Found problems: 3632
2014 Contests, 2
Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$
2025 AIME, 10
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$.
2008 Tuymaada Olympiad, 2
Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime?
[i]Author: L. Emelyanov[/i]
[hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$
may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]
2019 AMC 10, 18
Henry decides one morning to do a workout, and he walks $\tfrac{3}{4}$ of the way from his home to his gym. The gym is $2$ kilometers away from Henry's home. At that point, he changes his mind and walks $\tfrac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\tfrac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\tfrac{3}{4}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|$?
$\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 1\frac{1}{5} \qquad \textbf{(D) } 1\frac{1}{4} \qquad \textbf{(E) } 1\frac{1}{2}$
2007 AMC 10, 8
Triangles $ ABC$ and $ ADC$ are isosceles with $ AB \equal{} BC$ and $ AD \equal{} DC$. Point D is inside $ \triangle ABC$. $ \angle ABC \equal{} 40^\circ$, and $ \angle ADC \equal{} 140^\circ$. What is the degree measure of $ \angle BAD$?
$ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$
1976 AMC 12/AHSME, 16
In triangles $ABC$ and $DEF$, lengths $AC,~BC,~DF,$ and $EF$ are all equal. Length $AB$ is twice the length of the altitude of $\triangle DEF$ from $F$ to $DE$. Which of the following statements is (are) true?
$\textbf{I. }\angle ACB \text{ and }\angle DFE\text{ must be complementary.}$
$\textbf{II. }\angle ACB \text{ and }\angle DFE\text{ must be supplementary.}$
${\textbf{III. }\text{The area of }\triangle ABC\text{ must equal the area of }\triangle DEF.}$
${\textbf{IV. }\text{The area of }\triangle ABC\text{ must equal twice the area of }\triangle DEF.}$
$\textbf{(A) }\textbf{II. }\text{only}\qquad\textbf{(B) }\textbf{III. }\text{only}\qquad$
$\textbf{(C) }\textbf{IV. }\text{only}\qquad\textbf{(D) }\text{I. }\text{and }\textbf{III. }\text{only}\qquad \textbf{(E) }\textbf{II. }\text{and }\textbf{III. }\text{only}$
2016 AMC 10, 15
Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?
[asy]
draw(circle((0,0),3));
draw(circle((0,0),1));
draw(circle((1,sqrt(3)),1));
draw(circle((-1,sqrt(3)),1));
draw(circle((-1,-sqrt(3)),1)); draw(circle((1,-sqrt(3)),1));
draw(circle((2,0),1)); draw(circle((-2,0),1)); [/asy]
$\textbf{(A) } \sqrt{2} \qquad \textbf{(B) } 1.5 \qquad \textbf{(C) } \sqrt{\pi} \qquad \textbf{(D) } \sqrt{2\pi} \qquad \textbf{(E) } \pi$
2017 AMC 10, 4
Mia is “helping” her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time?
$\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5$
1987 Flanders Math Olympiad, 3
Find all continuous functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x)^3 = -\frac x{12}\cdot\left(x^2+7x\cdot f(x)+16\cdot f(x)^2\right),\ \forall x \in \mathbb{R}.\]
2012 AMC 8, 8
A shop advertises everything is "half price in today's sale." In addition, a coupon gives a $20\%$ discount on sale prices. Using the coupon, the price today represents what percentage off the original price?
$\textbf{(A)}\hspace{.05in}10 \qquad \textbf{(B)}\hspace{.05in}33 \qquad \textbf{(C)}\hspace{.05in}40 \qquad \textbf{(D)}\hspace{.05in}60 \qquad \textbf{(E)}\hspace{.05in}70 $
1994 AIME Problems, 10
In triangle $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D.$ The lengths of the sides of $\triangle ABC$ are integers, $BD=29^3,$ and $\cos B=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2021 AMC 12/AHSME Spring, 5
The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$?
$\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$
1971 AMC 12/AHSME, 16
After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The ratio of the second average to the true average was
$\textbf{(A) }1:1\qquad\textbf{(B) }35:36\qquad\textbf{(C) }36:35\qquad\textbf{(D) }2:1\qquad \textbf{(E) }\text{None of these}$
1992 AMC 12/AHSME, 9
Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair C=origin, N=dir(0), B=dir(20), A=dir(135), M=dir(180), P=(3/7)*dir(C--N);
draw((0,0)--(1,sqrt(3))--(2,0)--(3,sqrt(3))--(4,0)--(5,sqrt(3))--(6,0));
draw((1,0)--(2,sqrt(3))--(3,0)--(4,sqrt(3))--(5,0));
draw((-1.5,0)--(7.5,0));
[/asy]
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 10\sqrt{3}\qquad\textbf{(E)}\ 12\sqrt{3} $
1983 AIME Problems, 8
What is the largest 2-digit prime factor of the integer $n = \binom{200}{100}$?
1999 AIME Problems, 8
Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
2012 AIME Problems, 4
Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins walking as Sundance rides. When Sundance reaches the first of their hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at 6, 4, and 2.5 miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are $n$ miles from Dodge, and have been traveling for $t$ minutes. Find $n + t$.
2015 AMC 12/AHSME, 21
A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16$. The set of all possible values of $r$ is an interval $[a,b)$. What is $a+b$?
$\textbf{(A) }5\sqrt2+4\qquad\textbf{(B) }\sqrt{17}+7\qquad\textbf{(C) }6\sqrt2+3\qquad\textbf{(D) }\sqrt{15}+8\qquad\textbf{(E) }12$
2009 AMC 10, 4
Eric plans to compete in a triathlon. He can average $ 2$ miles per hour in the $ \tfrac{1}{4}$-mile swim and $ 6$ miles per hour in the $ 3$-mile run. His goal is to finish the triathlon in $ 2$ hours. To accomplish his goal what must his average speed, in miles per hour, be for the $ 15$-mile bicycle ride?
$ \textbf{(A)}\ \frac{120}{11} \qquad
\textbf{(B)}\ 11 \qquad
\textbf{(C)}\ \frac{56}{5} \qquad
\textbf{(D)}\ \frac{45}{4} \qquad
\textbf{(E)}\ 12$
2021 AMC 12/AHSME Spring, 7
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$
2013 AMC 10, 8
What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\]
$ \textbf{(A)}\ -1\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ 2013\qquad\textbf{(E)}\ 2^{4024} $
1979 AMC 12/AHSME, 27
An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each
such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^ 2 + bx + c = 0$ will
[i]not[/i] have distinct positive real roots?
$\textbf{(A) }\frac{106}{121}\qquad\textbf{(B) }\frac{108}{121}\qquad\textbf{(C) }\frac{110}{121}\qquad\textbf{(D) }\frac{112}{121}\qquad\textbf{(E) }\text{none of these}$
2012 AMC 12/AHSME, 4
Suppose that the euro is worth $1.30$ dollars. If Diana has $500$ dollars and Etienne has $400$ euros, by what percent is the value of Etienne's money greater than the value of Diana's money?
${{ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6.5\qquad\textbf{(D)}\ 8}\qquad\textbf{(E)}\ 13} $
2021 AMC 10 Fall, 14
Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$
$\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}$
2010 USAJMO, 4
A triangle is called a parabolic triangle if its vertices lie on a parabola $y = x^2$. Prove that for every nonnegative integer $n$, there is an odd number $m$ and a parabolic triangle with vertices at three distinct points with integer coordinates with area $(2^nm)^2$.