instruction_seed stringlengths 78 7.78k | source stringclasses 1
value | embeddings sequencelengths 1.02k 1.02k | mean_positive_score float64 0.79 0.87 | mean_negative_score float64 0.77 0.85 | difference_score float64 0.01 0.04 |
|---|---|---|---|---|---|
Let's consider a general polynomial of degree $n$ with complex coefficients:
$$p(z) = z^n + a_{n-1}z^{n-1} + a_{n-2}z^{n-2} + \cdots + a_1z + a_0$$
Given that $a_{n-1} = ia_{n-2}$ (where $i$ is the imaginary unit), find the upper bound of the product of the absolute values of the roots of $p(z)$.
Note: This new probl... | nvidia_math | [
0.033523183315992355,
0.031130162999033928,
-0.030332716181874275,
-0.017984483391046524,
0.011637646704912186,
-0.01381757017225027,
-0.016467196866869926,
0.05736098438501358,
0.03207225725054741,
-0.014265920966863632,
0.004269002005457878,
0.018001867458224297,
-0.04835791513323784,
-0... | 0.83778 | 0.808067 | 0.029713 |
Find a cubic polynomial with real coefficients, one of whose roots is $1 + \sqrt{2}$ and which satisfies the condition that the sum of its roots is 0. | nvidia_math | [
0.02202703058719635,
0.03701094537973404,
-0.0521540530025959,
-0.03143053501844406,
0.037524934858083725,
-0.026235824450850487,
0.0005780230858363211,
0.059913646429777145,
0.021108297631144524,
-0.021589700132608414,
0.016790496185421944,
0.012141788378357887,
-0.02038584090769291,
-0.0... | 0.845688 | 0.828365 | 0.017323 |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[a^2 + b^2 + c^2 + d^2 = 4.\]Find the maximum value of $a^3 + b^3 + c^3 + d^3.$
Here is another problem inspired by the given one:
Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[a^2 + b^2 + c^2 + d^2 = 4.\]Find the maximum value of $a^4 + b^4 + c^4 + d^4.$
... | nvidia_math | [
0.03938553109765053,
0.03599318116903305,
-0.03580751642584801,
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0.04945588484406471,
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0.0645613819360733,
0.039038144052028656,
-0.004799207206815481,
0.004842770751565695,
0.016371626406908035,
-0.044964373111724854,
-0.04... | 0.833872 | 0.80577 | 0.028102 |
Problem:
In the triangle $ABC$, the length of side $AB$ is $29$ units and the length of side $BC$ is $145$ units. Find the maximum possible value of the length of side $AC$.
Answer: As per the triangle inequality theorem, $AC < AB + BC$. Thus, the maximum value of $AC$ would be just less than $29 + 145 = 174$ units. T... | nvidia_math | [
0.02796694077551365,
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0.07000338286161423,
0.05448288843035698,
-0.0010867627570405602,
0.03103870525956154,
0.04823446646332741,
-0.037075355648994446,
-0.013... | 0.823999 | 0.803405 | 0.020594 |
A rectangular prism has a height of $h$ units and a square base with side length $s$ units. If the prism is rolled around the height to form a cylindrical shape, the volume is maximized. If the prism is then rolled around one of the sides of the square base to form another cylindrical shape, the volume is minimized. Wh... | nvidia_math | [
0.017578164115548134,
0.03355991095304489,
-0.04447397217154503,
-0.04046838358044624,
0.03355059772729874,
-0.014194738119840622,
-0.02384103089570999,
0.0755927562713623,
0.032683175057172775,
-0.002156686270609498,
0.020135898143053055,
0.04416344314813614,
-0.014426487497985363,
-0.004... | 0.845419 | 0.828955 | 0.016465 |
Given the function
\[g(x) =
\begin{cases}
-x^2 - 6x &\text{if } x<3, \\
3x &\text{if }x\ge 3,
\end{cases}
\]and a constant $c$, find the value of $c$ such that $g(g(x))$ is continuous for all real numbers $x$ if $g(x)$ is defined as above for $x<c$ and $g(x)=x^2$ for $x\ge c$. | nvidia_math | [
0.021351408213377,
0.03488057851791382,
-0.036555226892232895,
-0.03473368659615517,
0.04839722067117691,
-0.03923157975077629,
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0.08069901913404465,
0.026714205741882324,
-0.0330902561545372,
0.016978945583105087,
0.03863849863409996,
-0.022172097116708755,
-0.008885... | 0.819144 | 0.805176 | 0.013968 |
In the configuration of circles shown, $\angle BAC = 50^\circ$, $\angle ECA = 75^\circ$, and $\angle ACD = 90^\circ$. Find $\angle ADB$.
[asy]
draw(circle((0,0), 1), LLine); draw(circle((2,0), 1), LLine);
draw((0,0)--(0,-1)--(2,0));
draw((2,0)--(1,0.8)--(0,0));
draw((0,0)--(2,0), dashed);
label("$A$",(0,0), SW);
label... | nvidia_math | [
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0.023058751598000526,
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0.025462806224822998,
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0.049937654286623,
0.04162289947271347,
0.0013805802445858717,
0.015291976742446423,
0.011139200069010258,
-0.024283813312649727,
-0.0... | 0.829414 | 0.813329 | 0.016086 |
Let $a$ and $b$ be real numbers such that $a + 4i$ and $b + 5i$ are the roots of
\[z^2 - (10 + 9i) z + (4 + 46i) = 0.\]Enter the ordered pair $(a,b).$ | nvidia_math | [
0.05156571418046951,
0.02037729322910309,
-0.04387076944112778,
-0.041005413979291916,
0.035531651228666306,
-0.018880348652601242,
0.011280059814453125,
0.04497113823890686,
0.03877170383930206,
-0.011271592229604721,
0.02135847508907318,
0.021719496697187424,
-0.03788857161998749,
-0.016... | 0.847415 | 0.832893 | 0.014521 |
"Given a prime $p$ and an integer $a$, we say that $a$ is a $\\textit{primitive root} \\pmod p$ if t(...TRUNCATED) | nvidia_math | [0.047550804913043976,0.031040411442518234,-0.010021870024502277,-0.05044422298669815,0.033229716122(...TRUNCATED) | 0.843821 | 0.824585 | 0.019236 |
"A bookstore has a shelf of books of different heights, with the tallest book having a height of 30 (...TRUNCATED) | nvidia_math | [0.024621358141303062,0.00877715926617384,-0.04168121889233589,-0.036809857934713364,0.0435398891568(...TRUNCATED) | 0.818095 | 0.802462 | 0.015633 |
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