problem stringlengths 0 45.3k | solution stringlengths 0 26.8k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 9
values | question_type stringclasses 5
values | problem_is_valid stringclasses 5
values | solution_is_valid stringclasses 5
values | source stringclasses 11
values | synthetic bool 2
classes |
|---|---|---|---|---|---|---|---|---|
Determine the largest value of $x$ if $\frac{x}{4} + \frac{2}{3x} = \frac{5}{6}$. | 1. Start by eliminating the fractions: Multiply through by $12x$ (the least common multiple of 4 and 3x):
\[
3x^2 + 8 = 10x.
\]
2. Rearrange to form a quadratic equation:
\[
3x^2 - 10x + 8 = 0.
\]
3. Solve the quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{... | 2 | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
Find $97^{-1} \pmod{101}$, as a residue modulo 101. (Give an answer between 0 and 100, inclusive.) | To solve for $97^{-1} \pmod{101}$, notice that $97 \equiv -4 \pmod{101}$ since $101 - 97 = 4$. Thus, finding the inverse of 97 modulo 101 is equivalent to finding the inverse of -4 modulo 101.
We look for a number $x$ such that $(-4)x \equiv 1 \pmod{101}$. Multiplying both sides of the equation by -1 gives:
\[ 4x \equ... | 25 | Number Theory | math-word-problem | Yes | Yes | synthetic_math | true |
Given vectors $\mathbf{v} = \begin{pmatrix} 1 \\ y \end{pmatrix}$ and $\mathbf{w'} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$, the projection of $\mathbf{v}$ on $\mathbf{w'}$ is found to be $\begin{pmatrix} 2 \\ -2/3 \end{pmatrix}$. Determine the value of $y$. | First, compute $\mathbf{v} \cdot \mathbf{w'}$ and $\mathbf{w'} \cdot \mathbf{w'}$:
\[
\mathbf{v} \cdot \mathbf{w'} = \begin{pmatrix} 1 \\ y \end{pmatrix} \cdot \begin{pmatrix} -3 \\ 1 \end{pmatrix} = 1(-3) + y(1) = -3 + y
\]
\[
\mathbf{w'} \cdot \mathbf{w'} = \begin{pmatrix} -3 \\ 1 \end{pmatrix} \cdot \begin{pmatrix... | y = -\frac{11}{3} | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
Evaluate $\frac{8}{29} - \frac{5}{87}$ and reduce the result to its simplest form. | To solve the problem, we first need a common denominator. Noticing that 87 is a multiple of 29, we use 87 as the common denominator:
\[\frac{8}{29} = \frac{8 \cdot 3}{29 \cdot 3} = \frac{24}{87}\]
Now, subtract the fractions:
\[\frac{24}{87} - \frac{5}{87} = \frac{24 - 5}{87} = \frac{19}{87}\]
Next, we simplify $\frac{... | \frac{19}{87} | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
Let \(x_1, x_2, x_3\) be positive real numbers such that \(x_1 + 3x_2 + 4x_3 = 72\). Additionally, \(x_1 = 3x_2\). Find the smallest possible value of
\[
x_1^2 + x_2^2 + x_3^2.
\] | Using the Cauchy-Schwarz inequality, we have:
\[
(1^2 + 3^2 + 4^2)(x_1^2 + x_2^2 + x_3^2) \geq (x_1 + 3x_2 + 4x_3)^2.
\]
Plugging in the values,
\[
(1 + 9 + 16)(x_1^2 + x_2^2 + x_3^2) \geq 72^2,
\]
\[
26(x_1^2 + x_2^2 + x_3^2) \geq 5184,
\]
\[
x_1^2 + x_2^2 + x_3^2 \geq \frac{5184}{26} = 199.3846.
\]
Equality holds whe... | 347.04 | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
Let $p$, $q$, $r$, $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s$, and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q$. Additionally, $p + q + r + s = 201$. Find the value of $pq + rs$. | From Vieta's formulas, we have:
\begin{align*}
r + s &= 12p, \\
rs &= -13q, \\
p + q &= 12r, \\
pq &= -13s.
\end{align*}
Using the additional relationship $p + q + r + s = 201$, substitute $p + q = 12r$ and $r + s = 12p$:
\[ 12r + 12p = 201. \]
Therefore,
\[ 12(p + r) = 201 \]
\[ p + r = \frac{201}{12}. \]
Now using t... | -\frac{28743}{12} | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
A Senate committee has 10 Republicans and 8 Democrats. In how many ways can we form a subcommittee consisting of 4 Republicans and 3 Democrats? | 1. **Choose Republicans**: There are 10 Republicans available. We need to choose 4 of them for the subcommittee. The number of ways to choose 4 Republicans out of 10 is calculated by the binomial coefficient:
\[
\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \time... | 11760 | Combinatorics | math-word-problem | Yes | Yes | synthetic_math | true |
Let circle $\Theta$ be both the incircle of $\triangle DEF$ and the circumcircle of $\triangle PQR$. The point $P$ lies on $\overline{EF}$, point $Q$ on $\overline{DE}$, and point $R$ on $\overline{DF}$. Given $\angle D = 50^\circ$, $\angle E = 70^\circ$, and $\angle F = 60^\circ$, find the measure of the angle $\angle... | - With $\Theta$ as both the incircle of $\triangle DEF$ and the circumcircle of $\triangle PQR$, the centers align, making the distances from the center to vertices $DP=DQ=DR$. Thus, triangle $PQR$ is equilateral as all sides from the center are equal, simplifying the geometry.
- Using the interior angle sums of $\tr... | 60^\circ | Geometry | math-word-problem | Yes | Yes | synthetic_math | true |
Consider a geometric figure consisting of three triangles as follows: Triangle ABC, Triangle DEF, and Triangle GHI, where point D is on line segment AC, point F is on line segment BI, and point H is on line segment EG. If each triangle is labeled with three angles near the vertices A, B, C, D, E, F, G, H, I, such that ... | - Each triangle (ABC, DEF, GHI) has three interior angles.
- The sum of the internal angles of any triangle is always $180^\circ$.
- Thus, the sum of angles for each triangle is $180^\circ$.
Calculating for three individual triangles:
1. Triangle ABC: Angles 1, 2, 3 sum to $180^\circ$.
2. Triangle DEF: Angles 4, 5, 6 ... | 540^\circ | Geometry | math-word-problem | Yes | Yes | synthetic_math | true |
Calculate the positive difference between the solutions of the equation $|x - 3| = 15$. | Let's denote the two solutions to the equation $|x - 3| = 15$ as $x_1$ and $x_2$. The absolute value equation implies two scenarios:
1. \( x - 3 = 15 \)
2. \( x - 3 = -15 \)
Solving these:
1. \( x - 3 = 15 \Rightarrow x = 18 \)
2. \( x - 3 = -15 \Rightarrow x = -12 \)
To find the positive difference between $x_1$ and... | 30 | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
What is the \(45^{\text{th}}\) even positive integer? | Every even positive integer can be written in the form \(2x\), where \(x\) is a positive integer. The sequence of even positive integers thus starts with \(2 \cdot 1 = 2\), \(2 \cdot 2 = 4\), and so on.
To find the \(45^{\text{th}}\) even positive integer, we plug in \(x = 45\) into the formula:
\[
2 \cdot 45 = 90
\]
... | 90 | Number Theory | math-word-problem | Yes | Yes | synthetic_math | true |
In triangle $ABC$, $BC = 6$, $AC = 3$, and $\angle C = 30^\circ$. Altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. Find $AH:HD$. | First, we calculate $AB$ and the length of altitude $AD$. Since $\angle C = 30^\circ$, triangle $ABC$ is not a right triangle, and we use the Law of Cosines to find $AB$:
\[ AB^2 = AC^2 + BC^2 - 2 \times AC \times BC \times \cos(30^\circ) \]
\[ AB^2 = 3^2 + 6^2 - 2 \times 3 \times 6 \times \left(\frac{\sqrt{3}}{2}\righ... | AH:HD = 0 | Geometry | math-word-problem | Yes | Problem not solved | synthetic_math | true |
Brad has a cylindrical barrel with radius 8 inches and height 24 inches. He wants to fill it from a well, using a bucket in the shape of a hemisphere with a radius of 8 inches. How many trips must Brad make to the well in order to fill the barrel? | First, calculate the volume of the bucket. Since the bucket is a hemisphere with radius \( r = 8 \) inches:
\[ V_{\text{hemisphere}} = \frac{1}{2} \cdot \frac{4}{3} \pi r^3 = \frac{1}{2} \cdot \frac{4}{3} \pi (8)^3 = \frac{1}{2} \cdot \frac{4}{3} \pi \cdot 512 = \frac{2}{3} \pi \cdot 512 = \frac{1024}{3} \pi \text{ cub... | 5 | Geometry | math-word-problem | Yes | Yes | synthetic_math | true |
What value of $x$ will give the maximum value for the function $-2x^2 + 8x + 16$? | We begin by finding the vertex of the parabola defined by $f(x) = -2x^2 + 8x + 16$.
First, complete the square for $-2x^2 + 8x + 16$:
\begin{align*}
-2x^2 + 8x + 16 &= -2(x^2 - 4x) + 16 \\
&= -2(x^2 - 4x + (4/2)^2 - (4/2)^2) + 16 \\
&= -2((x-2)^2 - 2^2) + 16 \\
&= -2(x-2)^2 + 8 + 16 \\
&= -2(x-2)^2 + 24.
\end{align*}
... | 24 | Algebra | math-word-problem | Yes | Yes | synthetic_math | true |
An equilateral triangle has sides 10 units long. An equilateral triangle with sides 6 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction. | 1. **Area of the large equilateral triangle (side 10 units):**
\[
A_{\text{large}} = \frac{\sqrt{3}}{4} \times 10^2 = 25\sqrt{3} \text{ square units}
\]
2. **Area of the smaller equilateral triangle (side 6 units):**
\[
A_{\text{small}} = \frac{\sqrt{3}}{4} \times 6^2 = 9\sqrt{3} \text{ square units}
... | \frac{9}{16} | Geometry | math-word-problem | Yes | Yes | synthetic_math | true |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.