Found problems: 3632
2023 AMC 12/AHSME, 1
Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet?
$\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$
2012 AIME Problems, 7
At each of the sixteen circles in the network below stands a student. A total of 3360 coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally.
[asy]
import graph;
unitsize(1 cm);
pair[] O;
O[1] = (0,0);
O[2] = 0.6*dir(270);
O[3] = 0.6*dir(270 + 360/5);
O[4] = 0.6*dir(270 + 2*360/5);
O[5] = 0.6*dir(270 + 3*360/5);
O[6] = 0.6*dir(270 + 4*360/5);
O[7] = 1.2*dir(90);
O[8] = 1.2*dir(90 + 360/5);
O[9] = 1.2*dir(90 + 2*360/5);
O[10] = 1.2*dir(90 + 3*360/5);
O[11] = 1.2*dir(90 + 4*360/5);
O[12] = 2*dir(270);
O[13] = 2*dir(270 + 360/5);
O[14] = 2*dir(270 + 2*360/5);
O[15] = 2*dir(270 + 3*360/5);
O[16] = 2*dir(270 + 4*360/5);
draw(O[1]--O[2]);
draw(O[1]--O[3]);
draw(O[1]--O[4]);
draw(O[1]--O[5]);
draw(O[1]--O[6]);
draw(O[7]--O[5]--O[8]--O[6]--O[9]--O[2]--O[10]--O[3]--O[11]--O[4]--cycle);
draw(O[12]--O[10]--O[13]--O[11]--O[14]--O[7]--O[15]--O[8]--O[16]--O[9]--cycle);
draw(O[12]--O[13]--O[14]--O[15]--O[16]--cycle);
for(int i = 1; i <= 16; ++i) {
filldraw(Circle(O[i],0.2),white,black);
}
[/asy]
2007 AMC 10, 20
Suppose that the number $ a$ satisfies the equation $ 4 \equal{} a \plus{} a^{ \minus{} 1}$. What is the value of $ a^{4} \plus{} a^{ \minus{} 4}$?
$ \textbf{(A)}\ 164 \qquad \textbf{(B)}\ 172 \qquad \textbf{(C)}\ 192 \qquad \textbf{(D)}\ 194 \qquad \textbf{(E)}\ 212$
1967 AMC 12/AHSME, 22
For natural numbers, when $P$ is divided by $D$, the quotient is $Q$ and the remainder is $R$. When $Q$ is divided by $D'$, the quotient is $Q'$ and the remainder is $R'$. Then, when $P$ is divided by $DD'$, the remainder is:
$\textbf{(A)}\ R+R'D\qquad
\textbf{(B)}\ R'+RD\qquad
\textbf{(C)}\ RR'\qquad
\textbf{(D)}\ R\qquad
\textbf{(E)}\ R'$
2010 AMC 12/AHSME, 21
Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that
\[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\]
\[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\]
What is the smallest possible value of $ a$?
$ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$
1996 AMC 8, 2
Jose, Thuy, and Kareem each start with the number $10$. Jose subtracts $1$ from the number $10$, doubles his answer, and then adds $2$. Thuy doubles the number $10$, subtracts $1$ from her answer, and then adds $2$. Kareem subtracts $1$ from the number $10$, adds $2$ to his number, and then doubles the result. Who gets the largest final answer?
$\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kareem} \qquad \text{(D)}\ \text{Jose and Thuy} \qquad \text{(E)}\ \text{Thuy and Kareem}$
2014 Math Prize For Girls Problems, 2
Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$?
2000 AMC 8, 15
Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$?
[asy]
pair A,B,C,D,EE,F,G;
A = (4,0); B = (0,0); C = (2,2*sqrt(3)); D = (3,sqrt(3));
EE = (5,sqrt(3)); F = (5.5,sqrt(3)/2); G = (4.5,sqrt(3)/2);
draw(A--B--C--cycle);
draw(D--EE--A);
draw(EE--F--G);
label("$A$",A,S);
label("$B$",B,SW);
label("$C$",C,N);
label("$D$",D,NE);
label("$E$",EE,NE);
label("$F$",F,SE);
label("$G$",G,SE);
[/asy]
$\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 21$
2006 AIME Problems, 12
Equilateral $\triangle ABC$ is inscribed in a circle of radius 2. Extend $\overline{AB}$ through $B$ to point $D$ so that $AD=13$, and extend $\overline{AC}$ through $C$ to point $E$ so that $AE=11$. Through $D$, draw a line $l_1$ parallel to $\overline{AE}$, and through $E$, draw a line ${l}_2$ parallel to $\overline{AD}$. Let $F$ be the intersection of ${l}_1$ and ${l}_2$. Let $G$ be the point on the circle that is collinear with $A$ and $F$ and distinct from $A$. Given that the area of $\triangle CBG$ can be expressed in the form $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime, find $p+q+r$.
1969 AMC 12/AHSME, 14
The complete set of $x$-values satisfying the inequality $\dfrac{x^2-4}{x^2-1}>0$ is the set of all $x$ such that:
$\textbf{(A) }x>2\text{ or }x<-2\text{ or }-1<x<1\qquad\,
\textbf{(B) }x>2\text{ or }x<-2$
$\textbf{(C) }x>1\text{ or }x<-2\qquad\qquad\qquad\qquad\,\,\,\,\,\,\,
\textbf{(D) }x>1\text{ or }x<-2\qquad$
$\textbf{(E) }x\text{ is any real number except }1\text{ or }-1$
2023 AMC 12/AHSME, 22
A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ $$f(a + b) + f(a - b) = 2f(a) f(b).$$ Which one of the following cannot be the value of $f(1)?$
$
\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } -1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } -2$
2020 CHMMC Winter (2020-21), 4
Let $P(x) = x^3 - 6x^2 - 5x + 4$. Suppose that $y$ and $z$ are real numbers such that
\[ zP(y) = P(y - n) + P(y + n) \]
for all reals $n$. Evaluate $P(y)$.
2010 AIME Problems, 10
Let $ N$ be the number of ways to write $ 2010$ in the form \[2010 \equal{} a_3 \cdot 10^3 \plus{} a_2 \cdot 10^2 \plus{} a_1 \cdot 10 \plus{} a_0,\] where the $ a_i$'s are integers, and $ 0 \le a_i \le 99$. An example of such a representation is $ 1\cdot10^3 \plus{} 3\cdot10^2 \plus{} 67\cdot10^1 \plus{} 40\cdot10^0$. Find $ N$.
2022 USAJMO, 4
Let $ABCD$ be a rhombus, and let $K$ and $L$ be points such that $K$ lies inside the rhombus, $L$ lies outside the rhombus, and $KA = KB = LC = LD$. Prove that there exist points $X$ and $Y$ on lines $AC$ and $BD$ such that $KXLY$ is also a rhombus.
[i]Proposed by Ankan Bhattacharya[/i]
2020 AMC 10, 21
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\] What is $k?$
$\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$
2015 AMC 10, 8
The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?
[asy]
import cse5;pathpen=black;pointpen=black;
size(2cm);
D((0,-2)--MP("y",(0,7),N));
D((-3,0)--MP("x",(5,0),E));
D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5));
[/asy][asy]
import cse5;pathpen=black;pointpen=black;
unitsize(0.2cm);
D((0,-2)--MP("y",(0,7),N));
D(MP("\textbf{(A) }",(-3,0),W)--MP("x",(5,0),E));
D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5));
//
D((18,-2)--MP("y",(18,7),N));
D(MP("\textbf{(B) }",(13,0),W)--MP("x",(21,0),E));
D((17,0)--(17,2)--(16,2)--(16,3)--(17,3)--(17,4)--(15,4)--(15,5)--(18,5));
//
D((36,-2)--MP("y",(36,7),N));
D(MP("\textbf{(C) }",(29,0),W)--MP("x",(38,0),E));
D((31,0)--(31,1)--(33,1)--(33,2)--(34,2)--(34,1)--(35,1)--(35,3)--(36,3));
//
D((0,-17)--MP("y",(0,-8),N));
D(MP("\textbf{(D) }",(-3,-15),W)--MP("x",(5,-15),E));
D((3,-15)--(3,-14)--(1,-14)--(1,-13)--(2,-13)--(2,-12)--(1,-12)--(1,-10)--(0,-10));
//
D((15,-17)--MP("y",(15,-8),N));
D(MP("\textbf{(E) }",(13,-15),W)--MP("x",(22,-15),E));
D((15,-14)--(17,-14)--(17,-13)--(18,-13)--(18,-14)--(19,-14)--(19,-12)--(20,-12)--(20,-15));
[/asy]
2011 AMC 12/AHSME, 24
Let $P(z) = z^8 + (4\sqrt{3} + 6) z^4 - (4\sqrt{3}+7)$. What is the minimum perimeter among all the 8-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$?
$ \textbf{(A)}\ 4\sqrt{3}+4 \qquad
\textbf{(B)}\ 8\sqrt{2} \qquad
\textbf{(C)}\ 3\sqrt{2}+3\sqrt{6} \qquad
\textbf{(D)}\ 4\sqrt{2}+4\sqrt{3} \qquad
$
$\textbf{(E)}\ 4\sqrt{3}+6 $
1964 AMC 12/AHSME, 27
If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$, then:
$ \textbf{(A)}\ 0<a<.01\qquad\textbf{(B)}\ .01<a<1 \qquad\textbf{(C)}\ 0<a<1\qquad$
$\textbf{(D)}\ 0<a \le 1\qquad\textbf{(E)}\ a>1 $
2013 AMC 10, 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?
$\textbf{(A) }9\qquad
\textbf{(B) }\dfrac{28}{3}\qquad
\textbf{(C) }10\qquad
\textbf{(D) }\dfrac{31}{3}\qquad
\textbf{(E) }\dfrac{32}{3}\qquad$
2023 AMC 12/AHSME, 13
A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$?
$\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$
2013 AMC 8, 22
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?
[asy]
picture corner;
draw(corner,(5,0)--(35,0));
draw(corner,(0,-5)--(0,-35));
for (int i=0; i<3; ++i)
{
for (int j=0; j>-2; --j)
{
if ((i-j)<3)
{
add(corner,(50i,50j));
}
}
}
draw((5,-100)--(45,-100));
draw((155,0)--(185,0),dotted+linewidth(2));
draw((105,-50)--(135,-50),dotted+linewidth(2));
draw((100,-55)--(100,-85),dotted+linewidth(2));
draw((55,-100)--(85,-100),dotted+linewidth(2));
draw((50,-105)--(50,-135),dotted+linewidth(2));
draw((0,-105)--(0,-135),dotted+linewidth(2));[/asy]
$\textbf{(A)}\ 1920 \qquad \textbf{(B)}\ 1952 \qquad \textbf{(C)}\ 1980 \qquad \textbf{(D)}\ 2013 \qquad \textbf{(E)}\ 3932$
2017 AMC 10, 23
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$
2024 AMC 10, 11
In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$. Point $M$ lies $\overline{XY}$, point $A$ lies on $\overline{YZ}$, and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA$?
[asy]
pair X = (0, 0);
pair W = (0, 4);
pair Y = (8, 0);
pair Z = (8, 4);
label("$X$", X, dir(180));
label("$W$", W, dir(180));
label("$Y$", Y, dir(0));
label("$Z$", Z, dir(0));
draw(W--X--Y--Z--cycle);
dot(X);
dot(Y);
dot(W);
dot(Z);
pair M = (2, 0);
pair A = (8, 3);
label("$A$", A, dir(0));
dot(M);
dot(A);
draw(W--M--A--cycle);
markscalefactor = 0.05;
draw(rightanglemark(W, M, A));
label("$M$", M, dir(-90));
[/asy]
$
\textbf{(A) }13 \qquad
\textbf{(B) }14 \qquad
\textbf{(C) }15 \qquad
\textbf{(D) }16 \qquad
\textbf{(E) }17 \qquad
$
2015 AMC 10, 24
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$?
$ \textbf{(A) } (-22,-13)\qquad\textbf{(B) } (-13,-22)\qquad\textbf{(C) } (-13,22)\qquad\textbf{(D) } (13,-22)\qquad\textbf{(E) } (22,-13) $
2011 AIME Problems, 11
Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the remainder when $S$ is divided by $1000$.