Found problems: 98
2022 AMC 12/AHSME, 6
Consider the following $100$ sets of $10$ elements each:
\begin{align*}
&\{1,2,3,\cdots,10\}, \\
&\{11,12,13,\cdots,20\},\\
&\{21,22,23,\cdots,30\},\\
&\vdots\\
&\{991,992,993,\cdots,1000\}.
\end{align*}
How many of these sets contain exactly two multiples of $7$?
$\textbf{(A)} 40\qquad\textbf{(B)} 42\qquad\textbf{(C)} 43\qquad\textbf{(D)} 49\qquad\textbf{(E)} 50$
2022 AMC 12/AHSME, 2
The sum of three numbers is $96$. The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$
2022 AMC 12/AHSME, 16
A [i]triangular number[/i] is a positive integer that can be expressed in the form $t_n = 1 + 2 + 3 +\cdots + n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?
$\textbf{(A)} ~6 \qquad\textbf{(B)} ~9 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~18 \qquad\textbf{(E)} ~27 $
2022 AMC 10, 25
Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S$. See the figure (not drawn to scale).
[asy]
//kaaaaaaaaaante314
size(8cm);
import olympiad;
label(scale(.8)*"$y$", (0,60), N);
label(scale(.8)*"$x$", (60,0), E);
filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white);
label(scale(1.3)*"$R$", (55/2,55/2));
filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white);
label(scale(1.3)*"$S$",(-14,14));
filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white);
label(scale(1.3)*"$T$",(3.5,25/2));
draw((0,-10)--(0,60),EndArrow(TeXHead));
draw((-34,0)--(60,0),EndArrow(TeXHead));[/asy]
The fraction of lattice points in $S$ that are in $S \cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?
$\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340$
2022 AMC 10, 13
The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes?
$\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 16$
2022 AMC 12/AHSME, 9
On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.
"Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
$\textbf{(A) }7\qquad\textbf{(B) }12\qquad\textbf{(C) }21\qquad\textbf{(D) }27\qquad\textbf{(E) }31$
2022 AMC 12/AHSME, 24
How many strings of length $5$ formed from the digits $0$,$1$,$2$,$3$,$4$ are there such that for each $j\in\{1,2,3,4\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies the condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, at least $3$ digits less than $3$, and at least $4$ digits less than $4$. The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$.)
$\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296$
2022 AMC 10, 9
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy]
size(5.5cm);
draw((0,0)--(0,2)--(2,2)--(2,0)--cycle);
draw((2,0)--(8,0)--(8,2)--(2,2)--cycle);
draw((8,0)--(12,0)--(12,2)--(8,2)--cycle);
draw((0,2)--(6,2)--(6,4)--(0,4)--cycle);
draw((6,2)--(12,2)--(12,4)--(6,4)--cycle);
[/asy]
$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$
2022 AMC 12/AHSME, 15
One of the following numbers is not divisible by any prime number less than 10. Which is it?
(A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $
2022 AMC 12/AHSME, 25
A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$. What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$?
$\textbf{(A)} ~\frac{21}{5} \qquad\textbf{(B)} ~\frac{85}{13} \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~\frac{39}{5} \qquad\textbf{(E)} ~17 $
2022 AMC 12/AHSME, 16
Suppose $x$ and $y$ are positive real numbers such that
$x^y=2^{64}$ and $(\log_2{x})^{\log_2{y}}=2^{7}$.
What is the greatest possible value of $\log_2{y}$?
$\textbf{(A)}3~\textbf{(B)}4~\textbf{(C)}3+\sqrt{2}~\textbf{(D)}4+\sqrt{3}~\textbf{(E)}7$
2022 AMC 12/AHSME, 9
The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that
\[2^{a_7}=2^{27} \cdot a_7.\]
What is the minimum possible value of $a_2$?
$\textbf{(A)}8~\textbf{(B)}12~\textbf{(C)}16~\textbf{(D)}17~\textbf{(E)}22$
2022 AMC 10, 16
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?
[asy]
size(5cm);
filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray);
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1));
draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1));
label("$4$", (8,2), E);
label("$8$", (4,0), S);
label("$5$", (3,11/2), NW);
draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(.4));
draw((5,7)--(5+21/100,7-28/100)--(5-7/100,7-49/100)--(5-28/100,7-21/100)--cycle,linewidth(.4));
[/asy]
$\textbf{(A) } 15\dfrac{1}{8} \qquad \textbf{(B) } 15\dfrac{3}{8} \qquad \textbf{(C) } 15\dfrac{1}{2} \qquad \textbf{(D) } 15\dfrac{5}{8} \qquad \textbf{(E) } 15\dfrac{7}{8}$
2022 AMC 10, 17
How many three-digit positive integers $\underline{a}$ $\underline{b}$ $\underline{c}$ are there whose nonzero digits $a$, $b$, and $c$ satisfy
$$0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?$$
(The bar indicates repetition, thus $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ in the infinite repeating decimal $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$)
$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }13\qquad\textbf{(E) }14$
2022 AMC 12/AHSME, 25
Four regular hexagons surround a square with a 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. What is $m + n + p$?
[asy]
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)*polygon(6));
draw((0,1-sqrt(3))--(1,1-sqrt(3))--(3-sqrt(3),sqrt(3)-2)--(sqrt(3),0)--(sqrt(3),1)--(3-sqrt(3),3-sqrt(3))--(1,sqrt(3))--(0,sqrt(3))--(sqrt(3)-2,3-sqrt(3))--(1-sqrt(3),1)--(1-sqrt(3),0)--(sqrt(3)-2,sqrt(3)-2)--cycle,linewidth(2));
[/asy]
$\textbf{(A)}-12~\textbf{(B)}-4~\textbf{(C)} 4~\textbf{(D)}24~\textbf{(E)}32$
2022 AMC 10, 24
How many strings of length $5$ formed from the digits $0$,$1$,$2$,$3$,$4$ are there such that for each $j\in\{1,2,3,4\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies the condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, at least $3$ digits less than $3$, and at least $4$ digits less than $4$. The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$.)
$\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296$
2022 AMC 12/AHSME, 6
A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X$?
$\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$
2022 AMC 12/AHSME, 13
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?
[asy]
size(5cm);
filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray);
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1));
draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1));
label("$4$", (8,2), E);
label("$8$", (4,0), S);
label("$5$", (3,11/2), NW);
draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(.4));
draw((5,7)--(5+21/100,7-28/100)--(5-7/100,7-49/100)--(5-28/100,7-21/100)--cycle,linewidth(.4));
[/asy]
$\textbf{(A) } 15\dfrac{1}{8} \qquad \textbf{(B) } 15\dfrac{3}{8} \qquad \textbf{(C) } 15\dfrac{1}{2} \qquad \textbf{(D) } 15\dfrac{5}{8} \qquad \textbf{(E) } 15\dfrac{7}{8}$
2022 AMC 10, 23
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 $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $1$?
$\textbf{(A) }\frac{1}{3} \qquad \textbf{(B) }\frac{1}{2} \qquad \textbf{(C) }\frac{2}{3} \qquad \textbf{(D) }\frac{3}{4} \qquad \textbf{(E) }\frac{5}{6}$
2022 AMC 12/AHSME, 24
The figure below depicts a regular 7-gon inscribed in a unit circle.
[asy]
import geometry;
unitsize(3cm);
draw(circle((0,0),1),linewidth(1.5));
for (int i = 0; i < 7; ++i) {
for (int j = 0; j < i; ++j) {
draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5));
}
}
for(int i = 0; i < 7; ++i) {
dot(dir(i * 360/7),5+black);
}
[/asy]
What is the sum of the 4th powers of the lengths of all 21 of its edges and diagonals?
$\textbf{(A)}49~\textbf{(B)}98~\textbf{(C)}147~\textbf{(D)}168~\textbf{(E)}196$
2022 AMC 10, 18
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself?
$\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$
2022 AMC 10, 17
One of the following numbers is not divisible by any prime number less than 10. Which is it?
(A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $
2022 AMC 12/AHSME, 20
Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
$\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23$
2022 AMC 12/AHSME, 2
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$?
[asy]
import olympiad;
size(180);
real r = 3, s = 5, t = sqrt(r*r+s*s);
defaultpen(linewidth(0.6) + fontsize(10));
pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0);
draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D));
label("$A$",A,SW);
label("$B$", B, NW);
label("$C$",C,NE);
label("$D$",D,SE);
label("$P$",P,S);
[/asy]
$\textbf{(A) }3\sqrt 5 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }6\sqrt 5 \qquad
\textbf{(D) }20\qquad
\textbf{(E) }25$
2022 AMC 10, 21
A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of 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?
[asy]
size(200);
defaultpen(linewidth(0.8));
draw((342,-662) -- (600, -727) -- (757,-619) -- (967,-400) -- (1016,-300) -- (912,-116) -- (651,-46) -- (238,-90) -- (82,-204) -- (184, -388) -- (447,-458) -- (859,-410) -- (1016,-300));
draw((82,-204) -- (133,-490) -- (342, -662));
draw((652,-626) -- (600,-727));
draw((447,-458) -- (652,-626) -- (859,-410));
draw((133,-490) -- (184, -388));
draw((967,-400) -- (912,-116)^^(342,-662) -- (496, -545) -- (757,-619)^^(496, -545) -- (446, -262) -- (238, -90)^^(446, -262) -- (651, -46),linewidth(0.6)+linetype("5 5")+gray(0.4));
[/asy]
$\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }5+2\sqrt{2}\qquad\textbf{(D) }8\qquad\textbf{(E) }9$