Difference between revisions of "2015 AIME I Problems"
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+ | {{AIME Problems|year=2015|n=I}} | ||
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==Problem 1== | ==Problem 1== | ||
The expressions <math>A</math> = <math> 1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39 </math> and <math>B</math> = <math> 1 + 2 \times 3 + 4 \times 5 + \cdots + 36 \times 37 + 38 \times 39 </math> are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers <math>A</math> and <math>B</math>. | The expressions <math>A</math> = <math> 1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39 </math> and <math>B</math> = <math> 1 + 2 \times 3 + 4 \times 5 + \cdots + 36 \times 37 + 38 \times 39 </math> are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers <math>A</math> and <math>B</math>. | ||
[[2015 AIME I Problems/Problem 1|Solution]] | [[2015 AIME I Problems/Problem 1|Solution]] | ||
+ | |||
+ | |||
==Problem 2== | ==Problem 2== | ||
− | The nine delegates to the Economic Cooperation Conference include <math>2</math> officials from Mexico, <math>3</math> officials from Canada, and <math>4</math> officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is <math>\frac{m}{n}</math>, where m and n are relatively prime positive integers. Find <math>m+n</math>. | + | The nine delegates to the Economic Cooperation Conference include <math>2</math> officials from Mexico, <math>3</math> officials from Canada, and <math>4</math> officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |
[[2015 AIME I Problems/Problem 2|Solution]] | [[2015 AIME I Problems/Problem 2|Solution]] | ||
+ | |||
+ | |||
==Problem 3== | ==Problem 3== | ||
Line 13: | Line 19: | ||
[[2015 AIME I Problems/Problem 3|Solution]] | [[2015 AIME I Problems/Problem 3|Solution]] | ||
+ | |||
+ | |||
==Problem 4== | ==Problem 4== | ||
+ | Point <math>B</math> lies on line segment <math>\overline{AC}</math> with <math>AB=16</math> and <math>BC=4</math>. Points <math>D</math> and <math>E</math> lie on the same side of line <math>AC</math> forming equilateral triangles <math>\triangle ABD</math> and <math>\triangle BCE</math>. Let <math>M</math> be the midpoint of <math>\overline{AE}</math>, and <math>N</math> be the midpoint of <math>\overline{CD}</math>. The area of <math>\triangle BMN</math> is <math>x</math>. Find <math>x^2</math>. | ||
[[2015 AIME I Problems/Problem 4|Solution]] | [[2015 AIME I Problems/Problem 4|Solution]] | ||
+ | |||
+ | |||
==Problem 5== | ==Problem 5== | ||
− | In a drawer Sandy has <math>5</math> pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the <math>10</math> socks in the drawer. On Tuesday Sandy selects <math>2</math> of the remaining <math>8</math> socks at random and on Wednesday two of the remaining <math>6</math> socks at random. The probability that Wednesday is the first day Sandy selects matching socks is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers | + | In a drawer Sandy has <math>5</math> pairs of socks, each pair a different color. On Monday, Sandy selects two individual socks at random from the <math>10</math> socks in the drawer. On Tuesday Sandy selects <math>2</math> of the remaining <math>8</math> socks at random, and on Wednesday two of the remaining <math>6</math> socks at random. The probability that Wednesday is the first day Sandy selects matching socks is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |
[[2015 AIME I Problems/Problem 5|Solution]] | [[2015 AIME I Problems/Problem 5|Solution]] | ||
+ | |||
+ | |||
==Problem 6== | ==Problem 6== | ||
+ | Point <math>A,B,C,D,</math> and <math>E</math> are equally spaced on a minor arc of a circle. Points <math>E,F,G,H,I</math> and <math>A</math> are equally spaced on a minor arc of a second circle with center <math>C</math> as shown in the figure below. The angle <math>\angle ABD</math> exceeds <math>\angle AHG</math> by <math>12^\circ</math>. Find the degree measure of <math>\angle BAG</math>. | ||
+ | |||
+ | <asy> | ||
+ | pair A,B,C,D,E,F,G,H,I,O; | ||
+ | O=(0,0); | ||
+ | C=dir(90); | ||
+ | B=dir(70); | ||
+ | A=dir(50); | ||
+ | D=dir(110); | ||
+ | E=dir(130); | ||
+ | draw(arc(O,1,50,130)); | ||
+ | real x=2*sin(20*pi/180); | ||
+ | F=x*dir(228)+C; | ||
+ | G=x*dir(256)+C; | ||
+ | H=x*dir(284)+C; | ||
+ | I=x*dir(312)+C; | ||
+ | draw(arc(C,x,200,340)); | ||
+ | label("$A$",A,dir(0)); | ||
+ | label("$B$",B,dir(75)); | ||
+ | label("$C$",C,dir(90)); | ||
+ | label("$D$",D,dir(105)); | ||
+ | label("$E$",E,dir(180)); | ||
+ | label("$F$",F,dir(225)); | ||
+ | label("$G$",G,dir(260)); | ||
+ | label("$H$",H,dir(280)); | ||
+ | label("$I$",I,dir(315));</asy> | ||
[[2015 AIME I Problems/Problem 6|Solution]] | [[2015 AIME I Problems/Problem 6|Solution]] | ||
+ | |||
+ | |||
==Problem 7== | ==Problem 7== | ||
In the diagram below, <math>ABCD</math> is a square. Point <math>E</math> is the midpoint of <math>\overline{AD}</math>. Points <math>F</math> and <math>G</math> lie on <math>\overline{CE}</math>, and <math>H</math> and <math>J</math> lie on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, so that <math>FGHJ</math> is a square. Points <math>K</math> and <math>L</math> lie on <math>\overline{GH}</math>, and <math>M</math> and <math>N</math> lie on <math>\overline{AD}</math> and <math>\overline{AB}</math>, respectively, so that <math>KLMN</math> is a square. The area of <math>KLMN</math> is 99. Find the area of <math>FGHJ</math>. | In the diagram below, <math>ABCD</math> is a square. Point <math>E</math> is the midpoint of <math>\overline{AD}</math>. Points <math>F</math> and <math>G</math> lie on <math>\overline{CE}</math>, and <math>H</math> and <math>J</math> lie on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, so that <math>FGHJ</math> is a square. Points <math>K</math> and <math>L</math> lie on <math>\overline{GH}</math>, and <math>M</math> and <math>N</math> lie on <math>\overline{AD}</math> and <math>\overline{AB}</math>, respectively, so that <math>KLMN</math> is a square. The area of <math>KLMN</math> is 99. Find the area of <math>FGHJ</math>. | ||
− | + | <asy> | |
+ | pair A,B,C,D,E,F,G,H,J,K,L,M,N; | ||
+ | B=(0,0); | ||
+ | real m=7*sqrt(55)/5; | ||
+ | J=(m,0); | ||
+ | C=(7*m/2,0); | ||
+ | A=(0,7*m/2); | ||
+ | D=(7*m/2,7*m/2); | ||
+ | E=(A+D)/2; | ||
+ | H=(0,2m); | ||
+ | N=(0,2m+3*sqrt(55)/2); | ||
+ | G=foot(H,E,C); | ||
+ | F=foot(J,E,C); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | draw(C--E); | ||
+ | draw(G--H--J--F); | ||
+ | pair X=foot(N,E,C); | ||
+ | M=extension(N,X,A,D); | ||
+ | K=foot(N,H,G); | ||
+ | L=foot(M,H,G); | ||
+ | draw(K--N--M--L); | ||
+ | label("$A$",A,NW); | ||
+ | label("$B$",B,SW); | ||
+ | label("$C$",C,SE); | ||
+ | label("$D$",D,NE); | ||
+ | label("$E$",E,dir(90)); | ||
+ | label("$F$",F,NE); | ||
+ | label("$G$",G,NE); | ||
+ | label("$H$",H,W); | ||
+ | label("$J$",J,S); | ||
+ | label("$K$",K,SE); | ||
+ | label("$L$",L,SE); | ||
+ | label("$M$",M,dir(90)); | ||
+ | label("$N$",N,dir(180)); </asy> | ||
[[2015 AIME I Problems/Problem 7|Solution]] | [[2015 AIME I Problems/Problem 7|Solution]] | ||
+ | |||
+ | |||
==Problem 8== | ==Problem 8== | ||
Line 38: | Line 114: | ||
[[2015 AIME I Problems/Problem 8|Solution]] | [[2015 AIME I Problems/Problem 8|Solution]] | ||
+ | |||
+ | |||
==Problem 9== | ==Problem 9== | ||
+ | Let <math>S</math> be the set of all ordered triple of integers <math>(a_1,a_2,a_3)</math> with <math>1 \le a_1,a_2,a_3 \le 10</math>. Each ordered triple in <math>S</math> generates a sequence according to the rule <math>a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |</math> for all <math>n\ge 4</math>. Find the number of such sequences for which <math>a_n=0</math> for some <math>n</math>. | ||
[[2015 AIME I Problems/Problem 9|Solution]] | [[2015 AIME I Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
+ | |||
+ | Let <math>f(x)</math> be a third-degree polynomial with real coefficients satisfying | ||
+ | <cmath>|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.</cmath> Find <math>|f(0)|</math>. | ||
[[2015 AIME I Problems/Problem 10|Solution]] | [[2015 AIME I Problems/Problem 10|Solution]] | ||
+ | |||
+ | |||
==Problem 11== | ==Problem 11== | ||
+ | Triangle <math>ABC</math> has positive integer side lengths with <math>AB=AC</math>. Let <math>I</math> be the intersection of the bisectors of <math>\angle B</math> and <math>\angle C</math>. Suppose <math>BI=8</math>. Find the smallest possible perimeter of <math>\triangle ABC</math>. | ||
[[2015 AIME I Problems/Problem 11|Solution]] | [[2015 AIME I Problems/Problem 11|Solution]] | ||
+ | |||
+ | |||
==Problem 12== | ==Problem 12== | ||
− | Consider all 1000-element subsets of the set {1, 2, 3, ... , 2015}. From each such subset choose the least element. The arithmetic mean of all of these least elements is <math> \frac{p}{q} </math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p + q</math>. | + | Consider all 1000-element subsets of the set <math> \{ 1, 2, 3, ... , 2015 \} </math>. From each such subset choose the least element. The arithmetic mean of all of these least elements is <math> \frac{p}{q} </math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p + q</math>. |
[[2015 AIME I Problems/Problem 12|Solution]] | [[2015 AIME I Problems/Problem 12|Solution]] | ||
+ | |||
+ | |||
==Problem 13== | ==Problem 13== | ||
Line 60: | Line 149: | ||
[[2015 AIME I Problems/Problem 13|Solution]] | [[2015 AIME I Problems/Problem 13|Solution]] | ||
+ | |||
+ | |||
==Problem 14== | ==Problem 14== | ||
+ | |||
+ | For each integer <math>n \ge 2</math>, let <math>A(n)</math> be the area of the region in the coordinate plane defined by the inequalities <math>1\le x \le n</math> and <math>0\le y \le x \left\lfloor \sqrt x \right\rfloor</math>, where <math>\left\lfloor \sqrt x \right\rfloor</math> is the greatest integer not exceeding <math>\sqrt x</math>. Find the number of values of <math>n</math> with <math>2\le n \le 1000</math> for which <math>A(n)</math> is an integer. | ||
[[2015 AIME I Problems/Problem 14|Solution]] | [[2015 AIME I Problems/Problem 14|Solution]] | ||
+ | |||
+ | |||
==Problem 15== | ==Problem 15== | ||
A block of wood has the shape of a right circular cylinder with radius <math>6</math> and height <math>8</math>, and its entire surface has been painted blue. Points <math>A</math> and <math>B</math> are chosen on the edge of one of the circular faces of the cylinder so that <math>\overarc{AB}</math> on that face measures <math>120^\text{o}</math>. The block is then sliced in half along the plane that passes through point <math>A</math>, point <math>B</math>, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of these unpainted faces is <math>a\cdot\pi + b\sqrt{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>. | A block of wood has the shape of a right circular cylinder with radius <math>6</math> and height <math>8</math>, and its entire surface has been painted blue. Points <math>A</math> and <math>B</math> are chosen on the edge of one of the circular faces of the cylinder so that <math>\overarc{AB}</math> on that face measures <math>120^\text{o}</math>. The block is then sliced in half along the plane that passes through point <math>A</math>, point <math>B</math>, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of these unpainted faces is <math>a\cdot\pi + b\sqrt{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>. | ||
+ | |||
+ | <asy> | ||
+ | import three; import solids; | ||
+ | size(8cm); | ||
+ | currentprojection=orthographic(-1,-5,3); | ||
+ | |||
+ | picture lpic, rpic; | ||
+ | |||
+ | |||
+ | size(lpic,5cm); | ||
+ | draw(lpic,surface(revolution((0,0,0),(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8),Z,0,120)),gray(0.7),nolight); | ||
+ | draw(lpic,surface(revolution((0,0,0),(-3*sqrt(3),-3,8)..(-6,0,4)..(-3*sqrt(3),3,0),Z,0,90)),gray(0.7),nolight); | ||
+ | draw(lpic,surface((3,3*sqrt(3),8)..(-6,0,8)..(3,-3*sqrt(3),8)--cycle),gray(0.7),nolight); | ||
+ | draw(lpic,(3,-3*sqrt(3),8)..(-6,0,8)..(3,3*sqrt(3),8)); | ||
+ | draw(lpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0),dashed); | ||
+ | draw(lpic,(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0)--(-3,3*sqrt(3),0)..(-3*sqrt(3),3,0)..(-6,0,0),dashed); | ||
+ | draw(lpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(-3*sqrt(3),-3,0)..(-6,0,0)); | ||
+ | draw(lpic,(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),0)--(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),8)); | ||
+ | |||
+ | size(rpic,5cm); | ||
+ | draw(rpic,surface(revolution((0,0,0),(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0),Z,230,360)),gray(0.7),nolight); | ||
+ | draw(rpic,surface((-3,3*sqrt(3),0)..(6,0,0)..(-3,-3*sqrt(3),0)--cycle),gray(0.7),nolight); | ||
+ | draw(rpic,surface((-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--cycle),white,nolight); | ||
+ | draw(rpic,(-3,-3*sqrt(3),0)..(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)); | ||
+ | draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)..(-3,3*sqrt(3),0),dashed); | ||
+ | draw(rpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)); | ||
+ | draw(rpic,(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)..(3*sqrt(3),3,8)..(6,0,8)); | ||
+ | draw(rpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(0,-6,4)..(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(3*sqrt(3),-3,8)..(6,0,8)); | ||
+ | draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)--(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),8)); | ||
+ | label(rpic,"$A$",(-3,3*sqrt(3),0),W); | ||
+ | label(rpic,"$B$",(-3,-3*sqrt(3),0),W); | ||
+ | |||
+ | add(lpic.fit(),(0,0)); | ||
+ | add(rpic.fit(),(1,0)); </asy> | ||
[[2015 AIME I Problems/Problem 15|Solution]] | [[2015 AIME I Problems/Problem 15|Solution]] | ||
+ | |||
+ | |||
+ | {{AIME box|year=2015|n=I|before=[[2014 AIME II Problems]]|after=[[2015 AIME II Problems]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 16:03, 9 October 2024
2015 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
The expressions =
and
=
are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers
and
.
Problem 2
The nine delegates to the Economic Cooperation Conference include officials from Mexico,
officials from Canada, and
officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is
, where
and
are relatively prime positive integers. Find
.
Problem 3
There is a prime number such that
is the cube of a positive integer. Find
.
Problem 4
Point lies on line segment
with
and
. Points
and
lie on the same side of line
forming equilateral triangles
and
. Let
be the midpoint of
, and
be the midpoint of
. The area of
is
. Find
.
Problem 5
In a drawer Sandy has pairs of socks, each pair a different color. On Monday, Sandy selects two individual socks at random from the
socks in the drawer. On Tuesday Sandy selects
of the remaining
socks at random, and on Wednesday two of the remaining
socks at random. The probability that Wednesday is the first day Sandy selects matching socks is
, where
and
are relatively prime positive integers. Find
.
Problem 6
Point and
are equally spaced on a minor arc of a circle. Points
and
are equally spaced on a minor arc of a second circle with center
as shown in the figure below. The angle
exceeds
by
. Find the degree measure of
.
Problem 7
In the diagram below, is a square. Point
is the midpoint of
. Points
and
lie on
, and
and
lie on
and
, respectively, so that
is a square. Points
and
lie on
, and
and
lie on
and
, respectively, so that
is a square. The area of
is 99. Find the area of
.
Problem 8
For positive integer , let
denote the sum of the digits of
. Find the smallest positive integer satisfying
.
Problem 9
Let be the set of all ordered triple of integers
with
. Each ordered triple in
generates a sequence according to the rule
for all
. Find the number of such sequences for which
for some
.
Problem 10
Let be a third-degree polynomial with real coefficients satisfying
Find
.
Problem 11
Triangle has positive integer side lengths with
. Let
be the intersection of the bisectors of
and
. Suppose
. Find the smallest possible perimeter of
.
Problem 12
Consider all 1000-element subsets of the set . From each such subset choose the least element. The arithmetic mean of all of these least elements is
, where
and
are relatively prime positive integers. Find
.
Problem 13
With all angles measured in degrees, the product , where
and
are integers greater than 1. Find
.
Problem 14
For each integer , let
be the area of the region in the coordinate plane defined by the inequalities
and
, where
is the greatest integer not exceeding
. Find the number of values of
with
for which
is an integer.
Problem 15
A block of wood has the shape of a right circular cylinder with radius and height
, and its entire surface has been painted blue. Points
and
are chosen on the edge of one of the circular faces of the cylinder so that
on that face measures
. The block is then sliced in half along the plane that passes through point
, point
, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of these unpainted faces is
, where
,
, and
are integers and
is not divisible by the square of any prime. Find
.
2015 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2014 AIME II Problems |
Followed by 2015 AIME II Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.