Found problems: 3632
2022 AMC 10, 19
Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that $\frac{1}{1}+\frac{1}{2}+\frac{1}{3} \ldots +\frac{1}{17}=\frac{h}{L_{17}}$. What is the remainder when $h$ is divided by $17?$
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$
2013 Harvard-MIT Mathematics Tournament, 9
Let $z$ be a non-real complex number with $z^{23}=1$. Compute \[\sum_{k=0}^{22}\dfrac{1}{1+z^k+z^{2k}}.\]
1966 AMC 12/AHSME, 38
In triangle $ABC$ the medians $AM$ and $CN$ to sides $BC$ and $AB$, respectively, intersect in point $O$. $P$ is the midpoint of side $AC$, and $MP$ intersects $CN$ in $Q$. If the area of triangle $OMQ$ is $n$, then the area of triangle $ABC$ is:
$\text{(A)}\ 16n\qquad
\text{(B)}\ 18n\qquad
\text{(C)}\ 21n\qquad
\text{(D)}\ 24n\qquad
\text{(E)}\ 27n$
2013 AMC 8, 8
A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?
$\textbf{(A)}\ \frac18 \qquad \textbf{(B)}\ \frac14 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac34$
2017 AMC 10, 20
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$
2017 AMC 10, 5
Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating $10$ pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?
$\textbf{(A) } 10\qquad \textbf{(B) } 20\qquad \textbf{(C) } 30\qquad \textbf{(D) } 40\qquad \textbf{(E) } 50$
2023 AMC 10, 14
How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2=m^2n^2$?
$\textbf{(A) }7\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }5$
2021 AMC 12/AHSME Fall, 1
What is the value of $1234+2341+3412+4123$?
$\textbf{(A) } 10,000 \qquad \textbf{(B) }10,010 \qquad \textbf{(C) }10,110 \qquad \textbf{(D) }11,000 \qquad \textbf{(E) }11,110$
2000 AMC 8, 13
In triangle $CAT$, we have $\angle ACT = \angle ATC$ and $\angle CAT = 36^\circ$. If $\overline{TR}$ bisects $\angle ATC$, then $\angle CRT =$
[asy]
pair A,C,T,R;
C = (0,0); T = (2,0); A = (1,sqrt(5+sqrt(20))); R = (3/2 - sqrt(5)/2,1.175570);
draw(C--A--T--cycle);
draw(T--R);
label("$A$",A,N);
label("$T$",T,SE);
label("$C$",C,SW);
label("$R$",R,NW);
[/asy]
$\text{(A)}\ 36^\circ \qquad \text{(B)}\ 54^\circ \qquad \text{(C)}\ 72^\circ \qquad \text{(D)}\ 90^\circ \qquad \text{(E)}\ 108^\circ$
1963 AMC 12/AHSME, 26
[b]Form 1[/b]
Consider the statements:
$\textbf{(1)}\ p\text{ } \wedge\sim q\wedge r \qquad
\textbf{(2)}\ \sim p\text{ } \wedge\sim q\wedge r\qquad
\textbf{(3)}\ p\text{ } \wedge\sim q\text{ }\wedge \sim r \qquad
\textbf{(4)}\ \sim p\text{ } \wedge q\text{ }\wedge r $,
where $p,q,$ and $r$ are propositions. How many of these imply the truth of $(p\rightarrow q)\rightarrow r$?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
[b]Form 2[/b]
Consider the statements $(1)$ $p$ and $r$ are true and $q$ is false $(2)$ $r$ is true and $p$ and $q$ are false $(3)$ $p$ is true and $q$ and $r$ are false $(4)$ $q$ and $r$ are true and $p$ is false. How many of these imply the truth of the statement
"$r$ is implied by the statement that $p$ implies $q$"?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
1994 AMC 12/AHSME, 7
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is
[asy]
draw((0,0)--(10,0)--(10,10)--(0,10)--cycle);
draw((5,5)--(12,-2)--(5,-9)--(-2,-2)--cycle);
label("A", (0,0), W);
label("B", (10,0), E);
label("C", (10,10), NE);
label("D", (0,10), NW);
label("G", (5,5), N);
label("F", (12,-2), E);
label("E", (5,-9), S);
label("H", (-2,-2), W);
dot((-2,-2));
dot((5,-9));
dot((12,-2));
dot((0,0));
dot((10,0));
dot((10,10));
dot((0,10));
dot((5,5));
[/asy]
$ \textbf{(A)}\ 75 \qquad\textbf{(B)}\ 100 \qquad\textbf{(C)}\ 125 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 175 $
1966 AMC 12/AHSME, 29
The number of postive integers less than $1000$ divisible by neither $5$ nor $7$ is:
$\text{(A)}\ 688 \qquad
\text{(B)}\ 686\qquad
\text{(C)}\ 684 \qquad
\text{(D)}\ 658\qquad
\text{(E)}\ 630$
2008 AMC 10, 23
A rectangular floor measures $ a$ by $ b$ feet, where $ a$ and $ b$ are positive integers with $ b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $ 1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $ (a,b)$?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$
2014 AMC 10, 13
Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?
[asy]
import graph;
size(6cm);
pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps);
pair B = (0,0);
pair C = (1,0);
pair A = rotate(60,B)*C;
pair E = rotate(270,A)*B;
pair D = rotate(270,E)*A;
pair F = rotate(90,A)*C;
pair G = rotate(90,F)*A;
pair I = rotate(270,B)*C;
pair H = rotate(270,I)*B;
draw(A--B--C--cycle);
draw(A--E--D--B);
draw(A--F--G--C);
draw(B--I--H--C);
draw(E--F);
draw(D--I);
draw(I--H);
draw(H--G);
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,W);
label("$E$",E,W);
label("$F$",F,E);
label("$G$",G,E);
label("$H$",H,SE);
label("$I$",I,SW);
[/asy]
$ \textbf{(A)}\ \dfrac{12+3\sqrt3}4\qquad\textbf{(B)}\ \dfrac92\qquad\textbf{(C)}\ 3+\sqrt3\qquad\textbf{(D)}\ \dfrac{6+3\sqrt3}2\qquad\textbf{(E)}\ 6 $
2022 AMC 10, 3
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?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$
2005 AIME Problems, 7
Let \[x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.\] Find $(x+1)^{48}$.
1961 AMC 12/AHSME, 15
If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles (not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ days is:
${{ \textbf{(A)}\ \frac{x^3}{y^2} \qquad\textbf{(B)}\ \frac{y^3}{x^2} \qquad\textbf{(C)}\ \frac{x^2}{y^3} \qquad\textbf{(D)}\ \frac{y^2}{x^3} }\qquad\textbf{(E)}\ y} $
2011 USAJMO, 5
Points $A,B,C,D,E$ lie on a circle $\omega$ and point $P$ lies outside the circle. The given points are such that (i) lines $PB$ and $PD$ are tangent to $\omega$, (ii) $P, A, C$ are collinear, and (iii) $DE \parallel AC$. Prove that $BE$ bisects $AC$.
1971 AMC 12/AHSME, 4
After simple interest for two months at $5\%$ per annum was credited, a Boy Scout Troop had a total of $\textdollar 255.31$ in the Council Treasury. The interest credited was a number of dollars plus the following number of cents
$\textbf{(A) }11\qquad\textbf{(B) }12\qquad\textbf{(C) }13\qquad\textbf{(D) }21\qquad \textbf{(E) }31$
2007 AMC 10, 4
The larger of two consecutive odd integers is three times the smaller. What is their sum?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 20$
1988 AMC 12/AHSME, 9
An $8'\text{ X }10'$ table sits in the corner of a square room, as in Figure 1 below. The owners desire to move the table to the position shown in Figure 2. The side of the room is $S$ feet. What is the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart?
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=(0,0), B=(16,0), C=(16,16), D=(0,16), E=(32,0), F=(48,0), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16);
draw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N);
label("S", (18,8));
label("S", (50,8));
label("Figure 1", (A+B)/2, 2*S);
label("Figure 2", (E+F)/2, 2*S);
label("10'", (I+J)/2, S);
label("8'", (12,12));
label("8'", (L+M)/2, S);
label("10'", (42,11));
label("table", (5,12));
label("table", (36,11));
[/asy]
$ \textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15 $
2017 AMC 8, 4
When 0.000315 is multiplied by 7,928,564 the product is closest to which of the following?
$\textbf{(A) }210\qquad\textbf{(B) }240\qquad\textbf{(C) }2100\qquad\textbf{(D) }2400\qquad\textbf{(E) }24000$
2023 AIME, 13
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below.
[asy]
unitsize(2cm);
pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70);
draw(o--u--(u+v));
draw(o--v--(u+v), dotted);
draw(shift(w)*(o--u--(u+v)--v--cycle));
draw(o--w);
draw(u--(u+w));
draw(v--(v+w), dotted);
draw((u+v)--(u+v+w));
[/asy]
2021 AMC 10 Spring, 21
Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$, and the lines $BC$, $DE$, and $FA$ determine a triangle with area $324\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m + n\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$?
$\textbf{(A)}~47\qquad\textbf{(B)}~52\qquad\textbf{(C)}~55\qquad\textbf{(D)}~58\qquad\textbf{(E)}~63$
2006 AIME Problems, 1
In convex hexagon $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are right angles, and $\angle B$, $\angle C$, $\angle E$, and $\angle F$ are congruent. The area of the hexagonal region is $2116(\sqrt{2}+1)$. Find $AB$.