problem_id int64 1 978 | question stringlengths 86 2.11k | source stringlengths 19 76 | solution stringlengths 94 14.7k | asymptote_code stringlengths 44 17.8k | solution_image_url stringclasses 16
values |
|---|---|---|---|---|---|
895 | Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge... | 2021 AMC 12A Problem 23 | Imagine an infinite grid of $2$ by $2$ squares such that there is a $2$ by $2$ square centered at $(3x, 3y)$ for all ordered pairs of integers $(x, y).$
It is easy to see that the problem is equivalent to Frieda moving left, right, up, or down on this infinite grid starting at $(0, 0)$ (minus the teleportations). Si... | dot((0,0)); draw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle); draw((-4,-1)--(-4,1)--(-2,1)--(-2,-1)--cycle); draw((2,-1)--(2,1)--(4,1)--(4,-1)--cycle); draw((-1,2)--(-1,4)--(1,4)--(1,2)--cycle); draw((-1,-4)--(-1,-2)--(1,-2)--(1,-4)--cycle); draw((-4,2)--(-2,2)--(-2,4)--(-4,4)--cycle); draw((4,-2)--(2,-2)--(2,-4)--(4,-4)--c... | [] |
895 | Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge... | 2021 AMC 12A Problem 23 | Let us go through the frog’s jumps step by step to find the probability that it will land in a corner. Denote the frog’s position as $F$:
First notice that since the frog only moves up, down, left or right, it’s original hop will not matter by symmetry, because all first hops will bring the frog at the edge next to tw... | // Block 1
label("$F$",(0,0)); draw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle); draw((1,-1)--(1,1)--(3,1)--(3,-1)--cycle); draw((-1,1)--(-1,3)--(1,3)--(1,1)--cycle); draw((-3,-1)--(-3,1)--(-1,1)--(-1,-1)--cycle); draw((-1,-3)--(-1,-1)--(1,-1)--(1,-3)--cycle); draw((-3,-3)--(3,-3)--(3,3)--(-3,3)--cycle);
// Block 2
label("... | [] |
896 | Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ equals $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime... | 2021 AMC 12A Problem 24 | Let $O=\Gamma$ be the center of the semicircle and $X=\Omega$ be the center of the circle.
Applying the Extended Law of Sines to $\triangle PQR,$ we find the radius of $\odot X:$ \[XP=\frac{QR}{2\cdot\sin \angle QPR}=\frac{3\sqrt3}{2\cdot\frac{\sqrt3}{2}}=3.\]
Alternatively, by the Inscribed Angle Theorem, $\triangle... | // Block 1
/* Made by MRENTHUSIASM */
size(300);
pair O, X, A, B, P, Q, R, M, C, D;
O = (0,0);
X = (4,3);
A = (-7,0);
B = (7,0);
P = (4,0);
Q = intersectionpoints(Circle(O,7),Circle(X,3))[0];
R = intersectionpoints(Circle(O,7),Circle(X,3))[1];
M = midpoint(Q--R);
C = foot(P,Q,R);
D = foot(X,P,C);
fill(P--Q--R--cycle,ye... | [] |
896 | Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ equals $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime... | 2021 AMC 12A Problem 24 | Suppose we label the points as shown in the diagram above, where $C$ is the center of the semicircle and $O$ is the center of the circle tangent to $\overline{AB}$. Since $\angle QPR = 60^{\circ}$, we have $\angle QOR = 2\cdot 60^{\circ}=120^{\circ}$ and $\triangle QOR$ is a $30-30-120$ triangle, which can be split int... | // Block 1
size(150);
draw(circle((7,0),7));
pair A = (0, 0);
pair B = (14, 0);
draw(A--B);
draw(circle((11,3),3));
label("$C$", (7, 0), S);
label("$O$", (11, 3), E);
label("$P$", (11, 0), S);
pair C = (7, 0);
pair O = (11, 3);
pair P = (11, 0);
pair Q = intersectionpoints(circle(C, 7), circle(O, 3))[1];
pair R = inter... | [] |
896 | Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ equals $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime... | 2021 AMC 12A Problem 24 | Define points as shown above, where $N=\overleftrightarrow{PA}\cap\overleftrightarrow{QR}$. The area of $\triangle PQR$ is simply\[\dfrac{1}{2}PX\cdot QR=\dfrac{3\sqrt{3}}{2}PX;\]
it remains to compute the value of $PX$. Note that $PX$ is simply a weighted average of $BT$ and $CS;$ it is $\dfrac{CP}{BP}$ times closer t... | size(300); pair C = (7, 0); draw(arc(C, 7, 0, 180)); pair A = (0, 0), B = (14, 0); draw(A--B); draw(circle((11,3),3)); label("$A$", A, SSE); label("$B$", B, SSW); label("$C$", (A+B)/2, S); label("$O$", (11, 3), E); label("$P$", (11, 0), S); pair O = (11, 3), P = (11, 0), Q = intersectionpoints(circle(C, 7), circle(O, ... | [] |
897 | In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $\ell$ at the same point $A,$ but they may be on either side of $\ell$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$?
$\textbf{(A) }24\pi \qquad \textbf{(B)... | 2021 AMC 10B Problem 7 | Suppose that line $\ell$ is horizontal, and each circle lies either north or south to $\ell.$ We construct the circles one by one:
Without the loss of generality, we draw the circle with radius $7$ north to $\ell.$
To maximize the area of region $S,$ we draw the circle with radius $5$ south to $\ell.$
Now, we need to... | // Block 1
/* diagram made by samrocksnature, edited by MRENTHUSIASM */
pair A=(10,0);
pair B=(-10,0);
draw(A--B);
filldraw(circle((0,7),7),yellow);
filldraw(circle((0,-5),5),yellow);
filldraw(circle((0,-3),3),white);
filldraw(circle((0,-1),1),white);
dot((0,0));
label("$A$",(0,0),(0,1.5));
label("$\ell$",(10,0),(1.5,0... | [] |
898 | Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row f... | 2021 AMC 10B Problem 8 | In the diagram below, the red arrows indicate the progression of numbers. In the second row from the top, the greatest number and the least number are $D$ and $E,$ respectively. Note that the numbers in the yellow cells are consecutive odd perfect squares, as we can prove by induction.
By observations, we proceed as f... | // Block 1
/* Made by MRENTHUSIASM */
size(11.5cm);
for (real i=7.5; i<=14.5; ++i)
{
fill((i+0.5,i+0.5)--(i-0.5,i+0.5)--(i-0.5,i-0.5)--(i+0.5,i-0.5)--cycle,yellow);
}
fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green);
fill((1,14)--(0,14)--(0,13)--(1,13)--cycle,green);
label("$A$",(14.5,14.5));
label("$B$",(13.5,13... | [] |
898 | Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row f... | 2021 AMC 10B Problem 8 | In the diagram below, the red arrows indicate the progression of numbers. In the second row from the top, the greatest number and the least number are $C$ and $G,$ respectively.
By observations, we proceed as follows:
\begin{alignat*}{6} A=15^2=225 \quad &\implies \quad &B &= \hspace{1mm}&&A-14\hspace{1mm} &= 211& \\... | // Block 1
/* Made by MRENTHUSIASM */
size(11.5cm);
fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green);
fill((1,14)--(0,14)--(0,13)--(1,13)--cycle,green);
label("$A$",(14.5,14.5));
label("$B$",(0.5,14.5));
label("$C$",(0.5,13.5));
label("$D$",(0.5,0.5));
label("$E$",(14.5,0.5));
label("$F$",(14.5,13.5));
label("$G$",(... | [] |
898 | Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row f... | 2021 AMC 10B Problem 8 | From the full diagram below, the answer is $210+157=\boxed{\textbf{(A)} ~367}.$
This solution is extremely time-consuming and error-prone. Please do not try it in a real competition unless you have no other options.
~MRENTHUSIASM ~Taco12 | // Block 1
/* Made by MRENTHUSIASM */
size(11.5cm);
fill((2,14)--(1,14)--(1,13)--(2,13)--cycle,green);
fill((1,14)--(0,14)--(0,13)--(1,13)--cycle,green);
add(grid(15,15,linewidth(1.25)));
int adj = 1;
int curDown = 2;
int curLeft = 4;
int curUp = 6;
int curRight = 8;
label("$1$",(7.5,7.5));
for (int len = 3; len<=... | [] |
899 | Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,38,$ and $34$. What is the distance between two adjacent parallel lines?
$\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12$ | 2021 AMC 10B Problem 14 | Since two parallel chords have the same length ($38$), they must be equidistant from the center of the circle. Let the perpendicular distance of each chord from the center of the circle be $d$. Thus, the distance from the center of the circle to the chord of length $34$ is
\[2d + d = 3d\]
and the distance between ea... | size(8cm); pair O = (0, 0), A = (0, 3), B = (0, 9), R = (19, 3), L = (17, 9); draw(O--A--B); draw(O--R); draw(O--L); label("$A$", A, NE); label("$B$", B, N); label("$R$", R, NE); label("$L$", L, NE); label("$O$", O, S); label("$d$", O--A, W); label("$2d$", A--B, W); label("$r$", O--R, S); label("$r$", O--L, NW); dot(O)... | [] |
899 | Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,38,$ and $34$. What is the distance between two adjacent parallel lines?
$\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12$ | 2021 AMC 10B Problem 14 | If $d$ is the requested distance, and $r$ is the radius of the circle, Stewart's Theorem applied to $\triangle OCD$ with cevian $\overleftrightarrow{OP}$ gives \[19\cdot 38\cdot 19 + \tfrac{1}{2}d\cdot 38\cdot\tfrac{1}{2}d=19r^{2}+19r^{2}.\] This simplifies to $13718+\tfrac{19}{2}d^{2}=38r^{2}$. Similarly, another roun... | real r=sqrt(370); draw(circle((0, 0), r)); pair A = (-19, 3); pair B = (19, 3); draw(A--B); pair C = (-19, -3); pair D = (19, -3); draw(C--D); pair E = (-17, -9); pair F = (17, -9); draw(E--F); pair O = (0, 0); pair P = (0, -3); pair Q = (0, -9); draw(O--Q); draw(O--C); draw(O--D); draw(O--E); draw(O--F); label("$O$", ... | [] |
900 | The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$
$\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~2... | 2021 AMC 10B Problem 20 | Draw diagonals $AC$ and $AD$ to split the pentagon into three parts. We can compute the area for each triangle and sum them up at the end. For triangles $ABC$ and $ADE$, they each have area $2\cdot\frac{1}{2}\cdot\frac{4\sqrt{3}}{4}=\sqrt{3}$. For triangle $ACD$, we can see that $AC=AD=2\sqrt{3}$ and $CD=2$. Using Pyth... | /* Made by samrocksnature, adapted by Tucker, then adjusted by samrocksnature again, then adjusted by erics118 xD*/ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-1.8... | [] |
901 | Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | 2021 AMC 10B Problem 25 | Based on area ratios between a square of side length $30$ and a triangle with base $30$, we estimate that the slope $m$ of the line we want is approximately $\frac{2}{3}$. Following this estimate, we see if there are approximately $30 \cdot 30 - 300 = 600$ lattice points above the line $y=\frac{2}{3}x$.
Counting the n... | /* Created by Brendanb4321 */ import graph; size(16cm); defaultpen(fontsize(9pt)); xaxis(0,30,Ticks(1.0)); yaxis(0,25,Ticks(1.0)); draw((0,0)--(30,20)); draw((0,0)--(30,30/28*19), dotted); for (int i = 1; i<=30; ++i) { for (int j = 1; j<=2/3*i+1; ++j) { dot((i,j)); } } dot((28,19), red); label("$m=2/3$", (32,20)); lab... | [] |
901 | Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | 2021 AMC 10B Problem 25 | It's easier to calculate the number of lattice points inside a rectangle with vertices $(0,0)$, $(p,0)$, $(p,q)$, $(0,q)$. Those lattice points are divided by the diagonal $y = \frac{p}{q} \cdot x$ into $2$ halves. In this problem the number of lattice points on or below the diagonal and $x \ge 1$ is
$\frac{1}{2} [(p+... | /* Created by Brendanb4321, modified by isabelchen */ import graph; size(18cm); defaultpen(fontsize(9pt)); xaxis(0,31,Ticks(1.0)); yaxis(0,22,Ticks(1.0)); draw((0,0)--(30,20)); draw((0,0)--(30,30*19/28), dotted); draw((0,0)--(31,31*21/31), dotted); for (int i = 1; i<=31; ++i) { for (int j = 1; j<=2/3*i+1; ++j) { dot(... | [] |
901 | Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | 2021 AMC 10B Problem 25 | The lower bound of $m$ is $\frac23 = \frac{20}{30}$. Inside the rectangle with vertices $(0,0)$, $(30,0)$, $(30,20)$, $(0, 20)$ and diagonal $y = \frac23 x$, there are $(30-1)(20-1) = 551$ lattice points inside, not including the edges. There are $9$ lattice points on diagonal $y = \frac23 x$ inside the rectangle, $551... | /* Created by isabelchen */ import graph; size(8cm); defaultpen(fontsize(9pt)); xaxis(0,8); yaxis(0,6); draw((0,0)--(7,5)); draw((7,0)--(7,5)); draw((0,5)--(7,5)); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((5,0)); dot((6,0)); dot((7,0)); dot((8,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1))... | [] |
902 | Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,38,$ and $34$. What is the distance between two adjacent parallel lines?
$\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12$ | 2021 AMC 12B Problem 8 | Since two parallel chords have the same length ($38$), they must be equidistant from the center of the circle. Let the perpendicular distance of each chord from the center of the circle be $d$. Thus, the distance from the center of the circle to the chord of length $34$ is
\[2d + d = 3d\]
and the distance between ea... | // Block 1
size(8cm);
pair O = (0, 0), A = (0, 3), B = (0, 9), R = (19, 3), L = (17, 9);
draw(O--A--B);
draw(O--R);
draw(O--L);
label("$A$", A, NE);
label("$B$", B, N);
label("$R$", R, NE);
label("$L$", L, NE);
label("$O$", O, S);
label("$d$", O--A, W);
label("$2d$", A--B, W);
label("$r$", O--R, S);
label("$r$", O--L, ... | [] |
902 | Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,38,$ and $34$. What is the distance between two adjacent parallel lines?
$\textbf{(A) }5\frac12 \qquad \textbf{(B) }6 \qquad \textbf{(C) }6\frac12 \qquad \textbf{(D) }7 \qquad \textbf{(E) }7\frac12$ | 2021 AMC 12B Problem 8 | If $d$ is the requested distance, and $r$ is the radius of the circle, Stewart's Theorem applied to $\triangle OCD$ with cevian $\overleftrightarrow{OP}$ gives \[19\cdot 38\cdot 19 + \tfrac{1}{2}d\cdot 38\cdot\tfrac{1}{2}d=19r^{2}+19r^{2}.\] This simplifies to $13718+\tfrac{19}{2}d^{2}=38r^{2}$. Similarly, another roun... | // Block 1
real r=sqrt(370);
draw(circle((0, 0), r));
pair A = (-19, 3);
pair B = (19, 3);
draw(A--B);
pair C = (-19, -3);
pair D = (19, -3);
draw(C--D);
pair E = (-17, -9);
pair F = (17, -9);
draw(E--F);
pair O = (0, 0);
pair P = (0, -3);
pair Q = (0, -9);
draw(O--Q);
draw(O--C);
draw(O--D);
draw(O--E);
draw(O--F);
la... | [] |
903 | How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy \[1-3\sin\theta+5\cos3\theta = 0?\]
$\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8$ | 2021 AMC 12B Problem 13 | We rearrange to get \[5\cos3\theta = 3\sin\theta-1.\]
We can graph two functions in this case: $y=5\cos{3x}$ and $y=3\sin{x} -1$.
Using transformation of functions, we know that $5\cos{3x}$ is just a cosine function with amplitude $5$ and period $\frac{2\pi}{3}$. Similarly, $3\sin{x} -1$ is just a sine function with am... | // Block 1
import graph;
size(400,200,IgnoreAspect);
real Sin(real t) {return 3*sin(t) - 1;}
real Cos(real t) {return 5*cos(3*t);}
draw(graph(Sin,0, 2pi),red,"$3\sin{x} -1 $");
draw(graph(Cos,0, 2pi),blue,"$5\cos{3x}$");
xaxis("$x$",BottomTop,LeftTicks);
yaxis("$y$",LeftRight,RightTicks(trailingzero));
add(legend(... | [] |
904 | Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$
$\tex... | 2021 AMC 12B Problem 14 | Let $MA=a$ and $MD=d.$ It follows that $MC=a+2$ and $MB=a+4.$
As shown below, note that $\triangle MAD$ and $\triangle MBC$ are both right triangles.
By the Pythagorean Theorem, we have
\begin{alignat*}{6} AD^2 &= MA^2 - MD^2 &&= a^2 - d^2, \\ BC^2 &= MB^2 - MC^2 &&= (a+4)^2 - (a+2)^2. \end{alignat*}
Since $AD=BC$ in... | // Block 1
size(300);
import graph3;
import solids;
currentprojection=orthographic((0.5,-0.25,-0.5));
triple A, B, C, D, M;
A = (-2sqrt(10),0,0);
B = (-2sqrt(10),-6sqrt(2),0);
C = (0,-6sqrt(2),0);
D = (0,0,0);
M = (0,0,3);
draw(surface(M--A--D--cycle),yellow);
draw(surface(M--B--C--cycle),yellow);
draw(D--A--B--C^^A--... | [] |
904 | Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$
$\tex... | 2021 AMC 12B Problem 14 | First and foremost, the diagram must be drawn.
With the diagram, an impulse to utilize Pythagorean theorem is created. Thereby, the shaded right triangles may be used.
The following equations could be obtained from Pythagorean Theorem.
\begin{align*} I^2+a^2&=x^2 \\ I^2+b^2&=(x+2)^2 \\ I^2+a^2+b^2&=(x+4)... | // Block 1
import graph;
size(250);
pair A = (0,0);
pair B = (4.5,0);
pair C = (6,1.1);
pair D = (1.5,1.1);
pair M = (1.5,3.45);
draw(A--B);
draw(C--B);
draw(A--D, dashed);
draw(C--D, dashed);
draw(M--A);
draw(M--B);
draw(M--C);
draw(M--D, dashed);
draw((1.5,1.3)--(1.7,1.3)--(1.7,1.1));
dot(A,linewidth(4.5));
dot(... | [] |
905 | The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$
$\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~2... | 2021 AMC 12B Problem 15 | Draw diagonals $AC$ and $AD$ to split the pentagon into three parts. We can compute the area for each triangle and sum them up at the end. For triangles $ABC$ and $ADE$, they each have area $2\cdot\frac{1}{2}\cdot\frac{4\sqrt{3}}{4}=\sqrt{3}$. For triangle $ACD$, we can see that $AC=AD=2\sqrt{3}$ and $CD=2$. Using Pyth... | // Block 1
/* Made by samrocksnature, adapted by Tucker, then adjusted by samrocksnature again, then adjusted by erics118 xD*/
pair A=(-2.4638,4.10658);
pair B=(-4,2.6567453480756127);
pair C=(-3.47132,0.6335248637894945);
pair D=(-1.464483379039766,0.6335248637894945);
pair E=(-0.956630463955801,2.6567453480756127);
p... | [] |
906 | Let $S$ be the sum of all positive real numbers $x$ for which\[x^{2^{\sqrt2}}=\sqrt2^{2^x}.\]Which of the following statements is true?
$\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6$ | 2021 AMC 12B Problem 21 | We rewrite the right side without using square roots, then take the base-$2$ logarithm for both sides:
\begin{align*} x^{2^{\sqrt2}}&=\left(2^\frac12\right)^{2^x} \\ x^{2^{\sqrt2}}&=2^{\frac12\cdot2^x} \\ x^{2^{\sqrt2}}&=2^{2^{x-1}} \\ \log_2{\left(x^{2^{\sqrt2}}\right)}&=\log_2{\left(2^{2^{x-1}}\right)} \\ 2^{\sqrt2}\... | // Block 1
/* Made by MRENTHUSIASM */
size(1200,200);
int xMin = 0;
int xMax = 5;
int yMin = 0;
int yMax = 5;
//Draws the horizontal gridlines
void horizontalLines()
{
for (int i = yMin+1; i < yMax; ++i)
{
draw((xMin,i)--(xMax,i), mediumgray+linewidth(0.4));
}
}
//Draws the vertical gridlines
void vertica... | [] |
907 | Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
Suppose $PQ=6$ a... | 2021 AMC 12B Problem 24 | First and foremost, the condition given in the problem could be represented. Moreover, let $QB=x$ and $CQ=h$ for convenience.
Using the property of a parallelogram, it could be inferred that $\frac{h(3+x)}{2}=\frac{15}{4}$. Moreover, Pythagorean Theorem manifests that $\left(\frac{4}{3}h\right)^2+4^2=(3+x)^2$.
\begin{... | // Block 1
import olympiad;
size(10cm);
defaultpen(linewidth(1));
pair A = (-2.0833333333, -1.3020833333);
pair B = (3.35, 0.0);
pair C = (2.0833333333, 1.3020833333);
pair D = (-3.35, 0.0);
pair P = (-2.0833333333, 0.0);
pair Q = (2.0833333333, 0.0);
pair R = (2.4109375, 1.5052083333);
pair S = (-2.4109375, -1.505208... | [] |
908 | Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | 2021 AMC 12B Problem 25 | Based on area ratios between a square of side length $30$ and a triangle with base $30$, we estimate that the slope $m$ of the line we want is approximately $\frac{2}{3}$. Following this estimate, we see if there are approximately $30 \cdot 30 - 300 = 600$ lattice points above the line $y=\frac{2}{3}x$.
Counting the n... | /* Created by Brendanb4321 */ import graph; size(16cm); defaultpen(fontsize(9pt)); xaxis(0,30,Ticks(1.0)); yaxis(0,25,Ticks(1.0)); draw((0,0)--(30,20)); draw((0,0)--(30,30/28*19), dotted); for (int i = 1; i<=30; ++i) { for (int j = 1; j<=2/3*i+1; ++j) { dot((i,j)); } } dot((28,19), red); label("$m=2/3$", (32,20)); lab... | [] |
908 | Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | 2021 AMC 12B Problem 25 | It's easier to calculate the number of lattice points inside a rectangle with vertices $(0,0)$, $(p,0)$, $(p,q)$, $(0,q)$. Those lattice points are divided by the diagonal $y = \frac{p}{q} \cdot x$ into $2$ halves. In this problem the number of lattice points on or below the diagonal and $x \ge 1$ is
$\frac{1}{2} [(p+... | // Block 1
/* Created by Brendanb4321, modified by isabelchen */
import graph;
size(18cm);
defaultpen(fontsize(9pt));
xaxis(0,31,Ticks(1.0));
yaxis(0,22,Ticks(1.0));
draw((0,0)--(30,20));
draw((0,0)--(30,30*19/28), dotted);
draw((0,0)--(31,31*21/31), dotted);
for (int i = 1; i<=31; ++i) {
for (int j = 1; j<=2/3*i+1; ... | [] |
908 | Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | 2021 AMC 12B Problem 25 | The lower bound of $m$ is $\frac23 = \frac{20}{30}$. Inside the rectangle with vertices $(0,0)$, $(30,0)$, $(30,20)$, $(0, 20)$ and diagonal $y = \frac23 x$, there are $(30-1)(20-1) = 551$ lattice points inside, not including the edges. There are $9$ lattice points on diagonal $y = \frac23 x$ inside the rectangle, $551... | // Block 1
/* Created by isabelchen */
import graph;
size(8cm);
defaultpen(fontsize(9pt));
xaxis(0,8);
yaxis(0,6);
draw((0,0)--(7,5));
draw((7,0)--(7,5));
draw((0,5)--(7,5));
dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((5,0)); dot((6,0)); dot((7,0)); dot((8,0)); dot((0,1)); dot((1,1)); dot((2,1));... | [] |
909 | Daniel finds a rectangular index card and measures its diagonal to be $8$ centimeters.
Daniel then cuts out equal squares of side $1$ cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area o... | 2022 AMC 10A Problem 10 | Label the bottom left corner of the larger rectangle (without the square cut out) as $A$ and the top right as $D$. $w$ is the width of the rectangle and $\ell$ is the length. Now we have vertices $E, F, G, H$ as vertices of the irregular octagon created by cutting out the squares. Let $I, J$ be the two closest vertices... | // Block 1
/* Edited by MRENTHUSIASM */
size(250);
real x, y;
x = 6;
y = 3;
draw((0,0)--(x,0));
draw((0,0)--(0,y));
draw((0,y)--(x,y));
draw((x,0)--(x,y));
draw((0.5,0)--(0.5,0.5)--(0,0.5));
draw((x-0.5,y)--(x-0.5,y-0.5)--(x,y-0.5));
draw((0.5,0.5)--(x-0.5,y-0.5),dashed,Arrows());
draw((x,0)--(0,y),dashed,Arrows());
la... | [] |
910 | Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC.$ The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D.$ Suppose $BP=2$ and $PC=3.$ What is $AD?$
$\textbf{(A) } 8 \qquad \te... | 2022 AMC 10A Problem 13 | Since there is only one possible value of $AD$, we assume $\angle{B}=90^{\circ}$. By the angle bisector theorem, $\frac{AB}{AC}=\frac{2}{3}$, so $AB=2\sqrt{5}$ and $AC=3\sqrt{5}$. Now observe that $\angle{BAD}=90^{\circ}$. Let the intersection of $BD$ and $AP$ be $X$. Then $\angle{ABD}=90^{\circ}-\angle{BAX}=\angle{APB... | // Block 1
size(300);
pair A, B, C, P, XX, D;
B = (0,0);
P = (2,0);
C = (5,0);
A=(0,4.47214);
D = A + (10,0);
draw(A--B--C--cycle, linewidth(1));
dot("$A$", A, N);
dot("$B$", B, SW);
dot("$C$", C, E);
dot("$P$", P, S);
dot("$D$", D, E);
markscalefactor = 0.1;
draw(anglemark(B,A,P));
markscalefactor = 0.12;
draw(anglema... | [] |
911 | A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$. The edges of the adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?
$\textbf{(A) }6\qquad\textbf{(B... | 2022 AMC 10A Problem 21 | We extend line segments $\ell,m,$ and $n$ to their point of concurrency, as shown below:
We claim that lines $\ell,m,$ and $n$ are concurrent: In the lateral faces of the bowl, we know that lines $\ell$ and $m$ must intersect, and lines $\ell$ and $n$ must intersect. In addition, line $\ell$ intersects the top plane o... | // Block 1
/* Made by AoPS; edited by MRENTHUSIASM */
import three;
size(225);
currentprojection=
orthographic(camera=(-5.52541796301147,-2.61548797564715,1.6545450372312),
up=(0.00247902062334861,0.000877141782387748,0.00966536329192992),
target=(0,0,0),
zoom=0.5705885608... | [] |
911 | A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$. The edges of the adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?
$\textbf{(A) }6\qquad\textbf{(B... | 2022 AMC 10A Problem 21 | You could also think of this problem as folding the four triangles connecting the hexagon faces to cover parts of of the octagon:
So the octagon is made up of five unit squares and four $45-45-90$ triangles with legs of length $1.$ So the area is $\boxed{7}.$
~grogg007 | // Block 1
size(180);
import graph;
pair A = (0, 0);
pair B = A + dir(0)*1;
pair C = B + dir(-45)*sqrt(2);
pair D = C + dir(-90)*1;
pair E = D + dir(-135)*sqrt(2);
pair F = E + dir(180)*1;
pair G = F + dir(135)*sqrt(2);
pair H = G + dir(90)*1;
pair I1 = intersectionpoint(B--E, C--H);
pair I2 = intersectionpoint(B--E, ... | [] |
911 | A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$. The edges of the adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?
$\textbf{(A) }6\qquad\textbf{(B... | 2022 AMC 10A Problem 21 | Notice that $AD \parallel PQ$, $BC \parallel PQ$, so $AD \parallel BC$. Obviously, $AD = BC$, so $ABCD$ is a parallelogram, we get \[CD = AB = \sqrt{1^2+1^2} = \sqrt{2}.\]
Note that octagon has four pairs of parallel sides, and the successive side-lengths are $1,\sqrt2,1,\sqrt2,1,\sqrt2,1,\sqrt2,$ as shown below:
The... | // Block 1
import three;
size(225);
currentprojection=
orthographic(camera=(-5.52541796301147,-2.61548797564715,1.6545450372312),
up=(0.00247902062334861,0.000877141782387748,0.00966536329192992),
target=(0,0,0),
zoom=0.570588560870951);
currentpen = black+1.5bp;
triple A ... | [] |
912 | Suppose that $13$ cards numbered $1, 2, 3, \ldots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards $1, 2, 3$ are picked up on the first pass, $4$ and $5$ on the second pass, $6$ on the third pass, $7, 8, 9, 10$ on... | 2022 AMC 10A Problem 22 | For $1\leq k\leq 12,$ suppose that cards $1, 2, \ldots, k$ are picked up on the first pass. It follows that cards $k+1,k+2,\ldots,13$ are picked up on the second pass.
Once we pick the spots for the cards on the first pass, there is only one way to arrange all $\boldsymbol{13}$ cards.
For each value of $k,$ there are... | // Block 1
size(11cm);
draw((0,0)--(2,0)--(2,3)--(0,3)--cycle);
label("1", (1,1.5));
draw((3,0)--(5,0)--(5,3)--(3,3)--cycle);
label("2", (4,1.5));
draw((6,0)--(8,0)--(8,3)--(6,3)--cycle);
label("3", (7,1.5));
draw((9,0)--(11,0)--(11,3)--(9,3)--cycle);
label("4", (10,1.5));
draw((12,0)--(14,0)--(14,3)--(12,3)--cycle);
l... | [] |
913 | Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$
$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }... | 2022 AMC 10A Problem 23 | Consider the reflection $P^{\prime}$ of $P$ over the perpendicular bisector of $\overline{BC}$, creating two new isosceles trapezoids $DAPP^{\prime}$ and $CBPP^{\prime}$. Under this reflection, $P^{\prime}A=PD=4$, $P^{\prime}D=PA=1$, $P^{\prime}C=PB=2$, and $P^{\prime}B=PC=3$.
Since $DAPP'$ and $CBPP'$ are isosceles ... | // Block 1
size(300);
pair A = (0,0);
pair B = (1, 2);
pair C = (2,2);
pair D = (3,0);
label("$A$", A, SW);
label("$B$", B, NW);
label("$C$", C, NE);
label("$D$", D, SE);
draw(A--B--C--D--cycle, blue);
pair P = (0.8, 0.6);
dot("$P$", P, NW);
draw(P--A, magenta);
draw(P--B, magenta);
draw(P--C);
draw(P--D);
label("$1$",... | [] |
913 | Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$
$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }... | 2022 AMC 10A Problem 23 | Extend $AB$ and $CD$ to a point $Q$ as shown, and let $PQ = s$. Then let $BQ=CQ = a$ and $AB=DC = b$. Notice that $\frac{BC}{AD} = \frac{QC}{QD} = \frac{a}{a+b}$ by similar triangles.
By Stewart's theorem on $APQ$ and $DPQ$, we have \begin{align*} ab(a+b) + 9(a+b) &= 16a + s^2b \\ ab(a+b) + 4(a+b) &= a + s^2b \\ \end{... | // Block 1
size(7.5cm);
draw((0,0)--(4.2,0));
draw((0,0)--(1.4,2)--(2.8,2)--(4.2,0));
draw((1.4,2)--(2.1,3)--(2.8,2));
draw((0,0)--(1,0.5)--(1.4,2)--(1,0.5)--(2.8,2)--(1,0.5)--(4.2,0));
label("$A$",(0,0),SW);
label("$B$",(1.4,2),NW);
label("$C$",(2.8,2),NE);
label("$D$",(4.2,0),SE);
label("$P$",(1,0.5),NW);
label("$Q$"... | [] |
914 | Five rectangles, $A$, $B$, $C$, $D$, and $E$, are arranged in a square as shown below. These rectangles have dimensions $1\times6$, $2\times 4$, $5\times6$, $2\times7$, and $2\times3$, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?
$\textbf{(A) }A\qquad\... | 2022 AMC 12A Problem 3 | Note that rectangle $D$ must be on the edge. Without loss of generality, let the top-left rectangle be $D,$ as shown below:
It is clear that $x=1,$ so we can determine Rectangle $A.$
Continuing with a similar process, we can determine Rectangles $C,E,$ and $B,$ in this order. The answer is $\boxed{\textbf{(B) }B}$ as... | // Block 1
size(175);
currentpen = black+1.25bp;
fill((3,2.5)--(3,4.5)--(5.3,4.5)--(5.3,2.5)--cycle,gray);
draw((0,0)--(7,0)--(7,7)--(0,7)--(0,0));
draw((3,0)--(3,4.5));
draw((0,4.5)--(5.3,4.5));
draw((5.3,7)--(5.3,2.5));
draw((7,2.5)--(3,2.5));
label("$7$",midpoint((0,7)--(5.3,7)),N,blue);
label("$x$",midpoint((5.3,7)... | [] |
914 | Five rectangles, $A$, $B$, $C$, $D$, and $E$, are arranged in a square as shown below. These rectangles have dimensions $1\times6$, $2\times 4$, $5\times6$, $2\times7$, and $2\times3$, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?
$\textbf{(A) }A\qquad\... | 2022 AMC 12A Problem 3 | Let's label some points:
By finding the dimensions of the middle rectangle, we need to find the dimensions of the other four rectangles. We have a rule: \[AB + BC = CD + DE = EF + FG = GH + AH.\]
Let's make a list of all dimensions of the rectangles from the diagram. We have to fill in the dimensions from up above, bu... | // Block 1
size(175);
currentpen = black+1.25bp;
fill((3,2.5)--(3,4.5)--(5.3,4.5)--(5.3,2.5)--cycle,gray);
draw((0,0)--(7,0)--(7,7)--(0,7)--(0,0));
draw((3,0)--(3,4.5));
draw((0,4.5)--(5.3,4.5));
draw((5.3,7)--(5.3,2.5));
draw((7,2.5)--(3,2.5));
label("$A$",(0,0),SW);
label("$B$",(3,0),S);
label("$C$",(7,0),SE);
label... | [] |
915 | The taxicab distance between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \[|x_1 - x_2| + |y_1 - y_2|.\]
For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$?
$\textbf{(A)} \, 441 \qquad\textbf{(B)} \, 761 \qquad\text... | 2022 AMC 12A Problem 5 | This solution refers to the Diagram section.
The problem can be visualized as depicted on the right split equally into four "triangular" parts excluding the origin. The "triangular" parts are identical the ones that would be used in a visual proof of the formula for triangular numbers. Becuase of this the number of po... | // Block 1
/* Made by MRENTHUSIASM */
size(350);
for (int y = 20; y >= 1; --y) {
for (int x = 0; x <= 20-y; ++x) {
dot((x,y),green+linewidth(4));
}
}
for (int y = 0; y >= -20; --y) {
for (int x = 1; x <= y+20; ++x) {
dot((x,y),blue+linewidth(4));
}
}
for (int y = 20; y >= 0; --y) {
for (int x =... | [] |
915 | The taxicab distance between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \[|x_1 - x_2| + |y_1 - y_2|.\]
For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$?
$\textbf{(A)} \, 441 \qquad\textbf{(B)} \, 761 \qquad\text... | 2022 AMC 12A Problem 5 | This solution refers to the Diagram section.
As shown below, the taxicab distance between each red point and the origin is even, and the taxicab distance between each blue point and the origin is odd.
Note that the red array consists of $21^2=441$ points, and the blue array consists of $20^2=400$ points.
Together, t... | // Block 1
/* Made by MRENTHUSIASM */
size(350);
for (int y = 20; y >= 0; --y) {
for (int x = y-20; x <= 20-y; x+=2) {
dot((x,y),red+linewidth(4));
}
}
for (int y = -1; y >= -20; --y) {
for (int x = -y-20; x <= y+20; x+=2) {
dot((x,y),red+linewidth(4));
}
}
for (int y = 19; y >= 0; --y) {
for (... | [] |
916 | Let $\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. What integer is closest to the area of $\mathcal{R}$?
$\textb... | 2022 AMC 12A Problem 13 | If $z$ is a complex number and $z = a + bi$, then the magnitude (length) of $z$ is $\sqrt{a^2 + b^2}$. Therefore, $z_1$ has a magnitude of 5. If $z_2$ has a magnitude of at most one, that means for each point on the segment given by $z_1$, the bounds of the region $\mathcal{R}$ could be at most 1 away. Alone the line, ... | // Block 1
size(250);
import TrigMacros;
rr_cartesian_axes(-2,6,-2,6,complexplane=true, usegrid = true);
Label f;
f.p=fontsize(6);
xaxis(-1,5,Ticks(f, 1.0));
yaxis(-1,5,Ticks(f, 1.0));
dot((3,0));
dot((0,4));
draw((0,4)--(3,0), blue);
draw((0.8, 4.6)..(-.6,4.8)..(-.8, 3.4),red);
draw((-.8, 3.4)--(2.2, -0.6), red);
dr... | [] |
917 | Suppose that $13$ cards numbered $1, 2, 3, \ldots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards $1, 2, 3$ are picked up on the first pass, $4$ and $5$ on the second pass, $6$ on the third pass, $7, 8, 9, 10$ on... | 2022 AMC 12A Problem 19 | For $1\leq k\leq 12,$ suppose that cards $1, 2, \ldots, k$ are picked up on the first pass. It follows that cards $k+1,k+2,\ldots,13$ are picked up on the second pass.
Once we pick the spots for the cards on the first pass, there is only one way to arrange all $\boldsymbol{13}$ cards.
For each value of $k,$ there are... | size(11cm); draw((0,0)--(2,0)--(2,3)--(0,3)--cycle); label("1", (1,1.5)); draw((3,0)--(5,0)--(5,3)--(3,3)--cycle); label("2", (4,1.5)); draw((6,0)--(8,0)--(8,3)--(6,3)--cycle); label("3", (7,1.5)); draw((9,0)--(11,0)--(11,3)--(9,3)--cycle); label("4", (10,1.5)); draw((12,0)--(14,0)--(14,3)--(12,3)--cycle); label("5", (... | [] |
918 | Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$
$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }... | 2022 AMC 12A Problem 20 | Consider the reflection $P^{\prime}$ of $P$ over the perpendicular bisector of $\overline{BC}$, creating two new isosceles trapezoids $DAPP^{\prime}$ and $CBPP^{\prime}$. Under this reflection, $P^{\prime}A=PD=4$, $P^{\prime}D=PA=1$, $P^{\prime}C=PB=2$, and $P^{\prime}B=PC=3$.
Since $DAPP'$ and $CBPP'$ are isosceles ... | size(300); pair A = (0,0); pair B = (1, 2); pair C = (2,2); pair D = (3,0); label("$A$", A, SW); label("$B$", B, NW); label("$C$", C, NE); label("$D$", D, SE); draw(A--B--C--D--cycle, blue); pair P = (0.8, 0.6); dot("$P$", P, NW); draw(P--A, magenta); draw(P--B, magenta); draw(P--C); draw(P--D); label("$1$", P--A, NW);... | [] |
918 | Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$
$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }... | 2022 AMC 12A Problem 20 | Extend $AB$ and $CD$ to a point $Q$ as shown, and let $PQ = s$. Then let $BQ=CQ = a$ and $AB=DC = b$. Notice that $\frac{BC}{AD} = \frac{QC}{QD} = \frac{a}{a+b}$ by similar triangles.
By Stewart's theorem on $APQ$ and $DPQ$, we have \begin{align*} ab(a+b) + 9(a+b) &= 16a + s^2b \\ ab(a+b) + 4(a+b) &= a + s^2b \\ \end{... | size(7.5cm); draw((0,0)--(4.2,0)); draw((0,0)--(1.4,2)--(2.8,2)--(4.2,0)); draw((1.4,2)--(2.1,3)--(2.8,2)); draw((0,0)--(1,0.5)--(1.4,2)--(1,0.5)--(2.8,2)--(1,0.5)--(4.2,0)); label("$A$",(0,0),SW); label("$B$",(1.4,2),NW); label("$C$",(2.8,2),NE); label("$D$",(4.2,0),SE); label("$P$",(1,0.5),NW); label("$Q$",(2.1,3),N)... | [] |
919 | Let $c$ be a real number, and let $z_1$ and $z_2$ be the two complex numbers satisfying the equation
$z^2 - cz + 10 = 0$. Points $z_1$, $z_2$, $\frac{1}{z_1}$, and $\frac{1}{z_2}$ are the vertices of (convex) quadrilateral $\mathcal{Q}$ in the complex plane. When the area of $\mathcal{Q}$ obtains its maximum possible v... | 2022 AMC 12A Problem 22 | Note that $z=\dfrac c2\pm\dfrac{\sqrt{c^2-40}}2$, so let $z_1=\dfrac c2+\dfrac{\sqrt{c^2-40}}2$ and $z_2=\dfrac c2-\dfrac{\sqrt{c^2-40}}2$. Taking a look at the answer choices, they range between $c=4.5$ to $c=6.5$, and in that range, $c^2$ is always less than $40$. Thus, $c^2-40<0$ for our possible answer choices; we ... | // Block 1
import graph;
unitsize(0.5cm);
/*xaxis(Ticks, xmin=-1,xmax=8);
yaxis(Ticks, ymin=-11,ymax=11);*/
draw((-1,0)--(8,0));
draw((0,-11)--(0,11));
dot((7,10));dot((7,-10));dot((0.7,1));dot((0.7,-1));
draw((7,10)--(7,-10)--(0.7,-1)--(0.7,1)--cycle);
label("$z_1$", (7,10), E);
label("$z_2$", (7,-10), E);
label("... | [] |
920 | The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
$\textbf{(A) }15\dfrac{1}{8} \qquad \textbf{(B) }15\dfra... | 2022 AMC 10B Problem 16 | Let us label the points on the diagram.
By doing some angle chasing using the fact that $\angle ACE$ and $\angle CEG$ are right angles, we find that $\angle BAC = \angle DCE = \angle FEG$. Similarly, $\angle ACB = \angle CED = \angle EGF$. Therefore, $\triangle ABC \sim \triangle CDE \sim \triangle EFG$.
As we are ... | // Block 1
import olympiad;
size(200);
defaultpen(linewidth(1) + fontsize(10));
pair A = (0,0), B = (1,0), C = (4,0), D = (8,0), K = (0,4), F = (1,4), G = (7.25, 4), H = (8, 4), I = (8,3), J = (5, 7);
fill(F--G--I--C--F--cycle, grey);
markscalefactor=0.05;
draw(A--D--H--K--A^^B--F^^F--C--I--J--F^^rightanglemark(F,J,I)^... | [] |
920 | The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
$\textbf{(A) }15\dfrac{1}{8} \qquad \textbf{(B) }15\dfra... | 2022 AMC 10B Problem 16 | Same diagram as Solution 1, but added point $H$, which is $(4,7)$. I also renamed all the points to form coordinates using $B$ as the origin.
In order to find the area, point $G$'s coordinates must be found. Notice how $EH$ and $AG$ intercept at point $G$. This means that we need to find the equations for $EH$ and $... | // Block 1
import olympiad;
size(200);
defaultpen(linewidth(1) + fontsize(10));
pair A = (0,0), B = (1,0), C = (4,0), D = (8,0), K = (0,4), F = (1,4), G = (7.25, 4), H = (8, 4), I = (8,3), J = (5, 7);
fill(F--G--I--C--F--cycle, grey);
markscalefactor=0.05;
draw(A--D--H--K--A^^B--F^^F--C--I--J--F^^rightanglemark(F,J,I)^... | [] |
921 | Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:
Any filled square with two or three filled neighbors remains filled.
Any empty square with exactly t... | 2022 AMC 10B Problem 19 | There are two cases for the initial configuration:
The center square is filled.
Exactly two of the eight adjacent neighboring squares of the center are filled. Clearly, the only possibility is that the squares along one diagonal are filled, as shown below:
In this case, there are $2$ possible initial configurations... | // Block 1
import geometry;
unitsize(0.6cm);
void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,invisible);
}
ds((1,3));
ds((2,2));
ds((3,1));
for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
}
label("$2$ Configurations", (2.5,-1));
// Block 2
impor... | [] |
922 | Let $ABCD$ be a rhombus with $\angle ADC = 46^\circ$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point
on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle BFC$?
$\textbf{(A)}\ 110 \qquad\textbf{(B)}\ 111 \qquad\textbf{(C)}\ 112 \qq... | 2022 AMC 10B Problem 20 | This solution refers to the Diagram section.
We extend $AD$ and $BE$ to point $G$, as shown below:
We know that $AB=AD=2$ and $CE=DE=1$.
By AA Similarity, $\triangle ABG \sim \triangle DEG$ with a ratio of $2:1$. This implies that $2AD=AG$ and $AD \cong DG$, so $AG=2AD=2\cdot2=4$. That is, $D$ is the midpoint of $A... | // Block 1
/*
Made by ghfhgvghj10
Edited by MRENTHUSIASM
*/
size(300);
pair A, B, C, D, E, F, G;
D = origin;
A = 6*dir(46);
C = (6,0);
B = C + (A-D);
E = midpoint(C--D);
F = foot(A,B,E);
G = 6*dir(226);
dot("$A$",A,1.5*NW,linewidth(5));
dot("$B$",B,1.5*NE,linewidth(5));
dot("$C$",C,1.5*SE,linewidth(5));
dot("$D$",D,1.5... | [] |
923 | Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?
$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi... | 2022 AMC 10B Problem 22 | The circles match up as follows: Case $1$ is brown, Case $2$ is blue, Case $3$ is green, and Case 4 is red.
Let $x^2 + y^2 = 64$ be circle $O$, $x^2 + y^2 = 4$ be circle $P$, and $(x-5)^2 + y^2 = 3$ be circle $Q$.
All the circles in S are internally tangent to circle $O$.
There are four cases with two circles belonging... | // Block 1
import geometry;
unitsize(0.5cm);
void dc(pair x, pen p) {
pair y = intersectionpoints(circle((0,0),8),(0,0)--1000*x)[0];
draw(circle(x, abs(x-y)),p+linewidth(2));
}
pair O1 = (0,0),O2=(5,0),P1=intersectionpoints(circle(O1,5),circle(O2,3+sqrt(3)))[0],P2=interse... | [] |
924 | Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $... | 2022 AMC 10B Problem 23 | Let $x$ and $y$ be random variables that are independently and uniformly distributed in the interval $(0,1).$ Note that \[P(x+y\leq 1)=\frac{\frac12\cdot1^2}{1^2}=\frac12,\] as shown below:
Let $x,y,$ and $z$ be random variables that are independently and uniformly distributed in the interval $(0,1).$ Note that \[P(x+... | // Block 1
/* Made by MRENTHUSIASM */
size(200);
real xMin = -0.25;
real xMax = 1.25;
real yMin = -0.25;
real yMax = 1.25;
//Draws the horizontal ticks
void horizontalTicks()
{
for (real i = 1; i < yMax; ++i)
{
draw((-1/32,i)--(1/32,i), black+linewidth(1));
}
}
//Draws the vertical ticks
void verticalTick... | [] |
925 | The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
$\textbf{(A) }15\dfrac{1}{8} \qquad \textbf{(B) }15\dfra... | 2022 AMC 12B Problem 13 | Let us label the points on the diagram.
By doing some angle chasing using the fact that $\angle ACE$ and $\angle CEG$ are right angles, we find that $\angle BAC = \angle DCE = \angle FEG$. Similarly, $\angle ACB = \angle CED = \angle EGF$. Therefore, $\triangle ABC \sim \triangle CDE \sim \triangle EFG$.
As we are ... | import olympiad; size(200); defaultpen(linewidth(1) + fontsize(10)); pair A = (0,0), B = (1,0), C = (4,0), D = (8,0), K = (0,4), F = (1,4), G = (7.25, 4), H = (8, 4), I = (8,3), J = (5, 7); fill(F--G--I--C--F--cycle, grey); markscalefactor=0.05; draw(A--D--H--K--A^^B--F^^F--C--I--J--F^^rightanglemark(F,J,I)^^rightangle... | [] |
925 | The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
$\textbf{(A) }15\dfrac{1}{8} \qquad \textbf{(B) }15\dfra... | 2022 AMC 12B Problem 13 | Same diagram as Solution 1, but added point $H$, which is $(4,7)$. I also renamed all the points to form coordinates using $B$ as the origin.
In order to find the area, point $G$'s coordinates must be found. Notice how $EH$ and $AG$ intercept at point $G$. This means that we need to find the equations for $EH$ and $... | import olympiad; size(200); defaultpen(linewidth(1) + fontsize(10)); pair A = (0,0), B = (1,0), C = (4,0), D = (8,0), K = (0,4), F = (1,4), G = (7.25, 4), H = (8, 4), I = (8,3), J = (5, 7); fill(F--G--I--C--F--cycle, grey); markscalefactor=0.05; draw(A--D--H--K--A^^B--F^^F--C--I--J--F^^rightanglemark(F,J,I)^^rightangle... | [] |
926 | The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is $\tan(\angle ABC)$?
$\textbf{(A)}\ \frac{1}{7} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{3}{7} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{4}{7} \qquad$ | 2022 AMC 12B Problem 14 | From $x^2 + 2x - 15 = (x-3)(x+5)$, we may assume, without loss of generality, that $x$-intercepts of the given parabola are $A(-5,0)$ and $C(3,0)$. And, point $B$ has coordinates $(0,-15)$. Consider complex numbers $z = 3 + i$ and $w = 5 + i$ whose arguments are $\theta \coloneqq \angle OBA$ and $\phi \coloneqq \angle ... | // Block 1
/* Made by MRENTHUSIASM */
size(300);
real xMin = -15;
real xMax = 15;
real yMin = -17;
real yMax = 17;
draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5));
draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5));
label("$x$",(xMax,0),(2,0));
label("$y$",(0,yMax),(0,2));
real f(real x) { return x... | [] |
927 | Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:
Any filled square with two or three filled neighbors remains filled.
Any empty square with exactly t... | 2022 AMC 12B Problem 18 | There are two cases for the initial configuration:
The center square is filled.
Exactly two of the eight adjacent neighboring squares of the center are filled. Clearly, the only possibility is that the squares along one diagonal are filled, as shown below:
In this case, there are $2$ possible initial configurations... | // Block 1
import geometry; unitsize(0.6cm); void ds(pair x) { filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,invisible); } ds((1,3)); ds((2,2)); ds((3,1)); for (int i = 0; i <= 5; ++i) { draw((0,i)--(5,i)); draw((i,0)--(i,5)); } label("$2$ Configurations", (2.5,-1));
// Block 2
impor... | [] |
928 | Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?
$\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi... | 2022 AMC 12B Problem 21 | The circles match up as follows: Case $1$ is brown, Case $2$ is blue, Case $3$ is green, and Case 4 is red.
Let $x^2 + y^2 = 64$ be circle $O$, $x^2 + y^2 = 4$ be circle $P$, and $(x-5)^2 + y^2 = 3$ be circle $Q$.
All the circles in S are internally tangent to circle $O$.
There are four cases with two circles belonging... | import geometry; unitsize(0.5cm); void dc(pair x, pen p) { pair y = intersectionpoints(circle((0,0),8),(0,0)--1000*x)[0]; draw(circle(x, abs(x-y)),p+linewidth(2)); } pair O1 = (0,0),O2=(5,0),P1=intersectionpoints(circle(O1,5),circle(O2,3+sqrt(3)))[0],P2=intersectionpoints... | [] |
929 | Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $... | 2022 AMC 12B Problem 22 | Let $x$ and $y$ be random variables that are independently and uniformly distributed in the interval $(0,1).$ Note that \[P(x+y\leq 1)=\frac{\frac12\cdot1^2}{1^2}=\frac12,\] as shown below:
Let $x,y,$ and $z$ be random variables that are independently and uniformly distributed in the interval $(0,1).$ Note that \[P(x+... | // Block 1
/* Made by MRENTHUSIASM */ size(200); real xMin = -0.25; real xMax = 1.25; real yMin = -0.25; real yMax = 1.25; //Draws the horizontal ticks void horizontalTicks() { for (real i = 1; i < yMax; ++i) { draw((-1/32,i)--(1/32,i), black+linewidth(1)); } } //Draws the vertical ticks void verticalTick... | [] |
930 | Four regular hexagons surround a square with side length 1, each one sharing an edge with the square,
as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be
written as $m \sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime.
Wha... | 2022 AMC 12B Problem 25 | Refer to the diagram above.
Let the origin be at the center of the square, $A$ be the intersection of the top and right hexagons, $B$ be the intersection of the top and left hexagons, and $M$ and $N$ be the top points in the diagram.
By symmetry, $A$ lies on the line $y = x$. The equation of line $AN$ is $y = -x\sqrt... | // Block 1
import geometry;
unitsize(3cm);
draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle);
draw(shift((1/2,1-sqrt(3)/2))*polygon(6));
draw(shift((1/2,sqrt(3)/2))*polygon(6));
draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6));
draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*pol... | [] |
930 | Four regular hexagons surround a square with side length 1, each one sharing an edge with the square,
as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be
written as $m \sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime.
Wha... | 2022 AMC 12B Problem 25 | Begin by dividing the figure as shown above. Clearly, the entire figure has 8-fold symmetry. Therefore, we can calculate the area of $ODBA$ and multiply it by 8. We split $[ODBA]$ into $[ODBC]+[ABC]$.
Knowing the side length of the hexagon is $1$, we can use 30-60-90 triangles within the hexagon to find the total dist... | // Block 1
import geometry;
unitsize(3cm);
draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle);
draw(shift((1/2,1-sqrt(3)/2))*polygon(6));
draw(shift((1/2,sqrt(3)/2))*polygon(6));
draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6));
draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*pol... | [] |
930 | Four regular hexagons surround a square with side length 1, each one sharing an edge with the square,
as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be
written as $m \sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime.
Wha... | 2022 AMC 12B Problem 25 | Divide this polygon into a 'cross' and $8$ triangles like $\triangle ABC$. $BD = 2 \cdot \frac{ \sqrt{3} }{2} = \sqrt{3}$, $BE = 2BD - CD = 2\sqrt{3} - 1$, $EF = 1$, $[BEFG] = 1 \cdot (2\sqrt{3} - 1) = 2\sqrt{3} - 1$, $[\text{cross}] = 2\cdot [BEFG] - 1 = 2(2\sqrt{3} - 1)- 1 = 4\sqrt{3} - 3$
[Image: images/amc/2022_AM... | // Block 1
import geometry;
unitsize(3cm);
draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle);
draw(shift((1/2,1-sqrt(3)/2))*polygon(6));
draw(shift((1/2,sqrt(3)/2))*polygon(6));
draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6));
draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*pol... | ["https://artofproblemsolving.com/wiki/images/thumb/3/39/2022AMC12BP25.png/150px-2022AMC12BP25.png"] |
930 | Four regular hexagons surround a square with side length 1, each one sharing an edge with the square,
as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be
written as $m \sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime.
Wha... | 2022 AMC 12B Problem 25 | We calculate the area as the area of the red octagon minus the four purple congruent triangles:
We first find the important angles in the figure. We note that 2 adjacent hexagons are rotated $90^\circ$ with respect to the other, so the angles between any sides is $150^\circ$. In particular, as the purple triangles are... | // Block 1
import geometry;
unitsize(3cm);
draw((1-sqrt(3),1-sqrt(3))--(1-sqrt(3),sqrt(3))--(sqrt(3),sqrt(3))--(sqrt(3),1-sqrt(3))--cycle,dashed);
filldraw((0,1-sqrt(3))--(1,1-sqrt(3))--(sqrt(3),0)--(sqrt(3),1)--(1,sqrt(3))--(0,sqrt(3))--(1-sqrt(3),1)--(1-sqrt(3),0)--cycle,red*0.2+white,red);
filldr... | [] |
930 | Four regular hexagons surround a square with side length 1, each one sharing an edge with the square,
as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be
written as $m \sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime.
Wha... | 2022 AMC 12B Problem 25 | [Image: images/amc/2022_AMC_12B_Problem_25_0.jpeg]
Note that each of the green sections is a rectangle, so its interior angles are all $90^{\circ}.$ Since $\angle{ABC}=120^{\circ}$, every one of the orange sections is a $30-60-90$ right triangle.
Define $x$ to be the distance from the corner of the square with side l... | // Block 1
import geometry;
import fontsize;
defaultpen(fontsize(9pt));
unitsize(3cm);
draw(shift((1/2,1-sqrt(3)/2))*polygon(6));
draw((1,1-sqrt(3))--(1, 1)--cycle,linewidth(1));
draw((3/2,1-sqrt(3)/2)--(1, 1-sqrt(3)/2)--cycle,linewidth(1));
draw((1, 1-sqrt(3)/2)-... | ["https://artofproblemsolving.com/wiki/images/thumb/f/fb/2022AMC12B.jpeg/509px-2022AMC12B.jpeg"] |
930 | Four regular hexagons surround a square with side length 1, each one sharing an edge with the square,
as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be
written as $m \sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime.
Wha... | 2022 AMC 12B Problem 25 | Obviously, each of the hexagon have area of $\frac{3\sqrt{3}}{2}$, which gives a total of $6\sqrt{3}$ (without accounting for the overcounted overlapped areas). Now we calculate the area of the intersection of two hexagons. There are two cases :
Case 1: The hexagons are on the opposite sides
In this case, we see t... | // Block 1
pair[] A,B;
path h1, h2,h3,h4;
h1 = shift((sqrt(3)-1)/2*dir(90))*polygon(6);
h2 = shift((sqrt(3)-1)/2*dir(0))*rotate(90)*polygon(6);
h3 = shift((sqrt(3)-1)/2*dir(-90))*polygon(6);
h4 = shift((sqrt(3)-1)/2*dir(180))*rotate(90)*polygon(6);
B.cyclic=true;
B.push((0,0));
int indexSquare[] = sequence(4);
int inde... | [] |
931 | A quadrilateral has all integer sides lengths, a perimeter of $26$, and one side of length $4$. What is the greatest possible length of one side of this quadrilateral?
$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad\textbf{(E) }13$ | 2023 AMC 10A Problem 4 | This is an AMC 10 problem 4, so there is no need for any complex formulas. The largest singular side length from a quadrilateral comes from a trapezoid. So we can set the $2$ sides of the trapezoid equal to $4$. Next we can split the trapezoid into $5$ triangles, where each base length of the triangle equals $4$. So th... | // Block 1
size(120);
draw((0,0)--(2,6),red);
draw((2,6)--(5,6),red);
draw((5,6)--(7,0),red);
draw((0,0)--(7,0),red);
draw((2,6)--(2.67,0),red);
draw((2.67,0)--(3.5,6),red);
draw((3.5,6)--(4.67,0),red);
draw((4.67,0)--(5,6),red);
// Block 2
size(120); draw((0,0)--(2,6),red); draw((2,6)--(5,6),red); draw((5,6)--(7,0),re... | [] |
932 | Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$. Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}, \triangle{PCQ},$ and $\triangle{QDA}$ have integer lengths. What is the perimeter of $\triangle{APQ}$?
$\textbf{(A) } 84 \qquad \textbf{(B) } 86 \qquad... | 2023 AMC 10A Problem 17 | We know that all side lengths are integers, so we can test Pythagorean triples for all triangles.
First, we focus on $\triangle{ABP}$. The length of $AB$ is $30$, and the possible (small enough) Pythagorean triples $\triangle{ABP}$ can be are $(3, 4, 5), (5, 12, 13), (8, 15, 17),$ where the length of the longer leg i... | // Block 1
/* ~ItsMeNoobieboy */
size(200);
pair A, B, C, D, P, Q;
A = (0,28/30);
B = (1,28/30);
C = (1,0);
D = (0,0);
P = (1,12/30);
Q = (21/30,0);
draw(A--B--C--D--cycle);
draw(A--P--Q--cycle);
dot("$A$",A,NW,linewidth(4));
dot("$B$",B,NE,linewidth(4));
dot("$C$",C,SE,linewidth(4));
dot("$D$",D,SW,linewidth(4));
dot(... | [] |
933 | The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$. What is $|r-s|$?
$\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{2}{3} \qquad \textbf{(E) } 1$ | 2023 AMC 10A Problem 19 | To find the center of rotation, we find the intersection point of the perpendicular bisectors of $\overline{AA^\prime}$ and $\overline{BB^\prime}$. (this is because $\overline{PA}$ is equal to $\overline{PA^\prime}$ and same with $B$ and $B^\prime$
We can find that the equation of the line $\overline{AA^\prime}$ is $y... | // Block 1
pair A=(1,2);
pair B=(3,3);
pair A1=(3,1);
pair B1=(4,3);
dot("A",A,NW);
dot("B",B,S);
dot("A'",A1,S);
dot("B'",B1,E);
draw(A--A1);
draw(B--B1);
draw((3.5,0)--(3.5,6),BeginArrow(5),EndArrow(5));
draw((1,-0.5)--(4,5.5),BeginArrow(5),EndArrow(5));
pair P=(3.5,4.5);
dot("P",P,NW);
// Block 2
pair A=(1,2); pair ... | [] |
934 | Each square in a $3\times3$ grid of squares is colored red, white, blue, or green so that every $2\times2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?
$\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }... | 2023 AMC 10A Problem 20 | We can split this problem into $2$ cases as shown in solution 1. We can swap a set of equal colors for another set of equal colors to create a new square.
Square 1:
The first square can be rotated to create another square so we have to multiply the number of arrangements by $2$. We have $4! = 24$ arrangements without ... | // Block 1
unitsize(0.5cm, 0.5cm);
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((0,3)--(9,3));
draw((0,6)--(9,6));
draw((3,0)--(3,9));
draw((6,0)--(6,9));
label("R", (1.5,1.5));
label("B", (4.5,1.5));
label("R", (7.5,1.5));
label("G", (1.5,4.5));
label("W", (4.5,4.5));
label("G", (7.5,4.5));
label("B", (1.5,7.5));
... | [] |
934 | Each square in a $3\times3$ grid of squares is colored red, white, blue, or green so that every $2\times2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?
$\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }... | 2023 AMC 10A Problem 20 | Note that we could fill the 3 by 3 square with numbers of choices, rather than letters or color names. The top-left corner receives a 3 because there are 3 total choices to choose from: R, G and B. The squares bordering them has values of 2 and 1, regardless of order. 2 indicates that the small square can have any colo... | // Block 1
unitsize(0.5cm, 0.5cm);
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((0,3)--(9,3));
draw((0,6)--(9,6));
draw((3,0)--(3,9));
draw((6,0)--(6,9));
label("2", (1.5,1.5));
label("1", (4.5,1.5));
label("1", (7.5,1.5));
label("2", (1.5,4.5));
label("3", (4.5,4.5));
label("1", (7.5,4.5));
label("3", (1.5,7.5));
... | [] |
934 | Each square in a $3\times3$ grid of squares is colored red, white, blue, or green so that every $2\times2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?
$\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }... | 2023 AMC 10A Problem 20 | Let's name the cells $A,B,C,D,E,F,G,H,I$ from the top left to the bottom right.
Case 1. Cell $B$ and cell $H$ have the same color. The middle one cell $E$ has $4$ choices, cell $B$ has $3$ choices, then cell $E$ has $2$ choices and cell $F$ has $2$ choices, this gives $4\cdot 3\cdot 2\cdot 2=48$ ways.
Case 2. Cell ... | // Block 1
unitsize(0.5cm, 0.5cm);
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((0,3)--(9,3));
draw((0,6)--(9,6));
draw((3,0)--(3,9));
draw((6,0)--(6,9));
label("G", (1.5,1.5));
label("H", (4.5,1.5));
label("I", (7.5,1.5));
label("D", (1.5,4.5));
label("E", (4.5,4.5));
label("F", (7.5,4.5));
label("A", (1.5,7.5));
... | [] |
935 | Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$?
$\textbf{(A) }... | 2023 AMC 10A Problem 22 | Let $O$ be the center of the midpoint of the line segment connecting both the centers, say $A$ and $B$.
Let the point of tangency with the inscribed circle and the right larger circles be $T$.
Then $OT = BO + BT = BO + AT - \frac{1}{2} = \frac{1}{4} + 1 - \frac{1}{2} = \frac{3}{4}.$
Since $C_4$ is internally tangen... | import olympiad; size(10cm); draw(circle((0,0),0.75), gray(0.7)); draw(circle((-0.25,0),1), gray(0.7)); draw(circle((0.25,0),1), gray(0.7)); draw(circle((0,6/7),3/28), gray(0.7)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), EE = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118), G = (0,... | [] |
936 | Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is $\frac{3}{7}$ unit. What i... | 2023 AMC 10A Problem 24 | Examining the red isosceles trapezoid with $1$ and $\dfrac{3}{7}$ as two bases, we know that the side lengths are $\dfrac{4}{7}$ from $30-60-90$ triangle.
We can conclude that the big hexagon has side length 3.
Thus the target area is: area of the big hexagon - 6 * area of the small hexagon.
$\dfrac{3\sqrt{3}}{2}(3^2... | // Block 1
unitsize(1cm);
pair A, B, C, D, E, F, W,X,Y,Z;
real bigSide = 3;
real smallSide = 1;
real angle = 60; // Each external angle for the hexagon
real offset = 3/7; // Offset for the smaller hexagons
// Function to draw a hexagon given a starting point and side length
void drawHexagon(pair start, real side) {
... | [] |
936 | Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is $\frac{3}{7}$ unit. What i... | 2023 AMC 10A Problem 24 | Note that one can "slide' the small hexagons along their respective edges, and either by sliding them to the center or to the corners, and thus getting that the side length of the larger hexagon is 3. The rest proceeds the same as solution 1.
Solution 2.1 (Clarification)
Notice that when sliding the smaller hexagon al... | // Block 1
unitsize(1cm); draw(scale(3)*polygon(6)); filldraw(shift(dir(0)*2+dir(120)*0.4)*polygon(6), pink); filldraw(shift(dir(60)*2+dir(180)*0.4)*polygon(6), palered); filldraw(shift(dir(120)*2+dir(240)*0.4)*polygon(6), paleyellow); filldraw(shift(dir(180)*2+dir(300)*0.4)*polygon(6), palegreen); filldraw(shift(dir(2... | [] |
936 | Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is $\frac{3}{7}$ unit. What i... | 2023 AMC 10A Problem 24 | $(3^2-6\times 1^2)\times\frac{3\sqrt{3}}{2}=(9-6\times 1)\times\frac{3\sqrt{3}}{2}=(9-6)\times\frac{3\sqrt{3}}{2}=3\times\frac{3\sqrt{3}}{2}=\boxed{\textbf{(C) } \frac{9 \sqrt{3}}{2}}$
~~By afly
~(very minor change by Marshall_Huang) | // Block 1
unitsize(1cm);
unitsize(5cm);
draw(scale(3)*polygon(6));
filldraw(shift(dir(0)*2+dir(120)*3/7)*polygon(6), lightgray);
filldraw(shift(dir(60)*2+dir(180)*3/7)*polygon(6), lightgray);
filldraw(shift(dir(120)*2+dir(240)*3/7)*polygon(6), lightgray);
filldraw(shift(dir(180)*2+dir(300)*3/7)*polygon(6), lightgray);... | [] |
937 | Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta = \angle PAB=\angle QPC=\angle RQB=\cdots$... | 2023 AMC 12A Problem 15 | By "unfolding" $APQRS$ into a straight line, we get a right angled triangle $ABS'$.
It follows that
\begin{align*} \cos(\theta)&=\frac{100}{120} \\ \theta&=\boxed{\textbf{(A) } \arccos\left(\frac{5}{6}\right)}. \end{align*}
~lptoggled | // Block 1
import olympiad;
draw((-50,15)--(50,15));
draw((50,15)--(50,-15));
draw((50,-15)--(-50,-15));
draw((-50,-15)--(-50,15));
draw((-50,-15)--(-22.5,15));
draw((-22.5,15)--(5,-15));
draw((5,-15)--(32.5,15));
draw((32.5,15)--(50,-4.090909090909));
label("$\theta$", (-41.5,-10.5));
label("$\theta$", (-13,10.5));
la... | [] |
938 | Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)}~20\qquad\textbf{(B)}~21\qquad\textbf{(C)}~22\qquad\textbf{(D)}~23\q... | 2023 AMC 12A Problem 16 | We are given that $1+z+z^2=c$ where $c$ is some complex number with magnitude $4$. Rearranging the quadratic to standard form and applying the quadratic formula, we have
\[z=\frac{-1\pm \sqrt{1^2-4(1)(1-c)}}{2}=\frac{-1\pm\sqrt{4c-3}}{2}.\]
The imaginary part of $z$ is maximized when $c=-4$. (Why? See note below.)
Th... | // Block 1
size(250);
import TrigMacros;
rr_cartesian_axes(-22,19,-19,19,complexplane=true, usegrid = false);
Label f;
f.p=fontsize(4);
dot((-3, 0), red);
label("$(-3, 0)$", (-3, 0), NW);
dot((0,0));
draw(circle((-3, 0), 16), red);
dot((-19, 0), blue);
label("$w(-19, 0)$", (-19, 0), NW);
// Block 2
size(250); import T... | [] |
938 | Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)}~20\qquad\textbf{(B)}~21\qquad\textbf{(C)}~22\qquad\textbf{(D)}~23\q... | 2023 AMC 12A Problem 16 | We can write the given condition as \[\left|\left(z+\frac{1}{2}\right)^2 + \frac{3}{4}\right| = 4.\]
Letting $u = \left(z+\frac{1}{2}\right)^2$, the equation $\left|u + \frac{3}{4}\right| = 4$ equates to the circle centered at $-\frac{3}{4}$ with radius $4$ in the complex plane, call it $\omega$. Thus the locus of $\le... | // Block 1
size(250);
import TrigMacros;
rr_cartesian_axes(-6,5,-5,5,complexplane=true, usegrid = true);
Label f;
f.p=fontsize(6);
xaxis(-6,5,Ticks(f, 1.0));
yaxis(-5,5,Ticks(f, 1.0));
dot((0,0));
draw(circle((-3/4, 0), 4), red + dashed);
dot((-19/4, 0), blue);
label("$\phi$", (-19/4, 0), NW);
dot((0, 2.18), blue);
l... | [] |
939 | Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$?
$\textbf{(A) }... | 2023 AMC 12A Problem 18 | Let $O$ be the center of the midpoint of the line segment connecting both the centers, say $A$ and $B$.
Let the point of tangency with the inscribed circle and the right larger circles be $T$.
Then $OT = BO + BT = BO + AT - \frac{1}{2} = \frac{1}{4} + 1 - \frac{1}{2} = \frac{3}{4}.$
Since $C_4$ is internally tangen... | // Block 1
import olympiad;
size(10cm);
draw(circle((0,0),0.75), gray(0.7));
draw(circle((-0.25,0),1), gray(0.7));
draw(circle((0.25,0),1), gray(0.7));
draw(circle((0,6/7),3/28), gray(0.7));
pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), EE = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.7071067811... | [] |
940 | Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below.
Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digits of the sum of the 2023 numbers... | 2023 AMC 12A Problem 20 | Consider Pascal's triangle as the starting point. In the Pascal's triangle depicted below, the sum of the numbers in the $n$th row is $2^{(n-1)}$. For the 2023rd row in the Pascal's triangle, the sum of numbers is $2^{2022}$.
For the triangular array of integers in the problem, 1 is added to each interior entry, whi... | // Block 1
size(4.5cm);
label("$1$", (0,0));
label("$1$", (-0.5,-2/3));
label("$1$", (0.5,-2/3));
label("$1$", (-1,-4/3));
label("$2$", (0,-4/3));
label("$1$", (1,-4/3));
label("$1$", (-1.5,-2));
label("$3$", (-0.5,-2));
label("$3$", (0.5,-2));
label("$1$", (1.5,-2));
label("$1$", (-2,-8/3));
label("$4$", (-1,-8/3));
l... | [] |
940 | Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below.
Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digits of the sum of the 2023 numbers... | 2023 AMC 12A Problem 20 | The triangle provided by the question may be rewritten with Pascal's Triangle being the starting point.
The numbers in Pascal's Triangle could be written with combinations. Moreover, the following equation is true.
\[\sum_{k=0}^{n}{}_nC_k=2^n\]
Therefore, the sum of every number in $n^{th}$ row of a Pascal's Trian... | // Block 1
usepackage("color");
texpreamble("\usepackage{color}");
size(6cm); // Reduced from 8cm
// Reduced spacing
real dx = 1.2;
real dy = 0.6;
// Level 0
label(Label("$1$", align=Center), (0, 0));
// Level 1
label(Label("$1$", align=Center), (-dx/2, -dy));
label(Label("$1$", align=Center), ( dx/2, -dy));
// Le... | [] |
941 | You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a ... | 2023 AMC 10B Problem 10 | Notice that the $3\times3$ square grid has a total of $12$ possible $2\times1$ rectangles.
Suppose you choose the middle square for one of your turns. The middle square is covered by $4$ rectangles, and each of the remaining $8$ squares is covered by a maximum of $2$ uncounted rectangles. This means that the number of... | draw((0,0)--(0.5,0)--(0.5,0.5)--(0,0.5)--(0,0)); draw((0,1)--(0.5,1)--(0.5,1.5)--(0,1.5)--(0,1)); draw((0.5,0.5)--(1,0.5)--(1,1)--(0.5,1)--(0.5,0.5)); draw((1,0)--(1.5,0)--(1.5,0.5)--(1,0.5)--(1,0)); draw((1,1)--(1.5,1)--(1.5,1.5)--(1,1.5)--(1,1)); draw((0,0.5)--(0.5,0.5)--(0.5,1)--(0,1)--(0,0.5)); draw((0.5,0)--(1,0)-... | [] |
941 | You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a ... | 2023 AMC 10B Problem 10 | The $3 \times 3$ grid can be colored like a checkerboard with alternating black and white squares.
Let the top left square be white, and the rest of the squares alternate colors, as shown below:
Each $2 \times 1$ rectangle always covers $1$ white square and $1$ black square.
You can ensure that at least one of your gu... | /* Diagram by MRENTHUSIASM */ import olympiad; unitsize(25); fill((1,2)--(1,3)--(2,3)--(2,2)--cycle, mediumgray); fill((1,0)--(1,1)--(2,1)--(2,0)--cycle, mediumgray); fill((2,1)--(2,2)--(3,2)--(3,1)--cycle, mediumgray); fill((0,1)--(0,2)--(1,2)--(1,1)--cycle, mediumgray); for (int i = 0; i < 4; ++i) { draw((i,0)--(i,3)... | [] |
942 | 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 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 ... | 2023 AMC 10B Problem 17 | We can create three equations using the given information.
\[4a+4b+4c = 13\]
\[2ab+2ac+2bc=\frac{11}{2}\]
\[abc=\frac{1}{2}\]
We also know that we want $\sqrt{a^2 + b^2 + c^2}$ because that is the length that can be found from using the Pythagorean Theorem. We cleverly notice that $a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab+a... | import geometry; pair A = (-3, 4); pair B = (-3, 5); pair C = (-1, 4); pair D = (-1, 5); pair AA = (0, 0); pair BB = (0, 1); pair CC = (2, 0); pair DD = (2, 1); draw(D--AA,dashed); draw(A--B); draw(A--C); draw(B--D); draw(C--D); draw(A--AA); draw(B--BB); draw(C--CC); draw(D--DD); // Dotted vertices dot(A); dot... | [] |
943 | Sonya the frog chooses a point uniformly at random lying within the square
$[0, 6]$ $\times$ $[0, 6]$ in the coordinate plane and hops to that point. She then randomly
chooses a distance uniformly at random from $[0, 1]$ and a direction uniformly at
random from {north, south, east, west}. All of her choices are indepen... | 2023 AMC 10B Problem 19 | WLOG, we assume Sonya jumps $0.5$ units every time, since that is her expected value.
If Sonya is within $0.5$ blocks of an edge, she can jump off the board. Let us examine the region that is at most $0.5$ blocks from exactly one edge.
If Sonya starts in this region, she has a $\dfrac14$ chance of landing outside... | // Block 1
import graph;
Label f;
xaxis(0,6,Ticks(f, 6.0, 0.5));
yaxis(0,6,Ticks(f, 6.0, 0.5));
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
filldraw((0,0.5)--(0.5,0.5)--(0.5,5.5)--(0,5.5)--cycle,gray);
filldraw((0.5,0)--(0.5,0.5)--(5.5,0.5)--(5.5,0)--cycle,gray);
filldraw((6,0.5)--(5.5,0.5)--(5.5,5.5)--(6,5.5)--cycle,gr... | [] |
943 | Sonya the frog chooses a point uniformly at random lying within the square
$[0, 6]$ $\times$ $[0, 6]$ in the coordinate plane and hops to that point. She then randomly
chooses a distance uniformly at random from $[0, 1]$ and a direction uniformly at
random from {north, south, east, west}. All of her choices are indepen... | 2023 AMC 10B Problem 19 | We know that Sonya can only jump out if she is within 1 unit of the border. We can calculate the probability that Sonya can jump out. $\newline$
$\newline$
The total area of the colored regions is $20$, so the probability that Sonya lands in a colored region is $\frac{20}{36} = \frac{5}{9}$. We can calculate the pro... | // Block 1
import graph;
draw((0,0)--(0,6)--(6,6)--(6,0)--cycle);
filldraw((1,0)--(1,1)--(5,1)--(5,0)--cycle,blue);
filldraw((0,1)--(1,1)--(1,5)--(0,5)--cycle,blue);
filldraw((1,5)--(1,6)--(5,6)--(5,5)--cycle,blue);
filldraw((5,1)--(6,1)--(6,5)--(5,5)--cycle,blue);
filldraw((0,0)--(0,1)--(1,1)--(1,0)--cycle,red);
fill... | [] |
944 | What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$, $0\le v\le1,$ and $0\le w\le1$?
$\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16$ | 2023 AMC 10B Problem 24 | Notice that we are given a parametric form of the region, and $w$ is used in both $x$ and $y$. We first fix $u$ and $v$ to $0$, and graph $(-3w,4w)$ from $0\le w\le1$. When $w$ is $0$, we have the point $(0,0)$, and when $w$ is $1$, we have the point $(-3,4)$. We see that since this is a directly proportional function,... | // Block 1
import geometry;
pair A = (-3, 4);
pair B = (-3, 5);
pair C = (-1, 4);
pair D = (-1, 5);
pair AA = (0, 0);
pair BB = (0, 1);
pair CC = (2, 0);
pair DD = (2, 1);
//draw(A--B--D--C--cycle);
draw(A--B);
label("1",midpoint(A--B),W);
label("2",midpoint(D--B),N);
draw(A--C,dashed);
draw(B--D);
draw(C--D, d... | [] |
944 | What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$, $0\le v\le1,$ and $0\le w\le1$?
$\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16$ | 2023 AMC 10B Problem 24 | We can find the "boundary points" and work with our intuition to solve the problem. We set each of $u, v, w$ equal to $0, 1$ for a total of $8$ combinations in $u, v, w$. We now test each one.
Case 1: $u = 0, v = 0, w = 0 \implies (0, 0)$
Case 2: $u = 0, v = 0, w = 1 \implies (-3, 4)$
Case 3: $u = 0, v = 1, w = 0 \i... | // Block 1
import graph;
import geometry;
Label f;
size(5cm);
unitsize(0.7cm);
xaxis(-5,5,Ticks(f, 5.0, 1.0));
yaxis(-5,5,Ticks(f, 5.0, 1.0));
pair A = (0, 0);
dot (A);
pair B = (-3, 4);
dot (B);
pair C = (0, 1);
dot (C);
pair D = (-3, 5);
dot (D);
pair E = (2, 0);
dot (E);
pair F = (-1, 4);
dot (F);
pair G = (2, 1... | [] |
945 | A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D... | 2023 AMC 10B Problem 25 | We can find the area of the red pentagon by taking the area of the total pentagon and subtracting the area outside the red pentagon.
The area outside the red pentagon is the sum of the larger isosceles triangles, but this double counts the overlapping regions of the small isosceles triangles, so we have to subtract t... | // Block 1
unitsize(5cm); // Define the vertices of the pentagons pair A, B, C, D, E; pair F, G, H, I, J; // Calculate the vertices of the larger pentagon A = dir(90); B = dir(90 - 72); C = dir(90 - 2*72); D = dir(90 - 3*72); E = dir(90 - 4*72); // Draw the larger pentagon draw(A--B--C--D--E--cycle); pair O = (A+B... | [] |
945 | A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D... | 2023 AMC 10B Problem 25 | Let $r_b$ and $r_s$ be the circumradius of the big and small pentagon, respectively. Let $a_s$ be the apothem of the smaller pentagon and $A_s$ and $A_b$ be the areas of the smaller and larger pentagon, respectively.
From the diagram:
\begin{align*} \cos{36^\circ} &= \dfrac{a_s}{r_s} = \dfrac{\phi}{2} = \dfrac{\s... | unitsize(5cm); // Define the vertices of the pentagons pair A, B, C, D, E; pair F, G, H, I, J; // Calculate the vertices of the larger pentagon A = dir(90); B = dir(90 - 72); C = dir(90 - 2*72); D = dir(90 - 3*72); E = dir(90 - 4*72); // Draw the larger pentagon draw(A--B--C--D--E--cycle); pair O = (A+B+C+D+E)/5; p... | [] |
946 | You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a ... | 2023 AMC 12B Problem 5 | Notice that the $3\times3$ square grid has a total of $12$ possible $2\times1$ rectangles.
Suppose you choose the middle square for one of your turns. The middle square is covered by $4$ rectangles, and each of the remaining $8$ squares is covered by a maximum of $2$ uncounted rectangles. This means that the number of... | // Block 1
draw((0,0)--(0.5,0)--(0.5,0.5)--(0,0.5)--(0,0));
draw((0,1)--(0.5,1)--(0.5,1.5)--(0,1.5)--(0,1));
draw((0.5,0.5)--(1,0.5)--(1,1)--(0.5,1)--(0.5,0.5));
draw((1,0)--(1.5,0)--(1.5,0.5)--(1,0.5)--(1,0));
draw((1,1)--(1.5,1)--(1.5,1.5)--(1,1.5)--(1,1));
draw((0,0.5)--(0.5,0.5)--(0.5,1)--(0,1)--(0,0.5));
draw((0.5... | [] |
946 | You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a ... | 2023 AMC 12B Problem 5 | The $3 \times 3$ grid can be colored like a checkerboard with alternating black and white squares.
Let the top left square be white, and the rest of the squares alternate colors, as shown below:
Each $2 \times 1$ rectangle always covers $1$ white square and $1$ black square.
You can ensure that at least one of your gu... | // Block 1
/* Diagram by MRENTHUSIASM */
import olympiad;
unitsize(25);
fill((1,2)--(1,3)--(2,3)--(2,2)--cycle, mediumgray);
fill((1,0)--(1,1)--(2,1)--(2,0)--cycle, mediumgray);
fill((2,1)--(2,2)--(3,2)--(3,1)--cycle, mediumgray);
fill((0,1)--(0,2)--(1,2)--(1,1)--cycle, mediumgray);
for (int i = 0; i < 4; ++i) {
draw((... | [] |
947 | What is the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other?
$\textbf{(A) }\frac 54 \qquad \textbf{(B) } \frac 87 \qquad \textbf{(C)} \frac{5\sqrt2}4 \qquad \textbf{(D) } \frac 32 \qquad \textbf{(E) } \frac{3\sqrt3}4$ | 2023 AMC 12B Problem 11 | Let the length of the shorter base of the trapezoid be $2x$ and the height of the trapezoid be $y$.
Each leg has length $1$ if and only if $x^2+y^2=1$, where $x$ and $y$ are positive real numbers. The general solution to this equation is \[(x,y)=(\cos t,\sin t)\] for any number $0<t<\frac{\pi}{2}$ so that $x$ and $y... | unitsize(100); pair A=(-1, 0), B=(1, 0), C=(0.5, 0.5), D=(-0.5, 0.5); draw(A--B--C--D--cycle, black); label("$2x$",(0,0.58),(0,0)); label("$2x$",(0,-0.08),(0,0)); label("$x$",(-0.75,-0.08),(0,0)); label("$x$",(0.75,-0.08),(0,0)); draw(D--(-0.5,0),black); draw(C--(0.5,0),black); label("$y$",(0.58,0.25)); label("$y$",(... | [] |
948 | 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 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 ... | 2023 AMC 12B Problem 13 | We can create three equations using the given information.
\[4a+4b+4c = 13\]
\[2ab+2ac+2bc=\frac{11}{2}\]
\[abc=\frac{1}{2}\]
We also know that we want $\sqrt{a^2 + b^2 + c^2}$ because that is the length that can be found from using the Pythagorean Theorem. We cleverly notice that $a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab+a... | // Block 1
import geometry;
pair A = (-3, 4);
pair B = (-3, 5);
pair C = (-1, 4);
pair D = (-1, 5);
pair AA = (0, 0);
pair BB = (0, 1);
pair CC = (2, 0);
pair DD = (2, 1);
draw(D--AA,dashed);
draw(A--B);
draw(A--C);
draw(B--D);
draw(C--D);
draw(A--AA);
draw(B--BB);
draw(C--CC);
draw(D--DD);
// Dotted vertices
... | [] |
949 | A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D... | 2023 AMC 12B Problem 25 | We can find the area of the red pentagon by taking the area of the total pentagon and subtracting the area outside the red pentagon.
The area outside the red pentagon is the sum of the larger isosceles triangles, but this double counts the overlapping regions of the small isosceles triangles, so we have to subtract t... | // Block 1
unitsize(5cm);
// Define the vertices of the pentagons
pair A, B, C, D, E;
pair F, G, H, I, J;
// Calculate the vertices of the larger pentagon
A = dir(90);
B = dir(90 - 72);
C = dir(90 - 2*72);
D = dir(90 - 3*72);
E = dir(90 - 4*72);
// Draw the larger pentagon
draw(A--B--C--D--E--cycle);
pair O = (A+B... | [] |
949 | A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D... | 2023 AMC 12B Problem 25 | Let $r_b$ and $r_s$ be the circumradius of the big and small pentagon, respectively. Let $a_s$ be the apothem of the smaller pentagon and $A_s$ and $A_b$ be the areas of the smaller and larger pentagon, respectively.
From the diagram:
\begin{align*} \cos{36^\circ} &= \dfrac{a_s}{r_s} = \dfrac{\phi}{2} = \dfrac{\s... | // Block 1
unitsize(5cm);
// Define the vertices of the pentagons
pair A, B, C, D, E;
pair F, G, H, I, J;
// Calculate the vertices of the larger pentagon
A = dir(90);
B = dir(90 - 72);
C = dir(90 - 2*72);
D = dir(90 - 3*72);
E = dir(90 - 4*72);
// Draw the larger pentagon
draw(A--B--C--D--E--cycle);
pair O = (A+B+... | [] |
950 | One side of an equilateral triangle of height $24$ lies on line $\ell$. A circle of radius $12$ is tangent to line $\ l$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a \sqrt{b} - c \p... | 2024 AMC 10A Problem 14 | So, we're here to over-complicate things. Let's start with the circle. Recognize that since the triangle is equilateral, all angles measure $60^\circ$ Observe that the two tangent lines form a $120^\circ$ angle due to the Angle Sum Theorem. Therefore, the area within the circle is the area within a $60^\circ$ arc.
Let... | // Block 1
size(200);
pair A = (-48,0);
pair B = (-24,0);
pair C = (-36,24);
pair O = (-16.5,12); // center of the circle
draw(A--B--C--cycle, linewidth(1));
draw(circle(O,12), linewidth(1));
draw((-48,0) -- (10,0));
draw((-36,24) -- (-36, 0));
dot(O);
draw(O -- (-16.5,0));
pair dir = rotate(205)*(1,0);
draw(O -- O ... | [] |
951 | Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$, what is $\beta$?
$\textbf{(A) }\frac{\alpha}{3}\qquad \textbf{(B) }\alpha - \frac{\pi}{8}\qquad \textbf{(C) }\frac{\... | 2024 AMC 12A Problem 10 | We can scale the $3$-$4$-$5$ triangle up by a factor of $6$ to make its side lengths $18,24,$ and $30,$ then glue its side of length $24$ to the corresponding side in the $7$-$24$-$25$ triangle:
Angles $\angle DAB$ and $\angle BDA$ are complementary in $\triangle ABD,$ so $\angle DAB = \frac{\pi}{2} - \alpha.$ We al... | // Block 1
pair A = (0,0);
pair B = (18,0);
pair C = (25,0);
pair D = (18,24);
draw(A--C--D--cycle);
draw(B--D);
draw(rightanglemark(C,B,D,50));
label("A", A, SW);
label("B", B, S);
label("C", C, SE);
label("D", D, N);
label("18", A--B, S);
label("7", B--C, S);
label("25", C--D, NE);
label("30", D--A, NW);
label("2... | [] |
952 | The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $(-1,\tfrac{1}{2})$ over this axis?
$\textbf{(A) }\left(-1,-\frac{3}{2}\right)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\frac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\fr... | 2024 AMC 12A Problem 13 | Consider the graphs of $y=e^{x+1}-1$ and $y=e^{-x}-1$. A rough sketch will show that they intercept somewhere between -1 and 0 and the axis of symmetry is vertical. Thus, $\boxed{\textbf{(D) }\left(0,\dfrac12\right)}$ is the only possible answer.
Note: You can more rigorously think about the solution by noting that s... | // Block 1
unitsize(2cm);
real e = 2.71828;
real f1(real x) {return e^(x+1)-1;}
real f2(real x) {return e^(0-x)-1;}
draw(graph(f1,-1.5,0.5));
draw(graph(f2,-1.5,0.5));
xaxis(-2,1,Ticks());
yaxis(f2(0.5),f1(0.5),Ticks());
draw((-0.5,f2(0.5))--(-0.5,f1(0.5)),red+dashed);
/*graph by Technodoggo, 9 November 2024*/
// ... | [] |
953 | On top of a rectangular card with sides of length $1$ and $2+\sqrt{3}$, an identical card is placed so that two of their diagonals line up, as shown ($\overline{AC}$, in this case).
Continue the process, adding a third card to the second, and so on, lining up successive diagonals after rotating clockwise. In total, ... | 2024 AMC 12A Problem 18 | By the Pythagorean Theorem, $AC = \sqrt{(2 + \sqrt{3})^2 + 1} = \sqrt{8 + 4\sqrt{3}}$. So we have
\[\arcsin \angle BCA = \frac{1}{\sqrt{8 + 4\sqrt{3}}} = \frac{1}{\sqrt{(\sqrt{6} + \sqrt{2})^2}} = \frac{\sqrt{6} - \sqrt{2}}{4}.\]
We note that \[\sin 15^\circ = \cos 75^\circ = \cos(30^\circ + 45^\circ) = \cos30^\circ \... | // Block 1
size(150);
defaultpen(fontsize(10pt));
pair O = (0,0);
pair[] pts;
for (int i = 0; i < 12; ++i) {
pts[i] = dir(30 * i);
}
fill(pts[9]--pts[10]--pts[3]--pts[4]--cycle, paleblue);
fill(pts[8]--pts[9]--pts[2]--pts[3]--cycle, palegreen);
fill(pts[7]--pts[8]--pts[1]--pts[2]--cycle, palered);
draw(Circle(O,1)... | [] |
954 | Cyclic quadrilateral $ABCD$ has lengths $BC=CD=3$ and $DA=5$ with $\angle CDA=120^\circ$. What is the length of the shorter diagonal of $ABCD$?
$\textbf{(A) }\frac{31}7 \qquad \textbf{(B) }\frac{33}7 \qquad \textbf{(C) }5 \qquad \textbf{(D) }\frac{39}7 \qquad \textbf{(E) }\frac{41}7 \qquad$ | 2024 AMC 12A Problem 19 | ~diagram by erics118
First, $\angle CBA=60 ^\circ$ by properties of cyclic quadrilaterals.
Let $AC=u$. Apply the Law of Cosines on $\triangle ACD$:
\[u^2=3^2+5^2-2(3)(5)\cos120^\circ\]
\[u=7\]
Let $AB=v$. Apply the Law of Cosines on $\triangle ABC$:
\[7^2=3^2+v^2-2(3)(v)\cos60^\circ\]
\[v=\frac{3\pm13}{2}\]
\[v=8\]
... | // Block 1
import geometry;
size(200);
pair A = (-1.66, 0.33);
pair B = (-9.61277, 1.19799);
pair C = (-7.83974, 3.61798);
pair D = (-4.88713, 4.14911);
draw(circumcircle(A, B, C));
draw(A--C);
draw(A--D);
draw(C--D);
draw(B--C);
draw(A--B);
label("$A$", A, E);
label("$B$", B, W);
label("$C$", C, NW);
label("$D$",... | [] |
955 | Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline {AB}$ and $\overline{AC},$ respectively, of equilateral triangle $\triangle ABC.$ Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC?$
$\textbf{(A... | 2024 AMC 12A Problem 20 | Let $\overline{AP}=x$ and $\overline{AQ}=y$. Applying the sine formula for a triangle's area, we see that
\[[\Delta APQ]=\dfrac12\cdot x\cdot y\sin(\angle PAQ)=\dfrac{xy}2\sin(60^\circ)=\dfrac{\sqrt3}4xy.\]
Without loss of generality, we let $AB=BC=CA=1$, and thus $[\Delta ABC]=\dfrac{\sqrt3}4$; we therefore require ... | // Block 1
/*Asymptote visual by Technodoggo, 7 November 2024*/
unitsize(8cm);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
label("$0$",(-0.05,-0.05));
label("$1$",(1,-0.05),S);
label("$1$",(-0.05,1),W);
draw((-0.05,0)--(1,0)--(1,-0.05));
draw((0,-0.05)--(0,1)--(-0.05,1));
real f(real x) {return 1/(2*x);}
path c = graph... | [] |
956 | Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
$\textbf{(A)}~\frac{1}{9} \qquad\textbf{(B)}~... | 2024 AMC 10B Problem 21 | Notice that the sum of radii of two circles tangent to each other will equal to the distance from center to center. Set the center of the big circle be at $(0,1).$ Since both circles are tangent to a line (in this case, $y=0$), the y-coordinates of the centers are just its radius.
Hence, the center of the smaller cir... | // By Elephant200 size(8cm); draw(circle((0,1),1), linewidth(1.2)); draw((-1,0)--(3,0), linewidth(1.2)); draw(circle((1,1/4),1/4), linewidth(1.2)); draw(circle((2/3,1/9),1/9), red+linewidth(1.2)); draw(circle((2,1),1), red+linewidth(1.2)); | [] |
956 | Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
$\textbf{(A)}~\frac{1}{9} \qquad\textbf{(B)}~... | 2024 AMC 10B Problem 21 | We can immediately see that we can put a circle with radius $1$ outside of the two circles already drawn that are tangent to the circles. If you brought a ruler with you, you can confirm this. We now find the radius of the circle between the circles of radius $1$ and $\frac{1}{4}$.. By drawing out the diagram, a geomet... | // Block 1
// By Elephant200
size(8cm);
draw(circle((0,1),1), linewidth(1.2));
draw((-1,0)--(3,0), linewidth(1.2));
draw(circle((1,1/4),1/4), linewidth(1.2));
draw(circle((2/3,1/9),1/9), red+linewidth(1.2));
draw(circle((2,1),1), red+linewidth(1.2));
// Block 2
// By Elephant200 size(8cm); draw(circle((0,1),1), linewid... | [] |
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