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Difference between revisions of "2016 AMC 10B Problems"
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{{AMC10 Problems|year=2016|ab=B}}
 +
==Problem 1==
 +
What is the value of <math>\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}</math> when <math>a= \tfrac{1}{2}</math>?
 +
 
 +
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 1|Solution]]
 +
 
 +
==Problem 2==
 +
If <math>n\heartsuit m=n^3m^2</math>, what is <math>\frac{2\heartsuit 4}{4\heartsuit 2}</math>?
 +
 
 +
<math>\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 2|Solution]]
 +
 
 +
==Problem 3==
 +
Let <math>x=-2016</math>. What is the value of <cmath>\bigg| \big||x|-x\big|-|x| \bigg| -x?</cmath>
 +
 
 +
<math>\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 3|Solution]]
 +
 
 +
==Problem 4==
 +
Zoey read <math>15</math> books, one at a time. The first book took her <math>1</math> day to read, the second book took her <math>2</math> days to read, the third book took her <math>3</math> days to read, and so on, with each book taking her <math>1</math> more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day of the week did she finish her <math>15</math>th book?
 +
 
 +
<math>\textbf{(A)}\ \text{Sunday}\qquad\textbf{(B)}\ \text{Monday}\qquad\textbf{(C)}\ \text{Wednesday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 4|Solution]]
 +
 
 +
==Problem 5==
 +
The mean age of Amanda's <math>4</math> cousins is <math>8</math>, and their median age is <math>5</math>. What is the sum of the ages of Amanda's youngest and oldest cousins?
 +
 
 +
<math>\textbf{(A)}\ 13\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 25</math>
 +
 
 +
[[2016 AMC 10B Problems/Problem 5|Solution]]
 +
 
 +
==Problem 6==
 +
 
 +
Isaac added two three-digit positive integers. All six digits in these numbers are different. Isaac's sum is a three-digit number <math>S</math>. What is the smallest possible value for the sum of the digits of <math>S</math>?
 +
 
 +
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 6|Solution]]
 +
 
 +
==Problem 7==
 +
The ratio of the measures of two acute angles is <math>5:4</math>, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
 +
 
 +
<math>\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 7|Solution]]
 +
 
 +
==Problem 8==
 +
What is the tens digit of <math>2015^{2016}-2017?</math>
 +
 
 +
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 8</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 8|Solution]]
 +
 
 +
==Problem 9==
 +
All three vertices of <math>\bigtriangleup ABC</math> are lying on the parabola defined by <math>y=x^2</math>, with <math>A</math> at the origin and <math>\overline{BC}</math> parallel to the <math>x</math>-axis. The area of the triangle is <math>64</math>. What is the length of <math>BC</math>?
 +
 
 +
<math>\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 9|Solution]]
 +
 
 +
==Problem 10==
 +
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length <math>3</math> inches weighs <math>12</math> ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of <math>5</math> inches. Which of the following is closest to the weight, in ounces, of the second piece?
 +
 
 +
<math>\textbf{(A)}\ 14.0\qquad\textbf{(B)}\ 16.0\qquad\textbf{(C)}\ 20.0\qquad\textbf{(D)}\ 33.3\qquad\textbf{(E)}\ 55.6</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 10|Solution]]
 +
 
 +
==Problem 11==
 +
Sola decided to fence in his rectangular garden. He bought <math>20</math> fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly <math>4</math> yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Sola’s garden?
 +
 
 +
<math>\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 11|Solution]]
 +
 
 +
==Problem 12==
 +
Two different numbers are selected at random from <math>( 1, 2, 3, 4, 5)</math> and multiplied together. What is the probability that the product is even?
 +
 
 +
<math>\textbf{(A)}\ 0.2\qquad\textbf{(B)}\ 0.4\qquad\textbf{(C)}\ 0.5\qquad\textbf{(D)}\ 0.7\qquad\textbf{(E)}\ 0.8</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 12|Solution]]
 +
 
 +
==Problem 13==
 +
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for <math>1000</math> of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these <math>1000</math> babies were in sets of quadruplets?
 +
 
 +
<math>\textbf{(A)}\ 25\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 160</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 13|Solution]]
 +
 
 +
==Problem 14==
 +
How many squares whose sides are parallel to the axis and whose vertices have coordinates that are integers lie entirely within the region bounded by the line <math>y=\pi x</math>, the line <math>y=-0.1</math> and the line <math>x=5.1?</math>
 +
 
 +
<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 41 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 57</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 14|Solution]]
 +
 
 +
==Problem 15==
 +
All the numbers <math>1, 2, 3, 4, 5, 6, 7, 8, 9</math> are written in a <math>3\times3</math> array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to <math>18</math>. What is the number in the center?
 +
 
 +
<math>\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 15|Solution]]
 +
 
 +
==Problem 16==
 +
The sum of an infinite geometric series is a positive number <math>S</math>, and the second term in the series is <math>1</math>. What is the smallest possible value of <math>S?</math>
 +
 
 +
<math>\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt{5} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 16|Solution]]
 +
 
 +
==Problem 17==
 +
All the numbers <math>2, 3, 4, 5, 6, 7</math> are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
 +
 
 +
<math>\textbf{(A)}\ 312 \qquad \textbf{(B)}\ 343 \qquad \textbf{(C)}\ 625 \qquad \textbf{(D)}\ 729 \qquad \textbf{(E)}\ 1680</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 17|Solution]]
 +
 
 +
==Problem 18==
 +
In how many ways can <math>345</math> be written as the sum of an increasing sequence of two or more consecutive positive integers?
 +
 
 +
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 18|Solution]]
 +
 
 +
==Problem 19==
 +
Rectangle <math>ABCD</math> has <math>AB=5</math> and <math>BC=4</math>. Point <math>E</math> lies on <math>\overline{AB}</math> so that <math>EB=1</math>, point <math>G</math> lies on <math>\overline{BC}</math> so that <math>CG=1</math>. and point <math>F</math> lies on <math>\overline{CD}</math> so that <math>DF=2</math>. Segments <math>\overline{AG}</math> and <math>\overline{AC}</math> intersect <math>\overline{EF}</math> at <math>Q</math> and <math>P</math>, respectively. What is the value of <math>\frac{PQ}{EF}</math>?
 +
 
 +
 
 +
<asy> pair A1=(2,0),A2=(4,4);
 +
pair B1=(0,4),B2=(5,1);
 +
pair C1=(5,0),C2=(0,4);
 +
draw(A1--A2);
 +
draw(B1--B2);
 +
draw(C1--C2);
 +
draw((0,0)--B1--(5,4)--C1--cycle);
 +
dot((20/7,12/7));
 +
dot((3.07692307692,2.15384615384));
 +
label("$Q$",(3.07692307692,2.15384615384),N);
 +
label("$P$",(20/7,12/7),W);
 +
label("$A$",(0,4), NW);
 +
label("$B$",(5,4), NE);
 +
label("$C$",(5,0),SE);
 +
label("$D$",(0,0),SW);
 +
label("$F$",(2,0),S); label("$G$",(5,1),E);
 +
label("$E$",(4,4),N);
 +
 
 +
dot(A1); dot(A2);
 +
dot(B1); dot(B2);
 +
dot(C1); dot(C2);
 +
dot((0,0)); dot((5,4));</asy>
 +
 
 +
<math>\textbf{(A)}~\frac{\sqrt{13}}{16} \qquad
 +
\textbf{(B)}~\frac{\sqrt{2}}{13} \qquad
 +
\textbf{(C)}~\frac{9}{82} \qquad
 +
\textbf{(D)}~\frac{10}{91}\qquad
 +
\textbf{(E)}~\frac19</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 19|Solution]]
 +
 
 +
==Problem 20==
 +
A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius <math>2</math> centered at <math>A(2,2)</math> to the circle of radius <math>3</math> centered at <math>A’(5,6)</math>. What distance does the origin <math>O(0,0)</math>, move under this transformation?
 +
 
 +
<math>\textbf{(A)}\ 0\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ \sqrt{13}\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 20|Solution]]
 +
 
 +
==Problem 21==
 +
What is the area of the region enclosed by the graph of the equation <math>x^2+y^2=|x|+|y|?</math>
 +
 
 +
<math>\textbf{(A)}\ \pi+\sqrt{2}\qquad\textbf{(B)}\ \pi+2\qquad\textbf{(C)}\ \pi+2\sqrt{2}\qquad\textbf{(D)}\ 2\pi+\sqrt{2}\qquad\textbf{(E)}\ 2\pi+2\sqrt{2}</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 21|Solution]]
 +
 
 +
==Problem 22==
 +
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won <math>10</math> games and lost <math>10</math> games; there were no ties. How many sets of three teams <math>\{A, B, C\}</math> were there in which <math>A</math> beat <math>B</math>, <math>B</math> beat <math>C</math>, and <math>C</math> beat <math>A?</math>
 +
 
 +
<math>\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 22|Solution]]
 +
 
 +
==Problem 23==
 +
In regular hexagon <math>ABCDEF</math>, points <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</math> are chosen on sides <math>\overline{BC}</math>, <math>\overline{CD}</math>, <math>\overline{EF}</math>, and <math>\overline{FA}</math> respectively, so lines <math>AB</math>, <math>ZW</math>, <math>YX</math>, and <math>ED</math> are parallel and equally spaced. What is the ratio of the area of hexagon <math>WCXYFZ</math> to the area of hexagon <math>ABCDEF</math>?
 +
 
 +
<math>\textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{10}{27}\qquad\textbf{(C)}\ \frac{11}{27}\qquad\textbf{(D)}\ \frac{4}{9}\qquad\textbf{(E)}\ \frac{13}{27}</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 23|Solution]]
 +
 
 +
==Problem 24==
 +
How many four-digit integers <math>abcd</math>, with <math>a \neq 0</math>, have the property that the three two-digit integers <math>ab<bc<cd</math> form an increasing arithmetic sequence? One such number is <math>4692</math>, where <math>a=4</math>, <math>b=6</math>, <math>c=9</math>, and <math>d=2</math>.
 +
 
 +
<math>\textbf{(A)}\ 9\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 20</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 24|Solution]]
 +
 
 +
==Problem 25==
 +
Let <math>f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)</math>, where <math>\lfloor r \rfloor</math> denotes the greatest integer less than or equal to <math>r</math>. How many distinct values does <math>f(x)</math> assume for <math>x \ge 0</math>?
 +
 
 +
<math>\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ \text{infinitely many}</math>
 +
 
 +
[[2016 AMC 10B  Problems/Problem 25|Solution]]
 +
 
 +
==See also==
 +
{{AMC10 box|year=2016|ab=B|before=[[2016 AMC 10A Problems]]|after=[[2017 AMC 10A Problems]]}}
 +
* [[AMC 10]]
 +
* [[AMC 10 Problems and Solutions]]
 +
* [[2016 AMC 10A]]
 +
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Latest revision as of 16:50, 30 December 2023

2016 AMC 10B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \tfrac{1}{2}$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20$

Solution

Problem 2

If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$?

$\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$

Solution

Problem 3

Let $x=-2016$. What is the value of \[\bigg| \big||x|-x\big|-|x| \bigg| -x?\]

$\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$

Solution

Problem 4

Zoey read $15$ books, one at a time. The first book took her $1$ day to read, the second book took her $2$ days to read, the third book took her $3$ days to read, and so on, with each book taking her $1$ more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day of the week did she finish her $15$th book?

$\textbf{(A)}\ \text{Sunday}\qquad\textbf{(B)}\ \text{Monday}\qquad\textbf{(C)}\ \text{Wednesday}\qquad\textbf{(D)}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$

Solution

Problem 5

The mean age of Amanda's $4$ cousins is $8$, and their median age is $5$. What is the sum of the ages of Amanda's youngest and oldest cousins?

$\textbf{(A)}\ 13\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 25$

Solution

Problem 6

Isaac added two three-digit positive integers. All six digits in these numbers are different. Isaac's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$

Solution

Problem 7

The ratio of the measures of two acute angles is $5:4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?

$\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$

Solution

Problem 8

What is the tens digit of $2015^{2016}-2017?$

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 8$

Solution

Problem 9

All three vertices of $\bigtriangleup ABC$ are lying on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$

Solution

Problem 10

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?

$\textbf{(A)}\ 14.0\qquad\textbf{(B)}\ 16.0\qquad\textbf{(C)}\ 20.0\qquad\textbf{(D)}\ 33.3\qquad\textbf{(E)}\ 55.6$

Solution

Problem 11

Sola decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Sola’s garden?

$\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$

Solution

Problem 12

Two different numbers are selected at random from $( 1, 2, 3, 4, 5)$ and multiplied together. What is the probability that the product is even?

$\textbf{(A)}\ 0.2\qquad\textbf{(B)}\ 0.4\qquad\textbf{(C)}\ 0.5\qquad\textbf{(D)}\ 0.7\qquad\textbf{(E)}\ 0.8$

Solution

Problem 13

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets?

$\textbf{(A)}\ 25\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 160$

Solution

Problem 14

How many squares whose sides are parallel to the axis and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$

$\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 41 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 57$

Solution

Problem 15

All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center?

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$

Solution

Problem 16

The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$

$\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt{5} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution

Problem 17

All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

$\textbf{(A)}\ 312 \qquad \textbf{(B)}\ 343 \qquad \textbf{(C)}\ 625 \qquad \textbf{(D)}\ 729 \qquad \textbf{(E)}\ 1680$

Solution

Problem 18

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

Solution

Problem 19

Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$. and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?


[asy] pair A1=(2,0),A2=(4,4); pair B1=(0,4),B2=(5,1); pair C1=(5,0),C2=(0,4); draw(A1--A2); draw(B1--B2); draw(C1--C2); draw((0,0)--B1--(5,4)--C1--cycle); dot((20/7,12/7)); dot((3.07692307692,2.15384615384)); label("$Q$",(3.07692307692,2.15384615384),N); label("$P$",(20/7,12/7),W); label("$A$",(0,4), NW); label("$B$",(5,4), NE); label("$C$",(5,0),SE); label("$D$",(0,0),SW); label("$F$",(2,0),S); label("$G$",(5,1),E); label("$E$",(4,4),N); dot(A1); dot(A2); dot(B1); dot(B2); dot(C1); dot(C2); dot((0,0)); dot((5,4));[/asy]

$\textbf{(A)}~\frac{\sqrt{13}}{16} \qquad \textbf{(B)}~\frac{\sqrt{2}}{13} \qquad \textbf{(C)}~\frac{9}{82} \qquad \textbf{(D)}~\frac{10}{91}\qquad \textbf{(E)}~\frac19$

Solution

Problem 20

A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$. What distance does the origin $O(0,0)$, move under this transformation?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ \sqrt{13}\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution

Problem 21

What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$

$\textbf{(A)}\ \pi+\sqrt{2}\qquad\textbf{(B)}\ \pi+2\qquad\textbf{(C)}\ \pi+2\sqrt{2}\qquad\textbf{(D)}\ 2\pi+\sqrt{2}\qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$

Solution

Problem 22

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$

$\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$

Solution

Problem 23

In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$?

$\textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{10}{27}\qquad\textbf{(C)}\ \frac{11}{27}\qquad\textbf{(D)}\ \frac{4}{9}\qquad\textbf{(E)}\ \frac{13}{27}$

Solution

Problem 24

How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 20$

Solution

Problem 25

Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$?

$\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ \text{infinitely many}$

Solution

See also

2016 AMC 10B (Problems • Answer Key • Resources)
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All AMC 10 Problems and Solutions

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