | ||
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are <math>189</math>, <math>320</math>, <math>287</math>, <math>234</math>, <math>x</math>, and <math>y</math>. Find the greatest possible value of <math>x+y</math>. | A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are <math>189</math>, <math>320</math>, <math>287</math>, <math>234</math>, <math>x</math>, and <math>y</math>. Find the greatest possible value of <math>x+y</math>. | ||
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<math>\textbf{Problem 6}</math> | <math>\textbf{Problem 6}</math> | ||
Find the sum of all positive integers <math>n</math> such that <math>\sqrt{n^2+85n+2017}</math> is an integer. | Find the sum of all positive integers <math>n</math> such that <math>\sqrt{n^2+85n+2017}</math> is an integer. | ||
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<math>\textbf{Problem 7}</math> | <math>\textbf{Problem 7}</math> | ||
Find the number of integer values of <math>k</math> in the closed interval <math>[-500,500]</math> for which the equation <math>\log(kx)=2\log(x+2)</math> has exactly one real solution. | Find the number of integer values of <math>k</math> in the closed interval <math>[-500,500]</math> for which the equation <math>\log(kx)=2\log(x+2)</math> has exactly one real solution. | ||
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<math>\textbf{Problem 8}</math> | <math>\textbf{Problem 8}</math> | ||
Find the number of positive integers <math>n</math> less than <math>2017</math> such that <cmath>1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!}</cmath> is an integer. | Find the number of positive integers <math>n</math> less than <math>2017</math> such that <cmath>1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!}</cmath> is an integer. | ||
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<math>\textbf{Problem 9}</math> | <math>\textbf{Problem 9}</math> | ||
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 solors. 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 cardf 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 if <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | 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 solors. 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 cardf 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 if <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||
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<math>\textbf{Problem 10}</math> | <math>\textbf{Problem 10}</math> | ||
Rectangle <math>ABCD</math> has side lengths <math>AB=84</math> and <math>AD=42</math>. Point <math>M</math> is the midpoint of <math>\overline{AD}</math>, point <math>N</math> is the trisection point of <math>\overline{AB}</math> closer to <math>A</math>, and point <math>O</math> is the intersection of <math>\overline{CM}</math> and <math>\overline{DN}</math>. Point <math>P</math> lies on the quadrilateral <math>BCON</math>, and <math>\overline{BP}</math> bisects the area of <math>BCON</math>. Find the area of <math>\triangle CDP</math>. | Rectangle <math>ABCD</math> has side lengths <math>AB=84</math> and <math>AD=42</math>. Point <math>M</math> is the midpoint of <math>\overline{AD}</math>, point <math>N</math> is the trisection point of <math>\overline{AB}</math> closer to <math>A</math>, and point <math>O</math> is the intersection of <math>\overline{CM}</math> and <math>\overline{DN}</math>. Point <math>P</math> lies on the quadrilateral <math>BCON</math>, and <math>\overline{BP}</math> bisects the area of <math>BCON</math>. Find the area of <math>\triangle CDP</math>. | ||
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<math>\textbf{Problem 11}</math> | <math>\textbf{Problem 11}</math> | ||
Five towns are connected by a system of raods. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | Five towns are connected by a system of raods. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | ||
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<math>\textbf{Problem 12}</math> | <math>\textbf{Problem 12}</math> | ||
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label("<math>A_2</math>",(-55,20),E); | label("<math>A_2</math>",(-55,20),E); | ||
[/asy] | [/asy] | ||
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<math>\textbf{Problem 13}</math> | <math>\textbf{Problem 13}</math> | ||
For each integer <math>n\geq3</math>, let <math>f(n)</math> be the number of <math>3</math>-element subsets of the vertices of the regular <math>n</math>-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of <math>n</math> such that <math>f(n+1)=f(n)+78</math>. | For each integer <math>n\geq3</math>, let <math>f(n)</math> be the number of <math>3</math>-element subsets of the vertices of the regular <math>n</math>-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of <math>n</math> such that <math>f(n+1)=f(n)+78</math>. | ||
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<math>\textbf{Problem 14}</math> | <math>\textbf{Problem 14}</math> | ||
A <math>10\times10\times10</math> grid of points consists of all points in space of the form <math>(i,j,k)</math>, where <math>i</math>, <math>j</math>, and <math>k</math> are integers between <math>1</math> and <math>10</math>, inclusive. Find the number of different lines that contain exactly <math>8</math> of these points. | A <math>10\times10\times10</math> grid of points consists of all points in space of the form <math>(i,j,k)</math>, where <math>i</math>, <math>j</math>, and <math>k</math> are integers between <math>1</math> and <math>10</math>, inclusive. Find the number of different lines that contain exactly <math>8</math> of these points. | ||
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<math>\textbf{Problem 15}</math> | <math>\textbf{Problem 15}</math> | ||
Tetrahedron <math>ABCD</math> has <math>AD=BC=28</math>, <math>AC=BD=44</math>, and <math>AB=CD=52</math>. For any point <math>X</math> in space, define <math>f(X)=AX+BX+CX+DX</math>. The least possible value of <math>f(X)</math> can be expressed as <math>m\sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. | Tetrahedron <math>ABCD</math> has <math>AD=BC=28</math>, <math>AC=BD=44</math>, and <math>AB=CD=52</math>. For any point <math>X</math> in space, define <math>f(X)=AX+BX+CX+DX</math>. The least possible value of <math>f(X)</math> can be expressed as <math>m\sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. | ||
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Revision as of 12:34, 23 March 2017
Problem 1
Find the number of subsets of that are subsets of neither
nor
.
Theams
,
,
, and
are in the playoffs. In the semifinal matches,
plays
, and
plays
. The winners of those two matches will play each other in the final match to determine the champion. When
plays
, the probability that
wins is
, and the outcomes of all the matches are independent. The probability that
will be the champion is
, where
and
are relatively prime positive integers. Find
.
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
.
Find the number of positive integers less than or equal to
whose base-three representation contains no digit equal to
.
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are
,
,
,
,
, and
. Find the greatest possible value of
.
Find the sum of all positive integers
such that
is an integer.
Find the number of integer values of
in the closed interval
for which the equation
has exactly one real solution.
Find the number of positive integers
less than
such that
is an integer.
A special deck of cards contains
cards, each labeled with a number from
to
and colored with one of seven solors. 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 cardf with each number, the probability that Sharon can discard one of her cards and
have at least one card of each color and at least one card with each number if
, where
and
are relatively prime positive integers. Find
.
Rectangle
has side lengths
and
. Point
is the midpoint of
, point
is the trisection point of
closer to
, and point
is the intersection of
and
. Point
lies on the quadrilateral
, and
bisects the area of
. Find the area of
.
Five towns are connected by a system of raods. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).
Circle
has radius
, and the point
is a point on the circle. Circle
has radius
and is internally tangent to
at point
. Point
lies on circle
so that
is located
counterclockwise from
on
. Circle
has radius
and is internally tangent to
at point
. In this way a sequence of circles
and a sequence of points on the circles
are constructed, where circle
has radius
and is internally tangent to circle
at point
, and point
lies on
counterclockwise from point
, as shown in the figure below. There is one point
inside all of these circles. When
, the distance from the center
to
is
, where
and
are relatively prime positive integers. Find
.
[asy]
draw(Circle((0,0),125));
draw(Circle((25,0),100));
draw(Circle((25,20),80));
draw(Circle((9,20),64));
dot((125,0));
label("
",(125,0),E);
dot((25,100));
label("
",(25,100),SE);
dot((-55,20));
label("
",(-55,20),E);
[/asy]
For each integer
, let
be the number of
-element subsets of the vertices of the regular
-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of
such that
.
A
grid of points consists of all points in space of the form
, where
,
, and
are integers between
and
, inclusive. Find the number of different lines that contain exactly
of these points.
Tetrahedron
has
,
, and
. For any point
in space, define
. The least possible value of
can be expressed as
, where
and
are positive integers, and
is not divisible by the square of any prime. Find
.