data_source stringclasses 6
values | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict | problem stringlengths 11 4.82k |
|---|---|---|---|---|---|
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Suppose the point $(2,4)$ is on the graph of $y = 3f(x)$ where $f(x)$ is a function. Find the sum of the coordinates of one point that must lie on the graph of $y = \frac{1}{3}f^{-1}(x)$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "25",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In a right triangle with integer length sides, the hypotenuse has length 65 units. Determine the length of the shorter leg. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If real numbers \( x \) and \( y \) satisfy \((x+5)^{2} + (y-12)^{2} = 14^{2}\), find the minimum value of \( x^{2} + y^{2} \). |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x) = |x-a|$, where $a > 0$.
(1) When $a=1$, find the solution set for the inequality $f^2(x) \leq 2$;
(2) Given that the minimum value of the function $g(x) = f(2x+a) + 2f(x)$ is 4, find the value of the real number $a$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "-2x + 2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the remainder of the Euclidean division of \(x^{100} - 2 x^{51} + 1\) by \(x^{2} - 1\). |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "x = 6.5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Equilateral triangle $XYZ$ has a side length of 8, and a circle inscribed in this triangle touches side $XY$ at point $C$. A right triangle $ABC$ (hypotenuse $\overline{AB}$) is inscribed in triangle $XYZ$ such that point $C$ is also a vertex of triangle $ABC$. If $XC = 2$ and $CY = 3$, compute the length of $AQ$, wher... |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the quadratic polynomials \( f_{1}(x)=x^{2}-ax+2 \), \( f_{2}(x)=x^{2}+3x+b \), \( f_{3}(x)=3x^{2}+(3-2a)x+4+b \), and \( f_{4}(x)=3x^{2}+(6-a)x+2+2b \), let the differences of their roots be \( A, B, C \), and \( D \) respectively, with the condition \( |A| \neq |B| \). Find the ratio \( \frac{C^{2}-D^{2}}{A^{2}... |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "56\\%",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If a plane is tiled using congruent hexagons and congruent triangles, each large hexagonal tile subdivided into a grid of $4 \times 4$ smaller equal-sized triangles, and 9 of these smaller triangles in each large hexagon are part of larger triangles formed by combining multiple smaller triangles, calculate the percenta... |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "n = 2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the smallest positive integer $n$ such that
\[\sin^n x + \cos^n x \leq 2^{1-n}\] for all real numbers $x$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "e = \\sqrt{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a>0,b>0)$, whose left and right foci are ${{F}_{1}}$ and ${{F}_{2}}$, respectively. A line passing through ${{F}_{1}}$, perpendicular to the $x$-axis, intersects the left branch of the hyperbola at points $A$ and $B$. If $\mathrm{\Delta }A{{F}_{2}}B... |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "x^2-2=0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The equation $(x-1)(x+1)=1$ is converted into the general form of a quadratic equation as __ . |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\left( t= \\frac {1}{2},\\text{ the minimum value of } s\\text{ is } \\frac {\\pi}{6} \\right)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $y=\sin \left(x-\frac {\pi}{12}\right)$, translate the point $P\left( \frac {\pi}{4},t\right)$ to the left by $s(s > 0)$ units and determine the minimum value of $s$ such that the translated point $P'$ lies on the graph of the function $y=\sin 2x$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "108",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A square has an area of $36$. A rectangle has the same width as the square. The length of the rectangle is triple its width. What is the area of the rectangle? |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "14\\sqrt{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A parallelogram has its diagonals making an angle of \(60^{\circ}\) with each other. If two of its sides have lengths 6 and 8, find the area of the parallelogram. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "3a - 2b",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate: $4a+5b-a-7b$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "110^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the diagram below, lines $r$ and $s$ are parallel. Find the measure of angle $y$ in degrees.
[asy]
size(200);
import markers;
pair A = dir(-22)*(0,0);
pair B = dir(-22)*(4,0);
pair C = dir(-22)*(4,2);
pair D = dir(-22)*(0,2);
pair F = dir(-22)*(0,1.3);
pair G = dir(-22)*(4,1.3);
pair H = dir(-22)*(2,1);
markangle(... |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "(-\\infty, -4]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that for any real number $x$, the inequality $|x+3| \geq m+4$ always holds, determine the range of the real number $m$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{5}{9}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the mean of \(\frac{2}{3}\) and \(\frac{4}{9}\)?
A \(\frac{1}{2}\)
B \(\frac{2}{9}\)
C \(\frac{7}{9}\)
D \(\frac{3}{4}\)
E \(\frac{5}{9}\) |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "18",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Amy works for 40 hours per week for 12 weeks during the summer, making $\$4800$. If she receives a 10% pay raise and works for 36 weeks during the school year, how many hours per week must she work to earn another $\$7200$? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{2\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given vectors $\overrightarrow{a}=( \frac{1}{3},\tan \alpha )$ and $\overrightarrow{b}=(\cos \alpha,1)$, where $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the cosine value of the acute angle $\alpha$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "B = \\frac{n}{3}(4n^2-1)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine the following sums (as functions of $n$):
$$
\begin{aligned}
& A=2^{2}+4^{2}+6^{2}+\ldots+(2n-2)^{2} \\
& B=1^{2}+3^{2}+5^{2}+\ldots+(2n-1)^{2}
\end{aligned}
$$ |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\theta = \\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given vectors $\overrightarrow {a} = (1, y)$ and $\overrightarrow {b} = (1, -3)$, and it is known that $(2 \overrightarrow {a} + \overrightarrow {b})$ is perpendicular to $\overrightarrow {b}$.
(1) Determine the coordinates of vector $\overrightarrow {a}$;
(2) Find the angle between vector $\overrightarrow {a}$ and vec... |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Which of the following propositions is correct? Fill in the blank with the correct proposition number.
\\(①\\) Very small real numbers can form a set;
\\(②\\) The set \\(\\{y|y=x^{2}-1\\}\\) and the set \\(\\{(x,y)|y=x^{2}-1\\}\\) are the same set;
\\(③\\) The set consisting of \\(1\\), \\(\dfrac{3}{2}\\), \\(\df... |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "(-\\frac{3}{2}, +\\infty)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f\left(x\right)=|x-a|+|x+3|$.<br/>$(1)$ When $a=1$, find the solution set of the inequality $f\left(x\right)\geqslant 6$;<br/>$(2)$ If $f\left(x\right) \gt -a$, find the range of values for $a$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\operatorname{ctg}\\left(\\alpha - \\frac{\\pi}{4}\\right)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the value of the expression \(\frac{\sqrt{\operatorname{tg} \alpha}+\sqrt{\operatorname{ctg} \alpha}}{\sqrt{\operatorname{tg} \alpha}-\sqrt{\operatorname{ctg} \alpha}},\) given that \(0 < \alpha < \frac{\pi}{2}\) and \(\alpha \neq \frac{\pi}{4}\). |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\Delta \\text{Longest Day} = 51 \\text{ minutes } 2 \\text{ seconds }",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the distance between Budapest and Rozsnyó if the latitude and longitude of Budapest are $\varphi = 47^{\circ} 29^{\prime} 15^{\prime \prime}$ and $\lambda = 36^{\circ} 42^{\prime} 17^{\prime \prime}$, respectively, and the latitude and longitude of Rozsnyó are $\varphi^{\prime} = 40^{\circ} 39^{\prime} 2^{\prim... |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "36",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Simplify $\sqrt{18} \times \sqrt{72}$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2187",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A geometric sequence of positive integers is formed where the first term is 3 and the fifth term is 243. What is the seventh term of the sequence? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "31",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Factor the expression $81x^4 - 256y^4$ and find the sum of all integers in its complete factorization $(ax^2 + bxy + cy^2)(dx^2 + exy + fy^2)$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "0.41",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that $\sqrt{16.81}=4.1$, find $\sqrt{0.1681}$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "120",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In a sports event, six athletes — Alex, Ben, Carl, Danny, Emma, and Fiona — compete in a 100-meter dash. How many different outcomes are possible for the top three positions, assuming there are no ties? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{5}{8}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $\frac{a}{b} = 25$, $\frac{c}{b} = 5$, and $\frac{c}{d} = \frac{1}{8}$, then what is $\frac{a}{d}$? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate: $({-2023})^0+|{-\sqrt{2}}|-2\cos45°-\sqrt[3]{216}$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{7}{10}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the probability distribution list of the random variable $X$ as $P(X=i)=\frac{i}{2a}(i=1,2,3,4)$, find $P(2 < X\leqslant 4)$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the value of $\cos 120^{\circ}$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1078",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $[x]$ denote the greatest integer less than or equal to the real number $x$,
$$
\begin{array}{c}
S=\left[\frac{1}{1}\right]+\left[\frac{2}{1}\right]+\left[\frac{1}{2}\right]+\left[\frac{2}{2}\right]+\left[\frac{3}{2}\right]+ \\
{\left[\frac{4}{2}\right]+\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{... |
numina_aops_forum | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "12 + 4(m + n + p)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $m, n, p$ be fixed positive real numbers which satisfy $mnp = 8$ . Depending on these constants, find the minimum of $$ x^2+y^2+z^2+ mxy + nxz + pyz, $$ where $x, y, z$ are arbitrary positive real numbers satisfying $xyz = 8$ . When is the equality attained?
Solve the problem for:
[list=a][*] $m = n = p = ... |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "(-1, 0)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the functions $f(x) = \log_a(1+x)$ and $g(x) = \log_a(1-x)$, where $a>0$ and $a \neq 1$.
1. Find the domain of the function $f(x) - g(x)$.
2. Determine the parity of the function $f(x) - g(x)$.
3. Find the range of $x$ for which $f(x) - g(x) > 0$. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a square piece of paper is folded so that point (0,4) is matched with (4,0) using a fold that also involves a $45^\circ$ rotation, find the aligned coordinates of point $(8,6)$ and determine the sum of its coordinates. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?
The order of the selected cards does not matter: for example, selecting cards with numbers 5 and 16, as well as selecting cards with nu... |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Compute $(44^{1234} + 99^{567}) \mod 7$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "125",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the greatest integer less than 150 for which the greatest common factor of that integer and 30 is 5? |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{H^3 \\sqrt{3}}{4 \\sin^2 \\alpha}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The height of a regular triangular truncated pyramid is \( H \) and it is the geometric mean between the sides of the bases. The lateral edge makes an angle \( \alpha \) with the base. Find the volume of the pyramid. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1021",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Eight cubes, with volumes $1$, $8$, $27$, $64$, $125$, $216$, $343$, and $512$ cubic units, are stacked vertically to form a tower with decreasing volumes from the bottom to the top. Each cube except for the bottom one is placed perfectly on top of the cube below. Compute the total surface area of the tower.
- **A)** $... |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given two vectors $\overrightarrow{a} = (3,0)$ and $\overrightarrow{b} = (-5,5)$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "192",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the largest multiple of $8$ whose negation is greater than $-200$? |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "60:1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $ABC$ be an equilateral triangle with side length $s$. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ such that $BB' = 2s$, extend side $\overline{BC}$ beyond $C$ to a point $C'$ such that $CC' = 3s$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ such that $AA' = 4s$. Determine the ratio of the... |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\pi a^2 (2 - \\sqrt{2})",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the surface area formed by the rotation around the polar axis of the lemniscate $\rho = a \sqrt{\cos 2 \varphi}$ over the interval $0 \leq \varphi \leq \pi / 4$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "0.67",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} |
In a rectangular room, there is a point light source $A$ whose light falls only on a flat mirror $SS_{1}$ that occupies a part of one of the walls and spans the full height of the room. Determine the portion of the walls that remain unilluminated. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2970",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the arithmetic mean of the first nine powers of 3, starting from $3^1$ to $3^9$. What is this mean value if each power is summed first?
A) $2916$
B) $2970$
C) $3051$
D) $3123$ |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2, 3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Three statements are given below:
1. The converse of the statement "If $a^2 + b^2 = 0$, then $a$ and $b$ are both 0" is "If $a$ and $b$ are both nonzero, then $a^2 + b^2 \neq 0$".
2. "$m = \frac{1}{2}$" is a necessary but not sufficient condition for the lines $(m + 2)x + 3my + 1 = 0$ and $(m - 2)x + (m + 2)y - 3 = 0$ ... |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac {3 \\sqrt {3}}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given in $\triangle ABC$, $a=3 \sqrt {3}$, $c=2$, and $B=150^{\circ}$, find:
$(1)$ The length of side $b$;
$(2)$ The area of $\triangle ABC$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "399",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many integers from 1 to 2001 have a digit sum that is divisible by 5? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the solution to the equation $x|x| = 4x + 3$ which has the smallest value. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given four non-collinear points on a plane, O, A, B, and C, if \( \overrightarrow{OA} - 4 \overrightarrow{OB} + 3 \overrightarrow{OC} = 0, calculate \(\dfrac{|\overrightarrow{AB}|}{|\overrightarrow{BC}|}.\) |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that $f'(x)$ is the derivative function of $f(x)=x\sin x$, find the value of $f'(\dfrac{\pi}{2})$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the plane rectangular coordinate system $xOy$, the three vertices of $\triangle ABC$ are $A(-2,0)$, $B(0,4)$, $C(m,n)$, where point $C$ lies on the line $x-3y-3=0$. <br/>$(1)$ If $m=3$, find the equation of the median on side $AB$ of $\triangle ABC$; <br/>$(2)$ If $\triangle ABC$ is a right triangle, find the value ... |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\sqrt{2} - 1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that in the rectangular coordinate system $(xOy)$, the parametric equation of line $l$ is: $\begin{cases} & x=1-\frac{\sqrt{2}}{2}t, \ & y=2+\frac{\sqrt{2}}{2}t, \ \end{cases}$ ($t$ is the parameter). Establish a polar coordinate system with $Ox$ as the polar axis. The polar coordinate equation of circle $C$ is: ... |
numina_aops_forum | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "24",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Two circles have radius $2$ and $3$ , and the distance between their centers is $10$ . Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$ .
(A *common external tangent* is a tangent line to two circles such that th... |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "①②③④",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $\alpha$ and $\beta$ be two distinct planes, and let $m$ and $n$ be two distinct lines. Consider the following four propositions:
① If $m \perpendicular \alpha$ and $n \subset \alpha$, then $m \perpendicular n$;
② If $m \subset \alpha$, $n \subset \alpha$, $m \parallel \beta$, and $n \parallel \beta$, then $\al... |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2134",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $g(x) = x^3$ and $f(x) = 3x^2 - 2x + 1$, find the value of $f(g(3))$. |
numina_aops_forum | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2, 6, 42, 1806",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find all natural $n{}$ such that for every natural $a{}$ that is mutually prime with $n{}$ , the number $a^n - 1$ is divisible by $2n^2$ . |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "53",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many natural numbers greater than 10 but less than 100 are relatively prime to 21? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "(n-1)2^{n+1}+2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a sequence $\{a_n\}$ that satisfies $a_1=1$ and $a_{n+1}=2a_n+\lambda$ (where $\lambda$ is a constant).
(1) Investigate whether the sequence $\{a_n+\lambda\}$ is a geometric sequence and find $a_n$;
(2) When $\lambda=1$, find the sum of the first $n$ terms, $T_n$, of the sequence $\{n(a_n+\lambda)\}$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The midpoints of the sides of a regular octagon $ABCDEFGH$ are joined to form a smaller octagon. What fraction of the area of $ABCDEFGH$ is enclosed by the smaller octagon? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{4x^2}{9} - \\frac{y^2}{4} = 1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the equation of the hyperbola that has the same asymptotes as the hyperbola $$\frac{x^2}{9} - \frac{y^2}{16} = 1$$ and passes through the point (-3, $2\sqrt{3}$). |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "D) 256",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 24x^2 + 98x - 75$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{5}{s^3 - 24s^2 + 98s - 75} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\]for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?
A) 253
B... |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "5\\sqrt{2} \\text{ cm}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A right triangle \(ABC\) is divided by the altitude \(CD\), drawn to the hypotenuse, into two triangles \(BCD\) and \(ACD\). The radii of the circles inscribed in triangles \(BCD\) and \(ACD\) are 4 cm and 3 cm, respectively. Determine the distance between their centers. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If the sum of binomial coefficients in the expansion of $$(x^{2}+ \frac {1}{x})^{n}$$ is 64, find the value of $n$ and the constant term in this expansion. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | List all prime numbers \( p \) such that \( 2 \leq p \leq 100 \). |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "m=3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given two lines $l_1$: $x+my=-6$, $l_2$: $(m-2)x+3y+2m=0$, find the value of $m$ when the following conditions are met:
(1) $l_1$ and $l_2$ intersect;
(2) $l_1$ and $l_2$ coincide. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is $\log_{8}{1023}$ rounded to the nearest integer? |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{16661}{3240}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the value of the product
\[
\left(1+\frac{1}{1^2}\right)\cdot\left(1+\frac{1}{2^2}\right)\cdot\left(1+\frac{1}{3^2}\right)\cdot\left(1+\frac{1}{4^2}\right)\cdot\left(1+\frac{1}{5^2}\right)\cdot\left(1+\frac{1}{6^2}\right)?
\]
A) $\frac{16661}{3240}$
B) $\frac{20000}{3240}$
C) $\frac{14000}{3240}$
D) $\frac{1500... |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the units digit of $17^{2007}$? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In triangle $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $c=3$, $C=\frac{\pi}{3}$, and $a=2b$, find the value of $b$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "31",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $(x-2)^5 = a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0$, calculate the sum $a_1 + a_2 + a_3 + a_4 + a_5$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "42\\sqrt{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the area of the parallelogram formed by the vectors \( \mathbf{a} \) and \( \mathbf{b} \).
\[ \mathbf{a} = 3\mathbf{p} - \mathbf{q} \]
\[ \mathbf{b} = \mathbf{p} + 2\mathbf{q} \]
\[ |\mathbf{p}| = 3 \]
\[ |\mathbf{q}| = 4 \]
\[ (\widehat{\mathbf{p}, \mathbf{q}}) = \frac{\pi}{3} \] |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "S_3=21",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, and a geometric sequence $\{b_n\}$ with the sum of the first $n$ terms denoted as $T_n$, where $a_1=-1, b_1=1$, and $a_2+b_2=2$.
$(1)$ If $a_3+b_3=5$, find the general formula for $\{b_n\}$.
$(2)$ If $T_3=21$, find $S_3$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{6859\\pi}{48} \\text{ cubic inches}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A giant decorative sphere is designed to have a diameter of \(9 \frac{1}{2}\) inches. Calculate the surface area and the volume of this sphere. Express your answers in terms of \(\pi\) as a common fraction. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "56",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A pizza parlor now offers eight different toppings. What is the greatest number of three-topping pizzas that can be made such that no two pizzas have the same topping combination? |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\sqrt{S_1 S_2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | There are two triangles with corresponding parallel sides and areas $S_{1}$ and $S_{2}$, respectively. One of these triangles is inscribed in triangle $ABC$, and the other is circumscribed around it. Find the area of triangle $ABC$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "①②③",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For the function $f(x)=x|x|+px+q$, four propositions are provided:
$①$ When $q=0$, $f(x)$ is an odd function.
$②$ The graph of $y=f(x)$ is symmetric with respect to the point $(0,q)$.
$③$ When $p=0$ and $q > 0$, the equation $f(x)=0$ has exactly one real root.
$④$ The equation $f(x)=0$ has at most two real roots.
The c... |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "-\\frac{4}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $α \in (0, \pi)$, and $\sin α + \cos α = \frac{1}{5}$, calculate the value of $\tan α$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "-\\frac{7}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the vectors $\overrightarrow{a} = (\sqrt{3}, -1)$ and $\overrightarrow{b} = (\frac{1}{2}, \frac{\sqrt{3}}{2})$, suppose there exist non-zero real numbers $k$ and $t$ such that $\overrightarrow{x} = \overrightarrow{a} + (t^2 -3) \overrightarrow{b}$ and $\overrightarrow{y} = -k \overrightarrow{a} + t \overrightarro... |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{28}{153}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A box contains 8 white balls and 10 black balls. Two balls are drawn out of the box without replacement. What is the probability that both balls drawn are white? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "x_{1}=4, x_{2}=2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | $(1)$ Calculate: $\sin 45^{\circ}-3\tan 30^{\circ}+\sqrt{2}\cos 60^{\circ}$;
$(2)$ Solve the equation: $x^{2}-6x+8=0$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "120 \\text{ dollars}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A store sells a type of chocolate candies in boxes of 15 candies each, for $5 per box. They offer a special discount where boxes are priced at $4 each when a customer purchases at least 10 boxes at a time. How much does it cost to buy 450 chocolate candies with the discount? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the derivative of the function $f(x)=\frac{1}{x}$ and evaluate it at $x=1$. |
numina_aops_forum | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "105",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many ways are there to partition $7$ students into the groups of $2$ or $3$ ? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Tony has $4.90 in U.S. coins. He has the same number of quarters and dimes. What is the greatest number of quarters he could have? |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Solve the equation:
$$
\frac{8}{\{x\}}=\frac{9}{x}+\frac{10}{[x]}
$$
where $[x]$ is the greatest integer less than or equal to $x$, and $\{x\}=x-[x]$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\sqrt{17}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $C$ be the vertex of the graph of the equation $y = x^2 - 6x + 13$. Let $D$ be the vertex of the graph of the equation $y = x^2 + 2x + 4$. What is the distance between $C$ and $D$? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\frac{600}{7}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In a factory, workers produce gadgets and gizmos. Each type of product requires a constant but different amount of time per worker. In one hour, 150 workers can produce 450 gadgets and 300 gizmos. In two hours, 90 workers can produce 360 gadgets and 450 gizmos. In four hours, 75 workers can produce 300 gadgets and $n$ ... |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "-\\frac{32}{9}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A cubic polynomial $p(x)$ satisfies
\[p(1) = \frac{2}{1^2}, \, p(2) = \frac{2}{2^2}, \, p(3) = \frac{1}{3^2}, \, p(5) = \frac{1}{5^2}.\]
Find $p(6)$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "m \\lt \\frac{7}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If the solution of the system of equations $\left\{\begin{array}{l}{x-2y=1}\\{2x+y=4m}\end{array}\right.$ satisfies the inequality $x+3y \lt 6$, find the range of values for $m$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "2x - y - 1 = 0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the equation of the tangent line to the parabola $y=x^2$ that is parallel to the line $y=2x$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\dfrac{3 \\sqrt{2}}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $a+b=-6$ and $ab=8$, find the value of $\sqrt{\dfrac{b}{a}} +\sqrt{\dfrac{a}{b}}$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "(6,9]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If set $P = \{x | 3 < x \leq 22\}$, and non-empty set $Q = \{x | 2a + 1 \leq x < 3a - 5\}$, then the range of all real numbers $a$ that make $Q \subseteq (P \cap Q)$ true is ______. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "709.72",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $\triangle XYZ$ be a triangle in the plane, and let $W$ be a point outside the plane of $\triangle XYZ$, so that $WXYZ$ is a pyramid whose faces are all triangles.
Suppose that each edge of $WXYZ$ has length $13$ or $30$, but no face of $WXYZ$ is equilateral. Determine the surface area of $WXYZ$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]... | math | {
"ground_truth": "\\sin\\alpha= \\pm\\frac{4}{5}, \\tan\\alpha= \\mp\\frac{4}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given any angle $\alpha$ whose terminal side passes through point P(-3, m), and $\cos\alpha=-\frac{3}{5}$,
(1) Find the value of $m$.
(2) Find the values of $\sin\alpha$ and $\tan\alpha$. |
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