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One base of a trapezoid is 100 units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3.$ Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapez... | 1. Let \(ABCD\) be a trapezoid with \(AB\) and \(DC\) as the parallel bases, where \(DC = AB + 100\). The segment joining the midpoints of the legs of the trapezoid has a length equal to the average of the lengths of the two bases, which is \(\frac{AB + DC}{2} = \frac{AB + (AB + 100)}{2} = AB + 50\).
2. Given that the... | 181 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given that \[ \frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!} \] find the greatest integer that is less than $\frac N{100}.$ | 1. We start with the given equation:
\[
\frac{1}{2!17!} + \frac{1}{3!16!} + \frac{1}{4!15!} + \frac{1}{5!14!} + \frac{1}{6!13!} + \frac{1}{7!12!} + \frac{1}{8!11!} + \frac{1}{9!10!} = \frac{N}{1!18!}
\]
2. Multiply both sides by \(19!\):
\[
19! \left( \frac{1}{2!17!} + \frac{1}{3!16!} + \frac{1}{4!15!} ... | 137 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$ | 1. **Identify the given information and setup the problem:**
- Trapezoid \(ABCD\) with \( \overline{BC} \) perpendicular to bases \( \overline{AB} \) and \( \overline{CD} \).
- Diagonals \( \overline{AC} \) and \( \overline{BD} \) are perpendicular.
- Given \( AB = \sqrt{11} \) and \( AD = \sqrt{1001} \).
2. ... | 110 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ,$ find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}.$ | 1. **Express \( z \) in exponential form:**
Given \( z + \frac{1}{z} = 2 \cos 3^\circ \), we can write \( z \) as \( z = e^{i\theta} \). Therefore, \( \frac{1}{z} = e^{-i\theta} \).
2. **Simplify the given equation:**
\[
e^{i\theta} + e^{-i\theta} = 2 \cos 3^\circ
\]
Using Euler's formula, \( e^{i\theta... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107).$ The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum o f the absolute values of all possible slopes for $\overline{AB}$ is $m/n... | 1. Given the coordinates of vertices \( A = (20, 100) \) and \( D = (21, 107) \) of an isosceles trapezoid \(ABCD\) with \( \overline{AB} \parallel \overline{CD} \), we need to find the sum of the absolute values of all possible slopes for \( \overline{AB} \).
2. Let the slope of \( \overline{AB} \) and \( \overline{C... | 31 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square ... | 1. **Identify the given information and the goal:**
- Points \(A\), \(B\), and \(C\) lie on the surface of a sphere with center \(O\) and radius 20.
- The side lengths of triangle \(ABC\) are \(AB = 13\), \(BC = 14\), and \(CA = 15\).
- We need to find the distance from \(O\) to the plane containing \(\triangl... | 118 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \] | 1. We start with the given equation:
\[
\frac{1}{\sin 45^\circ \sin 46^\circ} + \frac{1}{\sin 47^\circ \sin 48^\circ} + \cdots + \frac{1}{\sin 133^\circ \sin 134^\circ} = \frac{1}{\sin n^\circ}
\]
2. Notice that \(\sin(180^\circ - x) = \sin x\). Therefore, we can rewrite the sines of angles greater than \(90^... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let $m$ be the smallest number of students who could study b... | 1. Let \( S \) be the number of students who study Spanish, \( F \) be the number of students who study French, and \( B \) be the number of students who study both languages.
2. We know the total number of students is 2001.
3. The number of students who study Spanish is between 80% and 85% of the school population:
... | 298 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $R=(8,6)$. The lines whose equations are $8y=15x$ and $10y=3x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. The length of $PQ$ equals $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. **Identify the equations of the lines and the coordinates of point \( R \):**
- The equations of the lines are \( 8y = 15x \) and \( 10y = 3x \).
- Point \( R \) is given as \( (8, 6) \).
2. **Express the equations of the lines in slope-intercept form:**
- For \( 8y = 15x \):
\[
y = \frac{15}{8}x... | 67 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A set of positive numbers has the $\text{triangle property}$ if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\{4, 5, 6, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest... | 1. **Understanding the Triangle Property**: A set of positive numbers has the triangle property if it contains three distinct elements \(a\), \(b\), and \(c\) such that they can form a triangle. This means they must satisfy the triangle inequality:
\[
a + b > c, \quad a + c > b, \quad b + c > a
\]
where \(a... | 253 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\overline{CD}$ and vertices $G$ and $H$ on the circle. The ratio of the area of square $EFGH$ to the area of square $ABCD$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers and $m<n$. Find $10n+m$... | 1. Let the side length of the larger square \(ABCD\) be \(s\). Since the square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. The diagonal of the square can be calculated using the Pythagorean theorem:
\[
\text{Diagonal} = s\sqrt{2}
\]
Therefore, the radius \(r... | 252 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A certain function $f$ has the properties that $f(3x)=3f(x)$ for all positive real values of $x$, and that $f(x)=1-\mid x-2 \mid$ for $1\leq x \leq 3$. Find the smallest $x$ for which $f(x)=f(2001)$. | 1. We start with the given properties of the function \( f \):
\[
f(3x) = 3f(x) \quad \text{for all positive real values of } x
\]
and
\[
f(x) = 1 - |x - 2| \quad \text{for } 1 \leq x \leq 3.
\]
2. First, we need to determine \( f(2001) \). To do this, we need to understand the behavior of \( f \)... | 429 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n}$, where $m$ and $n$ ... | To solve this problem, we need to calculate the probability that Club Truncator will finish the season with more wins than losses. Given that the probabilities of winning, losing, or tying any match are each $\frac{1}{3}$, we can use combinatorial methods and symmetry to find the desired probability.
1. **Define the p... | 341 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i+1}$, replace the midpoint triangle of every face of $P_{i}$ by an outward-pointing regular tet... | 1. **Initial Setup and Definitions:**
- We start with \( P_0 \), a regular tetrahedron with volume 1.
- To obtain \( P_{i+1} \), we replace the midpoint triangle of every face of \( P_i \) with an outward-pointing regular tetrahedron that has the midpoint triangle as a face.
2. **Volume Calculation for \( P_1 \)... | 27 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In quadrilateral $ABCD$, $\angle{BAD}\cong\angle{ADC}$ and $\angle{ABD}\cong\angle{BCD}$, $AB=8$, $BD=10$, and $BC=6$. The length $CD$ may be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. **Extend lines \(BC\) and \(AD\) to meet at point \(E\)**:
- Since \(\angle{BAD} \cong \angle{ADC}\) and \(\angle{ABD} \cong \angle{BCD}\), triangles \(ABD\) and \(DCE\) are similar by AA similarity criterion.
2. **Establish similarity and corresponding sides**:
- By similarity, \(\triangle ABD \sim \triangle... | 27 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC=8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI=EJ=EK=2$. A solid $S$ is obtained by drilling a tunnel through... | 1. **Understanding the Problem:**
- We have a cube with edge length \(EC = 8\).
- Points \(I\), \(J\), and \(K\) are on edges \(\overline{EF}\), \(\overline{EH}\), and \(\overline{EC}\) respectively, such that \(EI = EJ = EK = 2\).
- A tunnel is drilled through the cube, and we need to find the surface area of... | 90 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S}\cup\{1\}$ is 13 less than the mean of $\mathcal{S}$, and the mean of $\mathcal{S}\cup\{2001\}$ is 27 more than the mean of $\mathcal{S}.$ Find the mean of $\mathcal{S}.$ | 1. Let \( s \) be the sum of the members of the set \( \mathcal{S} \), and let \( n \) be the number of members in \( \mathcal{S} \). The mean of \( \mathcal{S} \) is then \( \frac{s}{n} \).
2. When we add 1 to the set \( \mathcal{S} \), the new set is \( \mathcal{S} \cup \{1\} \). The mean of this new set is given by... | 651 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, angles $A$ and $B$ measure 60 degrees and 45 degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$, and $AT=24.$ The area of triangle $ABC$ can be written in the form $a+b\sqrt{c},$ where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of... | 1. **Identify the given angles and side lengths:**
- In triangle \(ABC\), \(\angle A = 60^\circ\) and \(\angle B = 45^\circ\).
- The bisector of \(\angle A\) intersects \(\overline{BC}\) at \(T\), and \(AT = 24\).
2. **Determine the remaining angle:**
- Since the sum of angles in a triangle is \(180^\circ\), ... | 291 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac mn},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 1. **Transform the ellipse to a circle:**
The given ellipse is \( x^2 + 4y^2 = 4 \). We can rewrite it as:
\[
\frac{x^2}{4} + y^2 = 1
\]
This represents an ellipse with semi-major axis \( a = 2 \) and semi-minor axis \( b = 1 \).
To transform this ellipse into a circle, we perform a scaling transform... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N.$ For example, 51 is a 7-10 double because its base-7 representation is 102. What is the largest 7-10 double? | 1. **Understanding the Problem:**
We need to find the largest positive integer \( N \) such that the digits of its base-7 representation form a base-10 number that is twice \( N \).
2. **Base-7 Representation:**
Let \( N \) be represented in base-7 as \( \overline{abc} \), where \( a, b, c \) are the digits in ... | 315 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N+1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so tha... | 1. We need to establish a coordinate system to express both \( x_i \) and \( y_i \). Let \( P_i = (i, a_i) \), where \( i \) is the row number and \( a_i \) is the column number of point \( P_i \).
2. The number associated with \( P_i \) in the original numbering (left to right, top to bottom) is:
\[
x_i = (i-1)... | 149 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a certain circle, the chord of a $d$-degree arc is 22 centimeters long, and the chord of a $2d$-degree arc is 20 centimeters longer than the chord of a $3d$-degree arc, where $d<120.$ The length of the chord of a $3d$-degree arc is $-m+\sqrt{n}$ centimeters, where $m$ and $n$ are positive integers. Find $m+n.$ | 1. **Set up the problem using the Law of Cosines:**
- Let the radius of the circle be \( r \).
- The length of the chord subtending a \( d \)-degree arc is 22 cm.
- The length of the chord subtending a \( 2d \)-degree arc is 20 cm longer than the chord subtending a \( 3d \)-degree arc.
- Let the length of t... | 178 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are relati... | 1. **Identify the problem and the structure of the octahedron:**
- We have a regular octahedron with 8 faces.
- We need to place the numbers 1 through 8 on these faces such that no two consecutive numbers are on adjacent faces.
- Note that 1 and 8 are also considered consecutive.
2. **Calculate the total numb... | 85 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given that
\begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& \text{y is the number formed by reversing the digits of x; and}\\ &(3)& z=|x-y|. \end{eqnarray*}How many distinct values of $z$ are possible? | 1. **Express \( x \) and \( y \) in terms of their digits:**
Let \( x = \overline{abc} \), where \( a, b, \) and \( c \) are the digits of \( x \). Therefore, we can write:
\[
x = 100a + 10b + c
\]
Similarly, since \( y \) is the number formed by reversing the digits of \( x \), we have:
\[
y = 100... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
It is known that, for all positive integers $k,$
\[1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}. \]Find the smallest positive integer $k$ such that $1^{2}+2^{2}+3^{2}+\cdots+k^{2}$ is a multiple of $200.$ | 1. We start with the given formula for the sum of squares of the first \( k \) positive integers:
\[
1^{2} + 2^{2} + 3^{2} + \cdots + k^{2} = \frac{k(k+1)(2k+1)}{6}
\]
We need to find the smallest positive integer \( k \) such that this sum is a multiple of 200.
2. First, factorize 200:
\[
200 = 2^3 ... | 112 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{S}$ be the set $\{1,2,3,\ldots,10\}.$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}.$ (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000.$ | 1. We need to find the number of sets of two non-empty disjoint subsets of $\mathcal{S} = \{1, 2, 3, \ldots, 10\}$. Let's denote these two subsets as $A$ and $B$ such that $A \cap B = \emptyset$ and $A, B \neq \emptyset$.
2. For each element in $\mathcal{S}$, there are three choices: it can be in subset $A$, in subset ... | 501 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6),$ and the product of the radii is $68.$ The x-axis and the line $y=mx$, where $m>0,$ are tangent to both circles. It is given that $m$ can be written in the form $a\sqrt{b}/c,$ where $a,$ $b,$ and $c$ are positive integers, ... | 1. **Identify the properties of the circles:**
- The circles \(\mathcal{C}_1\) and \(\mathcal{C}_2\) intersect at \((9,6)\).
- The product of the radii of the circles is \(68\).
- Both circles are tangent to the x-axis and the line \(y = mx\).
2. **Consider a general circle tangent to the x-axis and the line ... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive in... | 1. Let \( d \) be Dick's current age and \( n \) be the number of years in the future. In \( n \) years, Dick's age will be \( d + n \) and Jane's age will be \( 25 + n \).
2. Let \( D = d + n \) and \( J = 25 + n \). Since \( D \) and \( J \) are two-digit numbers, we can express them as \( D = 10x + y \) and \( J = 1... | 25 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $ x, y,$ and $ r$ with $ |x| > |y|,$
\[ (x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \p... | 1. We start by using the Binomial Theorem for non-integer exponents. Given \( (x + y)^r \), the expansion is:
\[
(x + y)^r = x^r + rx^{r-1}y + \frac{r(r-1)}{2}x^{r-2}y^2 + \frac{r(r-1)(r-2)}{3!}x^{r-3}y^3 + \cdots
\]
Here, \( x = 10^{2002} \), \( y = 1 \), and \( r = \frac{10}{7} \).
2. Substitute \( x \) ... | 428 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12.$ A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P,$ which is 7 units from $\overline{BG}$ and 5 units from $\overline{BC}.$ The beam continues to be reflected off the faces of the cube. The length of the light path from the ti... | 1. **Understanding the Problem:**
- We have a cube with side length \( AB = 12 \).
- A beam of light emanates from vertex \( A \) and reflects off face \( BCFG \) at point \( P \).
- Point \( P \) is 7 units from \( \overline{BG} \) and 5 units from \( \overline{BC} \).
- We need to find the length of the l... | 230 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is ... | 1. Let the ordered sequence of the members of $\mathcal{S}$, ascending, be $a_1, a_2, a_3, \ldots, a_n$, where $n$ is the size of the set. Denote $N = a_1 + a_2 + \cdots + a_n$.
2. For every integer $x$ in $\mathcal{S}$, the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer.... | 30 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N.$ | 1. Let the three positive integers be \(a\), \(b\), and \(c\). Given that \(a = b + c\), we can set up the equation for the product \(N\) and the sum \(S\) as follows:
\[
N = abc \quad \text{and} \quad S = a + b + c
\]
We are given that \(N = 6000 \times S\). Therefore, we have:
\[
abc = 6000(a + b + ... | 232128000 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the greatest integer multiple of $8,$ no two of whose digits are the same. What is the remainder when $N$ is divided by $1000?$ | 1. **Understanding the Problem:**
We need to find the greatest integer multiple of 8, no two of whose digits are the same, and determine the remainder when this number is divided by 1000.
2. **Finding the Greatest Integer Multiple of 8:**
To maximize the integer, we should use the largest possible digits without... | 120 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define a $good~word$ as a sequence of letters that consists only of the letters $A,$ $B,$ and $C$ $-$ some of these letters may not appear in the sequence $-$ and in which $A$ is never immediately followed by $B,$ $B$ is never immediately followed by $C,$ and $C$ is never immediately followed by $A.$ How many seven-let... | 1. **Define the problem constraints**: We need to count the number of seven-letter sequences (good words) using the letters \(A\), \(B\), and \(C\) such that:
- \(A\) is never immediately followed by \(B\),
- \(B\) is never immediately followed by \(C\),
- \(C\) is never immediately followed by \(A\).
2. **Co... | 192 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 1. **Understanding the Problem:**
We need to find the ratio of the volume of a smaller tetrahedron, formed by the centers of the faces of a larger regular tetrahedron, to the volume of the larger tetrahedron. The ratio should be expressed in the form \( \frac{m}{n} \) where \( m \) and \( n \) are relatively prime p... | 35 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A cylindrical log has diameter $ 12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $ 45^\circ$ angle with the plane of the first cut. The intersection of these two planes has... | 1. **Determine the dimensions of the cylinder:**
- The diameter of the cylindrical log is given as \(12\) inches.
- Therefore, the radius \(r\) of the cylinder is:
\[
r = \frac{12}{2} = 6 \text{ inches}
\]
2. **Understand the cuts:**
- The first cut is perpendicular to the axis of the cylinder,... | 216 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC,$ $AB=13,$ $BC=14,$ $AC=15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area if the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$ | 1. **Understanding the Problem:**
We are given a triangle \(ABC\) with sides \(AB = 13\), \(BC = 14\), and \(AC = 15\). Point \(G\) is the centroid of the triangle, which is the intersection of the medians. Points \(A'\), \(B'\), and \(C'\) are the images of \(A\), \(B\), and \(C\) respectively after a \(180^\circ\)... | 140 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are... | 1. **Define the problem in terms of probabilities:**
- Let \( a_n \) be the probability that the bug is at the starting vertex after \( n \) moves.
- Let \( b_n \) be the probability that the bug is at one of the other two vertices after \( n \) moves.
2. **Establish initial conditions:**
- Initially, the bug... | 1195 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $A=(0,0)$ and $B=(b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB=120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct element... | 1. **Identify the coordinates of the vertices:**
- Given points \( A = (0,0) \) and \( B = (b,2) \).
- The y-coordinates of the vertices are distinct elements of the set \(\{0, 2, 4, 6, 8, 10\}\).
2. **Determine the side length of the hexagon:**
- Since \( ABCDEF \) is an equilateral hexagon, all sides are eq... | 51 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \textstyle \frac{1}{2} (\log_{10} n - 1)$, find $n$. | 1. Given the equation:
\[
\log_{10} \sin x + \log_{10} \cos x = -1
\]
Using the properties of logarithms, we can combine the logs:
\[
\log_{10} (\sin x \cos x) = -1
\]
This implies:
\[
\sin x \cos x = 10^{-1} = \frac{1}{10}
\]
2. Using the double-angle identity for sine, we know:
\[... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $(m + n \pi)/p$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p$. | 1. **Calculate the volume of the rectangular parallelepiped (box):**
The dimensions of the box are \(3 \times 4 \times 5\). Therefore, the volume \(V_{\text{box}}\) is:
\[
V_{\text{box}} = 3 \times 4 \times 5 = 60 \text{ cubic units}
\]
2. **Calculate the volume of the region within one unit of the box:**
... | 505 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The sum of the areas of all triangles whose vertices are also vertices of a $1\times 1 \times 1$ cube is $m+\sqrt{n}+\sqrt{p}$, where $m$, $n$, and $p$ are integers. Find $m+n+p$. | 1. **Counting the Total Number of Triangles:**
The total number of triangles that can be formed by choosing any three vertices from the 8 vertices of a cube is given by:
\[
\binom{8}{3} = 56
\]
2. **Case 1: Triangles on the Same Face:**
- Each face of the cube is a square with 4 vertices.
- We can fo... | 348 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21$. Point $D$ is not on $\overline{AC}$ so that $AD = CD$, and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s$. | 1. **Identify the given information and set up the problem:**
- Point \( B \) is on \( \overline{AC} \) with \( AB = 9 \) and \( BC = 21 \).
- Point \( D \) is not on \( \overline{AC} \) such that \( AD = CD \), and both \( AD \) and \( BD \) are integers.
- We need to find the sum of all possible perimeters o... | 380 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms. | 1. Denote by \( d \) the common difference in the arithmetic progression. The first three terms are then \( a \), \( a + d \), and \( a + 2d \).
2. The last term is equal to the third term multiplied by the ratio of the third term to the second term, or \(\frac{(a + 2d)^2}{a + d}\).
3. Given that the first and fourth t... | 129 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
An integer between 1000 and 9999, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there? | To solve the problem, we need to count the number of balanced integers between 1000 and 9999. A balanced integer is defined as an integer where the sum of its two leftmost digits equals the sum of its two rightmost digits.
1. **Define the integer and its digits:**
Let the integer be represented as \( \overline{abcd... | 570 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when $N$ is divided by 1000. | To solve the problem, we need to count the number of positive integers less than or equal to 2003 whose base-2 representation has more 1's than 0's. We will use a combinatorial approach to solve this problem.
1. **Extend the Range to a Power of 2**:
We will first consider the numbers less than \(2^{11} = 2048\) ins... | 179 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The decimal representation of $m/n$, where $m$ and $n$ are relatively prime positive integers and $m < n$, contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible. | 1. We start with the fraction \(\frac{m}{n}\) where \(m\) and \(n\) are relatively prime positive integers and \(m < n\). The decimal representation of \(\frac{m}{n}\) contains the digits 2, 5, and 1 consecutively, and in that order. We need to find the smallest value of \(n\) for which this is possible.
2. Without lo... | 127 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 1. **Define the problem and the constraints:**
- We have a rectangle \(ABCD\) with dimensions \(15 \times 36\).
- A circle of radius 1 is placed randomly within the rectangle such that it lies completely inside.
- We need to find the probability that the circle does not touch the diagonal \(AC\).
2. **Calcula... | 1259 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. | 1. **Calculate the probability that Terry picks two candies of the same color:**
- Probability that Terry picks 2 red candies:
\[
P(\text{Terry picks 2 red}) = \frac{10}{20} \times \frac{9}{19} = \frac{10 \times 9}{20 \times 19} = \frac{90}{380} = \frac{9}{38}
\]
- Probability that Terry picks 2 bl... | 396 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of $N$. | 1. Let the dimensions of the rectangular block be \( x, y, \) and \( z \). When viewed such that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. This means that the cubes that are not visible form a smaller rectangular block inside the original block with dimensions \( (x-1), (y-1), \) and... | 384 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third m... | 1. Let \( d \) be the total number of bananas in the pile, and let \( a \), \( b \), and \( c \) be the number of bananas taken by the first, second, and third monkeys, respectively. Thus, we have:
\[
d = a + b + c
\]
2. The first monkey keeps \(\frac{3}{4}a\) bananas and gives the remaining \(\frac{1}{4}a\) ... | 51 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S$, the probability that it is divisible by $9$ is $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 1. **Determine the total number of integers in \( S \):**
- The set \( S \) consists of integers between \( 1 \) and \( 2^{40} \) whose binary expansions have exactly two \( 1 \)'s.
- The binary representation of such numbers can be written as \( 2^i + 2^j \) where \( 0 \leq i < j \leq 39 \).
- The number of w... | 913 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCDE$ be a convex pentagon with $AB\parallel CE$, $BC\parallel AD$, $AC\parallel DE$, $\angle ABC=120^\circ$, $AB=3$, $BC=5$, and $DE=15$. Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. | 1. **Identify the given information and setup the problem:**
- We have a convex pentagon \(ABCDE\) with the following properties:
- \(AB \parallel CE\)
- \(BC \parallel AD\)
- \(AC \parallel DE\)
- \(\angle ABC = 120^\circ\)
- \(AB = 3\)
- \(BC = 5\)
- \(DE = 15\)
- We need to f... | 26 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider a string of $n$ 7's, $7777\cdots77$, into which $+$ signs are inserted to produce an arithmetic expression. For example, $7+77+777+7+7=875$ could be obtained from eight 7's in this way. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value 7000? | 1. **Understanding the Problem:**
We need to determine the number of values of \( n \) for which it is possible to insert \( + \) signs into a string of \( n \) 7's such that the resulting arithmetic expression equals 7000.
2. **Rewriting the Problem:**
We can rewrite the problem in terms of a simpler arithmetic... | 487 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A convex polyhedron $P$ has 26 vertices, 60 edges, and 36 faces, 24 of which are triangular, and 12 of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $P$ have? | 1. **Calculate the total number of line segments connecting any two vertices:**
The total number of line segments connecting any two vertices in a polyhedron with \(26\) vertices is given by the binomial coefficient:
\[
\binom{26}{2} = \frac{26 \times 25}{2} = 325
\]
2. **Subtract the number of edges:**
... | 241 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not atte... | 1. Let \( p \) be the number of points Beta attempted on the first day, and let \( q \) be the number of points Beta scored on the first day. Similarly, let \( 500 - p \) be the number of points Beta attempted on the second day, and let \( r \) be the number of points Beta scored on the second day.
2. We know that Alp... | 799 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that
$\bullet$ the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear,
$\bullet$ each of the $n$ line segments intersects at least one of the other line segments at a point other t... | 1. Consider a circle with $n$ equally spaced points. Label an arbitrary point as $P_1$. To ensure that all angles and lengths are the same, we must choose points such that $P_i$ and $P_{i+1}$ (with $P_{n+1} = P_1$) have the same number of points in between them for $1 \le i \le n$. This ensures equal chords and equal a... | 199 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$, in such a way that t... | 1. **Define the dimensions of the smaller cone \( C \):**
Let the smaller cone \( C \) have a base radius \( r_1 = 3x \), height \( h_1 = 4x \), and slant height \( l_1 = 5x \). The original cone has a base radius \( r = 3 \), height \( h = 4 \), and slant height \( l = 5 \).
2. **Surface area and volume of the ori... | 512 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and th... | 1. **Define the points and distances:**
- Let \( U \) be the point representing the unicorn.
- Let \( P \) be the point where the rope touches the tower.
- Let \( Q \) be the point directly above \( P \) at the height of the unicorn.
- Let \( C \) be the center of the cylinder at the height of the unicorn.
... | 813 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$. | 1. **Understanding the Geometry**:
- We have six smaller circles, each with radius \( r \), forming a ring.
- Each smaller circle is externally tangent to its two adjacent smaller circles.
- All smaller circles are internally tangent to a larger circle \( C \) with radius \( 30 \).
2. **Finding the Radius of ... | 942 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The $27$ cubes are randomly arranged to form a $3\times 3 \times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\frac{p^a}{q^br^c},$ where $p$,$q$, and $r$ are disti... | 1. **Identify the types of unit cube positions in the larger cube:**
- **Corner cubes:** These show three faces.
- **Edge cubes:** These show two faces.
- **Face-center cubes:** These show one face.
- **Center cube:** This shows no faces.
2. **Determine the probability that each type of cube shows only ora... | 160 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ lies in the Cartesian Plane and has an area of 70. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20)$, respectively, and the coordinates of $A$ are $(p,q)$. The line containing the median to side $BC$ has slope $-5$. Find the largest possible value of $p+q$. | 1. **Determine the midpoint \( M \) of \( \overline{BC} \):**
Given \( B = (12, 19) \) and \( C = (23, 20) \), the coordinates of the midpoint \( M \) are:
\[
M = \left( \frac{12 + 23}{2}, \frac{19 + 20}{2} \right) = \left( 17.5, 19.5 \right)
\]
2. **Find the equation of the line \( \overline{AM} \):**
... | 47 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A semicircle with diameter $d$ is contained in a square whose sides have length $8$. Given the maximum value of $d$ is $m- \sqrt{n}$, find $m+n$. | 1. **Understanding the Problem:**
- We have a semicircle with diameter \( d \) contained within a square with side length \( 8 \).
- We need to find the maximum value of \( d \) expressed as \( m - \sqrt{n} \) and then determine \( m + n \).
2. **Diagram and Setup:**
- Let the vertices of the square be \( A, ... | 544 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $n$, let $\tau (n)$ denote the number of positive integer divisors of $n$, including $1$ and $n$. For example, $\tau (1)=1$ and $\tau(6) =4$. Define $S(n)$ by \[S(n)=\tau(1)+ \tau(2) + ... + \tau(n).\] Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote... | 1. **Understanding the function $\tau(n)$:**
- $\tau(n)$ denotes the number of positive integer divisors of $n$. For example, $\tau(1) = 1$ and $\tau(6) = 4$.
- $\tau(n)$ is odd if and only if $n$ is a perfect square. This is because the divisors of a non-square number come in pairs, while a square number has a m... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A particle moves in the Cartesian Plane according to the following rules:
1. From any lattice point $ (a,b)$, the particle may only move to $ (a \plus{} 1,b)$, $ (a,b \plus{} 1)$, or $ (a \plus{} 1,b \plus{} 1)$.
2. There are no right angle turns in the particle's path.
How many different paths can the partic... | To solve this problem, we need to count the number of valid paths from \((0,0)\) to \((5,5)\) under the given movement rules and constraints.
1. **Understanding the Movement Rules:**
- The particle can move to \((a+1, b)\), \((a, b+1)\), or \((a+1, b+1)\).
- There are no right-angle turns allowed in the path.
... | 252 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the points $A(0,12)$, $B(10,9)$, $C(8,0)$, and $D(-4,7)$. There is a unique square $S$ such that each of the four points is on a different side of $S$. Let $K$ be the area of $S$. Find the remainder when $10K$ is divided by $1000$. | 1. **Identify the transformation:**
We are given points \( A(0,12) \), \( B(10,9) \), \( C(8,0) \), and \( D(-4,7) \). We need to find a square \( S \) such that each point lies on a different side of \( S \). We use a transformation \(\Phi\) defined by rotating \(90^\circ\) counterclockwise about the origin and tra... | 936 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The cards in a stack of $2n$ cards are numbered consecutively from $1$ through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A$. The remaining cards form pile $B$. The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A$, respectively. In this pr... | 1. **Understanding the Problem:**
- We have a stack of \(2n\) cards numbered from 1 to \(2n\).
- The top \(n\) cards are removed to form pile \(A\), and the remaining \(n\) cards form pile \(B\).
- The cards are then restacked by taking cards alternately from the tops of pile \(B\) and \(A\).
- We need to f... | 130 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3$. The radii of $C_1$ and $C_2$ are $4$ and $10$, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2$. Given that the length of the c... | 1. **Define the circles and their properties:**
- Let the radius of circle \( C_1 \) be \( r_1 = 4 \).
- Let the radius of circle \( C_2 \) be \( r_2 = 10 \).
- Let the radius of circle \( C_3 \) be \( R \).
2. **Determine the radius of \( C_3 \):**
- Since \( C_1 \) and \( C_2 \) are internally tangent to... | 19 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For how many positive integers $n$ less than or equal to $1000$ is \[(\sin t + i \cos t)^n=\sin nt + i \cos nt\] true for all real $t$? | To solve the problem, we need to determine for how many positive integers \( n \) less than or equal to \( 1000 \) the equation
\[
(\sin t + i \cos t)^n = \sin(nt) + i \cos(nt)
\]
holds for all real \( t \).
1. **Initial Observation**:
We start by testing \( t = 0 \). Substituting \( t = 0 \) into the equation, we... | 250 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O$, and that the ratio of the volume of $O$ to that of $C$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m+n$. | 1. **Set the side length of the octahedron to 1 for simplicity.**
- The volume of a regular octahedron with side length \( a \) is given by:
\[
V_O = \frac{\sqrt{2}}{3} a^3
\]
- For \( a = 1 \):
\[
V_O = \frac{\sqrt{2}}{3}
\]
2. **Determine the height of the octahedron.**
- An o... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ be a positive integer, and let $a_0, a_1,\ldots,a_m$ be a sequence of reals such that $a_0=37$, $a_1=72$, $a_m=0$, and \[a_{k+1}=a_{k-1}-\frac{3}{a_k}\] for $k=1,2, \dots, m-1$. Find $m$. | 1. Given the sequence \(a_0, a_1, \ldots, a_m\) with \(a_0 = 37\), \(a_1 = 72\), and \(a_m = 0\), and the recursive relation:
\[
a_{k+1} = a_{k-1} - \frac{3}{a_k}
\]
for \(k = 1, 2, \ldots, m-1\).
2. We start by examining the condition \(a_m = 0\). From the recursive relation, we have:
\[
a_m = a_{m-... | 889 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Square $ABCD$ has center $O$, $AB=900$, $E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F$, $m\angle EOF =45^\circ$, and $EF=400$. Given that $BF=p+q\sqrt{r}$, wherer $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$. | 1. **Identify the given information and set up the problem:**
- Square \(ABCD\) has center \(O\).
- Side length \(AB = 900\).
- Points \(E\) and \(F\) are on \(AB\) with \(AE < BF\) and \(E\) between \(A\) and \(F\).
- \(\angle EOF = 45^\circ\).
- \(EF = 400\).
2. **Determine the coordinates of the poin... | 307 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $w_{1}$ and $w_{2}$ denote the circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$, respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_{2}$ and internally tangent to $w_{1}$. Given that $m^{2}=p/q$, wh... | 1. **Identify the centers and radii of the circles \( w_1 \) and \( w_2 \):**
- The equation of circle \( w_1 \) is \( x^2 + y^2 + 10x - 24y - 87 = 0 \).
- Completing the square for \( x \) and \( y \):
\[
x^2 + 10x + y^2 - 24y = 87
\]
\[
(x + 5)^2 - 25 + (y - 12)^2 - 144 = 87... | 169 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
$\bullet$ Any cube may be the bottom cube in the tower.
$\bullet$ The cube immediately on top of a cube with edge-length $k$ must have ed... | To solve this problem, we need to determine the number of different towers that can be constructed using the given rules. We will use a recursive approach to find the number of valid stackings of \( n \) blocks.
1. **Define the problem recursively:**
Let \( a_n \) denote the number of valid stackings of \( n \) blo... | 458 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $\frac{1}{29}$ of the original integer. | 1. Let the original number be represented as \( N = 100a + 10b + c \), where \( a, b, \) and \( c \) are digits and \( a \neq 0 \) since \( N \) is a positive integer.
2. According to the problem, when the leftmost digit \( a \) is deleted, the resulting number is \( 10b + c \).
3. The problem states that this resultin... | 725 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In quadrilateral $ABCD, \angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD},$ $AB=18, BC=21,$ and $CD=14.$ Find the perimeter of $ABCD$. | 1. **Calculate the length of diagonal \( AC \):**
Since \(\angle B\) is a right angle, triangle \( \triangle ABC \) is a right triangle with \( AB \) and \( BC \) as the legs. We can use the Pythagorean theorem to find \( AC \):
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{18^2 + 21^2} = \sqrt{324 + 441} = \sqrt{765} = ... | 84 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$. | 1. **Determine the minimum sum \(X\):**
- The smallest 90-element subset of \(\{1, 2, 3, \ldots, 100\}\) is \(\{1, 2, 3, \ldots, 90\}\).
- The sum of the first 90 natural numbers is given by the formula for the sum of an arithmetic series:
\[
X = \sum_{i=1}^{90} i = \frac{90 \cdot (90 + 1)}{2} = \frac{9... | 901 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of consecutive 0's at the right end of the decimal representation of the product $1!\times2!\times3!\times4!\cdots99!\times100!.$ Find the remainder when $N$ is divided by 1000. | To find the number of consecutive 0's at the right end of the decimal representation of the product \(1! \times 2! \times 3! \times 4! \cdots 99! \times 100!\), we need to count the number of factors of 10 in this product. Since \(10 = 2 \times 5\), we need to count the number of pairs of factors 2 and 5. However, in f... | 54 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\mathcal{S}$. | 1. We start by understanding the set $\mathcal{S}$, which consists of numbers that can be written as repeating decimals of the form $0.\overline{abc}$, where $a, b, c$ are distinct digits. Each number in $\mathcal{S}$ can be expressed as a fraction:
\[
0.\overline{abc} = \frac{abc}{999}
\]
where $abc$ is a ... | 360 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r)$. | 1. Given that the sequence \(a_1, a_2, \ldots\) is geometric with \(a_1 = a\) and common ratio \(r\), we can express the terms of the sequence as:
\[
a_1 = a, \quad a_2 = ar, \quad a_3 = ar^2, \quad \ldots, \quad a_{12} = ar^{11}
\]
2. The logarithmic sum given in the problem is:
\[
\log_8 a_1 + \log_8 ... | 46 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For each even positive integer $x$, let $g(x)$ denote the greatest power of $2$ that divides $x$. For example, $g(20)=4$ and $g(16)=16$. For each positive integer $n$, let $S_n=\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than $1000$ such that $S_n$ is a perfect square. | 1. Consider the list of even integers \(2, 4, 6, \ldots, 2^n\). For each integer \(i\) such that \(1 \leq i \leq n-1\), we want to count how many integers in this list are divisible by \(2^i\) but not by \(2^{i+1}\).
2. The number of integers divisible by \(2^i\) is \(\frac{2^n}{2^i}\). However, some of these integers... | 511 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
This is the one with the 8 circles?
I made each circle into the square in which the circle is inscribed, then calculated it with that. It got the right answer but I don't think that my method is truly valid... | 1. **Understanding the Problem:**
We need to find the area of a specific region involving 8 circles. The problem suggests that the line passing through certain points splits the circles in half, and we need to find the equation of this line.
2. **Identifying Key Points:**
The line passes through the points \((1,... | 65 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit. | To find the number of ordered pairs of positive integers \((a, b)\) such that \(a + b = 1000\) and neither \(a\) nor \(b\) has a zero digit, we will use complementary counting. We will count the number of pairs \((a, b)\) where either \(a\) or \(b\) contains a zero digit and subtract this from the total number of pairs... | 801 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The lengths of the sides of a triangle with positive area are $\log_{10} 12$, $\log_{10} 75$, and $\log_{10} n$, where $n$ is a positive integer. Find the number of possible values for $n$. | 1. Given the side lengths of a triangle are $\log_{10} 12$, $\log_{10} 75$, and $\log_{10} n$, we need to ensure that these lengths satisfy the triangle inequality. The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the remaining side.
2. We start by writing ... | 893 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be the product of the first 100 positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k$. | To find the largest integer \( k \) such that the product \( P \) of the first 100 positive odd integers is divisible by \( 3^k \), we need to count the number of factors of 3 in \( P \). This can be done using the de Polignac's formula (Legendre's formula), which states that the exponent of a prime \( p \) in \( n! \)... | 48 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Circles $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$... | 1. **Identify the centers and radii of the circles:**
- Circle $\mathcal{C}_1$ has center at $(0,0)$ and radius $1$.
- Circle $\mathcal{C}_2$ has center at $(12,0)$ and radius $2$.
- Circle $\mathcal{C}_3$ has center at $(24,0)$ and radius $4$.
2. **Determine the slopes of the common internal tangents:**
-... | 27 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilate... | 1. **Determine the number of games each team plays:**
Each team plays every other team exactly once. With 7 teams, each team plays \(6\) games (since \(7 - 1 = 6\)).
2. **Calculate the probability distribution for the number of wins:**
Each game has a \(50\%\) chance of being won by either team, and the outcomes... | 831 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A sequence is defined as follows $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}=6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum^{28}_{k=1} a_k$ is divided by 1000. | 1. We start with the given sequence definition:
\[
a_1 = a_2 = a_3 = 1
\]
and for all positive integers \( n \),
\[
a_{n+3} = a_{n+2} + a_{n+1} + a_n
\]
2. We are given the values:
\[
a_{28} = 6090307, \quad a_{29} = 11201821, \quad a_{30} = 20603361
\]
3. We need to find the remainder w... | 489 | Other | math-word-problem | Yes | Yes | aops_forum | false |
How many integers $ N$ less than 1000 can be written as the sum of $ j$ consecutive positive odd integers from exactly 5 values of $ j\ge 1$? | To solve the problem, we need to determine how many integers \( N \) less than 1000 can be written as the sum of \( j \) consecutive positive odd integers from exactly 5 values of \( j \ge 1 \).
1. **Form of the Sum of Consecutive Odd Integers**:
The sum of \( j \) consecutive positive odd integers can be expressed... | 14 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer. | 1. **Understanding the problem**: We need to find the smallest positive integer \( n \) for which the sum of the reciprocals of the non-zero digits of the integers from 1 to \( 10^n \) inclusive, denoted as \( S_n \), is an integer.
2. **Excluding \( 10^n \)**: The number \( 10^n \) has only one non-zero digit, which ... | 63 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C=\frac{5}{9}(F-32)$. An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the neare... | 1. **Understanding the Conversion Formula:**
The formula for converting a Fahrenheit temperature \( F \) to the corresponding Celsius temperature \( C \) is:
\[
C = \frac{5}{9}(F - 32)
\]
We need to find the integer Fahrenheit temperatures \( T \) such that when converted to Celsius and then back to Fahr... | 539 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}|<\frac{1}{2}$. For example, $b(6) = 2$ and $b(23)=5$. If $S = \textstyle\sum_{p=1}^{2007}b(p)$, find the remainder when S is divided by 1000. | 1. **Understanding the function \( b(p) \)**:
- The function \( b(p) \) is defined such that \( b(p) = k \) where \( k \) is the unique positive integer satisfying \( |k - \sqrt{p}| < \frac{1}{2} \).
- This implies \( k - \frac{1}{2} < \sqrt{p} < k + \frac{1}{2} \).
- Squaring all parts of the inequality, we g... | 955 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA=5$ and $CD=2$. Point $E$ lies on side $CA$ such that $\angle DEF = 60^\circ$. The area of triangle $DEF$ is $14\sqrt{3}$. The two possible values of the length of side $AB$ are $p \pm q\sqrt{r}$, where ... | 1. **Label the Diagram and Define Variables:**
Let the side length of the equilateral triangle \( \triangle ABC \) be \( s \). Let \( AE = k \). Then, \( EC = s - k \).
2. **Area and Angle Conditions:**
Given that \( \angle DEF = 60^\circ \) and the area of \( \triangle DEF \) is \( 14\sqrt{3} \), we use the for... | 989 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In thr... | 1. Let \( x \) be the time (in hours) it takes for one worker to produce one widget.
2. Let \( y \) be the time (in hours) it takes for one worker to produce one whoosit.
Given:
- In one hour, 100 workers can produce 300 widgets and 200 whoosits.
- In two hours, 60 workers can produce 240 widgets and 300 whoosits.
- I... | 450 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $ 74$ kilometers after biking for $ 2$ hours, jogging for $ 3$ hours, and swimming for $ 4$ hours, while Sue covers $ 91$ kilometers after jogging for $ 2$ hours, swimming... | 1. Let's denote Ed's and Sue's biking, jogging, and swimming rates by \( b \), \( j \), and \( s \) respectively. According to the problem, we have the following equations based on their activities:
\[
2b + 3j + 4s = 74 \quad \text{(Equation 1)}
\]
\[
4b + 2j + 3s = 91 \quad \text{(Equation 2)}
\]
2.... | 314 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A right circular cone has base radius $ r$ and height $ h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first return... | 1. **Determine the slant height of the cone:**
The slant height \( l \) of a right circular cone with base radius \( r \) and height \( h \) is given by the Pythagorean theorem:
\[
l = \sqrt{r^2 + h^2}
\]
2. **Calculate the path length traced by the cone:**
As the cone rolls on the table, the point wher... | 14 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \overline{AB}$ be a diameter of circle $ \omega$. Extend $ \overline{AB}$ through $ A$ to $ C$. Point $ T$ lies on $ \omega$ so that line $ CT$ is tangent to $ \omega$. Point $ P$ is the foot of the perpendicular from $ A$ to line $ CT$. Suppose $ AB \equal{} 18$, and let $ m$ denote the maximum possible length o... | 1. Let $\overline{AB}$ be a diameter of circle $\omega$ with $AB = 18$. The center of the circle is $O$, and the radius is $r = \frac{AB}{2} = 9$.
2. Extend $\overline{AB}$ through $A$ to $C$. Let $AC = x$.
3. Point $T$ lies on $\omega$ such that line $CT$ is tangent to $\omega$. Since $CT$ is tangent to $\omega$ at $T... | 432 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S_i$ be the set of all integers $ n$ such that $ 100i\leq n < 100(i \plus{} 1)$. For example, $ S_4$ is the set $ {400,401,402,\ldots,499}$. How many of the sets $ S_0, S_1, S_2, \ldots, S_{999}$ do not contain a perfect square? | 1. **Identify the range of each set \( S_i \):**
Each set \( S_i \) contains integers \( n \) such that \( 100i \leq n < 100(i + 1) \). This means each set \( S_i \) contains exactly 100 integers.
2. **Determine the range of perfect squares:**
We need to find the perfect squares within the range of each set \( S... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the... | 1. **Identify the given information and setup the problem:**
- \(ABCD\) is an isosceles trapezoid with \(\overline{AD} \parallel \overline{BC}\).
- The angle at the longer base \(\overline{AD}\) is \(\frac{\pi}{3}\).
- The diagonals \(AC\) and \(BD\) have length \(10\sqrt{21}\).
- Point \(E\) is at distance... | 32 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that ... | 1. **Define the problem and the variables:**
We have ten identical crates, each with dimensions \(3 \text{ ft} \times 4 \text{ ft} \times 6 \text{ ft}\). The first crate is placed flat on the floor, and each of the remaining nine crates is placed flat on top of the previous crate with a random orientation. We need t... | 190 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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