Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "User:Thundius"
Line 97: Line 97:
  
 
<math> \textbf{(A)}\ 49\qquad\textbf{(B)}\ 65\qquad\textbf{(C)}\ 70\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 128 </math>
 
<math> \textbf{(A)}\ 49\qquad\textbf{(B)}\ 65\qquad\textbf{(C)}\ 70\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 128 </math>
 +
 +
==2014 AMC 10A Problem 22==
 +
 +
In rectangle <math>ABCD</math>, <math>AB=20</math> and <math>BC=10</math>. Let <math>E</math> be a point on <math>\overline{CD}</math> such that <math>\angle CBE=15^\circ</math>. What is <math>AE</math>?
 +
 +
<math> \textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20 </math>
 +
 +
[[2014 AMC 10A  Problems/Problem 22|Solution]]
 +
==2014 AMC 10A Problem 23==
 +
 +
A rectangular piece of paper whose length is <math>\sqrt3</math> times the width has area <math>A</math>. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area <math>B</math>. What is the ratio <math>B:A</math>?
 +
 +
<asy>
 +
import graph;
 +
size(6cm);
 +
 +
real L = 0.05;
 +
 +
pair A = (0,0);
 +
pair B = (sqrt(3),0);
 +
pair C = (sqrt(3),1);
 +
pair D = (0,1);
 +
 +
pair X1 = (sqrt(3)/3,0);
 +
pair X2= (2*sqrt(3)/3,0);
 +
pair Y1 = (2*sqrt(3)/3,1);
 +
pair Y2 = (sqrt(3)/3,1);
 +
 +
dot(X1);
 +
dot(Y1);
 +
 +
draw(A--B--C--D--cycle, linewidth(2));
 +
draw(X1--Y1,dashed);
 +
 +
draw(X2--(2*sqrt(3)/3,L));
 +
draw(Y2--(sqrt(3)/3,1-L));
 +
</asy>
 +
 +
<math> \textbf{(A)}\ 1:2\qquad\textbf{(B)}\ 3:5\qquad\textbf{(C)}\ 2:3\qquad\textbf{(D)}\ 3:4\qquad\textbf{(E)}\ 4:5 </math>
 +
 +
[[2014 AMC 10A  Problems/Problem 23|Solution]]
 +
 +
==2014 AMC 10A Problem 24==
 +
 +
A sequence of natural numbers is constructed by listing the first <math>4</math>, then skipping one, listing the next <math>5</math>, skipping <math>2</math>, listing <math>6</math>, skipping <math>3</math>, and, on the <math>n</math>th iteration, listing <math>n+3</math> and skipping <math>n</math>. The sequence begins <math>1,2,3,4,6,7,8,9,10,13</math>. What is the <math>500,000</math>th number in the sequence?
 +
 +
<math> \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996,507\qquad\textbf{(C)}\ 996,508\qquad\textbf{(D)}\ 996,509\qquad\textbf{(E)}\ 996,510 </math>
 +
 +
[[2014 AMC 10A  Problems/Problem 24|Solution]]
 +
 +
==2014 AMC 10A Problem 25==
 +
 +
The number <math>5^{867}</math> is between <math>2^{2013}</math> and <math>2^{2014}</math>.  How many pairs of integers <math>(m,n)</math> are there such that <math>1\leq m\leq 2012</math> and <cmath>5^n<2^m<2^{m+2}<5^{n+1}?</cmath>
 +
 +
<math>\textbf{(A) }278\qquad\textbf{(B) }279\qquad\textbf{(C) }280\qquad\textbf{(D) }281\qquad\textbf{(E) }282\qquad</math>
 +
 +
[[2014 AMC 10A  Problems/Problem 25|Solution]]

Revision as of 13:42, 1 January 2018

Going to put practice problems here.

2013 AMC 10A Problem 7

A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?


$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16$


Solution

2013 AMC 10A Problem 10

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?


$\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 70$


Solution

2013 AMC 10A Problem 13

How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit?


$\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 66 \qquad\textbf{(D)}\ 68 \qquad\textbf{(E)}\ 70$


Solution

2013 AMC 10A Problem 16

A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles?

$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ \frac{28}{3} \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ \frac{31}{3} \qquad\textbf{(E)}\ \frac{32}{3}$


Solution

2013 AMC 10A Problem 17

Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next 365-day period will exactly two friends visit her?

$\textbf{(A)}\ 48\qquad\textbf{(B)}\ 54\qquad\textbf{(C)}\ 60\qquad\textbf{(D)}\ 66\qquad\textbf{(E)}\ 72$

2013 AMC 10A Problem 18

Let points $A = (0, 0)$, $B = (1, 2)$, $C=(3, 3)$, and $D = (4, 0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\bigg(\frac{p}{q}, \frac{r}{s}\bigg)$, where these fractions are in lowest terms. What is $p+q+r+s$?


$\textbf{(A)}\ 54\qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 70\qquad\textbf{(E)}\ 75$


Solution

2013 AMC 10A Problem 20

A unit square is rotated $45^\circ$ about its center. What is the area of the region swept out by the interior of the square?


$\textbf{(A)}\ 1 - \frac{\sqrt2}{2} + \frac{\pi}{4}\qquad\textbf{(B)}\ \frac{1}{2} + \frac{\pi}{4} \qquad\textbf{(C)}\ 2 - \sqrt2 + \frac{\pi}{4}$

$\textbf{(D)}\ \frac{\sqrt2}{2} + \frac{\pi}{4} \qquad\textbf{(E)}\ 1 + \frac{\sqrt2}{4} + \frac{\pi}{8}$


Solution

2013 AMC 10A Problem 22

Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?


$\textbf{(A)}\ \sqrt2\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ \sqrt3\qquad\textbf{(E)}\ 2$


Solution

2013 AMC 10A Problem 23

In $\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC$?


$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 33\qquad\textbf{(D)}\ 61\qquad\textbf{(E)}\ 72$


Solution

2013 AMC 10A Problem 24

Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?

$\textbf{(A)}\ 540\qquad\textbf{(B)}\ 600\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 810\qquad\textbf{(E)}\ 900$


Solution

2013 AMC 10A Problem 25

All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?

$\textbf{(A)}\ 49\qquad\textbf{(B)}\ 65\qquad\textbf{(C)}\ 70\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 128$

2014 AMC 10A Problem 22

In rectangle $ABCD$, $AB=20$ and $BC=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $AE$?

$\textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20$

Solution

2014 AMC 10A Problem 23

A rectangular piece of paper whose length is $\sqrt3$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $B:A$?

[asy] import graph; size(6cm); real L = 0.05; pair A = (0,0); pair B = (sqrt(3),0); pair C = (sqrt(3),1); pair D = (0,1); pair X1 = (sqrt(3)/3,0); pair X2= (2*sqrt(3)/3,0); pair Y1 = (2*sqrt(3)/3,1); pair Y2 = (sqrt(3)/3,1); dot(X1); dot(Y1); draw(A--B--C--D--cycle, linewidth(2)); draw(X1--Y1,dashed); draw(X2--(2*sqrt(3)/3,L)); draw(Y2--(sqrt(3)/3,1-L)); [/asy]

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

Solution

2014 AMC 10A Problem 24

A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence?

$\textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996,507\qquad\textbf{(C)}\ 996,508\qquad\textbf{(D)}\ 996,509\qquad\textbf{(E)}\ 996,510$

Solution

2014 AMC 10A Problem 25

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]

$\textbf{(A) }278\qquad\textbf{(B) }279\qquad\textbf{(C) }280\qquad\textbf{(D) }281\qquad\textbf{(E) }282\qquad$

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=User:Thundius&oldid=89319"