Difference between revisions of "2013 AIME I Problems"
(→Problem 15) 

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== Problem 4 ==  == Problem 4 ==  
+  In the array of <math>13</math> squares shown below, <math>8</math> squares are colored red, and the remaining <math>5</math> squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated <math>90^{\circ}</math> around the central square is <math>\frac{1}{n}</math> , where <math>n</math> is a positive integer. Find <math>n</math>.  
+  
+  <asy>  
+  draw((0,0)(1,0)(1,1)(0,1)(0,0));  
+  draw((2,0)(2,2)(3,2)(3,0)(3,1)(2,1)(4,1)(4,0)(2,0));  
+  draw((1,2)(1,4)(0,4)(0,2)(0,3)(1,3)(1,3)(1,2)(1,2));  
+  draw((1,1)(3,1)(3,0)(1,0)(2,0)(2,1)(2,1)(1,1)(1,1));  
+  draw((0,1)(0,3)(1,3)(1,1)(1,2)(0,2)(2,2)(2,1)(0,1));  
+  size(100);</asy>  
+  
+  [[2013 AIME I Problems/Problem 4Solution]]  
== Problem 5 ==  == Problem 5 ==  
The real root of the equation <math>8x^33x^23x1=0</math> can be written in the form <math>\frac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers. Find <math>a+b+c</math>.  The real root of the equation <math>8x^33x^23x1=0</math> can be written in the form <math>\frac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers. Find <math>a+b+c</math>.  
+  
+  [[2013 AIME I Problems/Problem 5Solution]]  
+  
+  
+  ==Problem 6==  
+  Melinda has three empty boxes and <math>12</math> textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.  
+  
+  [[2013 AIME I Problems/Problem 6Solution]]  
+  
+  
+  ==Problem 7==  
+  A rectangular box has width <math>12</math> inches, length <math>16</math> inches, and height <math>\frac{m}{n}</math> inches, where <math>m</math> and <math>n</math> are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of <math>30</math> square inches. Find <math>m+n</math>.  
+  
+  [[2013 AIME I Problems/Problem 7Solution]]  
+  
+  
+  == Problem 8 ==  
+  The domain of the function <math>f(x) = \arcsin(\log_{m}(nx))</math> is a closed interval of length <math>\frac{1}{2013}</math> , where <math>m</math> and <math>n</math> are positive integers and <math>m>1</math>. Find the remainder when the smallest possible sum <math>m+n</math> is divided by <math>1000</math>.  
+  
+  [[2013 AIME I Problems/Problem 8Solution]]  
+  
+  ==Problem 9==  
+  A paper equilateral triangle <math>ABC</math> has side length <math>12</math>. The paper triangle is folded so that vertex <math>A</math> touches a point on side <math>\overline{BC}</math> a distance <math>9</math> from point <math>B</math>. The length of the line segment along which the triangle is folded can be written as <math>\frac{m\sqrt{p}}{n}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, <math>m</math> and <math>n</math> are relatively prime, and <math>p</math> is not divisible by the square of any prime. Find <math>m+n+p</math>.  
+  
+  <asy>  
+  import cse5;  
+  size(12cm);  
+  pen tpen = defaultpen + 1.337;  
+  real a = 39/5.0;  
+  real b = 39/7.0;  
+  pair B = MP("B", (0,0), dir(200));  
+  pair A = MP("A", (9,0), dir(80));  
+  pair C = MP("C", (12,0), dir(20));  
+  pair K = (6,10.392);  
+  pair M = (a*B+(12a)*K) / 12;  
+  pair N = (b*C+(12b)*K) / 12;  
+  draw(BMNCcycle, tpen);  
+  draw(MANcycle);  
+  fill(MANcycle, mediumgrey);  
+  pair shift = (20.13, 0);  
+  pair B1 = MP("B", B+shift, dir(200));  
+  pair A1 = MP("A", K+shift, dir(90));  
+  pair C1 = MP("C", C+shift, dir(20));  
+  draw(A1B1C1cycle, tpen);</asy>  
+  
+  [[2013 AIME I Problems/Problem 9Solution]]  
+  
+  ==Problem 10==  
+  There are nonzero integers <math>a</math>, <math>b</math>, <math>r</math>, and <math>s</math> such that the complex number <math>r+si</math> is a zero of the polynomial <math>P(x)={x}^{3}a{x}^{2}+bx65</math>. For each possible combination of <math>a</math> and <math>b</math>, let <math>{p}_{a,b}</math> be the sum of the zeros of <math>P(x)</math>. Find the sum of the <math>{p}_{a,b}</math>'s for all possible combinations of <math>a</math> and <math>b</math>.  
+  
+  [[2013 AIME I Problems/Problem 10Solution]]  
+  
+  
+  == Problem 11 ==  
+  Ms. Math's kindergarten class has 16 registered students. The classroom has a very large number, ''N'', of play blocks which satisfies the conditions:  
+  
+  (a) If 16, 15, or 14 students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and  
+  
+  (b) There are three integers <math>0 < x < y < z < 14</math> such that when <math>x</math>, <math>y</math>, or <math>z</math> students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.  
+  
+  Find the sum of the distinct prime divisors of the least possible value of ''N'' satisfying the above conditions.  
+  
+  [[2013 AIME I Problems/Problem 11Solution]]  
+  
+  == Problem 12 ==  
+  Let <math>\bigtriangleup PQR</math> be a triangle with <math>\angle P = 75^o</math> and <math>\angle Q = 60^o</math>. A regular hexagon <math>ABCDEF</math> with side length 1 is drawn inside <math>\triangle PQR</math> so that side <math>\overline{AB}</math> lies on <math>\overline{PQ}</math>, side <math>\overline{CD}</math> lies on <math>\overline{QR}</math>, and one of the remaining vertices lies on <math>\overline{RP}</math>. There are positive integers <math>a, b, c, </math> and <math>d</math> such that the area of <math>\triangle PQR</math> can be expressed in the form <math>\frac{a+b\sqrt{c}}{d}</math>, where <math>a</math> and <math>d</math> are relatively prime, and c is not divisible by the square of any prime. Find <math>a+b+c+d</math>.  
+  
+  [[2013 AIME I Problems/Problem 12Solution]]  
+  
+  == Problem 13 ==  
+  Triangle <math>AB_0C_0</math> has side lengths <math>AB_0 = 12</math>, <math>B_0C_0 = 17</math>, and <math>C_0A = 25</math>. For each positive integer <math>n</math>, points <math>B_n</math> and <math>C_n</math> are located on <math>\overline{AB_{n1}}</math> and <math>\overline{AC_{n1}}</math>, respectively, creating three similar triangles <math>\triangle AB_nC_n \sim \triangle B_{n1}C_nC_{n1} \sim \triangle AB_{n1}C_{n1}</math>. The area of the union of all triangles <math>B_{n1}C_nB_n</math> for <math>n\geq1</math> can be expressed as <math>\tfrac pq</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>q</math>.  
+  
+  [[2013 AIME I Problems/Problem 13Solution]]  
+  
+  == Problem 14 ==  
+  For <math>\pi \le \theta < 2\pi</math>, let  
+  
+  <cmath> P=\dfrac12\cos\theta\dfrac14\sin2\theta\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta\dfrac1{64}\sin6\theta\dfrac1{128}\cos7\theta+\ldots  
+  </cmath>  
+  
+  and  
+  
+  <cmath> Q=1\dfrac12\sin\theta\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta\dfrac1{32}\sin5\theta\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta  
+  +\ldots </cmath>  
+  
+  so that <math>\frac{P}{Q} = \frac{2\sqrt2}{7}</math>. Then <math>\sin\theta = \frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.  
+  
+  [[2013 AIME I Problems/Problem 14Solution]]  
+  
+  ==Problem 15==  
+  Let <math>N</math> be the number of ordered triples <math>(A,B,C)</math> of integers satisfying the conditions:  
+  
+  (a) <math>0\le A<B<C\le99</math>,  
+  
+  (b) there exist integers <math>a</math>, <math>b</math>, and <math>c</math>, and prime <math>p</math> where <math>0\le b<a<c<p</math>,  
+  
+  (c) <math>p</math> divides <math>Aa</math>, <math>Bb</math>, and <math>Cc</math>, and  
+  
+  (d) each ordered triple <math>(A,B,C)</math> and each ordered triple <math>(b,a,c)</math> form arithmetic sequences. Find <math>N</math>.  
+  
+  [[2013 AIME I Problems/Problem 15Solution]]  
+  
+  {{AIME boxyear=2013n=Ibefore=[[2012 AIME II Problems]]after=[[2013 AIME II Problems]]}}  
+  
+  {{MAA Notice}} 
Latest revision as of 10:57, 4 January 2021
2013 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 AIME Triathlon consists of a halfmile swim, a 30mile bicycle ride, and an eightmile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?
Problem 2
Find the number of fivedigit positive integers, , that satisfy the following conditions:

(a) the number is divisible by

(b) the first and last digits of are equal, and

(c) the sum of the digits of is divisible by
Problem 3
Let be a square, and let and be points on and respectively. The line through parallel to and the line through parallel to divide into two squares and two nonsquare rectangles. The sum of the areas of the two squares is of the area of square Find
Problem 4
In the array of squares shown below, squares are colored red, and the remaining squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated around the central square is , where is a positive integer. Find .
Problem 5
The real root of the equation can be written in the form , where , , and are positive integers. Find .
Problem 6
Melinda has three empty boxes and textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as , where and are relatively prime positive integers. Find .
Problem 7
A rectangular box has width inches, length inches, and height inches, where and are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of square inches. Find .
Problem 8
The domain of the function is a closed interval of length , where and are positive integers and . Find the remainder when the smallest possible sum is divided by .
Problem 9
A paper equilateral triangle has side length . The paper triangle is folded so that vertex touches a point on side a distance from point . The length of the line segment along which the triangle is folded can be written as , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .
Problem 10
There are nonzero integers , , , and such that the complex number is a zero of the polynomial . For each possible combination of and , let be the sum of the zeros of . Find the sum of the 's for all possible combinations of and .
Problem 11
Ms. Math's kindergarten class has 16 registered students. The classroom has a very large number, N, of play blocks which satisfies the conditions:
(a) If 16, 15, or 14 students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are three integers such that when , , or students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of N satisfying the above conditions.
Problem 12
Let be a triangle with and . A regular hexagon with side length 1 is drawn inside so that side lies on , side lies on , and one of the remaining vertices lies on . There are positive integers and such that the area of can be expressed in the form , where and are relatively prime, and c is not divisible by the square of any prime. Find .
Problem 13
Triangle has side lengths , , and . For each positive integer , points and are located on and , respectively, creating three similar triangles . The area of the union of all triangles for can be expressed as , where and are relatively prime positive integers. Find .
Problem 14
For , let
and
so that . Then where and are relatively prime positive integers. Find .
Problem 15
Let be the number of ordered triples of integers satisfying the conditions:
(a) ,
(b) there exist integers , , and , and prime where ,
(c) divides , , and , and
(d) each ordered triple and each ordered triple form arithmetic sequences. Find .
2013 AIME I (Problems • Answer Key • Resources)  
Preceded by 2012 AIME II Problems 
Followed by 2013 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.