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In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the $28$ students in the program, $25\%$ of the juniors and $10\%$ of the seniors are on the debate team. How many juniors are in the program?
$\textbf{(A)} ~5 \qquad\textbf{(B)... | Say there are $j$ juniors and $s$ seniors in the program. Converting percentages to fractions, $\frac{j}{4}$ and $\frac{s}{10}$ are on the debate team, and since an equal number of juniors and seniors are on the debate team, $\frac{j}{4} = \frac{s}{10}.$
Cross-multiplying and simplifying we get $5j=2s.$ Additionally, s... | 8 | Algebra | MCQ | Yes | Yes | amc_aime | false |
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 ... | The final image of $P$ is $(-6,3)$. We know the reflection rule for reflecting over $y=-x$ is $(x,y) \rightarrow (-y, -x)$. So before the reflection and after rotation the point is $(-3,6)$.
By definition of rotation, the slope between $(-3,6)$ and $(1,5)$ must be perpendicular to the slope between $(a,b)$ and $(1,5)$.... | 7 | Geometry | MCQ | Yes | Yes | amc_aime | false |
The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\text{nine}}.$ What is the remainder when $N$ is divided by $5?$
$\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) }4$ | Recall that $9\equiv-1\pmod{5}.$ We expand $N$ by the definition of bases:
\begin{align*} N&=27{,}006{,}000{,}052_9 \\ &= 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2 \\ &\equiv 2\cdot(-1)^{10} + 7\cdot(-1)^9 + 6\cdot(-1)^6 + 5\cdot(-1) + 2 &&\pmod{5} \\ &\equiv 2-7+6-5+2 &&\pmod{5} \\ &\equiv -2 &&\pmod{5} \\ &\... | 3 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations?
\begin{align*} x^2+3y&=9 \\ (|x|+|y|-4)^2 &= 1 \end{align*}
$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 7$ | The second equation is $(|x|+|y| - 4)^2 = 1$. We know that the graph of $|x| + |y|$ is a very simple diamond shape, so let's see if we can reduce this equation to that form: \[(|x|+|y| - 4)^2 = 1 \implies |x|+|y| - 4 = \pm 1 \implies |x|+|y| = \{3,5\}.\]
We now have two separate graphs for this equation and one graph f... | 5 | Algebra | MCQ | Yes | Yes | amc_aime | false |
The graph of \[f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|\] is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) }\text{the }y\text{-axis}\qquad \textbf{(B) }\text{the line }x = 1\qquad \textbf{(C) }\text{the origin}\qquad \textbf{(D) }\... | Note that \[f(1-x)=|\lfloor 1-x\rfloor|-|\lfloor x\rfloor|=-f(x),\]
so $f\left(\frac12+x\right)=-f\left(\frac12-x\right)$.
This means that the graph is symmetric about $\boxed{\textbf{(D) }\text{ the point }\left(\dfrac12, 0\right)}$. | 0 | Algebra | MCQ | Yes | Yes | amc_aime | false |
A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a+\frac{b\pi}{c}$, where $a,b$, and... | The side length of the inner square traced out by the disk with radius $1$ is $s-4.$ However, there is a piece at each corner (bounded by two line segments and one $90^\circ$ arc) where the disk never sweeps out. The combined area of these four pieces is $(1+1)^2-\pi\cdot1^2=4-\pi.$ As a result, we have \[A=s^2-(s-4)^2... | 10 | Geometry | MCQ | Yes | Yes | amc_aime | false |
For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions?
$\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16 \qquad$ | A quadratic equation does not have two distinct real solutions if and only if the discriminant is nonpositive. We conclude that:
Since $x^2+bx+c=0$ does not have real solutions, we have $b^2\leq 4c.$
Since $x^2+cx+b=0$ does not have real solutions, we have $c^2\leq 4b.$
Squaring the first inequality, we get $b^4\leq... | 6 | Algebra | MCQ | Yes | Yes | amc_aime | false |
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$ | First, we can test values that would make $f(x)=12$ true. For this to happen $x$ must have $6$ divisors, which means its prime factorization is in the form $pq^2$ or $p^5$, where $p$ and $q$ are prime numbers. Listing out values less than $50$ which have these prime factorizations, we find $12,18,20,28,44,45,50$ for $p... | 10 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$ | The volume of the cube is $V_{\text{cube}}=6^3=216,$ and the volume of a clay ball is $V_{\text{ball}}=\frac43\cdot\pi\cdot2^3=\frac{32}{3}\pi.$
Since the balls can be reshaped but not compressed, the maximum number of balls that can completely fit inside a cube is \[\left\lfloor\frac{V_{\text{cube}}}{V_{\text{ball}}}\... | 6 | Geometry | MCQ | Yes | Yes | amc_aime | false |
The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 9$ | First, modulo $2$ or $5$, $\underline{20210A} \equiv A$.
Hence, $A \neq 0, 2, 4, 5, 6, 8$.
Second modulo $3$, $\underline{20210A} \equiv 2 + 0 + 2 + 1 + 0 + A \equiv 5 + A$.
Hence, $A \neq 1, 4, 7$.
Third, modulo $11$, $\underline{20210A} \equiv A + 1 + 0 - 0 - 2 - 2 \equiv A - 3$.
Hence, $A \neq 3$.
Therefore, the ans... | 9 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in fee... | There are $41-1=40$ gaps between the $41$ telephone poles, so the distance of each gap is $5280\div40=132$ feet.
Each of Oscar's leaps covers $132\div12=11$ feet, and each of Elmer's strides covers $132\div44=3$ feet.
Therefore, Oscar's leap is $11-3=\boxed{\textbf{(B) }8}$ feet longer than Elmer's stride.
~MRENTHUSIAS... | 8 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is\[f(2) + f(3) + f(4) + f(5)+ f(6)?\]$(\textbf{A})\: 8\qquad(\textbf{B}) \: 9\qquad(\textbf{C}) \: 11\qquad(\textbf{D}) \: 22\qquad(\textbf{E}) \: 29$ | We can rewrite $N$ as $\frac{7}{9}\cdot 9999\ldots999 = \frac{7}{9}\cdot(10^{313}-1)$.
When approximating values, as we will shortly do, the minus one will become negligible so we can ignore it.
When we take the power of ten out of the square root, we’ll be multiplying by another power of ten, so the leading digit wil... | 8 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(... | The special fractions are \[\frac{1}{14},\frac{2}{13},\frac{3}{12},\frac{4}{11},\frac{5}{10},\frac{6}{9},\frac{7}{8},\frac{8}{7},\frac{9}{6},\frac{10}{5},\frac{11}{4},\frac{12}{3},\frac{13}{2},\frac{14}{1}.\]
We rewrite them in the simplest form: \[\frac{1}{14},\frac{2}{13},\frac{1}{4},\frac{4}{11},\frac{1}{2},\frac{2}... | 11 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$?
$\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22$ | We have
\begin{align*} 16383 & = 2^{14} - 1 \\ & = \left( 2^7 + 1 \right) \left( 2^7 - 1 \right) \\ & = 129 \cdot 127 \\ \end{align*}
Since $129$ is composite, $127$ is the largest prime which can divide $16383$. The sum of $127$'s digits is $1+2+7=\boxed{\textbf{(C) }10}$.
~Steven Chen (www.professorchenedu.com) ~NH14... | 10 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
Ted mistakenly wrote $2^m\cdot\sqrt{\frac{1}{4096}}$ as $2\cdot\sqrt[m]{\frac{1}{4096}}.$ What is the sum of all real numbers $m$ for which these two expressions have the same value?
$\textbf{(A) } 5 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$ | We are given that \[2^m\cdot\sqrt{\frac{1}{4096}} = 2\cdot\sqrt[m]{\frac{1}{4096}}.\]
Converting everything into powers of $2,$ we have
\begin{align*} 2^m\cdot(2^{-12})^{\frac12} &= 2\cdot (2^{-12})^{\frac1m} \\ 2^{m-6} &= 2^{1-\frac{12}{m}} \\ m-6 &= 1-\frac{12}{m}. \end{align*}
We multiply both sides by $m$, then rea... | 7 | Algebra | MCQ | Yes | Yes | amc_aime | false |
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 state... | Note that:
Truth-tellers would answer yes-no-no to the three questions in this order.
Liars would answer yes-yes-no to the three questions in this order.
Alternaters who responded truth-lie-truth would answer no-no-no to the three questions in this order.
Alternaters who responded lie-truth-lie would answer yes-yes-ye... | 7 | Logic and Puzzles | MCQ | Yes | Yes | amc_aime | false |
Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC.$ The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D.$ Suppose $BP=2$ and $PC=3.$ What is $AD?$
$\textbf{(A) } 8 \qquad \tex... | Suppose that $\overline{BD}$ intersects $\overline{AP}$ and $\overline{AC}$ at $X$ and $Y,$ respectively. By Angle-Side-Angle, we conclude that $\triangle ABX\cong\triangle AYX.$
Let $AB=AY=2x.$ By the Angle Bisector Theorem, we have $AC=3x,$ or $YC=x.$
By alternate interior angles, we get $\angle YAD=\angle YCB$ and $... | 10 | Geometry | MCQ | Yes | Yes | amc_aime | false |
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
\[\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{17}=\frac{h}{L_{17}}\]
What is the remainder when $h$ is divided by $17$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 ... | Notice that $L_{17}$ contains the highest power of every prime below $17$ since higher primes cannot divide $L_{17}$. Thus, $L_{17}=16\cdot 9 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17$.
When writing the sum under a common fraction, we multiply the denominators by $L_{17}$ divided by each denominator. However, since $L... | 5 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
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 \tex... | Let $x$ be the third number. It follows that the first number is $6x,$ and the second number is $x+40.$
We have \[6x+(x+40)+x=8x+40=96,\] from which $x=7.$
Therefore, the first number is $42,$ and the second number is $47.$ Their absolute value of the difference is $|42-47|=\boxed{\textbf{(E) } 5}.$
~MRENTHUSIASM | 5 | Algebra | MCQ | Yes | Yes | amc_aime | false |
The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$?
$\textbf{(A) } 3 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 12$ | Note that
\begin{align*} 18 &= 2\cdot3^2, \\ 180 &= 2^2\cdot3^2\cdot5, \\ 45 &= 3^2\cdot5 \\ 15 &= 3\cdot5. \end{align*}
Let $n = 2^a\cdot3^b\cdot5^c.$ It follows that:
From the least common multiple condition, we have \[\operatorname{lcm}(n,18) = \operatorname{lcm}(2^a\cdot3^b\cdot5^c,2\cdot3^2) = 2^{\max(a,1)}\cdot... | 6 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ ... | Let $M$ be the median. It follows that the two largest integers are both $M+2.$
Let $a$ and $b$ be the two smallest integers such that $a<b.$ The sorted list is \[a,b,M,M+2,M+2.\]
Since the median is $2$ greater than their arithmetic mean, we have $\frac{a+b+M+(M+2)+(M+2)}{5}+2=M,$ or \[a+b+14=2M.\]
Note that $a+b$ mus... | 11 | Algebra | MCQ | Yes | Yes | amc_aime | false |
A pair of fair $6$-sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$?
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$ | Rolling a pair of fair $6$-sided dice, the probability of getting a sum of $7$ is $\frac16:$ Regardless what the first die shows, the second die has exactly one outcome to make the sum $7.$ We consider the complement: The probability of not getting a sum of $7$ is $1-\frac16=\frac56.$ Rolling the pair of dice $n$ times... | 4 | Combinatorics | MCQ | Yes | Yes | amc_aime | false |
Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define
\[S_n = \sum_{k=0}^{n-1} x_k 2^k\]
Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$. What is the value of the sum
\[x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?\]
$\textbf{(A) } 6 \qq... | In binary numbers, we have \[S_n = (x_{n-1} x_{n-2} x_{n-3} x_{n-4} \ldots x_{2} x_{1} x_{0})_2.\]
It follows that \[8S_n = (x_{n-1} x_{n-2} x_{n-3} x_{n-4} \ldots x_{2} x_{1} x_{0}000)_2.\]
We obtain $7S_n$ by subtracting the equations:
\[\begin{array}{clccrccccccr} & (x_{n-1} & x_{n-2} & x_{n-3} & x_{n-4} & \ldots ... | 6 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
What is the value of \[\frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}?\]
$\textbf{(A)}\ \sqrt3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \sqrt{15} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{105}$ | We apply the difference of squares to the denominator, and then regroup factors:
\begin{align*} \frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}} &= \frac{\left(1+\frac13\right)\left(1+\frac15\right)\lef... | 2 | Algebra | MCQ | Yes | Yes | amc_aime | false |
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 16$ | Let $p$ and $q$ be the roots of $x^{2}+kx+36.$ By [Vieta's Formulas](https://artofproblemsolving.com/wiki/index.php/Vieta%27s_Formulas), we have $p+q=-k$ and $pq=36.$
It follows that $p$ and $q$ must be distinct factors of $36.$ The possibilities of $\{p,q\}$ are \[\pm\{1,36\},\pm\{2,18\},\pm\{3,12\},\pm\{4,9\}.\]
Each... | 8 | Algebra | MCQ | Yes | Yes | amc_aime | false |
A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$ | Note Euler's formula where $\text{Vertices}+\text{Faces}-\text{Edges}=2$. There are $12$ faces and the number of edges is $24$ because there are 12 faces each with four edges and each edge is shared by two faces. Now we know that there are $14$ vertices on the figure. Now note that the sum of the degrees of all the poi... | 8 | Geometry | MCQ | Yes | Yes | amc_aime | false |
The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$. What is $|r-s|$?
$\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{2}{3} \qquad \textbf{(E) } 1$ | Due to rotations preserving an equal distance, we can bash the answer with the distance formula. $D(A, P) = D(A', P)$, and $D(B, P) = D(B',P)$.
Thus we will square our equations to yield:
$(1-r)^2+(2-s)^2=(3-r)^2+(1-s)^2$, and $(3-r)^2+(3-s)^2=(4-r)^2+(3-s)^2$.
Canceling $(3-s)^2$ from the second equation makes it cle... | 1 | Geometry | MCQ | Yes | Yes | amc_aime | false |
A quadrilateral has all integer sides lengths, a perimeter of $26$, and one side of length $4$. What is the greatest possible length of one side of this quadrilateral?
$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad\textbf{(E) }13$ | Let's use the triangle inequality. We know that for a triangle, the sum of the 2 shorter sides must always be longer than the longest side. This is because if the longest side were to be as long as the sum of the other sides, or longer, we would only have a line.
Similarly, for a convex quadrilateral, the sum of the sh... | 12 | Geometry | MCQ | Yes | Yes | amc_aime | false |
How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2 = m^2n^2$?
$\textbf{(A) }7\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }5$ | Clearly, $m=0,n=0$ is 1 solution. However there are definitely more, so we apply [Simon's Favorite Factoring Expression](https://artofproblemsolving.comhttps://artofproblemsolving.com/wiki/index.php/Simon%27s_Favorite_Factoring_Trick) to get this:
\begin{align*} m^2+mn+n^2 &= m^2n^2\\ m^2+mn+n^2 +mn &= m^2n^2 +mn\\ (m+... | 3 | Algebra | MCQ | Yes | Yes | amc_aime | false |
How many distinct values of $x$ satisfy
$\lfloor{x}\rfloor^2-3x+2=0$, where $\lfloor{x}\rfloor$ denotes the largest integer less than or equal to $x$?
$\textbf{(A) } \text{an infinite number} \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 0$ | To further grasp at this equation, we rearrange the equation into
\[\lfloor{x}\rfloor^2=3x-2.\]
Thus, $3x-2$ is a perfect square and nonnegative. It is now much more apparent that $x \ge 2/3,$ and that $x = 2/3$ is a solution.
Additionally, by observing the RHS, $x 3\cdot4,\]
since squares grow quicker than linear func... | 4 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$. Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$. How many numbers are written on the blackboard?
$\textbf{(A) }... | Let there be $n$ numbers in the list of numbers, and let their sum be $S$. Then we have the following
\[S+3n=45\]
\[3S=45\]
From the second equation, $S=15$. So, $15+3n=45$ $\Rightarrow$ $n=\boxed{\textbf{(A) }10}.$
~Mintylemon66 (formatted atictacksh) | 10 | Algebra | MCQ | Yes | Yes | amc_aime | false |
What is the units digit of $2022^{2023} + 2023^{2022}$?
$\text{(A)}\ 7 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 3$ | $2022^{2023} + 2023^{2022} \equiv 2^3 + 3^2 \equiv 17 \equiv 7$ (mod 10). $\boxed{\text{A}}$
~andliu766 | 7 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
In [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) $ABC$, $AB = 13$, $BC = 14$, $AC = 15$. Let $D$ denote the [midpoint](https://artofproblemsolving.com/wiki/index.php/Midpoint) of $\overline{BC}$ and let $E$ denote the intersection of $\overline{BC}$ with the [bisector](https://artofproblemsolving.... | The answer is exactly $3$, choice $\mathrm{(C)}$.
We can find the area of triangle $ADE$ by using the simple formula $\frac{bh}{2}$. Dropping an altitude from $A$, we see that it has length $12$ (we can split the large triangle into a $9-12-15$ and a $5-12-13$ triangle). Then we can apply the [Angle Bisector Theorem](h... | 3 | Geometry | MCQ | Yes | Yes | amc_aime | false |
If $x,y,$ and $z$ are positive numbers satisfying
\[x + \frac{1}{y} = 4,\qquad y + \frac{1}{z} = 1, \qquad \text{and} \qquad z + \frac{1}{x} = \frac{7}{3}\]
Then what is the value of $xyz$ ?
$\text {(A)}\ \frac{2}{3} \qquad \text {(B)}\ 1 \qquad \text {(C)}\ \frac{4}{3} \qquad \text {(D)}\ 2 \qquad \text {(E)}\ \frac{7... | We multiply all given expressions to get:
\[(1)xyz + x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{xyz} = \frac{28}{3}\]
Adding all the given expressions gives that
\[(2) x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 4 + \frac{7}{3} + 1 = \frac{22}{3}\]
We subtract $(2)$ from $(1)$ to get that ... | 1 | Algebra | MCQ | Yes | Yes | amc_aime | false |
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common en... | Given any path on the surface, we can unfold the surface into a plane to get a path of the same length in the plane. Consider the net of a tetrahedron in the picture below. A pair of opposite points is marked by dots. It is obvious that in the plane the shortest path is just a segment that connects these two points. It... | 1 | Geometry | MCQ | Yes | Yes | amc_aime | false |
The polynomial $p(x) = x^3+ax^2+bx+c$ has the property that the average of its zeros, the product of its zeros, and the sum of its coefficients are all equal. The $y$-intercept of the graph of $y=p(x)$ is 2. What is $b$?
$(\mathrm{A})\ -11 \qquad (\mathrm{B})\ -10 \qquad (\mathrm{C})\ -9 \qquad (\mathrm{D})\ 1 \qquad (... | We are given $c=2$. So the product of the roots is $-c = -2$ by [Vieta's formulas](https://artofproblemsolving.com/wiki/index.php/Vieta%27s_formulas). These also tell us that $\frac{-a}{3}$ is the average of the zeros, so $\frac{-a}3=-2 \implies a = 6$. We are also given that the sum of the coefficients is $-2$, so $1+... | -11 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits
of the integer $n$. For example, $P(23) = 6$ and $S(23) = 5$. Suppose $N$ is a
two-digit number such that $N = P(N)+S(N)$. What is the units digit of $N$?
$\text{(A)}\ 2\qquad \text{(B)}\ 3\qquad \text{(C)}\ 6\qquad \text{(D)}\ 8\qquad \t... | Denote $a$ and $b$ as the tens and units digit of $N$, respectively. Then $N = 10a+b$. It follows that $10a+b=ab+a+b$, which implies that $9a=ab$. Since $a\neq0$, $b=9$. So the units digit of $N$ is $\boxed{\textbf{(E) }9}$. | 9 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
Points $A = (3,9)$, $B = (1,1)$, $C = (5,3)$, and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$ is a square. What is the sum of the coordinates of point $D$?
$\... | We already know two vertices of the square: $(A+B)/2 = (2,5)$ and $(B+C)/2 = (3,2)$.
There are only two possibilities for the other vertices of the square: either they are $(6,3)$ and $(5,6)$, or they are $(0,1)$ and $(-1,4)$. The second case would give us $D$ outside the first quadrant, hence the first case is the co... | 10 | Geometry | MCQ | Yes | Yes | amc_aime | false |
Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:
\[\begin{array}{rl} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{array}\]
What is $a-d$?
$\text{(A) }4 \qquad \text{(B) }6 \qquad \text{(C) }8 \qquad \text{(D) }10 \qquad \text{(E) }12$ | Using Simon's Favorite Factoring Trick, we can rewrite the three equations as follows:
\begin{align*} (a+1)(b+1) & = 525 \\ (b+1)(c+1) & = 147 \\ (c+1)(d+1) & = 105 \end{align*}
Let $(e,f,g,h)=(a+1,b+1,c+1,d+1)$. We get:
\begin{align*} ef & = 3\cdot 5\cdot 5\cdot 7 \\ fg & = 3\cdot 7\cdot 7 \\ gh & = 3\cdot 5\cdot ... | 10 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term $2001$ appear somewhere in the sequence?
$\text{(A) }1 \qquad \text{(B) }2 \qquad \text{(C) }3 \qq... | It never hurts to compute a few terms of the sequence in order to get a feel how it looks like. In our case, the definition is that $\forall$ (for all) $n>1:~ a_n = a_{n-1}a_{n+1} - 1$. This can be rewritten as $a_{n+1} = \frac{a_n +1}{a_{n-1}}$. We have $a_1=x$ and $a_2=2000$, and we compute:
\begin{align*} a_3 & = \... | 4 | Algebra | MCQ | Yes | Yes | amc_aime | false |
For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Let $N=f(11)+f(13)+f(14)$. Which of the following relations is true?
$\text{(A) }N2$ | First, note that $2002 = 11 \cdot 13 \cdot 14$.
Using the fact that for any base we have $\log a + \log b = \log ab$, we get that $N = \log_{2002} (11^2 \cdot 13^2 \cdot 14^2) = \log_{2002} 2002^2 = \boxed{(D) N=2}$. | 2 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?
$\text{(A) }3 \qquad \text{(B) }4 \qquad \text{(C) }5 \qquad \text{(D) }8 \qquad \text{(E) }9$ | Solution 1
The repeating decimal $0.\overline{ab}$ is equal to
\[\frac{10a+b}{100} + \frac{10a+b}{10000} + \cdots = (10a+b)\cdot\left(\frac 1{10^2} + \frac 1{10^4} + \cdots \right) = (10a+b) \cdot \frac 1{99} = \frac{10a+b}{99}\]
When expressed in the lowest terms, the denominator of this fraction will always be a di... | 5 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$?
$\mathrm{(A)}\ 4 \qquad\mathrm{(B)}\ 5 \qquad\mathrm{(C)}\ 6 \qquad\mathrm{(D)}\ 7 \qquad\mathrm{(E)}\ 8$ | Let $N = \overline{abcd} = 1000a + \overline{bcd}$, such that $\frac{N}{9} = \overline{bcd}$. Then $1000a + \overline{bcd} = 9\overline{bcd} \Longrightarrow 125a = \overline{bcd}$. Since $100 \le \overline{bcd} < 1000$, from $a = 1, \ldots, 7$ we have $7$ three-digit solutions, and the answer is $\mathrm{(D)}$. | 7 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
For all [integers](https://artofproblemsolving.com/wiki/index.php/Integer) $n$ greater than $1$, define $a_n = \frac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Then $b- c$ equals
$\mathrm{(A)}\ -2 \qquad\mathrm{(B)}\ -1 \qquad\mathrm{(C)}\ \frac{1}{2002} \qq... | By the [change of base](https://artofproblemsolving.com/wiki/index.php/Logarithms#Logarithmic_Properties) formula, $a_n = \frac{1}{\frac{\log 2002}{\log n}} = \left(\frac{1}{\log 2002}\right) \log n$. Thus
\begin{align*}b- c &= \left(\frac{1}{\log 2002}\right)(\log 2 + \log 3 + \log 4 + \log 5 - \log 10 - \log 11 - \l... | -1 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Two walls and the ceiling of a room meet at right angles at point $P.$ A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point $P$. How many meters is the fly from the ceiling?
$\text{(A) }\sqrt{13} \qquad \text{(B) }\sqrt{14} \qquad \text{(C) }\sqrt{15} \qquad \text{(D... | We can use the formula for the diagonal of the rectangle, or $d=\sqrt{a^2+b^2+c^2}$ The problem gives us $a=1, b=8,$ and $c=9.$ Solving gives us $9=\sqrt{1^2 + 8^2 + c^2} \implies c^2=9^2-8^2-1^2 \implies c^2=16 \implies c=\boxed{\textbf{(D) } 4}.$ | 4 | Geometry | MCQ | Yes | Yes | amc_aime | false |
If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$
$\mathrm{(A)}\ -2 \qquad \mathrm{(B)}\ 0 \qquad \mathrm{(C)}\ 2 \qquad \mathrm{(D)}\ 3 \qquad \mathrm{(E)}\ 4$ | Using logarithmic rules, we see that
\[\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a)\]
\[=2-\left(\log_{a}b+\frac {1}{\log_{a}b}\right)\]
Since $a$ and $b$ are both greater than $1$, using [AM-GM](https://artofproblemsolving.com/wiki/index.php/AM-GM) gives that the term in parentheses must be at lea... | 0 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Let $f(x)= \sqrt{ax^2+bx}$. For how many [ real](https://artofproblemsolving.com/wiki/index.php/Real_number) values of $a$ is there at least one [ positive](https://artofproblemsolving.com/wiki/index.php/Positive_number) value of $b$ for which the [domain](https://artofproblemsolving.com/wiki/index.php/Domain) of $f$ ... | The function $f(x) = \sqrt{x(ax+b)}$ has a [codomain](https://artofproblemsolving.com/wiki/index.php/Codomain) of all non-negative numbers, or $0 \le f(x)$. Since the domain and the range of $f$ are the same, it follows that the domain of $f$ also satisfies $0 \le x$.
The function has two zeroes at $x = 0, \frac{-b}{a}... | 2 | Algebra | MCQ | Yes | Yes | amc_aime | false |
A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$?
$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16$ | If $(2,3)$ is in $S$, then $(3,2)$ is also, and quickly we see that every point of the form $(\pm 2, \pm 3)$ or $(\pm 3, \pm 2)$ must be in $S$. Now note that these $8$ points satisfy all of the symmetry conditions. Thus the answer is $\boxed{\mathrm{(D)}\ 8}$. | 8 | Geometry | MCQ | Yes | Yes | amc_aime | false |
If $f(x) = ax+b$ and $f^{-1}(x) = bx+a$ with $a$ and $b$ real, what is the value of $a+b$?
$\mathrm{(A)}\ -2 \qquad\mathrm{(B)}\ -1 \qquad\mathrm{(C)}\ 0 \qquad\mathrm{(D)}\ 1 \qquad\mathrm{(E)}\ 2$ | Since $f(f^{-1}(x))=x$, it follows that $a(bx+a)+b=x$, which implies $abx + a^2 +b = x$. This equation holds for all values of $x$ only if $ab=1$ and $a^2+b=0$.
Then $b = -a^2$. Substituting into the equation $ab = 1$, we get $-a^3 = 1$. Then $a = -1$, so $b = -1$, and\[f(x)=-x-1.\]Likewise\[f^{-1}(x)=-x-1.\]These are ... | -2 | Algebra | MCQ | Yes | Yes | amc_aime | false |
A function $f$ is defined by $f(z) = i\overline{z}$, where $i=\sqrt{-1}$ and $\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$?
$\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 2 \qquad\mathrm{(D)}\ 4 \qquad\mathrm{(E)}\ 8$ | Solution 1
Let $z = a+bi$, so $\overline{z} = a-bi$. By definition, $z = a+bi = f(z) = i(a-bi) = b+ai$, which implies that all solutions to $f(z) = z$ lie on the line $y=x$ on the complex plane. The graph of $|z| = 5$ is a [circle](https://artofproblemsolving.com/wiki/index.php/Circle) centered at the origin, and there... | 2 | Algebra | MCQ | Yes | Yes | amc_aime | false |
A truncated cone has horizontal bases with radii $18$ and $2$. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
$\mathrm{(A)}\ 6 \qquad\mathrm{(B)}\ 4\sqrt{5} \qquad\mathrm{(C)}\ 9 \qquad\mathrm{(D)}\ 10 \qquad\mathrm{(E)}\ 6\sqrt{3}$ | Solution 1
Consider a [trapezoid](https://artofproblemsolving.com/wiki/index.php/Trapezoid) (label it $ABCD$ as follows) cross-section of the truncate cone along a diameter of the bases:
Above, $E,F,$ and $G$ are points of [tangency](https://artofproblemsolving.com/wiki/index.php/Tangent_(geometry)). By the Two Tange... | 6 | Geometry | MCQ | Yes | Yes | amc_aime | false |
If $x$ and $y$ are positive integers for which $2^x3^y=1296$, what is the value of $x+y$?
$(\mathrm {A})\ 8 \qquad (\mathrm {B})\ 9 \qquad (\mathrm {C})\ 10 \qquad (\mathrm {D})\ 11 \qquad (\mathrm {E})\ 12$ | $1296 = 2^4 3^4$ and $4+4=\boxed{8} \Longrightarrow \mathrm{(A)}$. | 8 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is $x - y$?
$(\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 20$ | $2 = \frac {1}{10}x \Longrightarrow x = 20,\quad 2 = \frac{1}{5}y \Longrightarrow y = 10,\quad x-y = 20 - 10=10 \mathrm{(D)}$. | 10 | Algebra | MCQ | Yes | Yes | amc_aime | false |
A wooden [cube](https://artofproblemsolving.com/wiki/index.php/Cube) $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$?
$(\mathrm {A}) \ 3 \qquad (\mathrm {B}) \ 4 \qquad (\mathrm {C})\ 5 \qquad... | There are $6n^3$ sides total on the unit cubes, and $6n^2$ are painted red.
$\dfrac{6n^2}{6n^3}=\dfrac{1}{4} \Rightarrow n=4 \rightarrow \mathrm {B}$ | 4 | Combinatorics | MCQ | Yes | Yes | amc_aime | false |
A [line](https://artofproblemsolving.com/wiki/index.php/Line) passes through $A\ (1,1)$ and $B\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?
$(\mathrm {A}) \ 0 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 3 \qquad (\mathrm {D}) \ 8 \qquad (\mathrm {E})\... | For convenience’s sake, we can transform $A$ to the origin and $B$ to $(99,999)$ (this does not change the problem). The line $AB$ has the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) $y = \frac{999}{99}x = \frac{111}{11}x$. The coordinates are integers if $11|x$, so the values of $x$ are $11, 22... | 8 | Geometry | MCQ | Yes | Yes | amc_aime | false |
In the five-sided star shown, the letters $A$, $B$, $C$, $D$ and $E$ are replaced by the
numbers 3, 5, 6, 7 and 9, although not necessarily in that order. The sums of the
numbers at the ends of the line segments $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$, and $\overline{EA}$ form an
arithmetic s... | Solution 1
$(A+B) + (B+C) + (C+D) + (D+E) + (E+A) = 2(A+B+C+D+E)$ (i.e., each number is counted twice). The sum $A + B + C + D + E$ will always be $3 + 5 + 6 + 7 + 9 = 30$, so the arithmetic sequence has a sum of $2 \cdot 30 = 60$. The middle term must be the average of the five numbers, which is $\frac{60}{5} = 12 \Lo... | 12 | Logic and Puzzles | MCQ | Yes | Yes | amc_aime | false |
Three [circles](https://artofproblemsolving.com/wiki/index.php/Circle) of [radius](https://artofproblemsolving.com/wiki/index.php/Radius) $s$ are drawn in the first [quadrant](https://artofproblemsolving.com/wiki/index.php/Quadrant) of the $xy$-[plane](https://artofproblemsolving.com/wiki/index.php/Plane). The first ci... | Solution 1
[2005 12A AMC-16b.png](https://artofproblemsolving.com/wiki/index.php/File:2005_12A_AMC-16b.png)
Set $s =1$ so that we only have to find $r$. Draw the segment between the center of the third circle and the large circle; this has length $r+1$. We then draw the [radius](https://artofproblemsolving.com/wiki/ind... | 9 | Geometry | MCQ | Yes | Yes | amc_aime | false |
How many ordered triples of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) $(a,b,c)$, with $a \ge 2$, $b\ge 1$, and $c \ge 0$, satisfy both $\log_a b = c^{2005}$ and $a + b + c = 2005$?
$\mathrm{(A)} \ 0 \qquad \mathrm{(B)} \ 1 \qquad \mathrm{(C)} \ 2 \qquad \mathrm{(D)} \ 3 \qquad \mathrm{(E)} \ 4$ | $a^{c^{2005}} = b$
[Casework](https://artofproblemsolving.com/wiki/index.php/Casework) upon $c$:
$c = 0$: Then $a^0 = b \Longrightarrow b = 1$. Thus we get $(2004,1,0)$.
$c = 1$: Then $a^1 = b \Longrightarrow a = b$. Thus we get $(1002,1002,1)$.
$c \ge 2$: Then the exponent of $a$ becomes huge, and since $a \ge 2$ the... | 2 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$?
$\mathrm{(A)}\ 8\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 14\qquad \mathrm{(E)}\ 16$ | Box P has dimensions $l$, $w$, and $h$.
Its surface area is \[2lw+2lh+2wh=384,\]
and the sum of all its edges is \[l + w + h = \dfrac{4l+4w+4h}{4} = \dfrac{112}{4} = 28.\]
The diameter of the sphere is the space diagonal of the prism, which is \[\sqrt{l^2 + w^2 +h^2}.\]
Notice that \[(l + w + h)^2 - (2lw + 2lh + 2wh)... | 10 | Geometry | MCQ | Yes | Yes | amc_aime | false |
Let $A,M$, and $C$ be digits with
\[(100A+10M+C)(A+M+C) = 2005\]
What is $A$?
$(\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 3 \qquad (\mathrm {D}) \ 4 \qquad (\mathrm {E})\ 5$ | Clearly the two quantities are both integers, so we check the [prime factorization](https://artofproblemsolving.com/wiki/index.php/Prime_factorization) of $2005 = 5 \cdot 401$. It is easy to see now that $(A,M,C) = (4,0,1)$ works, so the answer is $\mathrm{(D)}$. | 4 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
The sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Which of the following is not included among the eight digits?
$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$ | $221$ can be written as the sum of four two-digit numbers, let's say $\overline{ae}$, $\overline{bf}$, $\overline{cg}$, and $\overline{dh}$. Then $221= 10(a+b+c+d)+(e+f+g+h)$. The last digit of $221$ is $1$, and $10(a+b+c+d)$ won't affect the units digits, so $(e+f+g+h)$ must end with $1$. The smallest value $(e+f+g+h)... | 4 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?
$\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{3}}} \qquad \mathrm{(E)}\ {{{4}}}$ | We may let $n = 7^k \cdot m$, where $m$ is not divisible by 7. Using the fact that the number of divisors function $d(n)$ is multiplicative, we have $d(n) = d(7^k)d(m) = (k+1)d(m) = 60$. Also, $d(7n) = d(7^{k+1})d(m) = (k+2)d(m) = 80$. These numbers are in the ratio 3:4, so $\frac{k+1}{k+2} = \frac{3}{4} \implies k = 2... | 2 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
For how many values of $a$ is it true that the line $y = x + a$ passes through the
vertex of the parabola $y = x^2 + a^2$ ?
$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 2 \qquad \mathrm{(D)}\ 10 \qquad \mathrm{(E)}\ \text{infinitely many}$ | We see that the vertex of the quadratic function $y = x^2 + a^2$ is $(0,\,a^2)$. The y-intercept of the line $y = x + a$ is $(0,\,a)$. We want to find the values (if any) such that $a=a^2$. Solving for $a$, the only values that satisfy this are $0$ and $1$, so the answer is $\boxed{\mathrm{(C)}\ 2}$ | 2 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Mary is $20\%$ older than Sally, and Sally is $40\%$ younger than Danielle. The sum of their ages is $23.2$ years. How old will Mary be on her next birthday?
$\mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 8\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11$ | Let $m$ be Mary's age, let $s$ be Sally's age, and let $d$ be Danielle's age. We have $s=.6d$, and $m=1.2s=1.2(.6d)=.72d$. The sum of their ages is $m+s+d=.72d+.6d+d=2.32d$. Therefore, $2.32d=23.2$, and $d=10$. Then $m=.72(10)=7.2$. Mary will be $8$ on her next birthday. The answer is $\mathrm{(B)}$. | 8 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a [whole number](https://artofproblemsolving.com/wiki/index.php/Whole_number) of cents. What is the total cost, in cents, of one pencil and one eraser?
$\mathrm{(A)}\ 10\qquad\mathrm{(B)}\ 12\qquad\mathrm{(C)}\ 1... | Let the price of a pencil be $p$ and an eraser $e$. Then $13p + 3e = 100$ with $p > e > 0$. Since $p$ and $e$ are [positive integers](https://artofproblemsolving.com/wiki/index.php/Positive_integer), we must have $e \geq 1$ and $p \geq 2$.
Considering the [equation](https://artofproblemsolving.com/wiki/index.php/Equa... | 10 | Algebra | MCQ | Yes | Yes | amc_aime | false |
What is $( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}$?
$\text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } 1 \qquad \text {(E) } 2006$ | $(-1)^n=1$ if n is even and $-1$ if n is odd. So we have
$-1+1-1+1-\cdots-1+1=0+0+\cdots+0+0=0 \Rightarrow \text{(C)}$ | 0 | Algebra | MCQ | Yes | Yes | amc_aime | false |
The [parabola](https://artofproblemsolving.com/wiki/index.php/Parabola) $y=ax^2+bx+c$ has [vertex](https://artofproblemsolving.com/wiki/index.php/Vertex) $(p,p)$ and $y$-intercept $(0,-p)$, where $p\ne 0$. What is $b$?
$\text {(A) } -p \qquad \text {(B) } 0 \qquad \text {(C) } 2 \qquad \text {(D) } 4 \qquad \text {(E) ... | Substituting $(0,-p)$, we find that $y = -p = a(0)^2 + b(0) + c = c$, so our parabola is $y = ax^2 + bx - p$.
The x-coordinate of the vertex of a parabola is given by $x = p = \frac{-b}{2a} \Longleftrightarrow a = \frac{-b}{2p}$. Additionally, substituting $(p,p)$, we find that $y = p = a(p)^2 + b(p) - p \Longleftrigh... | 4 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is 24, and $\angle BAD = 60^\circ$. What is the area of rhombus $BFDE$?
$\textrm{(A) } 6 \qquad \textrm{(B) } 4\sqrt {3} \qquad \textrm{(C) } 8 \qquad \textrm{(D) } 9 \qquad \textrm{(E) } 6\sqrt {3}$ | The ratio of any length on $ABCD$ to a corresponding length on $BFDE^2$ is equal to the ratio of their areas. Since $\angle BAD=60$, $\triangle ADB$ and $\triangle DBC$ are equilateral. $DB$, which is equal to $AB$, is the diagonal of rhombus $ABCD$. Therefore, $AC=\frac{DB(2)}{2\sqrt{3}}=\frac{DB}{\sqrt{3}}$. $DB$ and... | 8 | Geometry | MCQ | Yes | Yes | amc_aime | false |
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and th... | First, The number of the plate is divisible by $9$ and in the form of
$aabb$, $abba$ or $abab$.
We can conclude straight away that $a+b= 9$ using the $9$ divisibility rule.
If $b=1$, the number is not divisible by $2$ (unless it's $1818$, which is not divisible by $4$), which means there are no $2$, $4$, $6$, or $8$ ... | 5 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
$\text {(A) } 10 \qquad \text {(B) } 14 \qquad \text {(C) } 17 \qquad \text {(D) } 20 \qquad \text {(E) } 24$ | Solution 1
If the Cougars won by a margin of 14 points, then the Panthers' score would be half of (34-14). That's 10 $\Rightarrow \boxed{\text{(A)}}$.
Solution 2
Let the Panthers' score be $x$. The Cougars then scored $x+14$. Since the teams combined scored $34$, we get $x+x+14=34 \\ \rightarrow 2x+14=34 \\ \rightarro... | 10 | Algebra | MCQ | Yes | Yes | amc_aime | false |
A piece of cheese is located at $(12,10)$ in a [coordinate plane](https://artofproblemsolving.com/wiki/index.php/Coordinate_plane). A mouse is at $(4,-2)$ and is running up the [line](https://artofproblemsolving.com/wiki/index.php/Line) $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese r... | The point $(a,b)$ is the foot of the perpendicular from $(12,10)$ to the line $y=-5x+18$. The perpendicular has slope $\frac{1}{5}$, so its equation is $y=10+\frac{1}{5}(x-12)=\frac{1}{5}x+\frac{38}{5}$. The $x$-coordinate at the foot of the perpendicular satisfies the equation $\frac{1}{5}x+\frac{38}{5}=-5x+18$, so $x... | 10 | Algebra | MCQ | Yes | Yes | amc_aime | false |
The polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0.$ What is $a + b + c + d?$
$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 4 \qquad \mathrm{(D)}\ 9 \qquad \mathrm{(E)}\ 16$ | A fourth degree polynomial has four [roots](https://artofproblemsolving.com/wiki/index.php/Root). Since the coefficients are real(meaning that complex roots come in conjugate pairs), the remaining two roots must be the [complex conjugates](https://artofproblemsolving.com/wiki/index.php/Complex_conjugate) of the two giv... | 9 | Algebra | MCQ | Yes | Yes | amc_aime | false |
An aquarium has a [rectangular base](https://artofproblemsolving.com/wiki/index.php/Rectangular_prism) that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By h... | The brick has volume $8000 cm^3$. The base of the aquarium has area $4000 cm^2$. For every inch the water rises, the volume increases by $4000 cm^3$; therefore, when the volume increases by $8000 cm^3$, the water level rises $2 cm \Rightarrow\fbox{D}$ | 2 | Geometry | MCQ | Yes | Yes | amc_aime | false |
The larger of two consecutive odd integers is three times the smaller. What is their sum?
$\mathrm{(A)}\ 4\qquad \mathrm{(B)}\ 8\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 16\qquad \mathrm{(E)}\ 20$ | Let $n$ be the smaller term. Then $n+2=3n \Longrightarrow 2n = 2 \Longrightarrow n=1$
Thus, the answer is $1+(1+2)=4 \mathrm{(A)}$ | 4 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour?
$\mathrm{(A)}\ 7\qquad \mathrm{(B)}\ 9\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 12\qquad \mathrm{(E)}\ 14$ | \[16 \cdot \frac{30}{60}+4\cdot\frac{90}{60}=14\]
\[\frac{14}2=7\Rightarrow\boxed{A}\] | 7 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Let $a, b, c, d$, and $e$ be five consecutive terms in an arithmetic sequence, and suppose that $a+b+c+d+e=30$. Which of $a, b, c, d,$ or $e$ can be found?
$\textrm{(A)} \ a\qquad \textrm{(B)}\ b\qquad \textrm{(C)}\ c\qquad \textrm{(D)}\ d\qquad \textrm{(E)}\ e$ | Let $f$ be the common difference between the terms.
$a=c-2f$
$b=c-f$
$c=c$
$d=c+f$
$e=c+2f$
$a+b+c+d+e=5c=30$, so $c=6$. But we can't find any more variables, because we don't know what $f$ is. So the answer is $\textrm{C}$. | 6 | Algebra | MCQ | Yes | Yes | amc_aime | false |
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters?
$\mathrm {(A)} 6\qquad \mathrm {(B)} 7\qquad \mathrm {(C)} 8\qquad \mathrm {(D)} 10\qquad \mathrm {(E)} 12$ | Let $a$ and $b$ be the two legs of the triangle.
We have $\frac{1}{2}ab = 3(a+b+c)$.
Then $ab=6 \left(a+b+\sqrt {a^2 + b^2}\right)$.
We can complete the square under the root, and we get, $ab=6 \left(a+b+\sqrt {(a+b)^2 - 2ab}\right)$.
Let $ab=p$ and $a+b=s$, we have $p=6 \left(s+ \sqrt {s^2 - 2p}\right)$.
After rearran... | 6 | Geometry | MCQ | Yes | Yes | amc_aime | false |
Points $A,B,C,D$ and $E$ are located in 3-dimensional space with $AB=BC=CD=DE=EA=2$ and $\angle ABC=\angle CDE=\angle DEA=90^o$. The plane of $\triangle ABC$ is parallel to $\overline{DE}$. What is the area of $\triangle BDE$?
$\mathrm {(A)} \sqrt{2}\qquad \mathrm {(B)} \sqrt{3}\qquad \mathrm {(C)} 2\qquad \mathrm {(D)... | [2007 AMC 12B Problem 25.png](https://artofproblemsolving.com/wiki/index.php/File:2007_AMC_12B_Problem_25.png)
Link to graph: [https://www.math3d.org/pHFSD6vRi](https://artofproblemsolving.comhttps://www.math3d.org/pHFSD6vRi)
Let $A=(0,0,0)$, and $B=(2,0,0)$. Since $EA=2$, we could let $C=(2,0,2)$, $D=(2,2,2)$, and $E... | 2 | Geometry | MCQ | Yes | Yes | amc_aime | false |
At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?
$\mathrm {(A)} 6\qquad \mathrm {(B)} 8\qquad \mathrm {(C)} 9\qquad \mathrm {(D)} 12\qquad \mathrm {(E)} 18$ | $18$ bananas cost the same as $12$ apples, and $12$ apples cost the same as $8$ oranges, so $18$ bananas cost the same as $8 \Rightarrow \mathrm {(B)}$ oranges. | 8 | Algebra | MCQ | Yes | Yes | amc_aime | false |
A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$?
$\mathrm{(A)}\ 7 \qquad \mathrm{(B)}\ 13 \qquad \mathrm{(C)}\ 31 \qquad \mathrm{(D)}\ 111 \qquad \mathrm{(E)}\ 211$ | $3x-1 =5 \implies x= 2$
$f(3(2)-1) = 2^2+2+1=7 \implies (A)$ | 7 | Algebra | MCQ | Yes | Yes | amc_aime | false |
What is the area of the region defined by the [inequality](https://artofproblemsolving.com/wiki/index.php/Inequality) $|3x-18|+|2y+7|\le3$?
$\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ \frac{7}{2}\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ \frac{9}{2}\qquad\mathrm{(E)}\ 5$ | Area is invariant under translation, so after translating left $6$ and up $7/2$ units, we have the inequality
\[|3x| + |2y|\leq 3\]
which forms a [diamond](https://artofproblemsolving.com/wiki/index.php/Rhombus) centered at the [origin](https://artofproblemsolving.com/wiki/index.php/Origin) and vertices at $(\pm 1, 0),... | 3 | Inequalities | MCQ | Yes | Yes | amc_aime | false |
A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$ | If the basketball player makes $x$ three-point shots and $5-x$ two-point shots, he scores $3x+2(5-x)=10+x$ points. Clearly every value of $x$ yields a different number of total points. Since he can make any number of three-point shots between $0$ and $5$ inclusive, the number of different point totals is $6 \Rightarrow... | 6 | Combinatorics | MCQ | Yes | Yes | amc_aime | false |
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $R$ be the region formed by the union of the... | [asy] real a = 1/2, b = sqrt(3)/2; draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,0)--(a,-b)--(1,0)--(1+b,a)--(1,1)--(a,1+b)--(0,1)--(-b,a)--(0,0)); draw((0,0)--(-1+a,-b)--(1+a,-b)--(1,0)--(1+b,-1+a)--(1+b,1+a)--(1,1)--(1+a,1+b)--(-1+a,1+b)--(0,1)--(-b,1+a)--(-b,-1+a)--(0,0)); filldraw((1+a,-b)--(1,0)--(1+b,-1+a)--cy... | 1 | Geometry | MCQ | Yes | Yes | amc_aime | false |
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 hal... | $A_{outer}=ab$
$A_{inner}=(a-2)(b-2)$
$A_{outer}=2A_{inner}$
$ab=2(a-2)(b-2)=2ab-4a-4b+8$
$0=ab-4a-4b+8$
By Simon's Favorite Factoring Trick:
$8=ab-4a-4b+16=(a-4)(b-4)$
Since $8=1\times8$ and $8=2\times4$ are the only positive factorings of $8$.
$(a,b)=(5,12)$ or $(a,b)=(6,8)$ yielding $\Rightarrow\textbf{(B)}$ $2$ sol... | 2 | Geometry | MCQ | Yes | Yes | amc_aime | false |
The sum of the base-$10$ logarithms of the divisors of $10^n$ is $792$. What is $n$?
$\text{(A)}\ 11\qquad \text{(B)}\ 12\qquad \text{(C)}\ 13\qquad \text{(D)}\ 14\qquad \text{(E)}\ 15$ | Solution 1
Every factor of $10^n$ will be of the form $2^a \times 5^b , a\leq n , b\leq n$. Not all of these base ten logarithms will be rational, but we can add them together in a certain way to make it rational. Recall the logarithmic property $\log(a \times b) = \log(a)+\log(b)$. For any factor $2^a \times 5^b$, the... | 11 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
A class collects $50$ dollars to buy flowers for a classmate who is in the hospital. Roses cost $3$ dollars each, and carnations cost $2$ dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$ dollars?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 9... | The class could send $25$ carnations and no roses, $22$ carnations and $2$ roses, $19$ carnations and $4$ roses, and so on, down to $1$ carnation and $16$ roses. There are 9 total possibilities (from 0 to 16 roses, incrementing by 2 at each step), $\Rightarrow \boxed{C}$ | 9 | Combinatorics | MCQ | Yes | Yes | amc_aime | false |
For real numbers $a$ and $b$, define $a\textdollar b = (a - b)^2$. What is $(x - y)^2\textdollar(y - x)^2$?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ x^2 + y^2 \qquad \textbf{(C)}\ 2x^2 \qquad \textbf{(D)}\ 2y^2 \qquad \textbf{(E)}\ 4xy$ | $\left[ (x-y)^2 - (y-x)^2 \right]^2$
$\left[ (x-y)^2 - (x-y)^2 \right]^2$
$[0]^2$
$0 \Rightarrow \textbf{(A)}$ | 0 | Algebra | MCQ | Yes | Yes | amc_aime | false |
How many positive integers less than $1000$ are $6$ times the sum of their digits?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 12$ | Solution 1
The sum of the digits is at most $9+9+9=27$. Therefore the number is at most $6\cdot 27 = 162$. Out of the numbers $1$ to $162$ the one with the largest sum of digits is $99$, and the sum is $9+9=18$. Hence the sum of digits will be at most $18$.
Also, each number with this property is divisible by $6$, ther... | 1 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
A circle with center $C$ is tangent to the positive $x$ and $y$-axes and externally tangent to the circle centered at $(3,0)$ with radius $1$. What is the sum of all possible radii of the circle with center $C$?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9... | Let $r$ be the radius of our circle. For it to be tangent to the positive $x$ and $y$ axes, we must have $C=(r,r)$. For the circle to be externally tangent to the circle centered at $(3,0)$ with radius $1$, the distance between $C$ and $(3,0)$ must be exactly $r+1$.
By the [Pythagorean theorem](https://artofproblemsolv... | 8 | Geometry | MCQ | Yes | Yes | amc_aime | false |
Let $a + ar_1 + ar_1^2 + ar_1^3 + \cdots$ and $a + ar_2 + ar_2^2 + ar_2^3 + \cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $r_1$, and the sum of the second series is $r_2$. What is $r_1 + r_2$?
$\textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac ... | Using the formula for the sum of a geometric series we get that the sums of the given two sequences are $\frac a{1-r_1}$ and $\frac a{1-r_2}$.
Hence we have $\frac a{1-r_1} = r_1$ and $\frac a{1-r_2} = r_2$.
This can be rewritten as $r_1(1-r_1) = r_2(1-r_2) = a$.
As we are given that $r_1$ and $r_2$ are distinct, the... | 1 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$. What is $a+b+c$?
$\textbf{(A)}\ -1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$ | As $f(x)=ax^2 + bx + c$, we have $f(1)=a\cdot 1^2 + b\cdot 1 + c = a+b+c$.
To compute $f(1)$, set $x=-2$ in the first formula. We get $f(1) = f(-2+3) = 3(-2)^2 + 7(-2) + 4 = 12 - 14 + 4 = \boxed{2}$. | 2 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\triangle ABC$?
$\mathrm{(A)}\ 6\qquad \mathrm{(B)}\ 8\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 12\qquad \mathrm{(E)}\ 14$ | Solution 1
Because the line $x + y = 7$ is parallel to $\overline {AB}$, the area of $\triangle ABC$ is independent of the location of $C$ on that line. Therefore it may be assumed that $C$ is $(7,0)$. In that case the triangle has base $AC = 4$ and altitude $3$, so its area is $\frac 12 \cdot 4 \cdot 3 = \boxed {6}$... | 6 | Geometry | MCQ | Yes | Yes | amc_aime | false |
Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 288 \qquad \textbf{(E)}\ 2009$ | Solution 1
Since $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1$, we can write the terms of each sequence as
\begin{align*}&\left(a_n\right) \Rightarrow \{1, x+1, 2x+1, 3x+1, ...\}\\ &\left(b_n\right) \Rightarrow \{1, y+1, 2y+1, 3y+1, ...\}\end{align*}
where $x$ and $y$ ($x\leq y$) are the... | 8 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
Rectangle $ABCD$, pictured below, shares $50\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$. What is $\frac{AB}{AD}$?
$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$ | If we shift $A$ to coincide with $E$, and add new horizontal lines to divide $EFGH$ into five equal parts:
This helps us to see that $AD=a/5$ and $AB=2a$, where $a=EF$.
Hence $\dfrac{AB}{AD}=\dfrac{2a}{a/5}=10$. | 10 | Geometry | MCQ | Yes | Yes | amc_aime | false |
In $\triangle ABC$, $\cos(2A-B)+\sin(A+B)=2$ and $AB=4$. What is $BC$?
$\textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ \sqrt{3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{3}$ | We note that $-1$ $\le$ $\sin x$ $\le$ $1$ and $-1$ $\le$ $\cos x$ $\le$ $1$.
Therefore, there is no other way to satisfy this equation other than making both $\cos(2A-B)=1$ and $\sin(A+B)=1$, since any other way would cause one of these values to become greater than 1, which contradicts our previous statement.
From t... | 2 | Geometry | MCQ | Yes | Yes | amc_aime | false |
A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$, and $a_3= \tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\cos x$?
$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$ | By the defintion of a geometric sequence, we have $\cos^2x=\sin x \tan x$. Since $\tan x=\frac{\sin x}{\cos x}$, we can rewrite this as $\cos^3x=\sin^2x$.
The common ratio of the sequence is $\frac{\cos x}{\sin x}$, so we can write
\[a_1= \sin x\]
\[a_2= \cos x\]
\[a_3= \frac{\cos^2x}{\sin x}\]
\[a_4=\frac{\cos^3x}{\s... | 8 | Algebra | MCQ | Yes | Yes | amc_aime | false |
Let $n$ be the smallest positive integer such that $n$ is divisible by $20$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$ | We know that $n^2 = k^3$ and $n^3 = m^2$. Cubing and squaring the equalities respectively gives $n^6 = k^9 = m^4$. Let $a = n^6$. Now we know $a$ must be a perfect $36$-th power because $lcm(9,4) = 36$, which means that $n$ must be a perfect $6$-th power. The smallest number whose sixth power is a multiple of $20$ is $... | 7 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<f(100)<5000(k+1)$ for some integer $k$. What is $k$?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$ | From $f(1) = 0$, we know that $a+b+c = 0$.
From the first inequality, we get $50 < 49a+7b+c < 60$. Subtracting $a+b+c = 0$ from this gives us $50 < 48a+6b < 60$, and thus $\frac{25}{3} < 8a+b < 10$. Since $8a+b$ must be an integer, it follows that $8a+b = 9$.
Similarly, from the second inequality, we get $70 < 64a+8b+c... | 3 | Algebra | MCQ | Yes | Yes | amc_aime | false |
A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \... | Since the frog always jumps in length $5$ and lands on a lattice point, the sum of its coordinates must change either by $5$ (by jumping parallel to the x- or y-axis), or by $3$ or $4$ (3-4-5 right triangle).
Because either $1$, $5$, or $7$ is always the change of the sum of the coordinates, the sum of the coordinates ... | 3 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
How many positive two-digit integers are factors of $2^{24}-1$?
$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14$
~ pi_is_3.14 | Repeating [difference of squares](https://artofproblemsolving.com/wiki/index.php/Difference_of_squares):
$2^{24}-1=(2^{12}+1)(2^{6}+1)(2^{3}+1)(2^{3}-1)$
$2^{24}-1=(2^{12}+1)\cdot65\cdot9\cdot7$
$2^{24}-1 = (2^{12} +1) * 5 * 13 * 3^2 * 7$
The sum of cubes formula gives us:
$2^{12}+1=(2^4+1)(2^8-2^4+1)$
$2^{12}+1 = 17\... | 12 | Number Theory | MCQ | Yes | Yes | amc_aime | false |
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