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Difference between revisions of "2016 AIME II Problems"

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==Problem 3==
 
==Problem 3==
 
Let <math>x,y,</math> and <math>z</math> be real numbers satisfying the system
 
Let <math>x,y,</math> and <math>z</math> be real numbers satisfying the system
−
<math>\log_2(xyz-3+\log_5 x)=5</math>
+
<cmath>
−
<math>\log_3(xyz-3+\log_5 y)=4</math>
+
\begin{align*}
−
<math>\log_4(xyz-3+\log_5 z)=4</math>
+
\log_2(xyz-3+\log_5 x)&=5,\\
 +
\log_3(xyz-3+\log_5 y)&=4,\\
 +
\log_4(xyz-3+\log_5 z)&=4.
 +
\end{align*}
 +
</cmath>
 
Find the value of <math>|\log_5 x|+|\log_5 y|+|\log_5 z|</math>.
 
Find the value of <math>|\log_5 x|+|\log_5 y|+|\log_5 z|</math>.
  
Line 24: Line 28:
 
[[2016 AIME II Problems/Problem 4 | Solution]]
 
[[2016 AIME II Problems/Problem 4 | Solution]]
 
==Problem 5==
 
==Problem 5==
−
Triangle <math>ABC_0</math> has a right angle at <math>C_0</math>. Its side lengths are pariwise relatively prime positive integers, and its perimeter is <math>p</math>. Let <math>C_1</math> be the foot of the altitude to <math>\overline{AB}</math>, and for <math>n \geq 2</math>, let <math>C_n</math> be the foot of the altitude to <math>\overline{C_{n-2}B}</math> in <math>\triangle C_{n-2}C_{n-1}B</math>. The sum <math>\sum_{i=1}^\infty C_{n-2}C_{n-1} = 6p</math>. Find <math>p</math>.
+
Triangle <math>ABC_0</math> has a right angle at <math>C_0</math>. Its side lengths are pairwise relatively prime positive integers, and its perimeter is <math>p</math>. Let <math>C_1</math> be the foot of the altitude to <math>\overline{AB}</math>, and for <math>n \geq 2</math>, let <math>C_n</math> be the foot of the altitude to <math>\overline{C_{n-2}B}</math> in <math>\triangle C_{n-2}C_{n-1}B</math>. The sum <math>\sum_{n=2}^\infty C_{n-2}C_{n-1} = 6p</math>. Find <math>p</math>.
  
 
[[2016 AIME II Problems/Problem 5 | Solution]]
 
[[2016 AIME II Problems/Problem 5 | Solution]]
 +
 
==Problem 6==
 
==Problem 6==
 
For polynomial <math>P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}</math>, define
 
For polynomial <math>P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}</math>, define
Line 34: Line 39:
 
[[2016 AIME II Problems/Problem 6 | Solution]]
 
[[2016 AIME II Problems/Problem 6 | Solution]]
 
==Problem 7==
 
==Problem 7==
−
Squares <math>ABCD</math> and <math>EFGH</math> have a common center at <math>\overline{AB} || \overline{EF}</math>. The area of <math>ABCD</math> is 2016, and the area of <math>EFGH</math> is a smaller positive integer. Square <math>IJKL</math> is constructed so that each of its vertices lies on a side of <math>ABCD</math> and each vertex of <math>EFGH</math> lies on a side of <math>IJKL</math>. Find the difference between the largest and smallest positive integer values for the area of <math>IJKL</math>.
+
Squares <math>ABCD</math> and <math>EFGH</math> have a common center and <math>\overline{AB} || \overline{EF}</math>. The area of <math>ABCD</math> is 2016, and the area of <math>EFGH</math> is a smaller positive integer. Square <math>IJKL</math> is constructed so that each of its vertices lies on a side of <math>ABCD</math> and each vertex of <math>EFGH</math> lies on a side of <math>IJKL</math>. Find the difference between the largest and smallest positive integer values for the area of <math>IJKL</math>.
  
 
[[2016 AIME II Problems/Problem 7 | Solution]]
 
[[2016 AIME II Problems/Problem 7 | Solution]]
  
 
==Problem 8==
 
==Problem 8==
−
Find the number of sets <math>{a,b,c}</math> of three distinct positive integers with the property that the product of <math>a,b,</math> and <math>c</math> is equal to the product of <math>11,21,31,41,51,61</math>.
+
Find the number of sets <math>\{a,b,c\}</math> of three distinct positive integers with the property that the product of <math>a,b,</math> and <math>c</math> is equal to the product of <math>11,21,31,41,51,</math> and <math>61</math>.
  
 
[[2016 AIME II Problems/Problem 8 | Solution]]
 
[[2016 AIME II Problems/Problem 8 | Solution]]
 +
 
==Problem 9==
 
==Problem 9==
 
The sequences of positive integers <math>1,a_2, a_3,...</math> and <math>1,b_2, b_3,...</math> are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let <math>c_n=a_n+b_n</math>. There is an integer <math>k</math> such that <math>c_{k-1}=100</math> and <math>c_{k+1}=1000</math>. Find <math>c_k</math>.
 
The sequences of positive integers <math>1,a_2, a_3,...</math> and <math>1,b_2, b_3,...</math> are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let <math>c_n=a_n+b_n</math>. There is an integer <math>k</math> such that <math>c_{k-1}=100</math> and <math>c_{k+1}=1000</math>. Find <math>c_k</math>.
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==Problem 12==
 
==Problem 12==
 
The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
 
The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
−
Insert figure of a smaller circle, a bigger circle, and 6 sections dividing the region between the concentric circles.
+
 
 +
<asy>
 +
draw(Circle((0,0), 4));
 +
draw(Circle((0,0), 3));
 +
draw((0,4)--(0,3));
 +
draw((0,-4)--(0,-3));
 +
draw((-2.598, 1.5)--(-3.4641, 2));
 +
draw((-2.598, -1.5)--(-3.4641, -2));
 +
draw((2.598, -1.5)--(3.4641, -2));
 +
draw((2.598, 1.5)--(3.4641, 2));
 +
</asy>
  
 
[[2016 AIME II Problems/Problem 12 | Solution]]
 
[[2016 AIME II Problems/Problem 12 | Solution]]
 +
 
==Problem 13==
 
==Problem 13==
−
Beatrix is going to place six rooks on a <math>6 \times 6</math> chessboard where both the rows and columns are labeled <math>1</math> to <math>6</math>; the rooks are placed so that no two rooks are in the same row or the same column. The <math>value</math> of a square is the sum of its row number and column number. The <math>score</math> of an arrangement of rooks is the least value of any occupied square.The average score over all valid configurations is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
+
Beatrix is going to place six rooks on a <math>6 \times 6</math> chessboard where both the rows and columns are labeled <math>1</math> to <math>6</math>; the rooks are placed so that no two rooks are in the same row or the same column. The ''value'' of a square is the sum of its row number and column number. The ''score'' of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
  
 
[[2016 AIME II Problems/Problem 13 | Solution]]
 
[[2016 AIME II Problems/Problem 13 | Solution]]
 +
 
==Problem 14==
 
==Problem 14==
 
Equilateral <math>\triangle ABC</math> has side length <math>600</math>. Points <math>P</math> and <math>Q</math> lie outside the plane of <math>\triangle ABC</math> and are on opposite sides of the plane. Furthermore, <math>PA=PB=PC</math>, and <math>QA=QB=QC</math>, and the planes of <math>\triangle PAB</math> and <math>\triangle QAB</math> form a <math>120^{\circ}</math> dihedral angle (the angle between the two planes). There is a point <math>O</math> whose distance from each of <math>A,B,C,P,</math> and <math>Q</math> is <math>d</math>. Find <math>d</math>.
 
Equilateral <math>\triangle ABC</math> has side length <math>600</math>. Points <math>P</math> and <math>Q</math> lie outside the plane of <math>\triangle ABC</math> and are on opposite sides of the plane. Furthermore, <math>PA=PB=PC</math>, and <math>QA=QB=QC</math>, and the planes of <math>\triangle PAB</math> and <math>\triangle QAB</math> form a <math>120^{\circ}</math> dihedral angle (the angle between the two planes). There is a point <math>O</math> whose distance from each of <math>A,B,C,P,</math> and <math>Q</math> is <math>d</math>. Find <math>d</math>.
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[[2016 AIME II Problems/Problem 14 | Solution]]
 
[[2016 AIME II Problems/Problem 14 | Solution]]
 
==Problem 15==
 
==Problem 15==
−
For <math>1 \leq i \leq 215</math> let <math>a_i = \dfrac{1}{2^{i}}</math> and <math>a_216 = \dfrac{1}{2^{215}}</math>. Let <math>x_1, x_2, ..., x_215</math> be positive real numbers such that <math>\sum_{i=1}^{215} x_i=1</math> and <math>\sum_{i \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}</math>. The maximum possible value of <math>x_2=\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
+
For <math>1 \leq i \leq 215</math> let <math>a_i = \dfrac{1}{2^{i}}</math> and <math>a_{216} = \dfrac{1}{2^{215}}</math>. Let <math>x_1, x_2, ..., x_{216}</math> be positive real numbers such that <math>\sum_{i=1}^{216} x_i=1</math> and <math>\sum_{1 \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}</math>. The maximum possible value of <math>x_2=\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
 
[[2016 AIME II Problems/Problem 15 | Solution]]
 
[[2016 AIME II Problems/Problem 15 | Solution]]
  
 +
 +
{{AIME box|year=2016|n=II|before=[[2016 AIME I Problems]]|after=[[2017 AIME I Problems]]}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 20:16, 3 March 2021

2016 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

Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.

Solution

Problem 2

There is a $40\%$ chance of rain on Saturday and a $30\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

Solution

Problem 3

Let $x,y,$ and $z$ be real numbers satisfying the system \begin{align*} \log_2(xyz-3+\log_5 x)&=5,\\ \log_3(xyz-3+\log_5 y)&=4,\\ \log_4(xyz-3+\log_5 z)&=4. \end{align*} Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.

Solution

Problem 4

An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Each of the $b$ layers of size $a \times 1 \times c$ parallel to the $(a \times c)$ faces of the box contains exactly $20$ green cubes, exactly $25$ yellow cubes, and some red cubes. Find the smallest possible volume of the box.

Solution

Problem 5

Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum_{n=2}^\infty C_{n-2}C_{n-1} = 6p$. Find $p$.

Solution

Problem 6

For polynomial $P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}$, define $Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\sum_{i=0}^{50} a_ix^{i}$. Then $\sum_{i=0}^{50} |a_i|=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 7

Squares $ABCD$ and $EFGH$ have a common center and $\overline{AB} || \overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest positive integer values for the area of $IJKL$.

Solution

Problem 8

Find the number of sets $\{a,b,c\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,$ and $61$.

Solution

Problem 9

The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.

Solution

Problem 10

Triangle $ABC$ is inscribed in circle $\omega$. Points $P$ and $Q$ are on side $\overline{AB}$ with $AP<AQ$. Rays $CP$ and $CQ$ meet $\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 11

For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.

Solution

Problem 12

The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.

[asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy]

Solution

Problem 13

Beatrix is going to place six rooks on a $6 \times 6$ chessboard where both the rows and columns are labeled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution

Problem 14

Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\triangle ABC$ and are on opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (the angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P,$ and $Q$ is $d$. Find $d$.

Solution

Problem 15

For $1 \leq i \leq 215$ let $a_i = \dfrac{1}{2^{i}}$ and $a_{216} = \dfrac{1}{2^{215}}$. Let $x_1, x_2, ..., x_{216}$ be positive real numbers such that $\sum_{i=1}^{216} x_i=1$ and $\sum_{1 \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}$. The maximum possible value of $x_2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution


2016 AIME II (Problems • Answer Key • Resources)
Preceded by
2016 AIME I Problems 		Followed by
2017 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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