problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 6
values | synthetic bool 1
class |
|---|---|---|---|---|---|---|---|---|
On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling ... | 1. **Define the problem and variables:**
- Let \( s \) be the speed of the cars in kilometers per hour.
- Each car is 4 meters long.
- The distance from the back of one car to the front of the next car is \( \frac{s}{15} \) car lengths.
- Therefore, the total distance occupied by one car and the gap to the ... | 375 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let
\[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3.
\]Suppose that
\begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minu... | 1. Given the polynomial:
\[
p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3
\]
and the conditions:
\[
p(0,0) = p(1,0) = p(-1,0) = p(0,1) = p(0,-1) = p(1,1) = p(1,-1) = p(2,2) = 0
\]
2. From \( p(0,0) = 0 \), we get:
\[
a_0 = 0
\]
3. From \( p(1,... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ N\equal{}100^2\plus{}99^2\minus{}98^2\minus{}97^2\plus{}96^2\plus{}\cdots\plus{}4^2\plus{}3^2\minus{}2^2\minus{}1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $ N$ is divided by $ 1000$. | 1. First, we rewrite the given expression \( N \) by grouping the terms in pairs:
\[
N = (100^2 + 99^2 - 98^2 - 97^2) + (96^2 + 95^2 - 94^2 - 93^2) + \cdots + (4^2 + 3^2 - 2^2 - 1^2)
\]
2. Notice that each pair can be simplified using the difference of squares formula:
\[
a^2 - b^2 = (a - b)(a + b)
\... | 796 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In trapezoid $ ABCD$ with $ \overline{BC}\parallel\overline{AD}$, let $ BC\equal{}1000$ and $ AD\equal{}2008$. Let $ \angle A\equal{}37^\circ$, $ \angle D\equal{}53^\circ$, and $ m$ and $ n$ be the midpoints of $ \overline{BC}$ and $ \overline{AD}$, respectively. Find the length $ MN$. | 1. **Extend lines and find intersection point:**
Extend $\overline{AD}$ and $\overline{BC}$ to meet at a point $E$. Since $\overline{BC} \parallel \overline{AD}$, the angles $\angle A$ and $\angle D$ sum up to $90^\circ$:
\[
\angle AED = 180^\circ - \angle A - \angle D = 180^\circ - 37^\circ - 53^\circ = 90^\c... | 504 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A particle is located on the coordinate plane at $ (5,0)$. Define a [i]move[/i] for the particle as a counterclockwise rotation of $ \pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$-direction. Given that the particle's position after $ 150$ moves is $ (p,q)$, find the greate... | 1. **Understanding the Move**:
- Each move consists of a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of 10 units in the positive $x$-direction.
- We need to determine the particle's position after 150 such moves.
2. **Rotation and Translation**:
- A rotation by $\pi... | 19 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are two distinguishable flagpoles, and there are $ 19$ flags, of which $ 10$ are identical blue flags, and $ 9$ are identical green flags. Let $ N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. ... | 1. **Case 1: One green flag on one pole and the rest on the other pole.**
- We have 10 blue flags and 8 green flags on one pole.
- The 10 blue flags create 11 spaces (gaps) where the green flags can be placed.
- We need to choose 8 out of these 11 spaces to place the green flags.
- The number of ways to ch... | 310 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $ R$ be the region outside the hexagon, and let $ S\equal{}\{\frac{1}{z}|z\in R\}$. Then the area of $ S$ has the form $ a\pi\plus{}\sqrt{b}$, where $ a$ ... | 1. **Understanding the Geometry of the Hexagon:**
- The hexagon is regular and centered at the origin in the complex plane.
- The distance between opposite sides is 1 unit.
- One pair of sides is parallel to the imaginary axis.
2. **Calculating the Side Length of the Hexagon:**
- The distance between oppos... | 31 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ ca... | 1. Consider the given system of equations:
\[
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
\]
We need to find the maximum possible value of \(\frac{a}{b}\) for which this system has a solution \((x, y)\) satisfying \(0 \le x < a\) and \(0 \le y < b\).
2. Let's analyze the geometric interpretation of the e... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest integer $ n$ satisfying the following conditions:
(i) $ n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $ 2n\plus{}79$ is a perfect square. | 1. **Condition (ii):** We start with the condition that \(2n + 79\) is a perfect square. Let:
\[
2n + 79 = (2k + 1)^2
\]
for some integer \(k \geq 0\). Expanding and solving for \(n\), we get:
\[
2n + 79 = 4k^2 + 4k + 1
\]
\[
2n = 4k^2 + 4k + 1 - 79
\]
\[
2n = 4k^2 + 4k - 78
\]
... | 181 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Six cards numbered 1 through 6 are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order. | To solve this problem, we need to find the number of arrangements of six cards numbered 1 through 6 such that removing one card leaves the remaining five cards in either ascending or descending order. We will break down the solution into detailed steps.
1. **Count the number of ways to remove one card and have the rem... | 40 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ and $n$ be positive integers satisfying the conditions
[list]
[*] $\gcd(m+n,210) = 1,$
[*] $m^m$ is a multiple of $n^n,$ and
[*] $m$ is not a multiple of $n$.
[/list]
Find the least possible value of $m+n$. | 1. **Initial Setup and Conditions:**
- We are given that $\gcd(m+n, 210) = 1$.
- $m^m$ is a multiple of $n^n$.
- $m$ is not a multiple of $n$.
- We need to find the least possible value of $m+n$.
2. **Prime Factor Considerations:**
- Since $\gcd(m+n, 210) = 1$ and $210 = 2 \cdot 3 \cdot 5 \cdot 7$, neit... | 407 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac mn,$ where $m$ and $n$ are relatively prime posi... | 1. **Identify the set \( S \)**:
The set \( S \) consists of all positive integer divisors of \( 20^9 \). Since \( 20 = 2^2 \cdot 5 \), we have:
\[
20^9 = (2^2 \cdot 5)^9 = 2^{18} \cdot 5^9
\]
Therefore, the divisors of \( 20^9 \) are of the form \( 2^m \cdot 5^n \) where \( 0 \leq m \leq 18 \) and \( 0 ... | 77 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT=CT=16$, $BC=22$, and $TX^2+TY^2+XY^2=1143$. Find $XY^2$. | 1. **Setup and Definitions:**
- Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$.
- The tangents to $\omega$ at $B$ and $C$ intersect at $T$.
- Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively.
- Given: $BT = CT = 16$, $BC = 22$, and $TX^2 + TY^2 + ... | 717 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\}=x-\lfloor x\rfloor$ to be the fractional part of $x$. For example, $\{3\}=0$ and $\{4.56\}=0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $... | 1. **Understanding the function \( f(x) \):**
\[
f(x) = x \{ x \} = x (x - \lfloor x \rfloor)
\]
Here, \( \{ x \} = x - \lfloor x \rfloor \) is the fractional part of \( x \).
2. **Analyzing the equation \( f(f(f(x))) = 17 \):**
We need to find the number of real-valued solutions to this equation for \(... | 10 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$.
| 1. **Identify the given information and the goal:**
- We have a convex pentagon \(ABCDE\) with side lengths \(AB = 5\), \(BC = CD = DE = 6\), and \(EA = 7\).
- The pentagon has an inscribed circle, meaning it is tangential to each side.
- We need to find the area of \(ABCDE\).
2. **Use Pitot's Theorem:**
-... | 60 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\cdots+n^3$ is divided by $n+5$, the remainder is $17.$
| 1. We start with the given problem: Find the sum of all positive integers \( n \) such that when \( 1^3 + 2^3 + 3^3 + \cdots + n^3 \) is divided by \( n+5 \), the remainder is 17.
2. We use the formula for the sum of cubes:
\[
\sum_{i=1}^n i^3 = \left( \frac{n(n+1)}{2} \right)^2
\]
We need to find \( n \) ... | 239 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, and $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$, where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$. | 1. **Finding the center of rotation:**
- The center of rotation lies on the intersection of the perpendicular bisectors of the segments \(AA'\), \(BB'\), and \(CC'\). We will find the perpendicular bisectors of \(AA'\) and \(BB'\) and solve for their intersection.
2. **Perpendicular bisector of \(AA'\):**
- Midp... | 108 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define a sequence of functions recursively by $f_1(x) = |x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n > 1$. Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500{,}000$.
| 1. Define the sequence of functions recursively:
\[
f_1(x) = |x-1|
\]
\[
f_n(x) = f_{n-1}(|x-n|) \quad \text{for integers } n > 1
\]
2. Calculate the sum of the zeros for the first few functions:
- For \( f_1(x) = |x-1| \), the root is \( x = 1 \), so \( S(1) = 1 \).
- For \( f_2(x) = f_1(|x-2|... | 101 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Two congruent right circular cones each with base radius $3$ and height $8$ have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies inside both cones. The maximum possible value for $r^2$ is $\frac mn$, where $... | 1. **Set up the problem in Cartesian coordinates:**
- Place the origin at the bottom-left point where the axes of symmetry intersect.
- The cones have their vertices at $(0, 0)$ and $(6, 0)$, and their bases at $y = 8$ and $x = 8$ respectively.
2. **Determine the equations of the lines forming the sides of the c... | 298 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_\text{four}) = 10 = 12_\text{eight}$, and $g(2020) = \text{the digit sum of } 12_\text{eight} = 3... | 1. **Identify the smallest value of \( g(n) \) that cannot be expressed using digits \( 0-9 \) in base 16:**
- The smallest number in base 16 that cannot be expressed using digits \( 0-9 \) is \( A_{16} \), which is \( 10 \) in decimal.
2. **Determine the smallest \( n \) such that \( g(n) = 10 \):**
- Since \( ... | 151 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each day, Jenny ate $ 20\%$ of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, $ 32$ remained. How many jellybeans were in the jar originally?
$ \textbf{(A)}\ 40\qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 60\qquad \textbf{(E)}\ 75$ | 1. Let \( x \) be the number of jellybeans in the jar originally.
2. Each day, Jenny eats \( 20\% \) of the jellybeans, which means \( 80\% \) of the jellybeans remain. Mathematically, this can be expressed as:
\[
\text{Remaining jellybeans at the end of the day} = 0.8 \times \text{jellybeans at the beginning of ... | 50 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
$ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$ | 1. Let the two different prime numbers be \( x \) and \( y \). We are given that these primes are between 4 and 18. The prime numbers in this range are: 5, 7, 11, 13, and 17.
2. We need to find the value of \( xy - (x + y) \). This can be rewritten as:
\[
xy - x - y = (x-1)(y-1) - 1
\]
3. We need to check whic... | 119 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Charlyn walks completely around the boundary of a square whose sides are each $5$ km long. From any point on her path she can see exactly $1$ km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the near... | 1. **Understanding the Problem:**
Charlyn walks around a square with side length \(5\) km. From any point on her path, she can see exactly \(1\) km horizontally in all directions. We need to find the total area that Charlyn can see during her walk.
2. **Visualizing the Problem:**
- The square has side length \(5... | 43 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=10$. What is the maximum value of $A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A$?
$\text{(A)}\ 49 \qquad\text{(B)}\ 59 \qquad\text{(C)}\ 69 \qquad\text{(D)}\ 79\qquad\text{(E)}\ 89$ | To solve the problem, we need to maximize the expression \( A \cdot M \cdot C + A \cdot M + M \cdot C + C \cdot A \) given that \( A + M + C = 10 \) and \( A, M, C \) are nonnegative integers.
1. **Identify the expression to maximize:**
\[
E = A \cdot M \cdot C + A \cdot M + M \cdot C + C \cdot A
\]
2. **Con... | 69 | Algebra | MCQ | Yes | Yes | aops_forum | false |
A street has parallel curbs $ 40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $ 15$ feet and each stripe is $ 50$ feet long. Find the distance, in feet, between the stripes.
$ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \te... | 1. **Identify the given information and the goal:**
- The distance between the parallel curbs is \(40\) feet.
- The length of the curb between the stripes is \(15\) feet.
- Each stripe is \(50\) feet long.
- We need to find the distance between the stripes.
2. **Visualize the problem:**
- Draw a diagram... | 12 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$ | 1. **Identify the roots using Vieta's formulas:**
By Vieta's formulas, for the quadratic equation \(x^2 - 63x + k = 0\), the sum of the roots \(a\) and \(b\) is given by:
\[
a + b = 63
\]
and the product of the roots is:
\[
ab = k
\]
2. **Determine the nature of the roots:**
Since both \(a\)... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $ \triangle{XOY}$ be a right-angled triangle with $ m\angle{XOY}\equal{}90^\circ$. Let $ M$ and $ N$ be the midpoints of legs $ OX$ and $ OY$, respectively. Given that $ XN\equal{}19$ and $ YM\equal{}22$, find $ XY$.
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 26 \qquad
\textbf{(C)}\ 28 \qquad
\textbf{(D)}\ 30 \q... | 1. Let \( OX = a \) and \( OY = b \). Since \( M \) and \( N \) are the midpoints of \( OX \) and \( OY \) respectively, we have \( OM = \frac{a}{2} \) and \( ON = \frac{b}{2} \).
2. Given \( XN = 19 \) and \( YM = 22 \), we can use the Pythagorean theorem in the triangles \( \triangle XON \) and \( \triangle YOM \).
... | 26 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Let $ \{a_k\}$ be a sequence of integers such that $ a_1 \equal{} 1$ and $ a_{m \plus{} n} \equal{} a_m \plus{} a_n \plus{} mn$, for all positive integers $ m$ and $ n$. Then $ a_{12}$ is
$ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 67 \qquad \textbf{(D)}\ 78 \qquad \textbf{(E)}\ 89$ | 1. We start with the given sequence definition and initial condition:
\[
a_1 = 1
\]
\[
a_{m+n} = a_m + a_n + mn \quad \text{for all positive integers } m \text{ and } n
\]
2. We need to find \(a_{12}\). To do this, we will first compute the initial terms of the sequence to identify a pattern.
3. Com... | 78 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Mary typed a six-digit number, but the two $1$s she typed didn't show. What appeared was $2002$. How many different six-digit numbers could she have typed?
$\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }15\qquad\textbf{(E) }20$ | To solve this problem, we need to determine the number of different six-digit numbers Mary could have typed, given that the two $1$s she typed didn't show up in the displayed number $2002$.
1. **Identify the positions for the missing $1$s:**
The displayed number is $2002$, which has four digits. We need to place tw... | 15 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $(a_n)_{n\geq 1}$ be a sequence such that $a_1=1$ and $3a_{n+1}-3a_n=1$ for all $n\geq 1$. Find $a_{2002}$.
$\textbf{(A) }666\qquad\textbf{(B) }667\qquad\textbf{(C) }668\qquad\textbf{(D) }669\qquad\textbf{(E) }670$ | 1. We start with the given recurrence relation:
\[
3a_{n+1} - 3a_n = 1
\]
Simplifying, we get:
\[
a_{n+1} - a_n = \frac{1}{3}
\]
2. To find a general formula for \(a_n\), we can express \(a_{n+1}\) in terms of \(a_n\):
\[
a_{n+1} = a_n + \frac{1}{3}
\]
3. We know the initial condition \(... | 668 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered triples of positive integers $(x,y,z)$ satisfy $(x^y)^z=64$?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$ | To solve the problem, we need to find all ordered triples \((x, y, z)\) of positive integers such that \((x^y)^z = 64\). We start by expressing 64 as a power of a prime number:
\[ 64 = 2^6 \]
This means we need to find all combinations of \(x\), \(y\), and \(z\) such that:
\[ (x^y)^z = 2^6 \]
This can be rewritten ... | 9 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$.
$\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$ | 1. Given the polynomial \( P(x) = kx^3 + 2k^2x^2 + k^3 \), we need to find the sum of all real numbers \( k \) for which \( x-2 \) is a factor of \( P(x) \).
2. If \( x-2 \) is a factor of \( P(x) \), then \( P(2) = 0 \). We substitute \( x = 2 \) into \( P(x) \):
\[
P(2) = k(2)^3 + 2k^2(2)^2 + k^3 = 8k + 8k^2 +... | -8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
What is the smallest integer $n$ for which any subset of $\{1,2,3,\ldots,20\}$ of size $n$ must contain two numbers that differ by $8$?
$\textbf{(A) }2\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }15$ | 1. **Identify the problem constraints**: We need to find the smallest integer \( n \) such that any subset of \(\{1, 2, 3, \ldots, 20\}\) of size \( n \) must contain two numbers that differ by 8.
2. **Consider the Pigeonhole Principle**: The Pigeonhole Principle states that if \( n \) items are put into \( m \) conta... | 9 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $f$ be a real-valued function such that \[f(x)+2f\left(\dfrac{2002}x\right)=3x\] for all $x>0$. Find $f(2)$.
$\textbf{(A) }1000\qquad\textbf{(B) }2000\qquad\textbf{(C) }3000\qquad\textbf{(D) }4000\qquad\textbf{(E) }6000$ | 1. Given the functional equation:
\[
f(x) + 2f\left(\dfrac{2002}{x}\right) = 3x \quad \text{for all } x > 0
\]
We need to find \( f(2) \).
2. Substitute \( \frac{2002}{x} \) for \( x \) in the given equation:
\[
f\left(\frac{2002}{x}\right) + 2f(x) = 3 \left(\frac{2002}{x}\right)
\]
Simplifying... | 2000 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Bertha has $ 6$ daughters and no sons. Some of her daughters have $ 6$ daughters and the rest have none. Bertha has a total of $ 30$ daughters and granddaughters, and no great-grand daughters. How many of Bertha's daughters and granddaughters have no daughters?
$ \textbf{(A)}\ 22\qquad
\textbf{(B)}\ 23\qquad
\text... | 1. Bertha has 6 daughters.
2. Let \( x \) be the number of Bertha's daughters who each have 6 daughters.
3. Therefore, the number of granddaughters is \( 6x \).
4. Bertha has a total of 30 daughters and granddaughters.
5. The equation representing the total number of daughters and granddaughters is:
\[
6 + 6x = 3... | 26 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?
$ \textbf{(A)}\ 8\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 18\qquad
\textbf{(E)}\ 24$ | 1. Let's denote the number of women at the party as \( w \).
2. Each man danced with exactly 3 women. Since there are 12 men, the total number of dances involving men is:
\[
12 \times 3 = 36
\]
3. Each woman danced with exactly 2 men. Therefore, the total number of dances involving women is:
\[
2w
\]
... | 18 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $ 100$ cans, how many rows does it contain?
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 11$ | 1. Let's denote the number of rows by \( n \). The number of cans in the \( k \)-th row is given by \( 2k - 1 \). Therefore, the total number of cans in the display is the sum of the first \( n \) odd positive integers.
2. The sum of the first \( n \) odd positive integers is given by the formula:
\[
1 + 3 + 5 + ... | 10 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The equations $ 2x \plus{} 7 \equal{} 3$ and $ bx\minus{}10 \equal{} \minus{}\!2$ have the same solution for $ x$. What is the value of $ b$?
$ \textbf{(A)}\minus{}\!8 \qquad
\textbf{(B)}\minus{}\!4 \qquad
\textbf{(C)}\minus{}\!2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$ | 1. First, solve the equation \(2x + 7 = 3\) for \(x\):
\[
2x + 7 = 3
\]
Subtract 7 from both sides:
\[
2x = 3 - 7
\]
Simplify the right-hand side:
\[
2x = -4
\]
Divide both sides by 2:
\[
x = \frac{-4}{2} = -2
\]
2. Now, substitute \(x = -2\) into the second equation \(bx -... | -4 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Forty slips are placed into a hat, each bearing a number $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, $ 8$, $ 9$, or $ 10$, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $ p$ be the probability that all four slips bear the same number. Let $ q$ be the prob... | 1. **Calculate the probability \( p \) that all four slips bear the same number:**
- There are 10 different numbers, each appearing on 4 slips.
- To draw 4 slips all bearing the same number, we must choose one of the 10 numbers, and then draw all 4 slips of that number.
- The number of ways to choose 4 slips ... | 162 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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$ | 1. The first term of the sequence is given as \(2005\).
2. To find the next term, we need to calculate the sum of the cubes of the digits of \(2005\):
\[
2^3 + 0^3 + 0^3 + 5^3 = 8 + 0 + 0 + 125 = 133
\]
So, the second term is \(133\).
3. Next, we calculate the sum of the cubes of the digits of \(133\):
... | 133 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Two farmers agree that pigs are worth $ \$300$ and that goats are worth $ \$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $ \$390$ debt could be paid with two pigs, with one goat received in change.) Wha... | 1. We are given that pigs are worth $ \$ 300 $ and goats are worth $ \$ 210 $. We need to find the smallest positive debt that can be resolved using these values.
2. To solve this, we need to find the smallest positive integer value that can be expressed as a linear combination of $ 300 $ and $ 210 $. This is equivale... | 30 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?
$ \textbf{(A)}\ 56 \qquad \textbf{(B)}\ 58 \qquad \textbf{(C)... | 1. Let the tour guides be \(a\) and \(b\). Each of the six tourists must choose one of the two guides, but each guide must have at least one tourist.
2. We need to count the number of ways to distribute the six tourists between the two guides such that neither guide is left without any tourists.
3. First, calculate the... | 62 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A sphere is inscribed in a cube that has a surface area of $ 24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$ | 1. **Determine the side length of the outer cube:**
- The surface area of the outer cube is given as \(24\) square meters.
- A cube has 6 faces, so the area of each face is:
\[
\frac{24}{6} = 4 \text{ square meters}
\]
- The side length \(s\) of the outer cube can be found by taking the square r... | 8 | Geometry | MCQ | Yes | Yes | aops_forum | false |
All sides of the convex pentagon $ ABCDE$ are of equal length, and $ \angle A \equal{} \angle B \equal{} 90^{\circ}$. What is the degree measure of $ \angle E$?
$ \textbf{(A)}\ 90 \qquad \textbf{(B)}\ 108 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$ | 1. We start by noting that all sides of the convex pentagon \(ABCDE\) are of equal length, and \(\angle A = \angle B = 90^\circ\). This implies that \(AB\) and \(BC\) are perpendicular to each other, forming a right angle at \(B\).
2. Since \(ABCDE\) is a convex pentagon with equal side lengths, we can infer that \(AB... | 150 | Geometry | MCQ | Yes | Yes | aops_forum | false |
On the trip home from the meeting where this AMC$ 10$ was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $ bbcac$, where $ 0 \le a < b < c \le 9$, and $ b$ was the average of $ a$ and $ c$. How many different five-digit numbers satisfy all these properties?
$ \textbf{(A)... | 1. Given the five-digit number is of the form \( bbcac \), where \( 0 \le a < b < c \le 9 \) and \( b \) is the average of \( a \) and \( c \). This implies:
\[
b = \frac{a + c}{2}
\]
Since \( b \) must be an integer, \( a + c \) must be even, meaning \( a \) and \( c \) must have the same parity (both even... | 20 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $ \left(u_n\right)$ is a sequence of real numbers satisfying $ u_{n \plus{} 2} \equal{} 2u_{n \plus{} 1} \plus{} u_{n}$, and that $ u_3 \equal{} 9$ and $ u_6 \equal{} 128$. What is $ u_5$?
$ \textbf{(A)}\ 40 \qquad \textbf{(B)}\ 53 \qquad \textbf{(C)}\ 68 \qquad \textbf{(D)}\ 88 \qquad \textbf{(E)}\ 104... | 1. Given the recurrence relation \( u_{n+2} = 2u_{n+1} + u_n \), we need to find \( u_5 \) given \( u_3 = 9 \) and \( u_6 = 128 \).
2. We start by expressing \( u_6 \) in terms of \( u_4 \) and \( u_5 \):
\[
u_6 = 2u_5 + u_4
\]
Given \( u_6 = 128 \), we have:
\[
128 = 2u_5 + u_4 \quad \text{(1)}
\... | 53 | Other | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers $n$ satisfy$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$ | 1. Let $\lfloor \sqrt{n} \rfloor = k$. This implies that $k \leq \sqrt{n} < k+1$, which translates to:
\[
k^2 \leq n < (k+1)^2
\]
2. Given the equation:
\[
\frac{n + 1000}{70} = k
\]
we can solve for $n$:
\[
n = 70k - 1000
\]
3. Substituting $n = 70k - 1000$ into the inequality $k^2 \leq n... | 6 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\rm nine}$. What is the remainder when $N$ is divided by $5?$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$ | 1. **Convert the base-nine number to its decimal (base-10) equivalent:**
The given number in base-nine is \(27{,}006{,}000{,}052_{\text{nine}}\). To convert this to base-10, we use the formula for base conversion:
\[
N = 2 \cdot 9^{10} + 7 \cdot 9^9 + 0 \cdot 9^8 + 0 \cdot 9^7 + 6 \cdot 9^6 + 0 \cdot 9^5 + 0 \... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Increasing the radius of a cylinder by $ 6$ units increased the volume by $ y$ cubic units. Increasing the altitude of the cylinder by $ 6$ units also increases the volume by $ y$ cubic units. If the original altitude is $ 2$, then the original radius is:
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6... | 1. Let the original radius of the cylinder be \( r \) and the original altitude (height) be \( h = 2 \).
2. The volume \( V \) of the cylinder is given by the formula:
\[
V = \pi r^2 h = \pi r^2 \cdot 2 = 2\pi r^2
\]
3. When the radius is increased by 6 units, the new radius is \( r + 6 \). The new volume \( V... | 6 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Given $ 12$ points in a plane no three of which are collinear, the number of lines they determine is:
$ \textbf{(A)}\ 24 \qquad\textbf{(B)}\ 54 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 66 \qquad\textbf{(E)}\ \text{none of these}$ | 1. To determine the number of lines formed by 12 points in a plane where no three points are collinear, we need to count the number of ways to choose 2 points out of the 12 points. This is because a line is uniquely determined by any two distinct points.
2. The number of ways to choose 2 points from 12 points is given... | 66 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
The hypotenuse of a right triangle is $ 10$ inches and the radius of the inscribed circle is $ 1$ inch. The perimeter of the triangle in inches is:
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 22 \qquad
\textbf{(C)}\ 24 \qquad
\textbf{(D)}\ 26 \qquad
\textbf{(E)}\ 30$ | 1. Let the sides of the right triangle be \(a\), \(b\), and \(c\) where \(c\) is the hypotenuse. Given \(c = 10\) inches.
2. The radius \(r\) of the inscribed circle is given as \(1\) inch.
3. The area \(A\) of the triangle can be expressed in two ways:
- Using the inradius: \(A = r \cdot s\), where \(s\) is the sem... | 24 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The number of distinct lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is:
$ \textbf{(A)}\ 9\qquad
\textbf{(B)}\ 7\qquad
\textbf{(C)}\ 6\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 3$ | 1. **Identify the properties of an isosceles triangle:**
- An isosceles triangle has two equal sides and two equal angles opposite those sides.
- The altitude, median, and angle bisector from the vertex angle (the angle between the two equal sides) are the same line.
2. **Count the lines for altitudes:**
- In... | 5 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The numbers $ x,\,y,\,z$ are proportional to $ 2,\,3,\,5$. The sum of $ x$, $ y$, and $ z$ is $ 100$. The number $ y$ is given by the equation $ y \equal{} ax \minus{} 10$. Then $ a$ is:
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ \frac{3}{2}\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ \frac{5}{2}\qquad
\textbf{(E)... | 1. Given that the numbers \( x, y, z \) are proportional to \( 2, 3, 5 \), we can write:
\[
x = 2k, \quad y = 3k, \quad z = 5k
\]
for some constant \( k \).
2. The sum of \( x, y, \) and \( z \) is given as 100:
\[
x + y + z = 100
\]
Substituting the proportional values, we get:
\[
2k + 3... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The base of the decimal number system is ten, meaning, for example, that $ 123 \equal{} 1\cdot 10^2 \plus{} 2\cdot 10 \plus{} 3$. In the binary system, which has base two, the first five positive integers are $ 1,\,10,\,11,\,100,\,101$. The numeral $ 10011$ in the binary system would then be written in the decimal syst... | To convert the binary number \(10011_2\) to its decimal equivalent, we need to express it as a sum of powers of 2. Each digit in the binary number represents a power of 2, starting from \(2^0\) on the right.
1. Write down the binary number and its corresponding powers of 2:
\[
10011_2 = 1 \cdot 2^4 + 0 \cdot 2^3... | 19 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Side $ AC$ of right triangle $ ABC$ is divide into $ 8$ equal parts. Seven line segments parallel to $ BC$ are drawn to $ AB$ from the points of division. If $ BC \equal{} 10$, then the sum of the lengths of the seven line segments:
$ \textbf{(A)}\ \text{cannot be found from the given information} \qquad
\textbf{(... | 1. Given a right triangle \( \triangle ABC \) with \( \angle C = 90^\circ \), side \( AC \) is divided into 8 equal parts. Seven line segments are drawn parallel to \( BC \) from these points of division to \( AB \). We need to find the sum of the lengths of these seven line segments, given \( BC = 10 \).
2. Let the l... | 35 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Given $ a_0 \equal{} 1$, $ a_1 \equal{} 3$, and the general relation $ a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n$ for $ n \ge 1$. Then $ a_3$ equals:
$ \textbf{(A)}\ \frac{13}{27}\qquad
\textbf{(B)}\ 33\qquad
\textbf{(C)}\ 21\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ \minus{}17$ | 1. We start with the given initial conditions and the recurrence relation:
\[
a_0 = 1, \quad a_1 = 3, \quad a_n^2 - a_{n-1}a_{n+1} = (-1)^n \quad \text{for} \quad n \geq 1
\]
2. For \( n = 1 \):
\[
a_1^2 - a_0 a_2 = (-1)^1
\]
Substituting the known values \( a_0 = 1 \) and \( a_1 = 3 \):
\[
... | 33 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Each edge of a cube is increased by $50 \%$. The percent of increase of the surface area of the cube is:
$ \textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 750 $ | 1. Let the original edge length of the cube be \( x \). The surface area \( A \) of a cube with edge length \( x \) is given by:
\[
A = 6x^2
\]
2. When the edge length is increased by \( 50\% \), the new edge length becomes:
\[
1.5x
\]
3. The new surface area \( A' \) of the cube with the new edge l... | 125 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The value of $x^2-6x+13$ can never be less than:
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 4.5 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 13 $ | 1. The given expression is \( x^2 - 6x + 13 \). This is a quadratic function of the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = -6 \), and \( c = 13 \).
2. To find the minimum value of a quadratic function \( ax^2 + bx + c \), we use the vertex formula. The x-coordinate of the vertex is given by \( x = -\frac{... | 4 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
A farmer divides his herd of $n$ cows among his four sons so that one son gets one-half the herd, a second son, one-fourth, a third son, one-fifth, and the fourth son, 7 cows. Then $n$ is:
$ \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 180\qquad\textbf{(E)}\ 240 $ | 1. Let \( n \) be the total number of cows.
2. According to the problem, the cows are divided as follows:
- The first son gets \(\frac{1}{2}n\) cows.
- The second son gets \(\frac{1}{4}n\) cows.
- The third son gets \(\frac{1}{5}n\) cows.
- The fourth son gets 7 cows.
3. The sum of all these parts must equ... | 140 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The logarithm of $.0625$ to the base $2$ is:
$ \textbf{(A)}\ .025 \qquad\textbf{(B)}\ .25\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ -4\qquad\textbf{(E)}\ -2 $ | 1. We start with the given logarithmic expression:
\[
\log_2 0.0625
\]
2. Convert \(0.0625\) to a fraction:
\[
0.0625 = \frac{1}{16}
\]
3. Rewrite the logarithmic expression using the fraction:
\[
\log_2 \left(\frac{1}{16}\right)
\]
4. Recall that \(\frac{1}{16}\) can be written as \(2^{-4}\... | -4 | Algebra | MCQ | Yes | Yes | aops_forum | false |
It is given that $x$ varies directly as $y$ and inversely as the square of $z$, and that $x=10$ when $y=4$ and $z=14$. Then, when $y=16$ and $z=7$, $x$ equals:
$ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 154\qquad\textbf{(D)}\ 140\qquad\textbf{(E)}\ 120 $ | 1. Given that \( x \) varies directly as \( y \) and inversely as the square of \( z \), we can write the relationship as:
\[
x = k \frac{y}{z^2}
\]
where \( k \) is a constant of proportionality.
2. We are given that \( x = 10 \) when \( y = 4 \) and \( z = 14 \). Substituting these values into the equati... | 160 | Algebra | MCQ | Yes | Yes | aops_forum | false |
On a examination of $n$ questions a student answers correctly $15$ of the first $20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be?
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf... | 1. Let \( n \) be the total number of questions on the examination.
2. The student answers 15 out of the first 20 questions correctly.
3. For the remaining \( n - 20 \) questions, the student answers one third correctly. Therefore, the number of correctly answered questions in the remaining part is \( \frac{1}{3}(n - 2... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression.
Let $S_n$ represent the sum of the first $n$ terms of the harmonic progression; for example $S_3$ represents the sum of the first three terms. If the first three terms of a harmonic progression are $3,4,6$, then:... | 1. **Identify the given terms and their reciprocals:**
The first three terms of the harmonic progression are \(3, 4, 6\). Their reciprocals are:
\[
\frac{1}{3}, \frac{1}{4}, \frac{1}{6}
\]
2. **Verify that the reciprocals form an arithmetic progression:**
Calculate the differences between consecutive te... | 25 | Other | MCQ | Yes | Yes | aops_forum | false |
The base of a triangle is $80$, and one side of the base angle is $60^\circ$. The sum of the lengths of the other two sides is $90$. The shortest side is:
$ \textbf{(A)}\ 45 \qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 12 $ | 1. Let \( \triangle ABC \) be a triangle with \( BC = 80 \) and \( \angle ABC = 60^\circ \). We are given that \( AB + AC = 90 \).
2. Draw the altitude \( AH \) from \( A \) to \( BC \), where \( H \in BC \). In \( \triangle ABH \), since \( \angle ABH = 60^\circ \), we have:
\[
BH = \frac{1}{2} AB \quad \text{a... | 17 | Geometry | MCQ | Yes | Yes | aops_forum | false |
In triangle $ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals:
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ \text{none of these} $ | 1. Given that $BD$ is a median in triangle $ABC$, it means that $D$ is the midpoint of $AC$. Therefore, $\overline{AD} = \overline{DC}$.
2. $CF$ intersects $BD$ at $E$ such that $\overline{BE} = \overline{ED}$. This implies that $E$ is the midpoint of $BD$.
3. Point $F$ is on $AB$ and $\overline{BF} = 5$.
4. We need to... | 15 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Given the polynomial $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$'s are integers or zero. Set $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$. [See example 25 for the meaning of $|x|$.] The number of polynomials with $h=3$ is:
$ \textbf{(A)}... | We need to find the number of polynomials of the form \(a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n\) such that \(h = n + a_0 + |a_1| + |a_2| + \cdots + |a_n| = 3\). Here, \(n\) is a non-negative integer, \(a_0\) is a positive integer, and the remaining \(a_i\) are integers or zero.
We will perform casework on \(n\)... | 5 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A club with $x$ members is organized into four committees in accordance with these two rules:
$ \text{(1)}\ \text{Each member belongs to two and only two committees}\qquad$
$\text{(2)}\ \text{Each pair of committees has one and only one member in common}$
Then $x$:
$\textbf{(A)} \ \text{cannont be determined} \... | 1. Let's denote the four committees as \( A, B, C, \) and \( D \).
2. According to the problem, each member belongs to exactly two committees. Therefore, each member can be represented as a pair of committees.
3. We need to determine the number of such pairs. The number of ways to choose 2 committees out of 4 is given ... | 6 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is:
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 $ | 1. Let \( m \) and \( n \) be any two odd numbers with \( n < m \). We need to find the largest integer that divides all possible numbers of the form \( m^2 - n^2 \).
2. We start by expressing \( m^2 - n^2 \) using the difference of squares formula:
\[
m^2 - n^2 = (m - n)(m + n)
\]
3. Since \( m \) and \( n ... | 8 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times... \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4, ...., 59$. let $N$ be the number of primes appearing in this sequence. Then $... | 1. First, we define \( P \) as the product of all prime numbers less than or equal to 61. Therefore,
\[
P = 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 31 \times 37 \times 41 \times 43 \times 47 \times 53 \times 59 \times 61
\]
2. We are given a sequence of 5... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If the graphs of $2y+x+3=0$ and $3y+ax+2=0$ are to meet at right angles, the value of $a$ is:
${{ \textbf{(A)}\ \pm \frac{2}{3} \qquad\textbf{(B)}\ -\frac{2}{3}\qquad\textbf{(C)}\ -\frac{3}{2} \qquad\textbf{(D)}\ 6}\qquad\textbf{(E)}\ -6} $ | 1. To determine the value of \( a \) such that the graphs of the lines \( 2y + x + 3 = 0 \) and \( 3y + ax + 2 = 0 \) meet at right angles, we need to find the slopes of these lines and use the property that the slopes of perpendicular lines are negative reciprocals of each other.
2. First, we find the slope of the li... | -6 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is:
${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3}\qquad\textbf{(E)}\ \text{More than three, but finite} } $ | To find the number of integer solutions to the equation \(2^{2x} - 3^{2y} = 55\), we start by factoring the left-hand side:
1. Rewrite the equation:
\[
2^{2x} - 3^{2y} = 55
\]
2. Notice that \(2^{2x} = (2^x)^2\) and \(3^{2y} = (3^y)^2\). Let \(a = 2^x\) and \(b = 3^y\). The equation becomes:
\[
a^2 - b... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If both $ x$ and $ y$ are both integers, how many pairs of solutions are there of the equation $ (x\minus{}8)(x\minus{}10) \equal{} 2^y?$
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ \text{more than 3}$ | To solve the problem, we need to find all integer pairs \((x, y)\) that satisfy the equation \((x-8)(x-10) = 2^y\).
1. **Expand and simplify the equation:**
\[
(x-8)(x-10) = x^2 - 18x + 80
\]
Therefore, we need to find integer values of \(x\) such that \(x^2 - 18x + 80 = 2^y\).
2. **Analyze the quadratic ... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$. The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$ | 1. Let's denote the lines as follows:
- \( l_1 \) and \( l_3 \) are parallel lines.
- \( l_2 \) intersects both \( l_1 \) and \( l_3 \).
2. Since \( l_3 \) is parallel to \( l_1 \), the distance between \( l_1 \) and \( l_3 \) is constant. Let this distance be \( d \).
3. Consider the line \( m \) that is equid... | 2 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Let $ n$ be the number of number-pairs $ (x,y)$ which satisfy $ 5y \minus{} 3x \equal{} 15$ and $ x^2 \plus{} y^2 \le 16$. Then $ n$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{more than two, but finite} \qquad \textbf{(E)}\ \text{greater than any finite number}$ | 1. **Rewrite the linear equation in slope-intercept form:**
\[
5y - 3x = 15 \implies 5y = 3x + 15 \implies y = \frac{3}{5}x + 3
\]
This is the equation of a line with slope \(\frac{3}{5}\) and y-intercept 3.
2. **Interpret the inequality \(x^2 + y^2 \leq 16\):**
This inequality represents a circle cente... | 2 | Geometry | MCQ | Yes | Yes | aops_forum | false |
When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$. When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$. If $ R_1 \equal{} R_2$ then $ m$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad... | 1. We start by using polynomial division to find the remainders when \( y^2 + my + 2 \) is divided by \( y - 1 \) and \( y + 1 \).
2. When \( y^2 + my + 2 \) is divided by \( y - 1 \):
\[
y^2 + my + 2 = (y - 1)Q_1(y) + R_1
\]
To find \( R_1 \), we substitute \( y = 1 \) into the polynomial:
\[
1^2 + ... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Of $ 28$ students taking at least one subject the number taking Mathematics and English only equals the number taking Mathematics only. No student takes English only or History only, and six students take Mathematics and History, but not English. The number taking English and History only is five times the number tak... | 1. Define the variables for the number of students taking each combination of subjects:
- Let \( x \) be the number of students taking Mathematics and English only.
- Let \( y \) be the number of students taking all three subjects (Mathematics, English, and History).
- Let \( z \) be the number of students tak... | 5 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }6$ | 1. Let \( a, b, c \) be the side lengths of the triangle. Let \( A \) be the area of the triangle. Let \( s = \frac{a+b+c}{2} \) be the semi-perimeter of the triangle.
2. We know that the area \( A \) of the triangle can be expressed in terms of the inradius \( r \) and the semi-perimeter \( s \) as:
\[
A = rs
... | 2 | Geometry | MCQ | Yes | Yes | aops_forum | false |
For each integer $N>1$, there is a mathematical system in which two or more positive integers are defined to be congruent if they leave the same non-negative remainder when divided by $N$. If $69,90,$ and $125$ are congruent in one such system, then in that same system, $81$ is congruent to
$\textbf{(A) }3\qquad\textb... | 1. We are given that the integers \(69\), \(90\), and \(125\) are congruent modulo \(N\). This means:
\[
69 \equiv 90 \equiv 125 \pmod{N}
\]
For these numbers to be congruent modulo \(N\), the differences between any pair of these numbers must be divisible by \(N\).
2. Calculate the differences:
\[
9... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If $(1.0025)^{10}$ is evaluated correct to $5$ decimal places, then the digit in the fifth decimal place is
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }5\qquad \textbf{(E) }8$ | To find the value of $(1.0025)^{10}$ correct to 5 decimal places, we can use the binomial theorem for approximation. The binomial theorem states that for any real number $x$ and integer $n$:
\[
(1 + x)^n \approx 1 + nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{6}x^3 + \cdots
\]
In this case, $x = 0.0025$ and $n = 10... | 2 | Calculus | MCQ | Yes | Yes | aops_forum | false |
A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get $4,289$. The sum of their ages was
$\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }56\qquad\textbf{(D) }59\qquad \textbf{(E) }64$ | 1. Let the father's age be denoted by $\overline{ab}$, where $a$ and $b$ are the digits of his age. Similarly, let the son's age be denoted by $\overline{1c}$, where $1$ is the tens digit and $c$ is the units digit (since he is a teenager).
2. When the boy writes his age after his father's, the new four-digit number c... | 59 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Nine lines parallel to the base of a triangle divide the other sides each into $10$ equal segments and the area into $10$ distinct parts. If the area of the largest of these parts is $38$, then the area of the original triangle is
$\textbf{(A) }180\qquad\textbf{(B) }190\qquad\textbf{(C) }200\qquad\textbf{(D) }210\qqu... | 1. Let $\triangle ABC$ have base $BC$, and let the sides $AB$ and $AC$ be divided into $10$ equal segments by $9$ lines parallel to $BC$. This means that each segment on $AB$ and $AC$ is of equal length.
2. Let the total area of $\triangle ABC$ be $A$. The $9$ lines parallel to $BC$ divide the triangle into $10$ small... | 200 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The sum of the digits in base ten of $ (10^{4n^2\plus{}8}\plus{}1)^2$, where $ n$ is a positive integer, is
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 4n \qquad
\textbf{(C)}\ 2\plus{}2n \qquad
\textbf{(D)}\ 4n^2 \qquad
\textbf{(E)}\ n^2\plus{}n\plus{}2$ | 1. Let's start by examining the expression \((10^{4n^2 + 8} + 1)^2\). We can rewrite it as:
\[
(10^X + 1)^2 \quad \text{where} \quad X = 4n^2 + 8
\]
2. Expanding the square, we get:
\[
(10^X + 1)^2 = 10^{2X} + 2 \cdot 10^X + 1
\]
3. Substituting \(X = 4n^2 + 8\), we have:
\[
10^{2(4n^2 + 8)} + 2... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If $p, q$ and $r$ are distinct roots of $x^3-x^2+x-2=0$, then $p^3+q^3+r^3$ equals
$ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{none of these} $ | Given the polynomial \( x^3 - x^2 + x - 2 = 0 \) with roots \( p, q, \) and \( r \), we need to find the value of \( p^3 + q^3 + r^3 \).
1. **Identify the coefficients:**
The polynomial is \( x^3 - x^2 + x - 2 = 0 \). The coefficients are:
\[
a_3 = 1, \quad a_2 = -1, \quad a_1 = 1, \quad a_0 = -2
\]
2. **... | -6 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many integers greater than $10$ and less than $100$, written in base-$10$ notation, are increased by $9$ when their digits are reversed?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ 10$ | 1. Let the two-digit number be represented as \(10x + y\), where \(x\) and \(y\) are the digits of the number, and \(x\) is the tens digit and \(y\) is the units digit. Given that the number is increased by 9 when its digits are reversed, we can write the equation:
\[
10x + y + 9 = 10y + x
\]
2. Simplify the ... | 8 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then the area of $\triangle AEF$ is
$\textbf{(A) }16\qquad\textbf{(B) }24\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad \textbf{(E) }48$ | 1. **Identify the midpoints and their implications:**
- \( D \) is the midpoint of \( AB \), so \( AD = DB \).
- \( E \) is the midpoint of \( DB \), so \( DE = EB \).
- \( F \) is the midpoint of \( BC \), so \( BF = FC \).
2. **Calculate the area of sub-triangles:**
- Since \( F \) is the midpoint of \( ... | 36 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The polynomial $x^{2n}+1+(x+1)^{2n}$ is not divisible by $x^2+x+1$ if $n$ equals
$\text{(A)} \ 17 \qquad \text{(B)} \ 20 \qquad \text{(C)} \ 21 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$ | To determine when the polynomial \( x^{2n} + 1 + (x+1)^{2n} \) is not divisible by \( x^2 + x + 1 \), we need to analyze the roots of the polynomial \( x^2 + x + 1 \). The roots of \( x^2 + x + 1 = 0 \) are the non-real cube roots of unity, denoted as \( \omega \) and \( \omega^2 \), where \( \omega = e^{2\pi i / 3} \)... | 21 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Suppose that at the end of any year, a unit of money has lost $10\%$ of the value it had at the beginning of that year. Find the smallest integer $n$ such that after $n$ years, the money will have lost at least $90\%$ of its value. (To the nearest thousandth $\log_{10}3=.477$.)
$\text{(A)}\ 14 \qquad \text{(B)}\ 16 ... | 1. Let the initial value of the money be \( V_0 \). After one year, the value of the money will be \( 0.9V_0 \) because it loses \( 10\% \) of its value.
2. After \( n \) years, the value of the money will be \( V_0 \times (0.9)^n \).
3. We need to find the smallest integer \( n \) such that the money has lost at least... | 22 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The number of real solutions to the equation \[ \frac{x}{100} = \sin x \] is
$\text{(A)} \ 61 \qquad \text{(B)} \ 62 \qquad \text{(C)} \ 63 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$ | 1. Consider the graphs of \( y = \frac{x}{100} \) and \( y = \sin x \) separately.
2. The function \( y = \sin x \) is periodic with period \( 2\pi \). This means that the sine function repeats its values every \( 2\pi \) units along the x-axis.
3. The function \( y = \frac{x}{100} \) is a straight line with a slope... | 63 | Calculus | MCQ | Yes | Yes | aops_forum | false |
By definition, $ r! \equal{} r(r \minus{} 1) \cdots 1$ and $ \binom{j}{k} \equal{} \frac {j!}{k!(j \minus{} k)!}$, where $ r,j,k$ are positive integers and $ k < j$. If $ \binom{n}{1}, \binom{n}{2}, \binom{n}{3}$ form an arithmetic progression with $ n > 3$, then $ n$ equals
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 7... | 1. We start by expressing the binomial coefficients $\binom{n}{1}$, $\binom{n}{2}$, and $\binom{n}{3}$ in terms of $n$:
\[
\binom{n}{1} = n
\]
\[
\binom{n}{2} = \frac{n(n-1)}{2}
\]
\[
\binom{n}{3} = \frac{n(n-1)(n-2)}{6}
\]
2. Since these coefficients form an arithmetic progression, the diff... | 7 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A vertical line divides the triangle with vertices $(0,0)$, $(1,1)$, and $(9,1)$ in the $xy\text{-plane}$ into two regions of equal area. The equation of the line is $x=$
$\textbf {(A) } 2.5 \qquad \textbf {(B) } 3.0 \qquad \textbf {(C) } 3.5 \qquad \textbf {(D) } 4.0\qquad \textbf {(E) } 4.5$ | 1. **Calculate the area of the triangle:**
The vertices of the triangle are \((0,0)\), \((1,1)\), and \((9,1)\). We can use the formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1... | 5 | Geometry | MCQ | Yes | Yes | aops_forum | false |
If $60^a = 3$ and $60^b = 5$, then $12^{[(1-a-b)/2(1-b)]}$ is
$\text{(A)} \ \sqrt{3} \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \sqrt{5} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \sqrt{12}$ | 1. Given the equations \(60^a = 3\) and \(60^b = 5\), we can express \(a\) and \(b\) in terms of logarithms:
\[
a = \frac{\log 3}{\log 60}
\]
\[
b = \frac{\log 5}{\log 60}
\]
2. We need to find the value of \(12^{\left(\frac{1-a-b}{2(1-b)}\right)}\). First, let's simplify the exponent \(\frac{1-a-b}{... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The volume of a certain rectangular solid is $ 8 \text{ cm}^3$, its total surface area is $ 32 \text{ cm}^3$, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is
$ \textbf{(A)}\ 28 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\... | 1. Let the three dimensions of the rectangular solid be \( \frac{a}{r}, a, ar \), where \(a\) is a common ratio and \(r\) is the geometric progression ratio.
2. Given the volume of the rectangular solid is \(8 \text{ cm}^3\), we have:
\[
\left(\frac{a}{r}\right) \cdot a \cdot (ar) = 8
\]
Simplifying, we get... | 32 | Geometry | MCQ | Yes | Yes | aops_forum | false |
A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains a... | To solve this problem, we need to determine the smallest number of socks that must be selected to guarantee that the selection contains at least 10 pairs. A pair of socks is defined as two socks of the same color.
1. **Identify the worst-case scenario**:
- In the worst-case scenario, we want to avoid forming pairs... | 24 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse of $50\%$ longer than the minor axis, the length of the major axis is
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ \frac{9}{4}\qquad\textbf{(E)}\ 3$ | 1. Consider a right circular cylinder with radius \( r = 1 \). The diameter of the base of the cylinder is \( 2r = 2 \).
2. When a plane intersects the cylinder, the resulting intersection is an ellipse. The minor axis of this ellipse is equal to the diameter of the base of the cylinder, which is \( 2 \).
3. Accordin... | 3 | Geometry | MCQ | Yes | Yes | aops_forum | false |
A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)?
$ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 8\sqrt{3} \qquad\textbf{(E)... | 1. Let the radius of the ball be \( r \) cm. The hole left in the ice is a circular segment of the sphere with a diameter of 24 cm and a depth of 8 cm.
2. The radius of the circular hole is half of the diameter, so the radius of the hole is \( 12 \) cm.
3. The depth of the hole is the distance from the top of the spher... | 13 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, the sum of the digits of $n$ is
$ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 19 \qquad\textbf{(D)}\ 21 \... | 1. We start with the given recursive sequence defined by:
\[
t_1 = 1
\]
For \( n > 1 \):
\[
t_n = 1 + t_{\frac{n}{2}} \quad \text{if } n \text{ is even}
\]
\[
t_n = \frac{1}{t_{n-1}} \quad \text{if } n \text{ is odd}
\]
2. We are given that \( t_n = \frac{19}{87} \). Since \( \frac{19}{87... | 15 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
If $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^3 + bx^2 + 1$, then $b$ is
$ \textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2 $ | 1. Let \( P(x) = ax^3 + bx^2 + 1 \). We are given that \( x^2 - x - 1 \) is a factor of \( P(x) \). This implies that \( P(x) \) can be written as \( P(x) = (x^2 - x - 1)Q(x) \) for some polynomial \( Q(x) \).
2. Since \( x^2 - x - 1 \) is a factor of \( P(x) \), the remainder when \( P(x) \) is divided by \( x^2 - x ... | -2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The six edges of a tetrahedron $ABCD$ measure $7$, $13$, $18$, $27$, $36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is
$ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 36 $ | To solve this problem, we need to use the triangle inequality theorem, which states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. We will apply this theorem to the tetrahedron $ABCD$.
1. **Identify the edges:**
The edges of the tetrahedron are ... | 13 | Geometry | MCQ | Yes | Yes | aops_forum | false |
An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$, and one of the base angles is $\arcsin(.8)$. Find the area of the trapezoid.
$ \textbf{(A)}\ 72\qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 80\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ \text{not uniquely determined} $ | 1. **Identify the given information and variables:**
- The longer base of the trapezoid is \(16\).
- One of the base angles is \(\arcsin(0.8)\).
- Let the shorter base be \(x\).
- Let the height of the trapezoid be \(h\).
2. **Use the property of the isosceles trapezoid circumscribed around a circle:**
... | 80 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Five people are sitting at a round table. Let $f \ge 0$ be the number of people sitting next to at least one female and $m \ge 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is
$ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ ... | 1. **Identify the possible gender arrangements:**
- We need to consider all possible gender arrangements of 5 people sitting in a circle. Since rotations and reflections are considered the same, we need to count distinct arrangements.
2. **Count the distinct arrangements:**
- **5 males (MMMMM):** There is only o... | 8 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.