Difference between revisions of "2014 AIME I Problems"
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==Problem 1== | ==Problem 1== | ||
+ | The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters. | ||
− | + | <asy> | |
+ | size(200); | ||
+ | defaultpen(linewidth(0.7)); | ||
+ | path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; | ||
+ | path laceR=reflect((75,0),(75,-240))*laceL; | ||
+ | draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); | ||
+ | for(int i=0;i<=3;i=i+1) | ||
+ | { | ||
+ | path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); | ||
+ | unfill(circ1); draw(circ1); | ||
+ | unfill(circ2); draw(circ2); | ||
+ | } | ||
+ | draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));</asy> | ||
[[2014 AIME I Problems/Problem 1|Solution]] | [[2014 AIME I Problems/Problem 1|Solution]] | ||
==Problem 2== | ==Problem 2== | ||
− | |||
+ | An urn contains <math>4</math> green balls and <math>6</math> blue balls. A second urn contains <math>16</math> green balls and <math>N</math> blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is <math>0.58</math>. Find <math>N</math>. | ||
[[2014 AIME I Problems/Problem 2|Solution]] | [[2014 AIME I Problems/Problem 2|Solution]] | ||
==Problem 3== | ==Problem 3== | ||
− | + | Find the number of rational numbers <math>r</math>, <math>0<r<1,</math> such that when <math>r</math> is written as a fraction in lowest terms, the numerator and the denominator have a sum of <math>1000</math>. | |
[[2014 AIME I Problems/Problem 3|Solution]] | [[2014 AIME I Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | + | Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at <math>20</math> miles per hour, and Steve rides west at <math>20</math> miles per hour. Two trains of equal length, traveling in opposite directions at constant but different speeds each pass the two riders. Each train takes exactly <math>1</math> minute to go past Jon. The westbound train takes <math>10</math> times as long as the eastbound train to go past Steve. The length of each train is <math>\tfrac{m}{n}</math> miles, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | |
[[2014 AIME I Problems/Problem 4|Solution]] | [[2014 AIME I Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
− | + | Let the set <math>S = \{P_1, P_2, \dots, P_{12}\}</math> consist of the twelve vertices of a regular <math>12</math>-gon. A subset <math>Q</math> of <math>S</math> is called communal if there is a circle such that all points of <math>Q</math> are inside the circle, and all points of <math>S</math> not in <math>Q</math> are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.) | |
[[2014 AIME I Problems/Problem 5|Solution]] | [[2014 AIME I Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
− | + | The graphs <math>y = 3(x-h)^2 + j</math> and <math>y = 2(x-h)^2 + k</math> have y-intercepts of <math>2013</math> and <math>2014</math>, respectively, and each graph has two positive integer x-intercepts. Find <math>h</math>. | |
[[2014 AIME I Problems/Problem 6|Solution]] | [[2014 AIME I Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
− | + | Let <math>w</math> and <math>z</math> be complex numbers such that <math>|w| = 1</math> and <math>|z| = 10</math>. Let <math>\theta = \arg \left(\tfrac{w-z}{z}\right) </math>. The maximum possible value of <math>\tan^2 \theta</math> can be written as <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. (Note that <math>\arg(w)</math>, for <math>w \neq 0</math>, denotes the measure of the angle that the ray from <math>0</math> to <math>w</math> makes with the positive real axis in the complex plane.) | |
[[2014 AIME I Problems/Problem 7|Solution]] | [[2014 AIME I Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | + | The positive integers <math>N</math> and <math>N^2</math> both end in the same sequence of four digits <math>abcd</math> when written in base 10, where digit <math>a</math> is not zero. Find the three-digit number <math>abc</math>. | |
[[2014 AIME I Problems/Problem 8|Solution]] | [[2014 AIME I Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
− | + | Let <math>x_1< x_2 < x_3</math> be the three real roots of the equation <math>\sqrt{2014} x^3 - 4029x^2 + 2 = 0</math>. Find <math>x_2(x_1+x_3)</math>. | |
[[2014 AIME I Problems/Problem 9|Solution]] | [[2014 AIME I Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
− | + | A disk with radius <math>1</math> is externally tangent to a disk with radius <math>5</math>. Let <math>A</math> be the point where the disks are tangent, <math>C</math> be the center of the smaller disk, and <math>E</math> be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of <math>360^\circ</math>. That is, if the center of the smaller disk has moved to the point <math>D</math>, and the point on the smaller disk that began at <math>A</math> has now moved to point <math>B</math>, then <math>\overline{AC}</math> is parallel to <math>\overline{BD}</math>. Then <math>\sin^2(\angle BEA)=\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | |
[[2014 AIME I Problems/Problem 10|Solution]] | [[2014 AIME I Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
− | + | A token starts at the point <math>(0,0)</math> of an <math>xy</math>-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of <math>|y|=|x|</math> is <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | |
[[2014 AIME I Problems/Problem 11|Solution]] | [[2014 AIME I Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | + | Let <math>A=\{1,2,3,4\}</math>, and <math>f</math> and <math>g</math> be randomly chosen (not necessarily distinct) functions from <math>A</math> to <math>A</math>. The probability that the range of <math>f</math> and the range of <math>g</math> are disjoint is <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m</math>. | |
[[2014 AIME I Problems/Problem 12|Solution]] | [[2014 AIME I Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
+ | On square <math>ABCD</math>, points <math>E,F,G</math>, and <math>H</math> lie on sides <math>\overline{AB},\overline{BC},\overline{CD},</math> and <math>\overline{DA},</math> respectively, so that <math>\overline{EG} \perp \overline{FH}</math> and <math>EG=FH = 34</math>. Segments <math>\overline{EG}</math> and <math>\overline{FH}</math> intersect at a point <math>P</math>, and the areas of the quadrilaterals <math>AEPH, BFPE, CGPF,</math> and <math>DHPG</math> are in the ratio <math>269:275:405:411.</math> Find the area of square <math>ABCD</math>. | ||
+ | <asy> | ||
+ | pair A = (0,sqrt(850)); | ||
+ | pair B = (0,0); | ||
+ | pair C = (sqrt(850),0); | ||
+ | pair D = (sqrt(850),sqrt(850)); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | dotfactor = 3; | ||
+ | dot("$A$",A,dir(135)); | ||
+ | dot("$B$",B,dir(215)); | ||
+ | dot("$C$",C,dir(305)); | ||
+ | dot("$D$",D,dir(45)); | ||
+ | pair H = ((2sqrt(850)-sqrt(306))/6,sqrt(850)); | ||
+ | pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); | ||
+ | dot("$H$",H,dir(90)); | ||
+ | dot("$F$",F,dir(270)); | ||
+ | draw(H--F); | ||
+ | pair E = (0,(sqrt(850)-6)/2); | ||
+ | pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); | ||
+ | dot("$E$",E,dir(180)); | ||
+ | dot("$G$",G,dir(0)); | ||
+ | draw(E--G); | ||
+ | pair P = extension(H,F,E,G); | ||
+ | dot("$P$",P,dir(60)); | ||
+ | label("$w$", intersectionpoint( A--P, E--H )); | ||
+ | label("$x$", intersectionpoint( B--P, E--F )); | ||
+ | label("$y$", intersectionpoint( C--P, G--F )); | ||
+ | label("$z$", intersectionpoint( D--P, G--H ));</asy> | ||
[[2014 AIME I Problems/Problem 13|Solution]] | [[2014 AIME I Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
+ | Let <math>m</math> be the largest real solution to the equation | ||
+ | <cmath> \dfrac{3}{x-3} + \dfrac{5}{x-5} + \dfrac{17}{x-17} + \dfrac{19}{x-19} = x^2 - 11x - 4</cmath> | ||
+ | |||
+ | There are positive integers <math>a, b,</math> and <math>c</math> such that <math>m = a + \sqrt{b + \sqrt{c}}</math>. Find <math>a+b+c</math>. | ||
[[2014 AIME I Problems/Problem 14|Solution]] | [[2014 AIME I Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
+ | In <math>\triangle ABC, AB = 3, BC = 4,</math> and <math>CA = 5</math>. Circle <math>\omega</math> intersects <math>\overline{AB}</math> at <math>E</math> and <math>B, \overline{BC}</math> at <math>B</math> and <math>D,</math> and <math>\overline{AC}</math> at <math>F</math> and <math>G</math>. Given that <math>EF=DF</math> and <math>\frac{DG}{EG} = \frac{3}{4},</math> length <math>DE=\frac{a\sqrt{b}}{c},</math> where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer not divisible by the square of any prime. Find <math>a+b+c</math>. | ||
− | [[2014 AIME I Problems/Problem | + | [[2014 AIME I Problems/Problem 15|Solution]] |
+ | |||
+ | {{AIME box|year=2014|n=I|before=[[2013 AIME II Problems]]|after=[[2014 AIME II Problems]]}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:59, 30 November 2021
2014 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 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
Problem 2
An urn contains green balls and
blue balls. A second urn contains
green balls and
blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is
. Find
.
Problem 3
Find the number of rational numbers ,
such that when
is written as a fraction in lowest terms, the numerator and the denominator have a sum of
.
Problem 4
Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at miles per hour, and Steve rides west at
miles per hour. Two trains of equal length, traveling in opposite directions at constant but different speeds each pass the two riders. Each train takes exactly
minute to go past Jon. The westbound train takes
times as long as the eastbound train to go past Steve. The length of each train is
miles, where
and
are relatively prime positive integers. Find
.
Problem 5
Let the set consist of the twelve vertices of a regular
-gon. A subset
of
is called communal if there is a circle such that all points of
are inside the circle, and all points of
not in
are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)
Problem 6
The graphs and
have y-intercepts of
and
, respectively, and each graph has two positive integer x-intercepts. Find
.
Problem 7
Let and
be complex numbers such that
and
. Let
. The maximum possible value of
can be written as
, where
and
are relatively prime positive integers. Find
. (Note that
, for
, denotes the measure of the angle that the ray from
to
makes with the positive real axis in the complex plane.)
Problem 8
The positive integers and
both end in the same sequence of four digits
when written in base 10, where digit
is not zero. Find the three-digit number
.
Problem 9
Let be the three real roots of the equation
. Find
.
Problem 10
A disk with radius is externally tangent to a disk with radius
. Let
be the point where the disks are tangent,
be the center of the smaller disk, and
be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of
. That is, if the center of the smaller disk has moved to the point
, and the point on the smaller disk that began at
has now moved to point
, then
is parallel to
. Then
, where
and
are relatively prime positive integers. Find
.
Problem 11
A token starts at the point of an
-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of
is
, where
and
are relatively prime positive integers. Find
.
Problem 12
Let , and
and
be randomly chosen (not necessarily distinct) functions from
to
. The probability that the range of
and the range of
are disjoint is
, where
and
are relatively prime positive integers. Find
.
Problem 13
On square , points
, and
lie on sides
and
respectively, so that
and
. Segments
and
intersect at a point
, and the areas of the quadrilaterals
and
are in the ratio
Find the area of square
.
Problem 14
Let be the largest real solution to the equation
There are positive integers and
such that
. Find
.
Problem 15
In and
. Circle
intersects
at
and
at
and
and
at
and
. Given that
and
length
where
and
are relatively prime positive integers, and
is a positive integer not divisible by the square of any prime. Find
.
2014 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2013 AIME II Problems |
Followed by 2014 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.