Olympiad problems

2020 AMC 10, 20

Tags: AMC , AMC 10 , geometry , 3D geometry , prism , 2020 AMC , 2020 AMC 10B

Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}?$ $\textbf{(A) } 6 \qquad\textbf{(B) } 19 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 38$

2022 AMC 10, 14

Tags: AMC , AMC 10 , 2022 AMC 10a , 2022 AMC , 2022 AMC 12A , AMC 12

What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number? $\textbf{(A) }108\qquad\textbf{(B) }120\qquad\textbf{(C) }126\qquad\textbf{(D) }132\qquad\textbf{(E) }144$

2005 AMC 12/AHSME, 10

Tags: modular arithmetic , algebra , AMC 10 , recursion , Sequences

The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence? $ \textbf{(A)}\ 29\qquad \textbf{(B)}\ 55\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 133\qquad \textbf{(E)}\ 250$

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 $5$? $\textbf{(A) } 8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) } 12$

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] $\textbf{(A) }1700\qquad\textbf{(B) }1702\qquad\textbf{(C) }1703\qquad\textbf{(D) }1713\qquad\textbf{(E) }1715$

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? $\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$

2018 AMC 12/AHSME, 18

Tags: AMC 10 , 2018 AMC , 2018 AMC 10B , 2018 AMC 12B , AMC 12

A function $f$ is defined recursively by $f(1)=f(2)=1$ and $$f(n)=f(n-1)-f(n-2)+n$$ for all integers $n \geq 3$. What is $f(2018)$? $\textbf{(A)} \text{ 2016} \qquad \textbf{(B)} \text{ 2017} \qquad \textbf{(C)} \text{ 2018} \qquad \textbf{(D)} \text{ 2019} \qquad \textbf{(E)} \text{ 2020}$

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$

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$

2009 AMC 10, 2

Tags: AMC 10 , AMC

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55$

2019 AMC 10, 21

Tags: 2019 AMC , AMC , AMC 10 , 2019 AMC 10B , probability

Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head? $\textbf{(A) } \frac{1}{36} \qquad \textbf{(B) } \frac{1}{24} \qquad \textbf{(C) } \frac{1}{18} \qquad \textbf{(D) } \frac{1}{12} \qquad \textbf{(E) } \frac{1}{6}$

2012 AMC 10, 20

Tags: AMC , AMC 12 , AMC 12 B , inequalities , AMC 10 , number theory , modular arithmetic

Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

2022 AMC 10, 9

Tags: AMC , AMC 10 , 2022 AMC , 2022 AMC 10B

The sum \[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\dots+\frac{2021}{2022!}\] can be expressed as $a-\frac{1}{b!}$, where $a$ and $b$ are positive integers. What is $a+b$? $ \textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024$

2022 AMC 10, 20

Tags: geometry , rhombus , AMC , AMC 10 , 2022 AMC , 2022 AMC 10B , Solvable without cyclic

Let $ABCD$ be a rhombus with $\angle{ADC} = 46^{\circ}$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle{BFC}$? $\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 111 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 113 \qquad \textbf{(E)}\ 114$

2022 AMC 12/AHSME, 10

Tags: AMC , AMC 10 , 2022 AMC 10a , 2022 AMC , 2022 AMC 12A , AMC 12

What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number? $\textbf{(A) }108\qquad\textbf{(B) }120\qquad\textbf{(C) }126\qquad\textbf{(D) }132\qquad\textbf{(E) }144$

2020 AMC 10, 7

Tags: 2020 AMC , 2020 AMC 12A , 2020 AMC 10A , AMC 10 , AMC 12 , AMC

The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? $\textbf{(A) }2 \qquad\textbf{(B) } 5\qquad\textbf{(C) } 10\qquad\textbf{(D) } 25\qquad\textbf{(E) } 50$

2019 AMC 12/AHSME, 9

Tags: AMC , AMC 10 , AMC 12 , 2019 AMC 12A , 2019 AMC 10A , 2019 AMC , recursion

A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and $$a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$$for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive inegers. What is $p+q ?$ $\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$

2022 AMC 10, 6

Tags: AMC 10 , AMC , 2022 AMC 10a , 2022 AMC , algebra , absolute value

Which expression is equal to $\left | a-2-\sqrt{(a-1)^2} \right|$ for $a<0$? $\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textbf{(C) } 1 \qquad \textbf{(D) } a+1 \qquad \textbf{(E) } 3$

2017 AMC 12/AHSME, 23

Tags: AMC , AMC 10 , AMC 12 , algebra , polynomial , 2017 AMC 10A , 2017 AMC 12A

For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$? $\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$

2017 AMC 10, 8

Tags: AMC , AMC 10 , AMC 10 B , analytic geometry , 2017 AMC 10B

Points $A(11,9)$ and $B(2,-3)$ are vertices of $\triangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(-1, 3)$. What are the coordinates of point $C$? $\textbf{(A) } (-8, 9)\qquad \textbf{(B) } (-4, 8)\qquad \textbf{(C) } (-4,9)\qquad \textbf{(D) } (-2, 3)\qquad \textbf{(E) } (-1, 0)$

2016 AMC 12/AHSME, 9

Tags: AMC , AMC 10 , 2016 AMC 10B , AMC 10 B

Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden? $\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$

2019 AMC 10, 3

Tags: AMC , AMC 10 , AMC 10 A , 2019 AMC 10A , 2019 AMC

Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$ $\textbf{(A) } 3 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 12 \qquad\textbf{(E) } 15$

2021 AMC 12/AHSME Fall, 6

Tags: AMC , AMC 10 , AMC 10 B , AMC 12 , AMC 12 B

The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor of $16383$? $\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }10\qquad\textbf{(D) }16\qquad\textbf{(E) }22$

2021 AMC 12/AHSME Fall, 3

Tags: AMC 10 B , AMC , AMC 10 , AMC 12 , AMC 12 B

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$ $(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$