This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 3632

2023 AMC 10, 3
Tags: AMC , AMC 10 , AMC 12 , 2023 AMC , 2023 amc 10A , 2023 AMC 12A
How many positive perfect squares less than $2023$ are divisible by $5$? $\textbf{(A) } 8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) } 12$

2014 AMC 12/AHSME, 3
Tags: AMC
Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip? $ \textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7} $

2006 AIME Problems, 6
Tags: AMC
Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\mathcal{S}$.

2014 AMC 10, 2
Tags: ceiling function , AMC
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box? ${ \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$

1987 AMC 12/AHSME, 13
Tags: AMC
A long piece of paper $5$ cm wide is made into a roll for cash registers by wrapping it $600$ times around a cardboard tube of diameter $2$ cm, forming a roll $10$ cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms $600$ concentric circles with diameters evenly spaced from $2$ cm to $10$ cm.) $ \textbf{(A)}\ 36\pi \qquad\textbf{(B)}\ 45\pi \qquad\textbf{(C)}\ 60\pi \qquad\textbf{(D)}\ 72\pi \qquad\textbf{(E)}\ 90\pi $

2024 AMC 12/AHSME, 12
Tags: complex numbers , geometry , AMC , AMC 12 , AMC 12 B , 2024 AMC , 2024 AMC 12B
Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$? $\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\displaystyle\frac{4}{3}\qquad\textbf{(D)}~\displaystyle\frac{3}{2}\qquad\textbf{(E)}~\displaystyle\frac{5}{3}$

2012 AMC 8, 22
Tags: AMC
Let $R$ be a set of nine distinct integers. Six of the elements are 2, 3, 4, 6, 9, and 14. What is the number of possible values of the median of $R$ ? $\textbf{(A)}\hspace{.05in}4 \qquad \textbf{(B)}\hspace{.05in}5 \qquad \textbf{(C)}\hspace{.05in}6 \qquad \textbf{(D)}\hspace{.05in}7 \qquad \textbf{(E)}\hspace{.05in}8 $

2024 AMC 10, 12
Tags: AMC , AMC 10 , 2024 AMC 10A , 2024 AMC , mean
Zelda played the [i]Adventures of Math[/I] game on August 1 and scored $1700$ points. She continued to play daily over the next $5$ days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was $1700 + 80 = 1780$ points.) What was Zelda's average score in points over the $6$ days? [img]https://cdn.artofproblemsolving.com/attachments/5/c/d246d9bf4002bfe23f859bd21605f882d8b7bc.png[/img] $\textbf{(A) }1700\qquad\textbf{(B) }1702\qquad\textbf{(C) }1703\qquad\textbf{(D) }1713\qquad\textbf{(E) }1715$

2013 AMC 12/AHSME, 13
Tags: geometry , trapezoid , AMC , AMC 12 , AMC 12 B , arithmetic sequence
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA=\angle DCB$ and $\angle ADB=\angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$? ${\textbf{(A)}\ 210\qquad\textbf{(B)}\ 220\qquad\textbf{(C)}\ 230\qquad\textbf{(D}}\ 240\qquad\textbf{(E)}\ 250$

2016 AMC 12/AHSME, 14
Tags: AMC 10 , geometry , 3D geometry , combinatorics , AMC , AMC 12 , 2016 AMC 10A
Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible? $\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$

1993 AMC 12/AHSME, 5
Tags: percent , percent increase , AMC
Last year a bicycle cost $\$160$ and a cycling helmet cost $ \$ 40$. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is $ \textbf{(A)}\ 6\% \qquad\textbf{(B)}\ 7\% \qquad\textbf{(C)}\ 7.5\% \qquad\textbf{(D)}\ 8\% \qquad\textbf{(E)}\ 15\% $

2008 AIME Problems, 1
Tags: AMC , AIME , AIME I
Of the students attending a school party, $ 60\%$ of the students are girls, and $ 40\%$ of the students like to dance. After these students are joined by $ 20$ more boy students, all of whom like to dance, the party is now $ 58\%$ girls. How many students now at the party like to dance?

1963 AMC 12/AHSME, 4
Tags: AMC
For what value(s) of $k$ does the pair of equations $y=x^2$ and $y=3x+k$ have two identical solutions? $\textbf{(A)}\ \dfrac{4}{9} \qquad \textbf{(B)}\ -\dfrac{4}{9} \qquad \textbf{(C)}\ \dfrac{9}{4} \qquad \textbf{(D)}\ -\dfrac{9}{4} \qquad \textbf{(E)}\ \pm\dfrac{9}{4}$

2010 Contests, 1
Tags: AMC , USA(J)MO , USAMO , USAJMO , geometry , rectangle
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

2020 AMC 12/AHSME, 5
Tags: 2020 AMC 12B , AMC 12 B , AMC , AMC 12 , 2020 AMC , counting
Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\tfrac{2}{3}$ of its games and team $B$ has won $\tfrac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ more games than team $A.$ How many games has team $A$ played? $\textbf{(A) } 21 \qquad \textbf{(B) } 27 \qquad \textbf{(C) } 42 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 63$

2013 AMC 10, 9
Tags: AMC
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on $20\%$ of her three-point shots and $30\%$ of her two-point shots. Shenille attempted $30$ shots. How many points did she score? $ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 36 $

2024 AMC 12/AHSME, 15
Tags: AMC , AMC 12 , 2024 AMC 12A , 2024 AMC , algebra , those who know
The roots of $x^3 + 2x^2 - x + 3$ are $p, q,$ and $r.$ What is the value of \[(p^2 + 4)(q^2 + 4)(r^2 + 4)?\] $\textbf{(A) } 64 \qquad \textbf{(B) } 75 \qquad \textbf{(C) } 100 \qquad \textbf{(D) } 125 \qquad \textbf{(E) } 144$

1988 AMC 12/AHSME, 29
Tags: analytic geometry , graphing lines , slope , AMC
You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points $(x_{1},y_{1})$, $(x_{2},y_{2})$, $(x_{3},y_{3})$. If \[x_{1} < x_{2} < x_{3}\quad\text{ and }\quad x_{3} - x_{2} = x_{2} - x_{1},\] which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line. $ \textbf{(A)}\ \frac{y_{3} - y_{1}}{x_{3} - x_{1}}\qquad\textbf{(B)}\ \frac{(y_{2} - y_{1}) - (y_{3} - y_{2})}{x_{3} - x_{1}}\qquad\textbf{(C)}\ \frac{2y_{3} - y_{1} - y_{2}}{2x_{3} - x_{1} - x_{2}}\qquad\textbf{(D)}\ \frac{y_{2} - y_{1}}{x_{2} - x_{1}} + \frac{y_{3} - y_{2}}{x_{3} - x_{2}}\qquad\textbf{(E)}\ \text{none of these} $

2021 AMC 12/AHSME Fall, 2
Tags: AMC , AMC 10 , AMC 10 A , amc12a
Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$

2002 AMC 10, 20
Tags: symmetry , AMC , AMC 10 , geometry , geometric transformation , rotation , linear algebra
Let $ a$, $ b$, and $ c$ be real numbers such that $ a \minus{} 7b \plus{} 8c \equal{} 4$ and $ 8a \plus{} 4b \minus{} c \equal{} 7$. Then $ a^2 \minus{} b^2 \plus{} c^2$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2015 AMC 10, 12
Tags: quadratics , algebra , polynomial , Vieta , quadratic formula , absolute value , AMC
Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$. What is $|a-b|$? $ \textbf{(A) }1\qquad\textbf{(B) }\dfrac{\pi}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{1+\pi}\qquad\textbf{(E) }1+\sqrt{\pi} $

2007 AMC 10, 23
Tags: AMC , AMC 10 , algebra , difference of squares , special factorizations
How many ordered pairs $ (m,n)$ of positive integers, with $ m > n$, have the property that their squares differ by $ 96$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$

1972 AMC 12/AHSME, 24
Tags: AMC
A man walked a certain distance at a constant rate. If he had gone $\textstyle\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\textstyle\frac{1}{2}$ mile per hour slower, he would have been $2\textstyle\frac{1}{2}$ hours longer on the road. The distance in miles he walked was $\textbf{(A) }13\textstyle\frac{1}{2}\qquad\textbf{(B) }15\qquad\textbf{(C) }17\frac{1}{2}\qquad\textbf{(D) }20\qquad \textbf{(E) }25$

2018 AIME Problems, 12
Tags: AMC , AIME , AIME I
For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$, let $s(T)$ be the sum of the elements of $T$, with $s(\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.

2006 AMC 10, 18
Tags: AMC
Let $ a_1, a_2, ...$ be a sequence for which \[a_1 \equal{} 2\,\hspace{.2in}a_2 \equal{} 3\, \hspace{.2in}\text{and}\hspace{.2in}a_n \equal{} \frac {a_{n \minus{} 1}}{a_{n \minus{} 2}} \text{ for each positive integer } n \ge 3.\]What is $ a_{2006}$? $\textbf{(A) } \frac 12 \qquad \textbf{(B) } \frac 23 \qquad \textbf{(C) } \frac 32 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } 3$