id stringlengths 19 19 | question stringlengths 73 887 | context_before_generation stringlengths 83 897 | generated_trace stringlengths 962 3.14k | generated_units listlengths 21 52 | input_x stringlengths 107 2.21k | target_y stringlengths 10 138 | pruning_depth int64 0 0 | metadata dict |
|---|---|---|---|---|---|---|---|---|
pt-fa9f84be87931528 | What is the range of the function $y = \frac{x^2 + 3x + 2}{x+1}$? (Express your answer using interval notation.) | Question:
What is the range of the function $y = \frac{x^2 + 3x + 2}{x+1}$? (Express your answer using interval notation.) | To find the range of the function $y = \frac{x^2 + 3x + 2}{x+1}$, we first simplify the expression for $y$.
Step 1: Factor the numerator.
$$x^2 + 3x + 2 = (x+1)(x+2)$$
Step 2: Substitute the factored form back into the function definition.
$$y = \frac{(x+1)(x+2)}{x+1}$$
Step 3: Determine the domain of the function.
... | [
"To find the range of the function $y = \\frac{x^2 + 3x + 2}{x+1}$, we first simplify the expression for $y$.",
"Step 1: Factor the numerator.",
"$$x^2 + 3x + 2 = (x+1)(x+2)$$",
"Step 2: Substitute the factored form back into the function definition.",
"$$y = \\frac{(x+1)(x+2)}{x+1}$$",
"Step 3: Determine... | Question:
What is the range of the function $y = \frac{x^2 + 3x + 2}{x+1}$? (Express your answer using interval notation.)
To find the range of the function $y = \frac{x^2 + 3x + 2}{x+1}$, we first simplify the expression for $y$. | $$x^2 + 3x + 2 = (x+1)(x+2)$$ | 0 | {
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"Step 1: Factor the numerator."
],
"removed_start_index": 1,
"removed_end_index": 1,
"decision_reason": "Unit 1 is a procedural step label (... |
pt-99befa3b894601c6 | Suppose $a$, $b,$ and $c$ are positive numbers satisfying: \begin{align*}
a^2/b &= 1, \\
b^2/c &= 2, \text{ and}\\
c^2/a &= 3.
\end{align*} Find $a$. | Question:
Suppose $a$, $b,$ and $c$ are positive numbers satisfying: \begin{align*}
a^2/b &= 1, \\
b^2/c &= 2, \text{ and}\\
c^2/a &= 3.
\end{align*} Find $a$. | Given the system of equations:
1) $a^2/b = 1$
2) $b^2/c = 2$
3) $c^2/a = 3$
Step 1: Rearrange equation (1) to express $b$ in terms of $a$.
$a^2/b = 1$
Multiply both sides by $b$:
$a^2 = b$
Step 2: Substitute the expression for $b$ from Step 1 into equation (2).
$b^2/c = 2$
$(a^2)^2/c = 2$
$a^4/c = 2$
This gives $c = ... | [
"Given the system of equations:",
"$a^2/b = 1$",
"$b^2/c = 2$",
"$c^2/a = 3$",
"Step 1: Rearrange equation (1) to express $b$ in terms of $a$.",
"$a^2/b = 1$",
"Multiply both sides by $b$:",
"$a^2 = b$",
"Step 2: Substitute the expression for $b$ from Step 1 into equation (2).",
"$b^2/c = 2$",
"... | Question:
Suppose $a$, $b,$ and $c$ are positive numbers satisfying: \begin{align*}
a^2/b &= 1, \\
b^2/c &= 2, \text{ and}\\
c^2/a &= 3.
\end{align*} Find $a$. | $a^2/b = 1$ | 0 | {
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"Given the system of equations:"
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pt-91cce1876436c0d1 | An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$ | Question:
An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$ | Let the first infinite geometric series be $S_1$.
The first term is $a_1 = 12$.
The second term is $a_2 = 4$.
The common ratio $r_1$ of the first series is given by $r_1 = \frac{a_2}{a_1}$.
$r_1 = \frac{4}{12} = \frac{1}{3}$.
The sum of an infinite geometric series is given by $S = \frac{a}{1-r}$, provided $|r| < 1$.... | [
"Let the first infinite geometric series be $S_1$.",
"The first term is $a_1 = 12$.",
"The second term is $a_2 = 4$.",
"The common ratio $r_1$ of the first series is given by $r_1 = \\frac{a_2}{a_1}$.",
"$r_1 = \\frac{4}{12} = \\frac{1}{3}$.",
"The sum of an infinite geometric series is given by $S = \\fr... | Question:
An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$ | The first term is $a_1 = 12$. | 0 | {
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"Let the first infinite geometric series be $S_1$."
],
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pt-e01532bd2ad8b4cf | What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A( -2, 0)$ and $B(-1,4)$? | Question:
What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A( -2, 0)$ and $B(-1,4)$? | Let $P$ be a point on the $y$-axis. The coordinates of $P$ are $(0, y)$.
Let $A$ be the point $(-2, 0)$ and $B$ be the point $(-1, 4)$.
The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
The distance between $P$ and $A$, denoted by... | [
"Let $P$ be a point on the $y$-axis. The coordinates of $P$ are $(0, y)$.",
"Let $A$ be the point $(-2, 0)$ and $B$ be the point $(-1, 4)$.",
"The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula $d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.",
"The distance between $P$... | Question:
What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A( -2, 0)$ and $B(-1,4)$?
Let $P$ be a point on the $y$-axis. The coordinates of $P$ are $(0, y)$. | The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. | 0 | {
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"removed_span": [
"Let $A$ be the point $(-2, 0)$ and $B$ be the point $(-1, 4)$."
],
"removed_start_index": 1,
"removed_end_index": 1,
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pt-282021bd4a662cd2 | A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | Question:
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | Let the legs of the right triangle be $a$ and $b$, where $a$ and $b$ are positive integers.
The area of the triangle is $A = \frac{1}{2}ab$.
The sum of the lengths of the legs is $S = a + b$.
The condition for a right triangle to be "cool" is that the number of square units in its area is equal to twice the number of ... | [
"Let the legs of the right triangle be $a$ and $b$, where $a$ and $b$ are positive integers.",
"The area of the triangle is $A = \\frac{1}{2}ab$.",
"The sum of the lengths of the legs is $S = a + b$.",
"The condition for a right triangle to be \"cool\" is that the number of square units in its area is equal t... | Question:
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?
Let the legs of the right triangle be $a$ and $b$, whe... | $$A = 2S$$ | 0 | {
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"removed_span": [
"The condition for a right triangle to be \"cool\" is that the number of square units in its area is equal to twice the number of units in the sum o... |
pt-1fa157c687b17fe4 | How many positive integers $n$ less than 100 have a corresponding integer $m$ divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | Question:
How many positive integers $n$ less than 100 have a corresponding integer $m$ divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | Let the roots of the quadratic equation $x^2 - nx + m = 0$ be consecutive positive integers.
Let the roots be $k$ and $k+1$, where $k$ is a positive integer.
By Vieta's formulas, we have:
Sum of the roots: $n = k + (k+1) = 2k + 1$.
Product of the roots: $m = k(k+1) = k^2 + k$.
We are given that $n$ is a positive integ... | [
"Let the roots of the quadratic equation $x^2 - nx + m = 0$ be consecutive positive integers.",
"Let the roots be $k$ and $k+1$, where $k$ is a positive integer.",
"By Vieta's formulas, we have:",
"Sum of the roots: $n = k + (k+1) = 2k + 1$.",
"Product of the roots: $m = k(k+1) = k^2 + k$.",
"We are given... | Question:
How many positive integers $n$ less than 100 have a corresponding integer $m$ divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers?
Let the roots of the quadratic equation $x^2 - nx + m = 0$ be consecutive positive integers.
Let the roots be $k$ and $k+1$, where $k$ is a positi... | $0 \le 2k \le 98$ | 0 | {
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"Subtract 1 from all parts:"
],
"removed_start_index": 8,
"removed_end_index": 8,
"decision_reason": "The unit 'Subtract 1 from all parts:' ... |
pt-e5c9729fc0616967 | What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 420 = 0$ has integral solutions? | Question:
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 420 = 0$ has integral solutions? | The quadratic equation is $10x^2 - mx + 420 = 0$.
For the equation to have integral solutions, the discriminant must be a perfect square, and the solutions must be integers.
Step 1: Calculate the discriminant, $\Delta$.
$\Delta = (-m)^2 - 4(10)(420)$
$\Delta = m^2 - 40(420)$
$\Delta = m^2 - 16800$
Step 2: Set the dis... | [
"The quadratic equation is $10x^2 - mx + 420 = 0$.",
"For the equation to have integral solutions, the discriminant must be a perfect square, and the solutions must be integers.",
"Step 1: Calculate the discriminant, $\\Delta$.",
"$\\Delta = (-m)^2 - 4(10)(420)$",
"$\\Delta = m^2 - 40(420)$",
"$\\Delta = ... | Question:
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 420 = 0$ has integral solutions?
The quadratic equation is $10x^2 - mx + 420 = 0$.
For the equation to have integral solutions, the discriminant must be a perfect square, and the solutions must be integers. | $\Delta = (-m)^2 - 4(10)(420)$ | 0 | {
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"Step 1: Calculate the discriminant, $\\Delta$."
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pt-d866b44f28623944 | Find the sum of all integral values of $c$ with $c\le 25$ for which the equation $y=x^2-7x-c$ has two rational roots. | Question:
Find the sum of all integral values of $c$ with $c\le 25$ for which the equation $y=x^2-7x-c$ has two rational roots. | The equation is $y = x^2 - 7x - c$.
The roots of the equation are found by setting $y=0$:
$x^2 - 7x - c = 0$.
For the quadratic equation $ax^2 + bx + d = 0$ to have rational roots, the discriminant $\Delta = b^2 - 4ad$ must be a perfect square of a rational number. Since the coefficients are integers, $\Delta$ must be... | [
"The equation is $y = x^2 - 7x - c$.",
"The roots of the equation are found by setting $y=0$:",
"$x^2 - 7x - c = 0$.",
"For the quadratic equation $ax^2 + bx + d = 0$ to have rational roots, the discriminant $\\Delta = b^2 - 4ad$ must be a perfect square of a rational number. Since the coefficients are intege... | Question:
Find the sum of all integral values of $c$ with $c\le 25$ for which the equation $y=x^2-7x-c$ has two rational roots.
The equation is $y = x^2 - 7x - c$.
The roots of the equation are found by setting $y=0$:
$x^2 - 7x - c = 0$.
For the quadratic equation $ax^2 + bx + d = 0$ to have rational roots, the discrim... | $x = \frac{-(-7) \pm \sqrt{49 + 4c}}{2(1)} = \frac{7 \pm \sqrt{k^2}}{2} = \frac{7 \pm k}{2}$. | 0 | {
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"The roots are given by the quadratic formula:"
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"removed_start_index": 8,
"removed_end_index": 8,
"decision_reason": "Unit 8 is a stateme... |
pt-59aa31e3f7faebb3 | If $f(x)=ax+b$ and $f^{-1}(x)=bx+a$ with $a$ and $b$ real, what is the value of $a+b$? | Question:
If $f(x)=ax+b$ and $f^{-1}(x)=bx+a$ with $a$ and $b$ real, what is the value of $a+b$? | Given the function $f(x) = ax+b$.
The inverse function is given as $f^{-1}(x) = bx+a$.
Step 1: Find the expression for $f^{-1}(x)$ by setting $y = f(x)$ and solving for $x$ in terms of $y$.
Let $y = ax+b$.
Step 2: Solve the equation for $x$:
$y - b = ax$
$x = \frac{y-b}{a}$
Step 3: Replace $y$ with $x$ to find the inv... | [
"Given the function $f(x) = ax+b$.",
"The inverse function is given as $f^{-1}(x) = bx+a$.",
"Step 1: Find the expression for $f^{-1}(x)$ by setting $y = f(x)$ and solving for $x$ in terms of $y$.",
"Let $y = ax+b$.",
"Step 2: Solve the equation for $x$:",
"$y - b = ax$",
"$x = \\frac{y-b}{a}$",
"Step... | Question:
If $f(x)=ax+b$ and $f^{-1}(x)=bx+a$ with $a$ and $b$ real, what is the value of $a+b$?
Given the function $f(x) = ax+b$.
The inverse function is given as $f^{-1}(x) = bx+a$. | Let $y = ax+b$. | 0 | {
"generator_model": "avreymi/gemma-4-E2B-it-reasoning-pruning",
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"Step 1: Find the expression for $f^{-1}(x)$ by setting $y = f(x)$ and solving for $x$ in terms of $y$."
],
"removed_start_index": 2,
"removed... |
pt-ddb020bdaf6c9a4e | For how many non-negative real values of $x$ is $\sqrt{144-\sqrt[3]{x}}$ an integer? | Question:
For how many non-negative real values of $x$ is $\sqrt{144-\sqrt[3]{x}}$ an integer? | Let $y = \sqrt{144-\sqrt[3]{x}}$. We are looking for the number of non-negative real values of $x$ such that $y$ is an integer.
Step 1: Determine the constraints on $x$.
Since $\sqrt[3]{x}$ is a real number, $x$ must be a real number.
Since the square root is defined, we must have $144 - \sqrt[3]{x} \ge 0$, which impl... | [
"Let $y = \\sqrt{144-\\sqrt[3]{x}}$. We are looking for the number of non-negative real values of $x$ such that $y$ is an integer.",
"Step 1: Determine the constraints on $x$.",
"Since $\\sqrt[3]{x}$ is a real number, $x$ must be a real number.",
"Since the square root is defined, we must have $144 - \\sqrt[3... | Question:
For how many non-negative real values of $x$ is $\sqrt{144-\sqrt[3]{x}}$ an integer?
Let $y = \sqrt{144-\sqrt[3]{x}}$. We are looking for the number of non-negative real values of $x$ such that $y$ is an integer. | Since $\sqrt[3]{x}$ is a real number, $x$ must be a real number. | 0 | {
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"Step 1: Determine the constraints on $x$."
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pt-393f10d1727243d6 | What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? | Question:
What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? | The given quadratic equation is $2x^2 - kx + 8 = 0$.
For the equation to have two distinct real solutions, the discriminant must be positive.
The discriminant $\Delta$ is given by $\Delta = (-k)^2 - 4(2)(8) = k^2 - 64$.
For two distinct real solutions, $\Delta > 0$, so $k^2 - 64 > 0$, which means $k^2 > 64$.
This impli... | [
"The given quadratic equation is $2x^2 - kx + 8 = 0$.",
"For the equation to have two distinct real solutions, the discriminant must be positive.",
"The discriminant $\\Delta$ is given by $\\Delta = (-k)^2 - 4(2)(8) = k^2 - 64$.",
"For two distinct real solutions, $\\Delta > 0$, so $k^2 - 64 > 0$, which means... | Question:
What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions?
The given quadratic equation is $2x^2 - kx + 8 = 0$.
For the equation to have two distinct real solutions, the discriminant must be positive.
The discriminant $\Delta$ is given by $\Delta = (-k)^2 - 4... | Sum $= 10 + (-10) = 0$. | 0 | {
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"The sum of all such values of $k$ is:"
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"removed_start_index": 32,
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pt-eace98a8ee45a8e9 | A two-row triangle is created with a total of 15 pieces: nine unit rods and six connectors, as shown. What is the total number of pieces that would be used to create an eight-row triangle?
[asy]
draw((0,0)--(4,0)--(2,2sqrt(3))--(0,0)--cycle,linewidth(1));
draw((2,0)--(3,sqrt(3))--(1,sqrt(3))--(2,0)--cycle,linewidth(1)... | Question:
A two-row triangle is created with a total of 15 pieces: nine unit rods and six connectors, as shown. What is the total number of pieces that would be used to create an eight-row triangle?
[asy]
draw((0,0)--(4,0)--(2,2sqrt(3))--(0,0)--cycle,linewidth(1));
draw((2,0)--(3,sqrt(3))--(1,sqrt(3))--(2,0)--cycle,li... | 1. Determine the number of pieces in the two-row triangle.
Number of unit rods in a two-row triangle is $1 + 2 = 3$.
Number of connectors in a two-row triangle is $2$.
Total pieces in a two-row triangle is $3 + 2 = 5$.
*Correction based on the problem statement:* The problem states the two-row triangle... | [
"Determine the number of pieces in the two-row triangle.",
"Number of unit rods in a two-row triangle is $1 + 2 = 3$.",
"Number of connectors in a two-row triangle is $2$.",
"Total pieces in a two-row triangle is $3 + 2 = 5$.",
"*Correction based on the problem statement:* The problem states the two-row tri... | Question:
A two-row triangle is created with a total of 15 pieces: nine unit rods and six connectors, as shown. What is the total number of pieces that would be used to create an eight-row triangle?
[asy]
draw((0,0)--(4,0)--(2,2sqrt(3))--(0,0)--cycle,linewidth(1));
draw((2,0)--(3,sqrt(3))--(1,sqrt(3))--(2,0)--cycle,li... | Number of unit rods in a two-row triangle is $1 + 2 = 3$. | 0 | {
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"Determine the number of pieces in the two-row triangle."
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pt-2475e9a5420ca3aa | A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.
What is the value of $8a-4b+2c-d$?
[asy]
import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=4.25,ymin=-9.25,ymax=4.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidt... | Question:
A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.
What is the value of $8a-4b+2c-d$?
[asy]
import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=4.25,ymin=-9.25,ymax=4.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen g... | The given function is $f(x)=ax^3+bx^2+cx+d$.
We need to find the value of the expression $8a-4b+2c-d$.
The expression to evaluate is $E = 8a-4b+2c-d$.
This expression can be related to the value of the function at a specific point by evaluating $f(x)$ for a specific $x$.
Consider $f(x) = ax^3+bx^2+cx+d$.
If we evaluat... | [
"The given function is $f(x)=ax^3+bx^2+cx+d$.",
"We need to find the value of the expression $8a-4b+2c-d$.",
"The expression to evaluate is $E = 8a-4b+2c-d$.",
"This expression can be related to the value of the function at a specific point by evaluating $f(x)$ for a specific $x$.",
"Consider $f(x) = ax^3+b... | Question:
A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.
What is the value of $8a-4b+2c-d$?
[asy]
import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=4.25,ymin=-9.25,ymax=4.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen g... | This expression can be related to the value of the function at a specific point by evaluating $f(x)$ for a specific $x$. | 0 | {
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"We need to find the value of the expression $8a-4b+2c-d$.",
"The expression to evaluate is $E = 8a-4b+2c-d$."
],
"removed_start_index": 1,
... |
pt-90f0207c5bb00ff8 | Find the equation whose graph is a parabola with vertex $(2,4)$, vertical axis of symmetry, and contains the point $(1,1)$. Express your answer in the form "$ax^2+bx+c$". | Question:
Find the equation whose graph is a parabola with vertex $(2,4)$, vertical axis of symmetry, and contains the point $(1,1)$. Express your answer in the form "$ax^2+bx+c$". | 1. The standard form of a parabola with a vertical axis of symmetry and vertex $(h,k)$ is $y = a(x-h)^2 + k$.
2. Given the vertex $(h,k) = (2,4)$, the equation becomes $y = a(x-2)^2 + 4$.
3. The parabola contains the point $(1,1)$. Substitute $x=1$ and $y=1$ into the equation to solve for $a$:
$1 = a(1-2)^2 + 4$
4. ... | [
"The standard form of a parabola with a vertical axis of symmetry and vertex $(h,k)$ is $y = a(x-h)^2 + k$.",
"Given the vertex $(h,k) = (2,4)$, the equation becomes $y = a(x-2)^2 + 4$.",
"The parabola contains the point $(1,1)$. Substitute $x=1$ and $y=1$ into the equation to solve for $a$:",
"$1 = a(1-2)^2 ... | Question:
Find the equation whose graph is a parabola with vertex $(2,4)$, vertical axis of symmetry, and contains the point $(1,1)$. Express your answer in the form "$ax^2+bx+c$".
The standard form of a parabola with a vertical axis of symmetry and vertex $(h,k)$ is $y = a(x-h)^2 + k$.
Given the vertex $(h,k) = (2,4)$... | $1 = a(-1)^2 + 4$ | 0 | {
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"removed_span": [
"Simplify the equation:"
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pt-70d5ddaa8624100b | For how many integer values of $x$ is $5x^{2}+19x+16 > 20$ not satisfied? | Question:
For how many integer values of $x$ is $5x^{2}+19x+16 > 20$ not satisfied? | We are asked to find the number of integer values of $x$ for which the inequality $5x^{2}+19x+16 > 20$ is not satisfied.
Step 1: Rewrite the inequality.
$$5x^{2}+19x+16 > 20$$
Step 2: Simplify the inequality by moving the constant term to the left side.
$$5x^{2}+19x+16 - 20 > 0$$
$$5x^{2}+19x-4 > 0$$
Step 3: Find th... | [
"We are asked to find the number of integer values of $x$ for which the inequality $5x^{2}+19x+16 > 20$ is not satisfied.",
"Step 1: Rewrite the inequality.",
"$$5x^{2}+19x+16 > 20$$",
"Step 2: Simplify the inequality by moving the constant term to the left side.",
"$$5x^{2}+19x+16 - 20 > 0$$",
"$$5x^{2}+... | Question:
For how many integer values of $x$ is $5x^{2}+19x+16 > 20$ not satisfied?
We are asked to find the number of integer values of $x$ for which the inequality $5x^{2}+19x+16 > 20$ is not satisfied. | $$5x^{2}+19x+16 > 20$$ | 0 | {
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"Step 1: Rewrite the inequality."
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"removed_start_index": 1,
"removed_end_index": 1,
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pt-c516ce1e00917abb | $k, a_2, a_3$ and $k, b_2, b_3$ are both nonconstant geometric sequences with different common ratios. We have $$a_3-b_3=3(a_2-b_2).$$Find the sum of the common ratios of the two sequences. | Question:
$k, a_2, a_3$ and $k, b_2, b_3$ are both nonconstant geometric sequences with different common ratios. We have $$a_3-b_3=3(a_2-b_2).$$Find the sum of the common ratios of the two sequences. | Let the first geometric sequence be $a_n$ with first term $a_1$ and common ratio $r_a$.
The terms are $a_n = a_1 r_a^{n-1}$.
We are given $a_2$ and $a_3$.
$a_2 = a_1 r_a$
$a_3 = a_1 r_a^2$
Let the second geometric sequence be $b_n$ with first term $b_1$ and common ratio $r_b$.
The terms are $b_n = b_1 r_b^{n-1}$.
We a... | [
"Let the first geometric sequence be $a_n$ with first term $a_1$ and common ratio $r_a$.",
"The terms are $a_n = a_1 r_a^{n-1}$.",
"We are given $a_2$ and $a_3$.",
"$a_2 = a_1 r_a$",
"$a_3 = a_1 r_a^2$",
"Let the second geometric sequence be $b_n$ with first term $b_1$ and common ratio $r_b$.",
"The ter... | Question:
$k, a_2, a_3$ and $k, b_2, b_3$ are both nonconstant geometric sequences with different common ratios. We have $$a_3-b_3=3(a_2-b_2).$$Find the sum of the common ratios of the two sequences.
Let the first geometric sequence be $a_n$ with first term $a_1$ and common ratio $r_a$.
The terms are $a_n = a_1 r_a^{... | For sequence $a_n$: | 0 | {
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"removed_span": [
"We are given that $k, a_2, a_3$ and $k, b_2, b_3$ are geometric sequences. This implies that the given terms are consecutive terms of their respect... |
pt-31b139cfb200ef20 | Real numbers $a$ and $b$ satisfy the equations $3^a=81^{b+2}$ and $125^b=5^{a-3}$. What is $ab$? | Question:
Real numbers $a$ and $b$ satisfy the equations $3^a=81^{b+2}$ and $125^b=5^{a-3}$. What is $ab$? | Given the equations:
1) $3^a = 81^{b+2}$
2) $125^b = 5^{a-3}$
Step 1: Rewrite the bases in terms of the same prime bases.
$81 = 3^4$
$125 = 5^3$
Step 2: Substitute the rewritten bases into the first equation.
$3^a = (3^4)^{b+2}$
Step 3: Apply the exponent rule $(x^m)^n = x^{mn}$ to the right side of the first equati... | [
"Given the equations:",
"$3^a = 81^{b+2}$",
"$125^b = 5^{a-3}$",
"Step 1: Rewrite the bases in terms of the same prime bases.",
"$81 = 3^4$",
"$125 = 5^3$",
"Step 2: Substitute the rewritten bases into the first equation.",
"$3^a = (3^4)^{b+2}$",
"Step 3: Apply the exponent rule $(x^m)^n = x^{mn}$ t... | Question:
Real numbers $a$ and $b$ satisfy the equations $3^a=81^{b+2}$ and $125^b=5^{a-3}$. What is $ab$? | $3^a = 81^{b+2}$ | 0 | {
"generator_model": "avreymi/gemma-4-E2B-it-reasoning-pruning",
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"Given the equations:"
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pt-58dd84c41e22fbc7 | On Monday, Jessica told two friends a secret. On Tuesday, each of those friends told the secret to two other friends. Each time a student heard the secret, he or she told the secret to two other friends the following day. On what day of the week will 1023 students know the secret? | Question:
On Monday, Jessica told two friends a secret. On Tuesday, each of those friends told the secret to two other friends. Each time a student heard the secret, he or she told the secret to two other friends the following day. On what day of the week will 1023 students know the secret? | Let $n$ be the number of students who know the secret on a given day.
Let $S_d$ be the set of students who know the secret on day $d$.
Let $N_d$ be the number of *new* students who learn the secret on day $d$.
The total number of students who know the secret on day $d+1$ is $n_{d+1} = n_d + N_d$.
Monday (Day 0): Jessi... | [
"Let $n$ be the number of students who know the secret on a given day.",
"Let $S_d$ be the set of students who know the secret on day $d$.",
"Let $N_d$ be the number of *new* students who learn the secret on day $d$.",
"The total number of students who know the secret on day $d+1$ is $n_{d+1} = n_d + N_d$.",
... | Question:
On Monday, Jessica told two friends a secret. On Tuesday, each of those friends told the secret to two other friends. Each time a student heard the secret, he or she told the secret to two other friends the following day. On what day of the week will 1023 students know the secret? | The total number of students who know the secret on day $d+1$ is $n_{d+1} = n_d + N_d$. | 0 | {
"generator_model": "avreymi/gemma-4-E2B-it-reasoning-pruning",
"generator_model_revision": null,
"decision_model": "gemini-3.1-flash-lite",
"removed_span": [
"Let $n$ be the number of students who know the secret on a given day.",
"Let $S_d$ be the set of students who know the secret on day $d$.",
... |
pt-fd8ae7d4457a6556 | In a certain city the rate of taxation is the following: $x\%$ tax is collected for an income of $x$ thousand dollars. What income, in dollars, will yield the greatest take home pay? (Take-home pay is the income minus the tax on that income.) | Question:
In a certain city the rate of taxation is the following: $x\%$ tax is collected for an income of $x$ thousand dollars. What income, in dollars, will yield the greatest take home pay? (Take-home pay is the income minus the tax on that income.) | Let $I$ be the income in dollars.
The tax rate is $x\%$ for an income of $x$ thousand dollars.
Step 1: Determine the tax rate $r$ as a function of income $I$.
The given information states that $x\%$ tax is collected for an income of $x$ thousand dollars.
Income in thousands of dollars is $\frac{I}{1000}$.
The tax rate... | [
"Let $I$ be the income in dollars.",
"The tax rate is $x\\%$ for an income of $x$ thousand dollars.",
"Step 1: Determine the tax rate $r$ as a function of income $I$.",
"The given information states that $x\\%$ tax is collected for an income of $x$ thousand dollars.",
"Income in thousands of dollars is $\\f... | Question:
In a certain city the rate of taxation is the following: $x\%$ tax is collected for an income of $x$ thousand dollars. What income, in dollars, will yield the greatest take home pay? (Take-home pay is the income minus the tax on that income.)
Let $I$ be the income in dollars.
The tax rate is $x\%$ for an inc... | The given information states that $x\%$ tax is collected for an income of $x$ thousand dollars. | 0 | {
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"generator_model_revision": null,
"decision_model": "gemini-3.1-flash-lite",
"removed_span": [
"Step 1: Determine the tax rate $r$ as a function of income $I$."
],
"removed_start_index": 2,
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pt-d17f360b4f3e451a | Find the sum of all possible positive integer values of $b$ such that the quadratic equation $2x^2 + 5x + b = 0$ has rational roots. | Question:
Find the sum of all possible positive integer values of $b$ such that the quadratic equation $2x^2 + 5x + b = 0$ has rational roots. | The given quadratic equation is $2x^2 + 5x + b = 0$.
For a quadratic equation $ax^2 + Bx + C = 0$ to have rational roots, the discriminant $\Delta = B^2 - 4AC$ must be a perfect square of a rational number. Since $a=2$, $B=5$, and $C=b$ are integers, the discriminant must be the square of an integer.
Step 1: Calculate... | [
"The given quadratic equation is $2x^2 + 5x + b = 0$.",
"For a quadratic equation $ax^2 + Bx + C = 0$ to have rational roots, the discriminant $\\Delta = B^2 - 4AC$ must be a perfect square of a rational number. Since $a=2$, $B=5$, and $C=b$ are integers, the discriminant must be the square of an integer.",
"St... | Question:
Find the sum of all possible positive integer values of $b$ such that the quadratic equation $2x^2 + 5x + b = 0$ has rational roots.
The given quadratic equation is $2x^2 + 5x + b = 0$.
For a quadratic equation $ax^2 + Bx + C = 0$ to have rational roots, the discriminant $\Delta = B^2 - 4AC$ must be a perfect... | $$\Delta = 5^2 - 4(2)(b) = 25 - 8b$$ | 0 | {
"generator_model": "avreymi/gemma-4-E2B-it-reasoning-pruning",
"generator_model_revision": null,
"decision_model": "gemini-3.1-flash-lite",
"removed_span": [
"Step 1: Calculate the discriminant $\\Delta$."
],
"removed_start_index": 2,
"removed_end_index": 2,
"decision_reason": "Unit 2 is a goal st... |
pt-88271b5fcbafb4e6 | Rationalize the denominator of $\frac{\sqrt{8}+\sqrt{3}}{\sqrt{2}+\sqrt{3}}$. Express your answer in simplest form. | Question:
Rationalize the denominator of $\frac{\sqrt{8}+\sqrt{3}}{\sqrt{2}+\sqrt{3}}$. Express your answer in simplest form. | To rationalize the denominator of $\frac{\sqrt{8}+\sqrt{3}}{\sqrt{2}+\sqrt{3}}$, we multiply the numerator and the denominator by the conjugate of the denominator.
Step 1: Identify the denominator and its conjugate.
The denominator is $\sqrt{2}+\sqrt{3}$.
The conjugate of the denominator is $\sqrt{2}-\sqrt{3}$.
Step ... | [
"To rationalize the denominator of $\\frac{\\sqrt{8}+\\sqrt{3}}{\\sqrt{2}+\\sqrt{3}}$, we multiply the numerator and the denominator by the conjugate of the denominator.",
"Step 1: Identify the denominator and its conjugate.",
"The denominator is $\\sqrt{2}+\\sqrt{3}$.",
"The conjugate of the denominator is $... | Question:
Rationalize the denominator of $\frac{\sqrt{8}+\sqrt{3}}{\sqrt{2}+\sqrt{3}}$. Express your answer in simplest form.
To rationalize the denominator of $\frac{\sqrt{8}+\sqrt{3}}{\sqrt{2}+\sqrt{3}}$, we multiply the numerator and the denominator by the conjugate of the denominator. | The denominator is $\sqrt{2}+\sqrt{3}$. | 0 | {
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"decision_model": "gemini-3.1-flash-lite",
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"Step 1: Identify the denominator and its conjugate."
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"removed_start_index": 1,
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