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Difference between revisions of "2018 AIME II Problems"

m (Protected "2018 AIME II Problems" ([Edit=Allow only administrators] (expires 18:44, 21 March 2018 (UTC)) [Move=Allow only administrators] (expires 18:44, 21 March 2018 (UTC))))
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(18 intermediate revisions by 9 users not shown)
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==Problem 1==
 
==Problem 1==
 +
 +
Points <math>A</math>, <math>B</math>, and <math>C</math> lie in that order along a straight path where the distance from <math>A</math> to <math>C</math> is <math>1800</math> meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at <math>A</math> and running toward <math>C</math>, Paul starting at <math>B</math> and running toward <math>C</math>, and Eve starting at <math>C</math> and running toward <math>A</math>. When Paul meets Eve, he turns around and runs toward <math>A</math>. Paul and Ina both arrive at <math>B</math> at the same time. Find the number of meters from <math>A</math> to <math>B</math>.
  
 
[[2018 AIME II Problems/Problem 1 | Solution]]
 
[[2018 AIME II Problems/Problem 1 | Solution]]
Line 7: Line 9:
 
==Problem 2==
 
==Problem 2==
  
 +
Let <math>a_{0} = 2</math>, <math>a_{1} = 5</math>, and <math>a_{2} = 8</math>, and for <math>n > 2</math> define <math>a_{n}</math> recursively to be the remainder when <math>4(a_{n-1} + a_{n-2} + a_{n-3})</math> is divided by <math>11</math>. Find <math>a_{2018} \cdot a_{2020} \cdot a_{2022}</math>.
  
 
[[2018 AIME II Problems/Problem 2 | Solution]]
 
[[2018 AIME II Problems/Problem 2 | Solution]]
Line 12: Line 15:
 
==Problem 3==
 
==Problem 3==
  
 +
Find the sum of all positive integers <math>b < 1000</math> such that the base-<math>b</math> integer <math>36_{b}</math> is a perfect square and the base-<math>b</math> integer <math>27_{b}</math> is a perfect cube.
  
 
[[2018 AIME II Problems/Problem 3 | Solution]]
 
[[2018 AIME II Problems/Problem 3 | Solution]]
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==Problem 4==
 
==Problem 4==
  
 +
In equiangular octagon <math>CAROLINE</math>, <math>CA = RO = LI = NE =</math> <math>\sqrt{2}</math> and <math>AR = OL = IN = EC = 1</math>. The self-intersecting octagon <math>CORNELIA</math> encloses six non-overlapping triangular regions. Let <math>K</math> be the area enclosed by <math>CORNELIA</math>, that is, the total area of the six triangular regions. Then <math>K = \frac{a}{b}</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers. Find <math>a + b</math>.
  
 
[[2018 AIME II Problems/Problem 4 | Solution]]
 
[[2018 AIME II Problems/Problem 4 | Solution]]
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==Problem 5==
 
==Problem 5==
  
 +
Suppose that <math>x</math>, <math>y</math>, and <math>z</math> are complex numbers such that <math>xy = -80 - 320i</math>, <math>yz = 60</math>, and <math>zx = -96 + 24i</math>, where <math>i</math> <math>=</math> <math>\sqrt{-1}</math>. Then there are real numbers <math>a</math> and <math>b</math> such that <math>x + y + z = a + bi</math>. Find <math>a^2 + b^2</math>.
  
 
[[2018 AIME II Problems/Problem 5 | Solution]]
 
[[2018 AIME II Problems/Problem 5 | Solution]]
Line 27: Line 33:
 
==Problem 6==
 
==Problem 6==
  
 +
A real number <math>a</math> is chosen randomly and uniformly from the interval <math>[-20, 18]</math>. The probability that the roots of the polynomial
 +
 +
<cmath>x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2</cmath>
 +
 +
are all real can be written in the form <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
  
 
[[2018 AIME II Problems/Problem 6 | Solution]]
 
[[2018 AIME II Problems/Problem 6 | Solution]]
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==Problem 7==
 
==Problem 7==
  
 +
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.
  
 
[[2018 AIME II Problems/Problem 7 | Solution]]
 
[[2018 AIME II Problems/Problem 7 | Solution]]
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==Problem 8==
 
==Problem 8==
  
 +
A frog is positioned at the origin of the coordinate plane. From the point <math>(x, y)</math>, the frog can jump to any of the points <math>(x + 1, y)</math>, <math>(x + 2, y)</math>, <math>(x, y + 1)</math>, or <math>(x, y + 2)</math>. Find the number of distinct sequences of jumps in which the frog begins at <math>(0, 0)</math> and ends at <math>(4, 4)</math>.
  
 
[[2018 AIME II Problems/Problem 8 | Solution]]
 
[[2018 AIME II Problems/Problem 8 | Solution]]
  
−
==Problem 9==
+
==Problem 9==  
 +
Octagon <math>ABCDEFGH</math> with side lengths <math>AB = CD = EF = GH = 10</math> and <math>BC = DE = FG = HA = 11</math> is formed by removing 6-8-10 triangles from the corners of a <math>23</math> <math>\times</math> <math>27</math> rectangle with side <math>\overline{AH}</math> on a short side of the rectangle, as shown. Let <math>J</math> be the midpoint of <math>\overline{AH}</math>, and partition the octagon into 7 triangles by drawing segments <math>\overline{JB}</math>, <math>\overline{JC}</math>, <math>\overline{JD}</math>, <math>\overline{JE}</math>, <math>\overline{JF}</math>, and <math>\overline{JG}</math>. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.
  
 +
<asy>
 +
unitsize(6);
 +
pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0);
 +
pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23),  G = (8, 23), J = (0, 23/2), H = (0, 17);
 +
draw(P--Q--R--SS--cycle);
 +
draw(J--B);
 +
draw(J--C);
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draw(J--D);
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draw(J--EE);
 +
draw(J--F);
 +
draw(J--G);
 +
draw(A--B);
 +
draw(H--G);
 +
real dark = 0.6;
 +
filldraw(A--B--P--cycle, gray(dark));
 +
filldraw(H--G--Q--cycle, gray(dark));
 +
filldraw(F--EE--R--cycle, gray(dark));
 +
filldraw(D--C--SS--cycle, gray(dark));
 +
dot(A);
 +
dot(B);
 +
dot(C);
 +
dot(D);
 +
dot(EE);
 +
dot(F);
 +
dot(G);
 +
dot(H);
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dot(J);
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dot(H);
 +
defaultpen(fontsize(10pt));
 +
real r = 1.3;
 +
label("$A$", A, W*r);
 +
label("$B$", B, S*r);
 +
label("$C$", C, S*r);
 +
label("$D$", D, E*r);
 +
label("$E$", EE, E*r);
 +
label("$F$", F, N*r);
 +
label("$G$", G, N*r);
 +
label("$H$", H, W*r);
 +
label("$J$", J, W*r);
 +
</asy>
  
 
[[2018 AIME II Problems/Problem 9 | Solution]]
 
[[2018 AIME II Problems/Problem 9 | Solution]]
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==Problem 10==
 
==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>.
  
 
[[2018 AIME II Problems/Problem 10 | Solution]]
 
[[2018 AIME II Problems/Problem 10 | Solution]]
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==Problem 11==
 
==Problem 11==
  
 +
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>.
  
 
[[2018 AIME II Problems/Problem 11 | Solution]]
 
[[2018 AIME II Problems/Problem 11 | Solution]]
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==Problem 12==
 
==Problem 12==
  
 +
Let <math>ABCD</math> be a convex quadrilateral with <math>AB = CD = 10</math>, <math>BC = 14</math>, and <math>AD = 2\sqrt{65}</math>. Assume that the diagonals of <math>ABCD</math> intersect at point <math>P</math>, and that the sum of the areas of triangles <math>APB</math> and <math>CPD</math> equals the sum of the areas of triangles <math>BPC</math> and <math>APD</math>. Find the area of quadrilateral <math>ABCD</math>.
  
 
[[2018 AIME II Problems/Problem 12 | Solution]]
 
[[2018 AIME II Problems/Problem 12 | Solution]]
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==Problem 13==
 
==Problem 13==
  
 +
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is <math>\dfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
 
[[2018 AIME II Problems/Problem 13 | Solution]]
 
[[2018 AIME II Problems/Problem 13 | Solution]]
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==Problem 14==
 
==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>.
  
 
[[2018 AIME II Problems/Problem 14 | Solution]]
 
[[2018 AIME II Problems/Problem 14 | Solution]]
Line 72: Line 131:
 
==Problem 15==
 
==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
 +
 +
<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>.
  
 
[[2018 AIME II Problems/Problem 15 | Solution]]
 
[[2018 AIME II Problems/Problem 15 | Solution]]
  
−
{{AIME box|year=2018|n=II|before=[[2018 AIME I]]|after=[[2019 AIME I]]}}
+
{{AIME box|year=2018|n=II|before=[[2018 AIME I Problems]]|after=[[2019 AIME I Problems]]}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 23:26, 20 January 2024

2018 AIME II (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$.

Solution

Problem 2

Let $a_{0} = 2$, $a_{1} = 5$, and $a_{2} = 8$, and for $n > 2$ define $a_{n}$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$. Find $a_{2018} \cdot a_{2020} \cdot a_{2022}$.

Solution

Problem 3

Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_{b}$ is a perfect square and the base-$b$ integer $27_{b}$ is a perfect cube.

Solution

Problem 4

In equiangular octagon $CAROLINE$, $CA = RO = LI = NE =$ $\sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, the total area of the six triangular regions. Then $K = \frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.

Solution

Problem 5

Suppose that $x$, $y$, and $z$ are complex numbers such that $xy = -80 - 320i$, $yz = 60$, and $zx = -96 + 24i$, where $i$ $=$ $\sqrt{-1}$. Then there are real numbers $a$ and $b$ such that $x + y + z = a + bi$. Find $a^2 + b^2$.

Solution

Problem 6

A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial

\[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\]

are all real can be written in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Problem 7

Triangle $ABC$ has side lengths $AB = 9$, $BC =$ $5\sqrt{3}$, and $AC = 12$. Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$, and points $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C$ are on segment $\overline{AC}$ with $Q_{k}$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1, 2, ..., 2449$. Furthermore, each segment $\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2449$, is parallel to $\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions has the same area. Find the number of segments $\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2450$, that have rational length.

Solution

Problem 8

A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y)$, $(x + 2, y)$, $(x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$.

Solution

Problem 9

Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\times$ $27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{AH}$, and partition the octagon into 7 triangles by drawing segments $\overline{JB}$, $\overline{JC}$, $\overline{JD}$, $\overline{JE}$, $\overline{JF}$, and $\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.

[asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label("$A$", A, W*r); label("$B$", B, S*r); label("$C$", C, S*r); label("$D$", D, E*r); label("$E$", EE, E*r); label("$F$", F, N*r); label("$G$", G, N*r); label("$H$", H, W*r); label("$J$", J, W*r); [/asy]

Solution

Problem 10

Find the number of functions $f(x)$ from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1, 2, 3, 4, 5\}$.

Solution

Problem 11

Find the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k$ with $1$ $\leq$ $k$ $\leq$ $5$, at least one of the first $k$ terms of the permutation is greater than $k$.

Solution

Problem 12

Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$.

Solution

Problem 13

Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 14

The incircle $\omega$ of triangle $ABC$ is tangent to $\overline{BC}$ at $X$. Let $Y \neq X$ be the other intersection of $\overline{AX}$ with $\omega$. Points $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $\overline{PQ}$ is tangent to $\omega$ at $Y$. Assume that $AP = 3$, $PB = 4$, $AC = 8$, and $AQ = \dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 15

Find the number of functions $f$ from $\{0, 1, 2, 3, 4, 5, 6\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and

\[|x - y| \leq |f(x) - f(y)| \leq 3|x - y|\]

for all $x$ and $y$ in $\{0, 1, 2, 3, 4, 5, 6\}$.

Solution

2018 AIME II (Problems • Answer Key • Resources)
Preceded by
2018 AIME I Problems 		Followed by
2019 AIME I Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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