Found problems: 3632
2022 AMC 10, 11
All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of
the choices below is logically equivalent to the statement “No school bigger than Euclid HS sold more
T-shirts than Euclid HS”?
$(\textbf{A})$ All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.
$(\textbf{B})$ No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.
$(\textbf{C})$ All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.
$(\textbf{D})$ All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.
$(\textbf{E})$ All schools smaller than Euclid HS sold more T-shirts than Euclid HS.
1977 AMC 12/AHSME, 11
For each real number $x$, let $\textbf{[}x\textbf{]}$ be the largest integer not exceeding $x$ (i.e., the integer $n$ such that $n\le x<n+1$). Which of the following statements is (are) true?
$\textbf{I. [}x+1\textbf{]}=\textbf{[}x\textbf{]}+1\text{ for all }x$
$\textbf{II. [}x+y\textbf{]}=\textbf{[}x\textbf{]}+\textbf{[}y\textbf{]}\text{ for all }x\text{ and }y$
$\textbf{III. [}xy\textbf{]}=\textbf{[}x\textbf{]}\textbf{[}y\textbf{]}\text{ for all }x\text{ and }y$
$\textbf{(A) }\text{none}\qquad\textbf{(B) }\textbf{I }\text{only}\qquad\textbf{(C) }\textbf{I}\text{ and }\textbf{II}\text{ only}\qquad\textbf{(D) }\textbf{III }\text{only}\qquad \textbf{(E) }\text{all}$
1959 AMC 12/AHSME, 19
With the use of three different weights, namely 1 lb., 3 lb., and 9 lb., how many objects of different weights can be weighed, if the objects is to be weighed and the given weights may be placed in either pan of the scale?
$ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 7 $
2012 AIME Problems, 9
Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
2014 AMC 10, 24
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is [i]bad[/i] if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
$ \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 $
2016 AMC 12/AHSME, 6
All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$
2023 AMC 10, 13
What is the area of the region in the coordinate plane defined by the inequality \[\left||x|-1\right|+\left||y|-1\right|\leq 1?\]
$\textbf{(A)}~4\qquad\textbf{(B)}~8\qquad\textbf{(C)}~10\qquad\textbf{(D)}~12\qquad\textbf{(E)}~15$
2014 AMC 12/AHSME, 6
The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?
$\textbf{(A) }44\qquad
\textbf{(B) }55\qquad
\textbf{(C) }77\qquad
\textbf{(D) }99\qquad
\textbf{(E) }110$
2009 AMC 12/AHSME, 15
Assume $ 0 < r < 3$. Below are five equations for $ x$. Which equation has the largest solution $ x$?
$ \textbf{(A)}\ 3(1 \plus{} r)^x \equal{} 7\qquad \textbf{(B)}\ 3(1 \plus{} r/10)^x \equal{} 7\qquad \textbf{(C)}\ 3(1 \plus{} 2r)^x \equal{} 7$
$ \textbf{(D)}\ 3(1 \plus{} \sqrt {r})^x \equal{} 7\qquad \textbf{(E)}\ 3(1 \plus{} 1/r)^x \equal{} 7$
2010 Contests, 2
Find the remainder when \[9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}\] is divided by $ 1000$.
1996 AMC 8, 24
The measure of angle $ABC$ is $50^\circ $, $\overline{AD}$ bisects angle $BAC$, and $\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is
[asy]
pair A,B,C,D;
A = (0,0); B = (9,10); C = (10,0); D = (6.66,3);
dot(A); dot(B); dot(C); dot(D);
draw(A--B--C--cycle);
draw(A--D--C);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,N);
label("$50^\circ $",(9.4,8.8),SW);
[/asy]
$\text{(A)}\ 90^\circ \qquad \text{(B)}\ 100^\circ \qquad \text{(C)}\ 115^\circ \qquad \text{(D)}\ 122.5^\circ \qquad \text{(E)}\ 125^\circ $
2024 AMC 12/AHSME, 24
A $\textit{disphenoid}$ is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
$\textbf{(A) }\sqrt{3}\qquad\textbf{(B) }3\sqrt{15}\qquad\textbf{(C) }15\qquad\textbf{(D) }15\sqrt{7}\qquad\textbf{(E) }24\sqrt{6}$
2012 AMC 10, 1
Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms?
${{ \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72}\qquad\textbf{(E)}\ 80} $
1992 AMC 12/AHSME, 29
An "unfair" coin has a $2/3$ probability of turning up heads. If this coin is tossed $50$ times, what is the probability that the total number of heads is even?
$ \textbf{(A)}\ 25\left(\frac{2}{3}\right)^{50}\qquad\textbf{(B)}\ \frac{1}{2}\left(1 - \frac{1}{3^{50}}\right)\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{1}{2}\left(1 + \frac{1}{3^{50}}\right)\qquad\textbf{(E)}\ \frac{2}{3} $
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?
$\textbf{(A) }7\qquad\textbf{(B) }12\qquad\textbf{(C) }21\qquad\textbf{(D) }27\qquad\textbf{(E) }31$
1978 AMC 12/AHSME, 8
If $x\neq y$ and the sequences $x,a_1,a_2,y$ and $x,b_1,b_2,b_3,y$ each are in arithmetic progression, then $(a_2-a_1)/(b_2-b_1)$ equals
$\textbf{(A) }\frac{2}{3}\qquad\textbf{(B) }\frac{3}{4}\qquad\textbf{(C) }1\qquad\textbf{(D) }\frac{4}{3}\qquad \textbf{(E) }\frac{3}{2}$
2013 AMC 10, 20
A unit square is rotated $45^\circ$ about its center. What is the area of the region swept out by the interior of the square?
$ \textbf{(A)}\ 1-\frac{\sqrt2}2+\frac\pi4\qquad\textbf{(B)}\ \frac12+\frac\pi4\qquad\textbf{(C)}\ 2-\sqrt2+\frac\pi4\qquad\textbf{(D)}\ \frac{\sqrt2}2+\frac\pi4\qquad\textbf{(E)}\ 1+\frac{\sqrt2}4+\frac\pi8 $
2015 AMC 10, 4
Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?
$ \textbf{(A) }\dfrac{1}{12}\qquad\textbf{(B) }\dfrac{1}{6}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad\textbf{(E) }\dfrac{1}{2} $
2017 AMC 10, 7
Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
$\textbf{(A)}\ 2.0 \qquad \textbf{(B)}\ 2.2\qquad \textbf{(C)}\ 2.8 \qquad \textbf{(D)}\ 3.4 \qquad \textbf{(E)}\ 4.4$
2010 AMC 12/AHSME, 12
For what value of $ x$ does
\[ \log_{\sqrt{2}} \sqrt{x} \plus{} \log_2 x \plus{} \log_4 (x^2) \plus{} \log_8 (x^3) \plus{} \log_{16} (x^4) \equal{} 40?\]
$ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 256 \qquad \textbf{(E)}\ 1024$
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 $2$.)
$\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296$
2018 AMC 10, 10
In the rectangular parallelpiped shown, $AB = 3, BC= 1,$ and $CG = 2.$ Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$?
[asy]
size(250);
defaultpen(fontsize(10pt));
pair A =origin;
pair B = (4.75,0);
pair E1=(0,3);
pair F = (4.75,3);
pair G = (5.95,4.2);
pair C = (5.95,1.2);
pair D = (1.2,1.2);
pair H= (1.2,4.2);
pair M = ((4.75+5.95)/2,3.6);
draw(E1--M--H--E1--A--B--E1--F--B--M--C--G--H);
draw(B--C);
draw(F--G);
draw(A--D--H--C--D,dashed);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,E);
label("$D$",D,W);
label("$E$",E1,W);
label("$F$",F,SW);
label("$G$",G,NE);
label("$H$",H,NW);
label("$M$",M,N);
dot(A);
dot(B);
dot(E1);
dot(F);
dot(G);
dot(C);
dot(D);
dot(H);
dot(M);
label("3",A/2+B/2,S);
label("2",C/2+G/2,E);
label("1",C/2+B/2,SE);[/asy]
$\textbf{(A) } 1 \qquad \textbf{(B) } \frac{4}{3} \qquad \textbf{(C) } \frac{3}{2} \qquad \textbf{(D) } \frac{5}{3} \qquad \textbf{(E) } 2$
2013 Tuymaada Olympiad, 7
Points $A_1$, $A_2$, $A_3$, $A_4$ are the vertices of a regular tetrahedron of edge length $1$. The points $B_1$ and $B_2$ lie inside the figure bounded by the plane $A_1A_2A_3$ and the spheres of radius $1$ and centres $A_1$, $A_2$, $A_3$.
Prove that $B_1B_2 < \max\{B_1A_1, B_1A_2, B_1A_3, B_1A_4\}$.
[i] A. Kupavsky [/i]
2011 AIME Problems, 12
Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent.
2014 AMC 8, 23
Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of you two uniform numbers is today's date.
What number does Caitlin wear?
$\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$