Found problems: 3632
1964 AMC 12/AHSME, 4
The expression
\[ \frac{P+Q}{P-Q}-\frac{P-Q}{P+Q} \]
where $P=x+y$ and $Q=x-y$, is equivalent to:
${ \textbf{(A)}\ \frac{x^2-y^2}{xy}\qquad\textbf{(B)}\ \frac{x^2-y^2}{2xy}\qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac{x^2+y^2}{xy} \qquad\textbf{(E)}\ \frac{x^2+y^2}{2xy} } $
1975 AMC 12/AHSME, 25
A woman, her brother, her son and her daughter are chess players (all relations by birth). The worst player's twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are the same age. Who is the worst player?
$ \textbf{(A)}\ \text{the woman} \qquad\textbf{(B)}\ \text{her son} \qquad\textbf{(C)}\ \text{her brother} \qquad\textbf{(D)}\ \text{her daughter} \\ \qquad\textbf{(E)}\ \text{No solution is consistent with the given information} $
1960 AMC 12/AHSME, 37
The base of a triangle is of length $b$, and the latitude is of length $h$. A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:
$ \textbf{(A)}\ \frac{bx}{h}(h-x)\qquad\textbf{(B)}\ \frac{hx}{b}(b-x)\qquad\textbf{(C)}\ \frac{bx}{h}(h-2x)\qquad$
$\textbf{(D)}\ x(b-x)\qquad\textbf{(E)}\ x(h-x) $
2018 AMC 12/AHSME, 15
How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3?
$\textbf{(A) } 96 \qquad \textbf{(B) } 97 \qquad \textbf{(C) } 98 \qquad \textbf{(D) } 102 \qquad \textbf{(E) } 120 $
1970 AMC 12/AHSME, 22
If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is
$\textbf{(A) }300\qquad\textbf{(B) }350\qquad\textbf{(C) }400\qquad\textbf{(D) }450\qquad \textbf{(E) }600$
2006 AMC 10, 15
Rhombus $ ABCD$ is similar to rhombus $ BFDE$. The area of rhombus $ ABCD$ is 24, and $ \angle BAD \equal{} 60^\circ$. What is the area of rhombus $ BFDE$?
[asy]
size(180);
defaultpen(linewidth(0.7)+fontsize(11));
pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3));
pair point=(3/2, sqrt(3)/2);
draw(B--C--D--A--B--F--D--E--B);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));[/asy]
$ \textbf{(A) } 6 \qquad \textbf{(B) } 4\sqrt {3} \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 6\sqrt {3}$
1992 AMC 12/AHSME, 23
What is the size of the largest subset, $S$, of $\{1, 2, 3, \ldots, 50\}$ such that no pair of distinct elements of $S$ has a sum divisible by $7$?
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 14\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 23 $
2013 AMC 12/AHSME, 7
Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying "$1$", so Blair follows by saying "$1$, $2$". Jo then says "$1$, $2$, $3$", and so on. What is the $53$rd number said?
$ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $
2021 AMC 12/AHSME Fall, 4
The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$
$(\textbf{A})\: 1\qquad(\textbf{B}) \: 3\qquad(\textbf{C}) \: 5 \qquad(\textbf{D}) \: 7\qquad(\textbf{E}) \: 9$
2024 AMC 12/AHSME, 11
There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$?
$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21$
2016 AMC 10, 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 $A?$
$\textbf{(A)}\ 385 \qquad
\textbf{(B)}\ 665 \qquad
\textbf{(C)}\ 945 \qquad
\textbf{(D)}\ 1140 \qquad
\textbf{(E)}\ 1330$
2013 AIME Problems, 10
Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be written in the form $\tfrac{a-b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
2009 AMC 12/AHSME, 21
Ten women sit in $ 10$ seats in a line. All of the $ 10$ get up and then reseat themselves using all $ 10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?
$ \textbf{(A)}\ 89\qquad
\textbf{(B)}\ 90\qquad
\textbf{(C)}\ 120\qquad
\textbf{(D)}\ 2^{10}\qquad
\textbf{(E)}\ 2^2 3^8$
2000 AMC 8, 4
In $1960$ only $5\%$ of the working adults in Carlin City worked at home. By $1970$ the "at-home" work force increased to $8\%$. In $1980$ there were approximately $15\%$ working at home, and in $1990$ there were $30\%$. The graph that best illustrates this is
[asy]
unitsize(13);
draw((0,4)--(0,0)--(7,0));
draw((0,1)--(.2,1)); draw((0,2)--(.2,2)); draw((0,3)--(.2,3));
draw((2,0)--(2,.2)); draw((4,0)--(4,.2)); draw((6,0)--(6,.2));
for (int a = 1; a < 4; ++a)
{
for (int b = 1; b < 4; ++b)
{
draw((2*a,b-.1)--(2*a,b+.1));
draw((2*a-.1,b)--(2*a+.1,b));
}
}
label("1960",(0,0),S); label("1970",(2,0),S); label("1980",(4,0),S); label("1990",(6,0),S);
label("10",(0,1),W); label("20",(0,2),W); label("30",(0,3),W);
label("$\%$",(0,4),N);
draw((12,4)--(12,0)--(19,0));
draw((12,1)--(12.2,1)); draw((12,2)--(12.2,2)); draw((12,3)--(12.2,3));
draw((14,0)--(14,.2)); draw((16,0)--(16,.2)); draw((18,0)--(18,.2));
for (int a = 1; a < 4; ++a)
{
for (int b = 1; b < 4; ++b)
{
draw((2*a+12,b-.1)--(2*a+12,b+.1));
draw((2*a+11.9,b)--(2*a+12.1,b));
}
}
label("1960",(12,0),S); label("1970",(14,0),S); label("1980",(16,0),S); label("1990",(18,0),S);
label("10",(12,1),W); label("20",(12,2),W); label("30",(12,3),W);
label("$\%$",(12,4),N);
draw((0,12)--(0,8)--(7,8));
draw((0,9)--(.2,9)); draw((0,10)--(.2,10)); draw((0,11)--(.2,11));
draw((2,8)--(2,8.2)); draw((4,8)--(4,8.2)); draw((6,8)--(6,8.2));
for (int a = 1; a < 4; ++a)
{
for (int b = 1; b < 4; ++b)
{
draw((2*a,b+7.9)--(2*a,b+8.1));
draw((2*a-.1,b+8)--(2*a+.1,b+8));
}
}
label("1960",(0,8),S); label("1970",(2,8),S); label("1980",(4,8),S); label("1990",(6,8),S);
label("10",(0,9),W); label("20",(0,10),W); label("30",(0,11),W);
label("$\%$",(0,12),N);
draw((12,12)--(12,8)--(19,8));
draw((12,9)--(12.2,9)); draw((12,10)--(12.2,10)); draw((12,11)--(12.2,11));
draw((14,8)--(14,8.2)); draw((16,8)--(16,8.2)); draw((18,8)--(18,8.2));
for (int a = 1; a < 4; ++a)
{
for (int b = 1; b < 4; ++b)
{
draw((2*a+12,b+7.9)--(2*a+12,b+8.1));
draw((2*a+11.9,b+8)--(2*a+12.1,b+8));
}
}
label("1960",(12,8),S); label("1970",(14,8),S); label("1980",(16,8),S); label("1990",(18,8),S);
label("10",(12,9),W); label("20",(12,10),W); label("30",(12,11),W);
label("$\%$",(12,12),N);
draw((24,12)--(24,8)--(31,8));
draw((24,9)--(24.2,9)); draw((24,10)--(24.2,10)); draw((24,11)--(24.2,11));
draw((26,8)--(26,8.2)); draw((28,8)--(28,8.2)); draw((30,8)--(30,8.2));
for (int a = 1; a < 4; ++a)
{
for (int b = 1; b < 4; ++b)
{
draw((2*a+24,b+7.9)--(2*a+24,b+8.1));
draw((2*a+23.9,b+8)--(2*a+24.1,b+8));
}
}
label("1960",(24,8),S); label("1970",(26,8),S); label("1980",(28,8),S); label("1990",(30,8),S);
label("10",(24,9),W); label("20",(24,10),W); label("30",(24,11),W);
label("$\%$",(24,12),N);
draw((0,9)--(2,9.25)--(4,10)--(6,11));
draw((12,8.5)--(14,9)--(16,10)--(18,10.5));
draw((24,8.5)--(26,8.8)--(28,10.5)--(30,11));
draw((0,0.5)--(2,1)--(4,2.8)--(6,3));
draw((12,0.5)--(14,.8)--(16,1.5)--(18,3));
label("(A)",(-1,12),W);
label("(B)",(11,12),W);
label("(C)",(23,12),W);
label("(D)",(-1,4),W);
label("(E)",(11,4),W);[/asy]
1984 AIME Problems, 7
The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$.
2003 AMC 10, 1
Which of the following is the same as
\[ \frac{2\minus{}4\plus{}6\minus{}8\plus{}10\minus{}12\plus{}14}{3\minus{}6\plus{}9\minus{}12\plus{}15\minus{}18\plus{}21}?
\]$ \textbf{(A)}\ \minus{}1 \qquad
\textbf{(B)}\ \minus{}\frac23 \qquad
\textbf{(C)}\ \frac23 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ \frac{14}{3}$
2014 AIME Problems, 1
The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
[asy]
size(200);
defaultpen(linewidth(0.7));
path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin;
path laceR=reflect((75,0),(75,-240))*laceL;
draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray);
for(int i=0;i<=3;i=i+1)
{
path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5);
unfill(circ1); draw(circ1);
unfill(circ2); draw(circ2);
}
draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]
2021 AMC 12/AHSME Spring, 19
Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is $\frac{3}{4}$ of the probability of rolling a sum of 10 and the probability of rolling a sum of 12 is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined?
$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 19 \qquad\textbf{(E)}\ 20$
1988 AMC 12/AHSME, 14
For any real number $a$ and positive integer $k$, define \[ {a \choose k} = \frac{a(a-1)(a-2)\cdots(a-(k-1))}{k(k-1)(k-2)\cdots(2)(1)}. \]What is \[{-\frac{1}{2} \choose 100} \div {\frac{1}{2} \choose 100}?\]
$ \textbf{(A)}\ -199\qquad\textbf{(B)}\ -197\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ 197\qquad\textbf{(E)}\ 199 $
2024 AMC 10, 24
A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^+, A^-, B^+, B^-, C^+$, and $C^-$ is rolled. Suppose the bee occupies the point $(a, b, c)$. If the die shows $A^+$, then the bee moves to the point $(a+1, b, c)$ and if the die shows $A^-$, then the bee moves to the point $(a-1, b, c)$. Analogous moves are made with the other four outcomes. Suppose the bee starts at the point $(0, 0, 0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
$
\textbf{(A) }\frac{1}{54} \qquad
\textbf{(B) }\frac{7}{54} \qquad
\textbf{(C) }\frac{1}{6} \qquad
\textbf{(D) }\frac{5}{18} \qquad
\textbf{(E) }\frac{2}{5} \qquad
$
2011 AMC 10, 19
In $1991$ the population of a town was a perfect square. Ten years later, after an increase of $150$ people, the population was $9$ more than a perfect square. Now, in $2011$, with an increase of another $150$ people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period?
$ \textbf{(A)}\ 42 \qquad\textbf{(B)}\ 47 \qquad\textbf{(C)}\ 52\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 62 $
2008 AIME Problems, 1
Let $ N\equal{}100^2\plus{}99^2\minus{}98^2\minus{}97^2\plus{}96^2\plus{}\cdots\plus{}4^2\plus{}3^2\minus{}2^2\minus{}1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $ N$ is divided by $ 1000$.
2013 AMC 12/AHSME, 4
What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\]
$ \textbf{(A)}\ -1\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ 2013\qquad\textbf{(E)}\ 2^{4024} $
1998 AMC 8, 4
How many triangles are in this figure? (Some triangles may overlap other triangles.)
[asy]
draw((0,0)--(42,0)--(14,21)--cycle);
draw((14,21)--(18,0)--(30,9));[/asy]
$ \text{(A)}\ 9\qquad\text{(B)}\ 8\qquad\text{(C)}\ 7\qquad\text{(D)}\ 6\qquad\text{(E)}\ 5 $
2018 AMC 10, 18
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?
$\textbf{(A)} \text{ 60} \qquad \textbf{(B)} \text{ 72} \qquad \textbf{(C)} \text{ 92} \qquad \textbf{(D)} \text{ 96} \qquad \textbf{(E)} \text{ 120}$