instruction_seed stringlengths 78 7.78k | source stringclasses 1
value | embeddings listlengths 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 | [
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-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,
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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,
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0.03103870525956154,
0.04823446646332741,
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-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 | [
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0.03355991095304489,
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0.0755927562713623,
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0.020135898143053055,
0.04416344314813614,
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-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,
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0.04839722067117691,
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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 | [
0.010896642692387104,
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,
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0.011280059814453125,
0.04497113823890686,
0.03877170383930206,
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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 the set $\{a,a^2,a^3,\ldots,a^{p-1}\}$ contains exactly one element congruent to each of $1,2,3,\ldots,p-1\pmod p$.
Find the product of all integers in the set $\{1,2,3,4,5,6,7,8,9,10\}$ that are primitive roots $\pmod{11}$... | nvidia_math | [
0.047550804913043976,
0.031040411442518234,
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0.0052345567382872105,
0.05518431216478348,
0.037195708602666855,
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0.009359198622405529,
0.019424250349402428,
-0.052848368883132935,
-... | 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 cm and the shortest book having a height of 10 cm. Tom wants to arrange the books in a special way: he wants to stack the books in a vertical column, with each book aligned at the bottom, and he wants the total height of ... | nvidia_math | [
0.024621358141303062,
0.00877715926617384,
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0.04353988915681839,
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0.0730108916759491,
0.033874042332172394,
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0.04609330743551254,
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0.006... | 0.818095 | 0.802462 | 0.015633 |
Let $a$ and $b$ be real numbers, with $a>b$. Find the value of the following expression:
\[\left|\sum_{k=0}^{\left\lfloor \frac{a-b}{\pi}\right\rfloor}(-1)^k\left(\binom{\left\lfloor a/\pi\right\rfloor}{k}\sin(\pi(a-b-k))+\binom{\left\lfloor b/\pi\right\rfloor}{k}\sin(\pi(b-a-k))\right)\right|.\] | nvidia_math | [
0.008967550471425056,
0.04247100651264191,
-0.04850183054804802,
-0.02508540078997612,
0.033968307077884674,
0.007203924003988504,
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0.05366067215800285,
0.050593890249729156,
0.0012536501744762063,
0.031179256737232208,
0.037591710686683655,
-0.04092653468251228,
-0.0... | 0.858264 | 0.841096 | 0.017168 |
Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that
\[z + \frac{1}{z} = r.\]Find $|z|.$ | nvidia_math | [
0.0198233500123024,
0.038057561963796616,
-0.032064300030469894,
-0.03802686557173729,
0.038131628185510635,
0.0013733532978221774,
0.003453509183600545,
0.04627476632595062,
0.0379815511405468,
-0.022760484367609024,
0.010493581183254719,
0.02702280879020691,
-0.03076906129717827,
-0.0176... | 0.844188 | 0.825497 | 0.018691 |
Let $ABCD$ be a regular tetrahedron. The point $P$ is the center of the face $ABC$ and $P'$ is the center of the face $ABD$. A regular octahedron has one vertex on each of the segments $\overline{AP}$, $\overline{BP}$, $\overline{CP}$, $\overline{AP'}$, $\overline{BP'}$, and $\overline{DP'}$. Determine the ratio of the... | nvidia_math | [
0.01790064387023449,
0.024133069440722466,
-0.05680447071790695,
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0.02081386186182499,
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0.06194620579481125,
0.03709305077791214,
0.007891934365034103,
0.007781794294714928,
0.03273454308509827,
-0.031155629083514214,
0.0102... | 0.832215 | 0.809592 | 0.022623 |
The real numbers $a,$ $b,$ $c,$ and $d$ satisfy
\[a^2 + b^2 + c^2 + 1 = d + \sqrt{a + b + c - d}.\]Find $d.$
The real numbers $a,$ $b,$ $c,$ and $d$ satisfy
\[a^2 + b^2 + c^2 + 1 = d + \sqrt{a + b + c - d}.\]Find $d.$
Let's try to create a problem that requires a different approach to solve.
The real numbers $x,$ $y... | nvidia_math | [
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0.002876914106309414,
0.029596583917737007,
-0.036698151379823685,
-0.05... | 0.845624 | 0.821369 | 0.024254 |
The vectors $\mathbf{v}_1=(1,-1)$ and $\mathbf{v}_2=(-1,2)$ define a parallelogram whose vertices are the endpoints of these vectors, their origin, and the tip of $\mathbf{v}_1+\mathbf{v}_2$. Find the area of the triangle whose vertices are the same as the parallelogram except for the tip of $\mathbf{v}_1$. | nvidia_math | [
0.02541026473045349,
0.03585224598646164,
-0.048135336488485336,
-0.0516168549656868,
0.04415259510278702,
0.00865344051271677,
-0.014684035442769527,
0.07603694498538971,
0.02627362497150898,
-0.005581425502896309,
0.008196959272027016,
0.023465586826205254,
-0.013805740512907505,
-0.0173... | 0.835635 | 0.818273 | 0.017361 |
Let $g$ be a function that maps real numbers to real numbers, such that $g(0) = 0$ and
\[g(x + y) - g(x - y) = \frac{g(2x) - g(2y)}{x + y}\]for all real numbers $x \neq -y$. Find all possible values of $g(2)$. | nvidia_math | [
0.03184790164232254,
0.035731296986341476,
-0.04664459452033043,
-0.03759776055812836,
0.03428373485803604,
-0.01941676251590252,
-0.011495079845190048,
0.0562351755797863,
0.039912573993206024,
-0.025289149954915047,
0.011906972154974937,
0.042571019381284714,
-0.026642300188541412,
-0.00... | 0.854299 | 0.8358 | 0.018499 |
In the equation
$$5x^2-kx+1=0$$
determine $k$ such that the difference of the roots be equal to unity.
Write another problem inspired by this one:
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$.
In the equation
$$5x^2-kx+1=0$$
determine $k$ such that the... | nvidia_math | [
0.03023463487625122,
0.04092938080430031,
-0.038976505398750305,
-0.07215240597724915,
0.03875241056084633,
-0.03532736748456955,
0.0022228467278182507,
0.07830516248941422,
0.038576170802116394,
-0.009283337742090225,
0.015145279467105865,
0.004183810669928789,
-0.019763527438044548,
-0.0... | 0.843604 | 0.821932 | 0.021672 |
A right pyramid with a square base of side length $a$ units is inscribed in a cube of edge length $2a$ units. The total surface area, in square units, of the pyramid is $S$. Find $S$, as a common fraction. | nvidia_math | [
0.030900198966264725,
0.03592411056160927,
-0.05600012466311455,
-0.039094228297472,
0.020682448521256447,
-0.033840809017419815,
-0.019669121131300926,
0.0661296620965004,
0.03323661908507347,
-0.00784377008676529,
0.03074263036251068,
0.03127150237560272,
-0.01080809161067009,
-0.0051844... | 0.834184 | 0.81615 | 0.018033 |
Consider the line $y = mx + b$. A point $(x_1, y_1)$ is reflected across this line to a new point $(x_2, y_2)$. The midpoint of these two points is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$ and it lies on the line of reflection. In this case, the midpoint is $(\frac{-3+5}{2}, \frac{-1+3}{2}) = (1, 1)$.
The slope of the... | nvidia_math | [
0.014907626435160637,
0.024993468075990677,
-0.05089918524026871,
-0.052668653428554535,
0.05617212876677513,
-0.014213469810783863,
-0.018417075276374817,
0.0609864816069603,
0.04039909318089485,
-0.014853582717478275,
0.014829992316663265,
0.029373090714216232,
-0.025811685249209404,
-0.... | 0.8414 | 0.8245 | 0.016899 |
Let $O$ be the origin, and let $(a,b,c)$ be a fixed point. A plane passes through $(a,b,c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the center of the sphere passing through $A,$ $B,$ $C,$ and $O.$ Find $\frac{a}{p} + \frac{b}{q} ... | nvidia_math | [
0.033198919147253036,
0.04091019183397293,
-0.043612685054540634,
-0.030764654278755188,
0.027991466224193573,
-0.004979255143553019,
-0.01697589084506035,
0.04321784898638725,
0.04105427861213684,
-0.007648302707821131,
0.009233922697603703,
0.029086457565426826,
-0.02810998074710369,
-0.... | 0.825076 | 0.803321 | 0.021755 |
The ellipse $E$ is given by the equation $\ds\frac{x^2}{6}+\frac{y^2}{3}=1$. The normal to the ellipse $E$ at the point $A(0,\sqrt 3)$ intersects the ellipse again at $B$. Determine $B$. | nvidia_math | [
0.0145656643435359,
0.02395901270210743,
-0.026913190260529518,
-0.031201329082250595,
0.04425087571144104,
-0.036483921110630035,
-0.017181219533085823,
0.04333944246172905,
0.05546018108725548,
-0.023554213345050812,
0.040394093841314316,
0.04548822343349457,
-0.017193416133522987,
-0.02... | 0.822524 | 0.808905 | 0.013618 |
The roots of the quadratic equation $ax^2+bx+c=0$ are the same as the solutions to the inequality $|x-3|>2$. Determine the ordered triple $(a, b, c)$, assuming $a$ is positive. | nvidia_math | [
0.03395817056298256,
0.009715504944324493,
-0.053912293165922165,
-0.04383816942572594,
0.020763080567121506,
-0.025080468505620956,
0.007224319968372583,
0.07409904897212982,
0.03376712277531624,
-0.015888668596744537,
0.009719094261527061,
0.022045914083719254,
-0.039940059185028076,
-0.... | 0.836391 | 0.818833 | 0.017559 |
Determine the smallest possible value of $n$ for which there exists a polynomial $p(x)$ of degree $n$ with integer coefficients such that $p(0) = 0$, $p(1) = 1$, and $p(x) \equiv 0 \mod{x^2 - 2}$ for all integers $x$. | nvidia_math | [
0.014677311293780804,
0.029376721009612083,
-0.0340825617313385,
-0.03235636278986931,
0.025798318907618523,
-0.029000889509916306,
-0.008851525373756886,
0.050661902874708176,
0.035904839634895325,
-0.008027410134673119,
0.009958107955753803,
0.03094838559627533,
-0.019201533868908882,
-0... | 0.85129 | 0.830348 | 0.020942 |
Let $a$, $b$, and $c$ be integers such that the polynomial $ax^3 + bx^2 + cx + 3$ has roots equal to $-1/2$, $3/4$, and $-2/3$. If the sum of the coefficients of the polynomial is a negative integer, find the largest possible value of $a$. | nvidia_math | [
0.03908107802271843,
0.019744260236620903,
-0.044691961258649826,
-0.03489157184958458,
0.02346920780837536,
-0.025142474099993706,
0.002192889805883169,
0.05527020990848541,
0.03235698863863945,
-0.00661084707826376,
0.013639579527080059,
0.04660067334771156,
-0.043589748442173004,
-0.039... | 0.849016 | 0.834466 | 0.014551 |
The sequence $(b_n)$ satisfies $b_0=1$ and $b_{n + 1} = \frac{1}{2}b_n + \frac{1}{2}\sqrt{b_n^2 + 4^{n-1}}$ for $n \geq 0$. Find the limit of $b_n$ as $n$ approaches infinity. | nvidia_math | [
0.004933970049023628,
0.05297661945223808,
-0.015998240560293198,
-0.03452594205737114,
0.033806122839450836,
-0.03392576426267624,
-0.026157403364777565,
0.07090336829423904,
0.04360248148441315,
-0.010149798355996609,
0.011035474017262459,
0.02663596160709858,
-0.02030530385673046,
-0.01... | 0.843378 | 0.822606 | 0.020772 |
Consider the set of all points $(x,y)$ in the coordinate plane for which one of the coordinates is exactly three times the other. If we were to plot all such points, into how many regions would the resulting graph split up the plane? | nvidia_math | [
0.01962057314813137,
0.016800081357359886,
-0.06155708432197571,
-0.031035883352160454,
0.029211154207587242,
-0.002128125634044409,
-0.01631348952651024,
0.04165279120206833,
0.036477502435445786,
-0.026850802823901176,
0.0038458972703665495,
0.04596022143959999,
-0.03361784666776657,
-0.... | 0.849424 | 0.835892 | 0.013533 |
Given the sequences $a_n = n^2 + 2n - 1$ and $b_n = n^2 + n + 1$, where $n$ is a positive integer, find the sum of all possible values of the greatest common divisor of $a_n$ and $b_n$. | nvidia_math | [
0.02671022154390812,
0.030707839876413345,
-0.020551934838294983,
-0.04897639900445938,
0.03773004189133644,
-0.023432273417711258,
-0.027130352333188057,
0.057248275727033615,
0.02942546457052231,
-0.031326472759246826,
0.01737099327147007,
0.019865047186613083,
-0.030916767194867134,
-0.... | 0.844905 | 0.825124 | 0.019782 |
Consider three points on the plane whose coordinates are $(x_i, y_i)$ for $i=1,2,3$. These points must satisfy $|x_i|\le 1$ and $|y_i|\le 1$. Given that $\sum_{i=1}^{3} x_i = 0$, $\sum_{i=1}^{3} y_i = 0$, and $\sum_{i=1}^{3} x_i^2 + y_i^2 = 1$, find the maximum value of $\sum_{i=1}^{3} \sqrt{x_i^2 + y_i^2 + 1}$. | nvidia_math | [
0.017200028523802757,
0.05114864557981491,
-0.05203491076827049,
-0.04940230771899223,
0.03947661444544792,
-0.0012255668407306075,
-0.016988780349493027,
0.05507185682654381,
0.028226466849446297,
-0.016994493082165718,
0.0057359980419278145,
0.038681503385305405,
-0.028699956834316254,
-... | 0.855017 | 0.820945 | 0.034073 |
Let $P$ be a point inside triangle $XYZ$ such that
$\angle XPY + \angle XZP = 80^\circ$ and
$\angle YPZ + \angle YXP = 80^\circ$.
Find $\angle ZPZ$. | nvidia_math | [
0.018487242981791496,
0.014510129578411579,
-0.028919395059347153,
-0.051873743534088135,
0.021638920530676842,
0.0028452968690544367,
-0.0045785438269376755,
0.029694732278585434,
0.04088149964809418,
-0.02450036257505417,
0.017379937693476677,
0.029971681535243988,
-0.04053152725100517,
... | 0.837881 | 0.815609 | 0.022272 |
Problem:
Consider a regular hexagon with side length 1 inscribed in a circle of radius $\frac{\sqrt{3}}{3}$.
Let P be a point inside or on the hexagon, and let $f(P)$ be the sum of the distances from P to the vertices of the hexagon.
Determine the smallest possible value of $f(P)$. | nvidia_math | [
0.0025575330946594477,
0.04115518182516098,
-0.03633427619934082,
-0.03720741346478462,
0.05030565336346626,
-0.022531714290380478,
-0.024858536198735237,
0.044142041355371475,
0.03450554981827736,
0.0031391363590955734,
0.020260419696569443,
0.035860270261764526,
-0.019286323338747025,
-0... | 0.846881 | 0.828755 | 0.018126 |
Let $\alpha$ be a root of $x^4 + x^3 + x^2 + x + 1 = 0.$ Let $b_1,$ $b_2,$ $\dots,$ $b_m$ be real numbers such that
\[\frac{1}{b_1 + \alpha} + \frac{1}{b_2 + \alpha} + \dots + \frac{1}{b_m + \alpha} = 3 - 2i.\]Compute
\[\frac{2b_1 + 1}{b_1^4 + b_1^3 + b_1^2 + b_1 + 1} + \frac{2b_2 + 1}{b_2^4 + b_2^3 + b_2^2 + b_2 + 1} ... | nvidia_math | [
0.036445096135139465,
0.04075931757688522,
-0.04345415160059929,
-0.041958730667829514,
0.041044268757104874,
-0.024721337482333183,
-0.015809310600161552,
0.06705165654420853,
0.045605048537254333,
-0.006382230203598738,
0.002184957033023238,
0.012210799381136894,
-0.0400397814810276,
-0.... | 0.853805 | 0.835756 | 0.01805 |
Given that $\triangle ABC$ is an isosceles triangle with base $BC$, where $\angle ACB=30^\circ$, find the measure of $\angle ABC$ if $AB=AD$ and $\triangle ADB$ is equilateral. | nvidia_math | [
0.01863134279847145,
0.005481080152094364,
-0.035606056451797485,
-0.04571168124675751,
0.024003269150853157,
-0.015633651986718178,
-0.0013492730213329196,
0.047546062618494034,
0.02971922606229782,
-0.01746707782149315,
0.03700551018118858,
0.034716662019491196,
-0.03695348650217056,
-0.... | 0.825385 | 0.808068 | 0.017317 |
Let $A$, $B$, and $C$ be real numbers such that
$$(x-A)^{x-B}+(x-B)^{x-C}+(x-C)^{x-A}=3$$ has a unique real solution. Determine the value of $A+B+C$.
The best way to create a new problem is to create one that is still related to the original problem, but has new and unique components that requires the solver to appr... | nvidia_math | [
0.03548519313335419,
0.012012121267616749,
-0.04651269689202309,
-0.03473430871963501,
0.036423198878765106,
0.0016889466205611825,
-0.0012796444352716208,
0.08257532864809036,
0.03643171861767769,
-0.021055128425359726,
0.009344743564724922,
0.021143000572919846,
-0.03879919648170471,
-0.... | 0.856516 | 0.829987 | 0.026529 |
Let $P$ be a point inside triangle $ABC$ such that $PA=2000$, $PB=2001$ and $PC=2002$. How many triangles can be formed with side lengths that are consecutive integers if $PA$, $PB$, and $PC$ are the sides of a triangle and $PA$ is the shortest side? | nvidia_math | [
0.005978369154036045,
0.016509560868144035,
-0.051279690116643906,
-0.033728402107954025,
0.01863538660109043,
0.004785493016242981,
-0.022618617862462997,
0.04185771197080612,
0.03751681372523308,
-0.005321649834513664,
0.021549183875322342,
0.0459400899708271,
-0.022357892245054245,
-0.0... | 0.848647 | 0.832944 | 0.015702 |
If $\sin a + \sin b = 1$ and $\cos a + \cos b = \frac{3}{2},$ then find all possible values of $\tan \frac{a+b}{2}.$ | nvidia_math | [
0.0268145389854908,
0.03764352202415466,
-0.04216868057847023,
-0.03581651672720909,
0.03558366745710373,
-0.012411955744028091,
-0.0061693815514445305,
0.04962105304002762,
0.03955205902457237,
-0.00972115620970726,
0.03249461576342583,
0.021355152130126953,
-0.025171641260385513,
-0.0195... | 0.83591 | 0.820787 | 0.015124 |
In the figure, $ABCD$ is a rectangle with $AB = 6$ and $BC = 3$. Points $E$, $F$, and $G$ are the midpoints of $\overline{AD}$, $\overline{CD}$, and $\overline{BE}$, respectively. If $EF$ is extended to meet the line containing $\overline{BC}$ at $H$, find the area of quadrilateral $BHFH$. [asy]
defaultpen(linewidth(0.... | nvidia_math | [
0.009207520633935928,
0.013848373666405678,
-0.03111656755208969,
-0.0526418536901474,
0.023961249738931656,
-0.0421389639377594,
-0.0082357507199049,
0.05739380419254303,
0.045488081872463226,
0.008986832574009895,
0.011988714337348938,
0.04173757880926132,
-0.01310846209526062,
-0.027943... | 0.833178 | 0.814387 | 0.018791 |
Let $y=px^2-12x+4$, where $p$ is a nonzero constant. The graph of $y$ is a parabola opening upward or downward, and it intersects the $x$-axis at exactly one point. Find the value of $p$ that minimizes the $y$-coordinate of the vertex of the parabola. | nvidia_math | [
0.017280375584959984,
0.022003689780831337,
-0.03690025210380554,
-0.024751029908657074,
0.03988661989569664,
-0.015291077084839344,
-0.013942659832537174,
0.07179627567529678,
0.04652848839759827,
-0.01387354638427496,
0.015347779728472233,
0.024988815188407898,
-0.013724838383495808,
-0.... | 0.849382 | 0.832408 | 0.016975 |
Let $\mathcal{R}$ be the set of all $2\times 2$ invertible matrices with real entries.
Is there a matrix $A\in \mathcal{R}$ such that for every $B\in\mathcal{R}$ we have that $AB+BA$ is invertible? | nvidia_math | [
0.03131217136979103,
0.013561014086008072,
-0.03205404058098793,
-0.024890050292015076,
0.029903260990977287,
-0.014071059413254261,
-0.003451110329478979,
0.08048097789287567,
0.033612512052059174,
-0.025836559012532234,
0.010427030734717846,
0.028942491859197617,
-0.03747311979532242,
-0... | 0.833292 | 0.816817 | 0.016474 |
Let $f(x)$ be a quadratic function such that $f(1) = 0$ and $f(-2) = 0$. Determine the value of $f(f(f(x)))$ when $x = 3$.
Note: To make the problem have a unique solution, the coefficient of $x^2$ in $f(x)$ can be specified to be 1.
See you in the next problem generating task! | nvidia_math | [
0.01760391891002655,
0.028198136016726494,
-0.061325255781412125,
-0.035210780799388885,
0.034605126827955246,
-0.006844873074442148,
-0.004054654855281115,
0.08079390227794647,
0.027729559689760208,
-0.005893708672374487,
0.021607721224427223,
0.022010592743754387,
-0.03308013826608658,
-... | 0.841063 | 0.827061 | 0.014001 |
The polynomial $x^8 - 98x^4 + 1 = 0$ has eight distinct complex roots. Let $S$ denote the sum of all the four roots that lie in the first quadrant of the complex plane. If $S = r(\cos{\alpha}+i\sin{\alpha})$, where $r > 0$ and $0^\circ \leq \alpha < 360^\circ$, find $\alpha$. | nvidia_math | [
0.02055145427584648,
0.02899075672030449,
-0.04553000256419182,
-0.03937363252043724,
0.009441939182579517,
-0.019192006438970566,
-0.00405150605365634,
0.05749896913766861,
0.05272923409938812,
0.002867496106773615,
0.010584354400634766,
0.024276429787278175,
-0.04280627891421318,
-0.0329... | 0.835378 | 0.8165 | 0.018878 |
Determine $a$ such that the line specified by
$\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 0 \\ a \end{pmatrix}$ intersects the curve
$x^2+y^2=5$. | nvidia_math | [
0.023580368608236313,
0.026245472952723503,
-0.04820495843887329,
-0.04738105088472366,
0.059729304164648056,
-0.01839076168835163,
-0.025998996570706367,
0.09148263186216354,
0.03321981430053711,
-0.024574024602770805,
0.012128080241382122,
0.025788655504584312,
-0.028400614857673645,
-0.... | 0.833528 | 0.815491 | 0.018037 |
Let $a$, $b$ and $c$ be positive real numbers satisfying the equations
\begin{align*}
a^{bc} + b^{ac} + c^{ab} & = 3,\\
a^{bc} \cdot b^{ac} \cdot c^{ab} & = 1,\\
a^{ac} + b^{ab} + c^{bc} & = 3.
\end{align*}
Determine the possible values of $a$, $b$ and $c$ that also satisfy
\begin{align*}
(a^{bc} - 1) \cdot (b^{ac} ... | nvidia_math | [
0.054525621235370636,
0.03968197479844093,
-0.04420909658074379,
-0.03352724388241768,
0.045368731021881104,
-0.01865328848361969,
-0.0006354559445753694,
0.06292993575334549,
0.030099105089902878,
-0.016418403014540672,
-0.0036041480489075184,
0.02784441038966179,
-0.04787759855389595,
-0... | 0.843429 | 0.812591 | 0.030838 |
Given a fixed triangle $\triangle ABC$, let $x,$ $y,$ and $z$ be the distances from the vertices $A,$ $B,$ and $C$ to their respective angle bisectors. Find the maximum value of $\frac{xyz}{(x+y+z)^3}.$ | nvidia_math | [
0.04039471223950386,
0.01621084287762642,
-0.037353262305259705,
-0.029434066265821457,
0.024945015087723732,
-0.017410989850759506,
-0.020869096741080284,
0.027374228462576866,
0.05440184846520424,
-0.004106445703655481,
0.031109552830457687,
0.0396144799888134,
-0.037712883204221725,
-0.... | 0.841446 | 0.816313 | 0.025133 |
Let $\mathbf{u}, \mathbf{v}, \mathbf{w}$ be vectors in $\mathbb{R}^n$ such that $\|\mathbf{u}\| = \|\mathbf{v}\| = 1$, $\|\mathbf{w}\| = 2$, $\|\mathbf{u} + \mathbf{v} + \mathbf{w}\| = 4$, and $\|\mathbf{u} - \mathbf{v}\| = 2$.
Determine the maximum possible value of the magnitude of $\mathbf{u} \times \mathbf{v}$, wh... | nvidia_math | [
0.032331500202417374,
0.042748577892780304,
-0.047786083072423935,
-0.04053680971264839,
0.0485425628721714,
0.009896734729409218,
-0.03282178193330765,
0.07921427488327026,
0.030321795493364334,
0.004548692610114813,
0.00599310640245676,
0.03721404820680618,
-0.02249613031744957,
-0.03324... | 0.832164 | 0.811745 | 0.020418 |
In the complex plane, the graph of $|z - 2i| = 3|z + 2i|$ intersects the graph of $|z| = m$ in exactly one point. Find all possible values of $m$. | nvidia_math | [
0.041097335517406464,
0.02487465925514698,
-0.03982267156243324,
-0.061776261776685715,
0.04753797501325607,
-0.01696530543267727,
-0.02092025987803936,
0.03700884431600571,
0.05099556967616081,
-0.02865506149828434,
0.033396147191524506,
0.02815447375178337,
-0.015874996781349182,
-0.0322... | 0.842799 | 0.826931 | 0.015868 |
A parabola has focus $(2,4)$ and directrix $2x - 5y = 12.$ Express the equation of the parabola in the form
\[ax^2 + bxy + cy^2 + dx + ey + f = 0,\]where $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are integers, $a$ is a positive integer, and $\gcd(|a|,|b|,|c|,|d|,|e|,|f|) = 1.$ | nvidia_math | [
0.03376214951276779,
0.03137044608592987,
-0.048357442021369934,
-0.052962642163038254,
0.049330051988363266,
-0.013225723057985306,
-0.013440292328596115,
0.06975759565830231,
0.04335619881749153,
0.009650695137679577,
0.005568395368754864,
0.022832388058304787,
-0.0016420027241110802,
-0... | 0.838141 | 0.82212 | 0.016021 |
Six congruent circles are drawn such that each circle is tangent to two sides of a regular hexagon. The circles have radius 1 and the area of their union is $m/n\pi,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ | nvidia_math | [
0.002303440822288394,
0.05052248015999794,
-0.03029431402683258,
-0.034360598772764206,
0.04876168444752693,
-0.017310885712504387,
-0.03047432005405426,
0.03106829896569252,
0.034290704876184464,
-0.008211323991417885,
0.008535362780094147,
0.018711045384407043,
-0.02130943164229393,
-0.0... | 0.851253 | 0.824323 | 0.02693 |
Let $\alpha \neq 1$ be a complex number such that the distance from $\alpha^2$ to 1 is twice the distance from $\alpha$ to 1, while the distance from $\alpha^4$ to 1 is four times the distance from $\alpha$ to 1. Enter all possible values of $\alpha,$ separated by commas.
A complex number $\beta$ satisfies the equati... | nvidia_math | [
0.030555149540305138,
0.029304945841431618,
-0.040898118168115616,
-0.039467308670282364,
0.042029522359371185,
-0.019283924251794815,
-0.017544517293572426,
0.0483982227742672,
0.053746242076158524,
-0.01846231333911419,
0.010715074837207794,
0.03661510348320007,
-0.03245288133621216,
-0.... | 0.861153 | 0.834609 | 0.026544 |
The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r).$
| nvidia_math | [
0.016208119690418243,
0.039667997509241104,
-0.04639741778373718,
-0.02434663288295269,
0.014248637482523918,
0.0013050433481112123,
-0.04896083474159241,
0.0626937672495842,
0.03166916221380234,
-0.02112189307808876,
0.009601311758160591,
0.030953381210565567,
-0.03516804799437523,
-0.016... | 0.848637 | 0.834232 | 0.014406 |
Find the largest possible value of the constant term of the polynomial
\[P(x) = x^n + ax^{n-1} + bx^{n-2} + cx^{n-3} + \dots\]
such that \(P(x)\) divides the polynomial
\[x^{2n} + (2a+1)x^{2n-1} + (2b+a+1)x^{2n-2} + (2c+b+2)x^{2n-3} + \dots + x^2 + 2x + 2.\] | nvidia_math | [
0.047407977283000946,
0.03180130198597908,
-0.048147961497306824,
-0.042307425290346146,
0.016805414110422134,
-0.03963199630379677,
-0.013102508150041103,
0.059959474951028824,
0.025433991104364395,
-0.014651523903012276,
0.003185990732163191,
0.03888232633471489,
-0.04125171899795532,
-0... | 0.839447 | 0.821696 | 0.017751 |
The first square below is in position ABCD. After rotating the square 90 degrees clockwise about its center point, the second square is in position DABC, as shown. Next, square DABC is reflected over its vertical line of symmetry, resulting in the third square in position CBAD. If the pattern of alternately rotating 90... | nvidia_math | [
0.02478012442588806,
0.027659861370921135,
-0.030564583837985992,
-0.047179222106933594,
0.008370955474674702,
-0.022326644510030746,
-0.016589069738984108,
0.0574270635843277,
0.03802289068698883,
-0.008142383769154549,
0.022582102566957474,
0.018182817846536636,
-0.032858431339263916,
-0... | 0.831345 | 0.818091 | 0.013254 |
A polynomial product of the form
\[(1 - w)^{c_1} (1 - w^2)^{c_2} (1 - w^3)^{c_3} (1 - w^4)^{c_4} (1 - w^5)^{c_5} \dotsm (1 - w^{25})^{c_{25}},\]
where the $c_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $w$ to a power larger than 25, what is left is ... | nvidia_math | [
0.04517391324043274,
0.041233550757169724,
-0.03460799157619476,
-0.04137880355119705,
0.03513336926698685,
-0.032930463552474976,
-0.025420619174838066,
0.0684228390455246,
0.036259088665246964,
-0.008709495887160301,
0.0035755587741732597,
0.03197069093585014,
-0.04527152329683304,
-0.02... | 0.848451 | 0.83324 | 0.015211 |
The sequences of positive integers $1,a_2, a_3, \dots$ and $1,b_2, b_3, \dots$ are a decreasing arithmetic sequence and a decreasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=2000$ and $c_{k+1}=200$. Find $c_k$. | nvidia_math | [
0.02286817878484726,
0.04932088404893875,
-0.05462200939655304,
-0.033040016889572144,
0.02021065354347229,
-0.0004958143690600991,
-0.03079960122704506,
0.07863849401473999,
0.053889643400907516,
-0.0010313790990039706,
0.013930064626038074,
0.013791807927191257,
-0.05109354108572006,
0.0... | 0.855153 | 0.831404 | 0.023749 |
The Fibonacci sequence is defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$. The sequence $a_1,$ $a_2,$ $a_3,$ $\dots$ satisfies $a_1 = 1,$ $a_2 = 2,$ and for all $n \ge 3,$ $a_n$ is the harmonic mean of $F_n$ and $F_{n-1}$. Find $a_{10}$. | nvidia_math | [
0.03644653037190437,
0.02218041941523552,
-0.04849613830447197,
-0.026635395362973213,
0.03774969279766083,
-0.01852971501648426,
-0.03989289700984955,
0.06827236711978912,
0.02062860131263733,
-0.01964280940592289,
0.026250064373016357,
0.025361819192767143,
-0.046382807195186615,
-0.0111... | 0.838305 | 0.817747 | 0.020559 |
Consider two functions defined as follows:
\[F(z) = \frac{z^3 - iz^2 - z + i}{z^2 + 1}\]
and
\[G(z) = \frac{Az^3 + Bz^2 + Cz + D}{z^2 + 4}.\]
Given that:
* Both $F(z)$ and $G(z)$ have the same type of singularities in the complex plane.
* The polynomial parts of $F(z)$ and $G(z)$ are such that their zeros form an ... | nvidia_math | [
0.018218714743852615,
0.033918850123882294,
-0.033989131450653076,
-0.06513180583715439,
0.035174328833818436,
-0.00043966149678453803,
0.0014267096994444728,
0.06304948776960373,
0.03824642673134804,
0.005467199254781008,
0.022622672840952873,
0.01012241467833519,
-0.0276302769780159,
-0.... | 0.82543 | 0.796106 | 0.029324 |
Let $f(x)=ax^2+bx+c$, where $a\neq 0$. The function $f^{-1}(x)$ is defined as follows:
$$f^{-1}(x)=\left\{\begin{array}{lll}
\ds\sqrt{\ds\f{x-b}{a}-c},&\qquad&\mbox{if}\quad x\geq b+ac\\
-\ds\sqrt{\ds\f{x-b}{a}-c},&\qquad&\mbox{if}\quad x<b+ac.
\end{array}\right.$$
Given that the graph of $f^{-1}(x)$ is symmetric with... | nvidia_math | [
0.031298864632844925,
0.04636358097195625,
-0.07544387876987457,
-0.04214845970273018,
0.04133296757936478,
-0.019580937922000885,
-0.01554141379892826,
0.09366953372955322,
0.03922329470515251,
0.0068481480702757835,
-0.0006239028298296034,
0.038442324846982956,
-0.03151858225464821,
-0.0... | 0.83827 | 0.819719 | 0.018552 |
Consider the set of all points $(x,y)$ in the coordinate plane for which the distance from the origin to the point is equal to the sum of the coordinates. If we were to plot all such points, into how many regions would the resulting graph split up the plane? | nvidia_math | [
0.01427973061800003,
0.01425626128911972,
-0.06594338268041611,
-0.036430396139621735,
0.031601399183273315,
-0.009265843778848648,
-0.016219770535826683,
0.031277984380722046,
0.04518648982048035,
-0.02837696485221386,
0.006216588895767927,
0.04194333031773567,
-0.03768565133213997,
-0.01... | 0.852929 | 0.837413 | 0.015516 |
Let $f(x)$ be defined on $0 \le x \le 1$ such that $$f(x) = \begin{cases} x &\quad x\text{ is rational} \\ f(x - \frac{1}{n}) + \frac{1}{n^2} &\quad \text{otherwise} \end{cases}$$ where $n$ is the smallest positive integer such that $x - \frac{1}{n}$ is rational. Express the range of $f$ in interval notation. | nvidia_math | [
0.023138534277677536,
0.03172842413187027,
-0.032504819333553314,
-0.029768865555524826,
0.03484838083386421,
-0.021081093698740005,
-0.014635024592280388,
0.06687011569738388,
0.04501507058739662,
-0.01035917829722166,
0.02305545285344124,
0.0334622859954834,
-0.0270849522203207,
-0.01061... | 0.829683 | 0.811025 | 0.018658 |
Consider a regular hexagon and the set $T$ of all its sides and diagonals. Two elements of $T$ are chosen randomly without replacement. Find the probability that the two selected segments have different lengths. | nvidia_math | [
0.010319884866476059,
0.015026971697807312,
-0.04802324250340462,
-0.0288676954805851,
0.03801804035902023,
-0.030062131583690643,
-0.028217840939760208,
0.05130835622549057,
0.026528390124440193,
-0.026144493371248245,
0.03464169800281525,
0.023521050810813904,
-0.018854156136512756,
-0.0... | 0.84588 | 0.82414 | 0.02174 |
In a triangle ABC, five circles are inscribed, each tangent to two sides of the triangle and one of its medians: two circles for each median. Find the ratio of the radius of the circle tangent to AB and the median from C to the radius of the circle tangent to BC and the median from A. | nvidia_math | [
0.033841170370578766,
0.015941638499498367,
-0.046993885189294815,
-0.025609463453292847,
0.03413550183176994,
-0.020525487139821053,
-0.029190856963396072,
0.04009536653757095,
0.02661760523915291,
-0.00789416953921318,
0.018775638192892075,
0.019896583631634712,
-0.016040533781051636,
-0... | 0.821841 | 0.805824 | 0.016017 |
Given a 3x3 matrix $\mathbf{A},$ find a 3x3 matrix $\mathbf{B}$ such that $\mathbf{AB} = \mathbf{C},$ where $\mathbf{C}$ is obtained by rotating the elements of $\mathbf{A}$ clockwise by $90^\circ.$ In other words,
\[\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \mathbf{B} = \begin{pmatrix} g & d & ... | nvidia_math | [
0.0363902784883976,
0.02478352189064026,
-0.04448208957910538,
-0.027912687510252,
0.029941963031888008,
-0.024459440261125565,
-0.014328382909297943,
0.06766165047883987,
0.023574380204081535,
-0.013714751228690147,
0.011214192025363445,
0.01784818433225155,
-0.054982990026474,
-0.0264743... | 0.812435 | 0.793916 | 0.018519 |
In a triangle $ABC$, the altitude from vertex $A$ to side $BC$ intersects side $BC$ at point $D$. The lengths of $AD$, $BD$, and $CD$ are 10, 15, and 25, respectively. A point is selected at random from the region determined by triangle $ABC$. What is the probability that the point is in the area of the triangle bounde... | nvidia_math | [
0.020378965884447098,
0.01635623350739479,
-0.04495721310377121,
-0.039839085191488266,
0.025593601167201996,
-0.032810281962156296,
-0.008950232528150082,
0.051762860268354416,
0.03162551671266556,
0.0016223801067098975,
0.007409864105284214,
0.02940039336681366,
-0.0226297490298748,
-0.0... | 0.842244 | 0.820968 | 0.021276 |
Let $f(x)$ be a polynomial function of degree $n$, where $n$ is a positive integer. It is known that $f(x)$ has $n+1$ distinct roots $r_1, r_2, ..., r_{n+1}$ such that $f(r_i) = 0$ for all $i$. If the polynomial $g(x) = f(x^2)$ has exactly $2n+1$ distinct roots, determine the number of distinct roots of the polynomial ... | nvidia_math | [
0.03231646120548248,
0.03345621004700661,
-0.04966605827212334,
-0.03869560360908508,
0.016294734552502632,
-0.035567037761211395,
-0.01756332442164421,
0.055125657469034195,
0.03077966347336769,
-0.0014457463985309005,
-0.005181486252695322,
0.012815574184060097,
-0.02688087522983551,
-0.... | 0.842547 | 0.814562 | 0.027986 |
A sector with acute central angle $\theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is
$\textbf{(A)}\ 3\cos\theta \qquad \textbf{(B)}\ 3\sec\theta \qquad \textbf{(C)}\ 3 \cos \frac12 \theta \qquad \textbf{(D)}\ 3 \sec \frac12 \theta \qquad \textbf{(E)}\ 3$
| nvidia_math | [
0.02384471334517002,
0.03881997615098953,
-0.04373866319656372,
-0.03276354447007179,
0.03230651095509529,
-0.010632949881255627,
-0.028250662609934807,
0.04864777252078056,
0.02809729240834713,
-0.0007095862529240549,
0.03297153115272522,
0.031486667692661285,
-0.03901221603155136,
-0.003... | 0.823739 | 0.809653 | 0.014086 |
An element is randomly chosen from among the first $15$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?
Note: The 1 at the top is often labelled the "zeroth" row of Pascal's Triangle, by convention. So to count a total of 15 rows, use rows 0 through 14. | nvidia_math | [
0.027315668761730194,
0.01943838968873024,
-0.038016606122255325,
-0.03339139372110367,
0.04231585934758186,
-0.021257884800434113,
-0.004211058374494314,
0.05260205641388893,
0.018401464447379112,
0.010583114810287952,
0.025219358503818512,
0.013649316504597664,
-0.021048085764050484,
-0.... | 0.835304 | 0.821066 | 0.014238 |
Let $S$ be the set of all positive integers whose digits are digits of a strictly increasing sequence (for example, $1567$ and $2348$ are such numbers, but $1456$ and $4367$ are not). Find the units digit of the sum of the cubes of all the numbers in $S$. | nvidia_math | [
0.015618180856108665,
0.02086387202143669,
-0.058867380023002625,
-0.0279496181756258,
0.02391868270933628,
-0.016684100031852722,
-0.017605092376470566,
0.05158550664782524,
0.030533824115991592,
-0.01952131651341915,
0.026954220607876778,
0.02056785859167576,
-0.04048563912510872,
0.0034... | 0.857016 | 0.838364 | 0.018652 |
Let $z$ be a complex number such that $|z| = 1$. Find the number of ordered pairs $(a,b)$ of integers such that $|a \sin z + b \cos z| = 1$. | nvidia_math | [
0.02208343707025051,
0.03097875975072384,
-0.04770198464393616,
-0.036959320306777954,
0.04051303118467331,
0.00764210382476449,
-0.011006161570549011,
0.029831402003765106,
0.02613150142133236,
-0.002914480399340391,
0.03227248042821884,
0.018472081050276756,
-0.032996080815792084,
-0.018... | 0.851336 | 0.836351 | 0.014985 |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}$ be unit vectors such that $\mathbf{a}$ and $\mathbf{b}$ are parallel, and the angle between $\mathbf{a}$ and $\mathbf{c}$ is $\theta$. If $\mathbf{c} \cdot (\mathbf{d} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{d} \times \mathbf{b}) = \frac{1}{\sqrt{2}}... | nvidia_math | [
0.01826334185898304,
0.05501203238964081,
-0.03872613236308098,
-0.040317352861166,
0.03095691278576851,
0.011514495126903057,
-0.022670060396194458,
0.06089521944522858,
0.03410442918539047,
-0.01391470804810524,
0.009424030780792236,
0.017218997702002525,
-0.014683323912322521,
-0.024227... | 0.856662 | 0.832476 | 0.024186 |
Let $f(x) = \frac{x + 6}{x}.$ The sequence $(f_n)$ of functions is defined by $f_1 = f$ and
\[f_n = f \circ f_{n - 1}\]for all $n \ge 2.$ For example,
\[f_2(x) = f(f(x)) = \frac{\frac{x + 6}{x} + 6}{\frac{x + 6}{x}} = \frac{7x + 6}{x + 6}\]and
\[f_3(x) = f(f_2(x)) = \frac{\frac{7x + 6}{x + 6} + 6}{\frac{7x + 6}{x + 6... | nvidia_math | [
0.039969295263290405,
0.042666561901569366,
-0.04844025894999504,
-0.022814916446805,
0.04405937343835831,
-0.021423421800136566,
-0.026179591193795204,
0.0833650752902031,
0.02808123640716076,
-0.0073373145423829556,
0.007857619784772396,
0.02932683750987053,
-0.03163456171751022,
-0.0159... | 0.843598 | 0.826735 | 0.016863 |
Given that angle $P$ of quadrilateral $ABCD$ is $60^\circ$ and angles $B$, $C$, and $D$ are congruent, find $\angle A$, given that $AB = 3AD$, and the perimeter of $ABCD$ equals $28$ cm. | nvidia_math | [
0.010941972956061363,
0.010717744007706642,
-0.022709228098392487,
-0.04313674941658974,
0.023508591577410698,
-0.02809317596256733,
0.0015709843719378114,
0.0438075065612793,
0.02459045685827732,
-0.010915837250649929,
0.008887341246008873,
0.02972007915377617,
-0.04342075437307358,
-0.02... | 0.825545 | 0.811578 | 0.013966 |
Let $P(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients such that $P(-3)=0$. Suppose that the quadratic polynomial $Q(x)=P(x)/x+3$ has exactly one solution. If $c=9$ and $a+b=-6$, find the ordered pair $(a,b)$. | nvidia_math | [
0.022980691865086555,
0.024122299626469612,
-0.07877634465694427,
-0.035550471395254135,
0.026318049058318138,
-0.026041856035590172,
0.0066298991441726685,
0.05379955470561981,
0.036021724343299866,
-0.006152195855975151,
0.012188628315925598,
0.027526535093784332,
-0.030218061059713364,
... | 0.850553 | 0.828681 | 0.021872 |
In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\angle A = 37^\circ$, $\angle D = 53^\circ$, and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $MN$. | nvidia_math | [
0.020377475768327713,
0.003896876936778426,
-0.0489240437746048,
-0.0519004687666893,
0.010294726118445396,
-0.013408340513706207,
-0.03791486844420433,
0.03935781121253967,
0.03373253718018532,
-0.001365077798254788,
0.021020924672484398,
0.026782484725117683,
-0.0398331955075264,
-0.0152... | 0.836885 | 0.820364 | 0.016521 |
The function $f(x)$ satisfies
\[f(2^x) + xf(2^{-x}) = 1\]for all real numbers $x.$ Find $f(2).$ | nvidia_math | [
0.04889954626560211,
0.047601502388715744,
-0.04191310331225395,
-0.03982895612716675,
0.030271606519818306,
-0.02351325750350952,
-0.006699582561850548,
0.0721055269241333,
0.04026389494538307,
-0.006953360978513956,
0.01641625165939331,
0.02991439402103424,
-0.019279638305306435,
-0.0136... | 0.840756 | 0.826319 | 0.014437 |
A matrix $\mathbf{A}$ is said to be an orthogonal projection matrix if it satisfies the condition $\mathbf{A}^{2}=\mathbf{A}$ and $\mathbf{A}^{T}=\mathbf{A}$. Given the vectors $\mathbf{u}_{1}=(\cos \theta,\sin \theta )^{T}$ and $\mathbf{u}_{2}=(-\sin \theta ,\cos \theta )^{T}$, find the orthogonal projection matrix $\... | nvidia_math | [
0.00034609000431373715,
0.04259716346859932,
-0.039688386023044586,
-0.043373171240091324,
0.05397849529981613,
-0.01534945797175169,
-0.003561733989045024,
0.072176493704319,
0.04546273127198219,
0.007548642344772816,
0.012718112207949162,
0.015107478015124798,
-0.02703467383980751,
-0.01... | 0.812527 | 0.789565 | 0.022962 |
Let $f(x)$ be a function defined on the interval $[-\pi, \pi]$, such that its graph is symmetric with respect to the origin. Let $g(x) = |f(x)|$. Determine which of the statements are necessarily true:
A. The graph of $g(x)$ is symmetric with respect to the $y$-axis.
B. The graph of $g(x)$ is symmetric with respect t... | nvidia_math | [
0.009150821715593338,
0.03515772148966789,
-0.05147283151745796,
-0.030888797715306282,
0.04192868992686272,
-0.025720203295350075,
-0.02568790502846241,
0.07807084918022156,
0.053041212260723114,
-0.013894263654947281,
0.007939541712403297,
0.010037624277174473,
-0.022058837115764618,
-0.... | 0.833708 | 0.810809 | 0.022899 |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a+b+c=1.$ Find the maximum value of
\[\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}.\] | nvidia_math | [
0.04351049289107323,
0.03489089012145996,
-0.04292469099164009,
-0.0284343920648098,
0.03185835853219032,
-0.01060467679053545,
-0.0008447322179563344,
0.0557083860039711,
0.026915255934000015,
-0.029142776504158974,
0.0039030604530125856,
0.03580276295542717,
-0.035879574716091156,
-0.027... | 0.839591 | 0.820881 | 0.01871 |
Let $A$, $B$, $C$ be three dihedral angles formed by three edges of a tetrahedron and faces that don't share those edges.
Determine the maximum volume of a tetrahedron for which
$\cos A + \cos B + \cos C = -\frac{11}{8}$. | nvidia_math | [
0.03919076547026634,
0.01985878497362137,
-0.03639482706785202,
-0.03354724124073982,
0.03712465241551399,
-0.01324903778731823,
-0.030252179130911827,
0.05423184856772423,
0.015330065041780472,
0.01759960502386093,
0.022801274433732033,
0.04443471133708954,
-0.04593685269355774,
-0.006016... | 0.831849 | 0.814281 | 0.017568 |
Let $a_n$ be the sequence defined recursively by $a_n = \frac{n(n + 1)}{(n + 3)^2}a_{n - 1}$ for $n \geq 2$, where $a_1 = 1$.
Compute $\lim_{n \to \infty} a_n$. | nvidia_math | [
0.020264042541384697,
0.048670101910829544,
-0.023455588147044182,
-0.033992283046245575,
0.01843692921102047,
-0.016271619126200676,
-0.02848426252603531,
0.07020127773284912,
0.01999502256512642,
-0.03010058030486107,
0.007982022128999233,
0.0339641347527504,
-0.03362102434039116,
-0.006... | 0.846685 | 0.825432 | 0.021253 |
Let $\mathbf{A}$ be a $3\times 3$ matrix consisting only of 1s and -1s and having the property that its entries are symmetric with respect to the main diagonal. Calculate the maximum value of det $\mathbf{A}$. | nvidia_math | [
0.05373677238821983,
0.012224588543176651,
-0.06163239851593971,
-0.036004189401865005,
0.04199956730008125,
-0.01049612183123827,
-0.022402461618185043,
0.06054428592324257,
0.026527952402830124,
-0.015084614977240562,
0.010537289083003998,
0.035475220531225204,
-0.06428202986717224,
-0.0... | 0.834671 | 0.816723 | 0.017949 |
A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the fie... | nvidia_math | [
0.01870562881231308,
0.02135223150253296,
-0.05568957328796387,
-0.04865230247378349,
0.006873545236885548,
-0.031560495495796204,
-0.029925953596830368,
0.05113409459590912,
0.03735817223787308,
-0.019927728921175003,
-0.008176969364285469,
0.026265310123562813,
-0.030636385083198547,
0.0... | 0.841335 | 0.824841 | 0.016495 |
Let $a,$ $b,$ and $c$ be positive real numbers. Find the maximum value of
\[\frac{a^2b^2 + b^2c^2 + c^2a^2}{abc(a + b + c)}.\] | nvidia_math | [
0.03841135650873184,
0.030230438336730003,
-0.045847438275814056,
-0.029156897217035294,
0.03181474655866623,
-0.026299260556697845,
-0.0008998574339784682,
0.042599763721227646,
0.03276116028428078,
-0.03449604660272598,
0.002497822744771838,
0.03829415887594223,
-0.03763267397880554,
-0.... | 0.847226 | 0.829494 | 0.017732 |
A square and four circles, each with a radius of 5 inches, are arranged as shown. What is the area, in square inches, of the square?
[asy]
unitsize(1mm);
defaultpen(linewidth(0.7pt));
draw((0,0)--(20,0)--(20,20)--(0,20)--cycle);
draw(Circle((5,5),5));
draw(Circle((15,5),5));
draw(Circle((5,15),5));
draw(Circle((15,15... | nvidia_math | [
0.017421919852495193,
0.042902298271656036,
-0.043351948261260986,
-0.042977601289749146,
0.0571524016559124,
-0.017461737617850304,
-0.030557412654161453,
0.05421982333064079,
0.029414566233754158,
0.005895289126783609,
0.01302789431065321,
0.027186311781406403,
-0.04108615592122078,
-0.0... | 0.818957 | 0.805465 | 0.013492 |
Let $\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})$, where $0<r$ and $0\leq \theta <360$. Find $\theta$. | nvidia_math | [
0.01722201704978943,
0.03732543811202049,
-0.036036085337400436,
-0.0379926972091198,
0.036351773887872696,
-0.020645746961236,
-0.008016188628971577,
0.049838244915008545,
0.028395984321832657,
-0.004331278149038553,
0.011026034131646156,
0.012011322192847729,
-0.04271926358342171,
-0.037... | 0.842852 | 0.824315 | 0.018537 |
The figure shows a square inscribed in a right triangle, where the legs of the triangle are $a$ and $b$, and the side length of the square is $x$. What is the maximum value of the ratio $\frac{x^2}{ab}$? [asy]
draw((0,0)--(4,0)--(0,4)--cycle);
fill((1,0)--(1,3)--(3,3)--(3,0)--cycle, gray);
label("$x$", (2,0.5), N);
lab... | nvidia_math | [
0.032899439334869385,
0.03200552612543106,
-0.04540439322590828,
-0.03940746560692787,
0.0248713456094265,
-0.025288140401244164,
-0.009865617379546165,
0.06655703485012054,
0.040250398218631744,
0.007978260517120361,
0.030180741101503372,
0.03976580128073692,
-0.004692458547651768,
-0.006... | 0.83416 | 0.820743 | 0.013416 |
A circle of radius 3 and a rectangle with dimensions 4x5 are placed inside a larger rectangle without overlapping. The sides of the larger rectangle are parallel to the sides of the given rectangle. What is the smallest possible perimeter of the larger rectangle? | nvidia_math | [
0.012675944715738297,
0.04167473316192627,
-0.029188521206378937,
-0.035945720970630646,
0.03156895563006401,
0.0021126214414834976,
-0.028260502964258194,
0.05395941063761711,
0.04251851513981819,
0.00022505341621581465,
0.019731828942894936,
0.0395674854516983,
-0.029434729367494583,
-0.... | 0.837001 | 0.823439 | 0.013563 |
Given that $0\le x_3 \le x_2 \le x_1\le 1$ and $(1-x_1)^2+(x_1-x_2)^2+(x_2-x_3)^2+x_3^2=\frac{1}{4},$ find $x_1.$ | nvidia_math | [
0.0052564810030162334,
0.03479429706931114,
-0.03937211260199547,
-0.043565914034843445,
0.05317259579896927,
-0.01633046194911003,
-0.028575416654348373,
0.06964783370494843,
0.031950633972883224,
-0.012734737247228622,
0.008632016368210316,
0.025971725583076477,
-0.04244309291243553,
-0.... | 0.852063 | 0.835508 | 0.016555 |
Let $\mathcal{E}$ be the ellipse in the plane determined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Suppose a circle $\mathcal{C}$ intersects $\mathcal{E}$ at four distinct points. If three of these points are $(-a,0),$ $(0,b),$ and $(a,0),$ find the area of the quadrilateral formed by all four of the int... | nvidia_math | [
0.009497249498963356,
0.03495320305228233,
-0.039057713001966476,
-0.03701165318489075,
0.03953827917575836,
-0.01844097673892975,
-0.013353927992284298,
0.046876199543476105,
0.047011472284793854,
-0.009336184710264206,
0.004043206572532654,
0.03210601583123207,
-0.02326289936900139,
-0.0... | 0.850399 | 0.81871 | 0.031688 |
Three mutually orthogonal unit vectors $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ satisfy the equations
\[\mathbf{u}\cdot(\mathbf{v}+\mathbf{w}) = \frac{1}{2}\qquad\text{and}\qquad(\mathbf{u} + \mathbf{v})\cdot\mathbf{w} = \frac{\sqrt{6}}{4}.\]Let $\theta$ denote the angle between $\mathbf{v}$ and $\mathbf{w},$ wher... | nvidia_math | [
0.03134812042117119,
0.041928138583898544,
-0.05300033092498779,
-0.04597575217485428,
0.0418563038110733,
0.004618257284164429,
-0.01991472765803337,
0.06671327352523804,
0.04205317050218582,
-0.006755753420293331,
0.01521000824868679,
0.01605675183236599,
-0.012277512811124325,
-0.037119... | 0.82933 | 0.812969 | 0.016361 |
For a positive integer $p$, define the positive integer $n$ to be $p$-secure if $n$ differs in absolute value by more than $3$ from all multiples of $p$. For example, the set of $8$-secure numbers is $\{ 4, 5, 6, 7, 12, 13, 14, 15, 20, \ldots\}$. Find the number of positive integers less than or equal to $20,000$ which... | nvidia_math | [
0.02554832026362419,
0.03552709519863129,
-0.042417701333761215,
-0.03599754348397255,
0.029648130759596825,
-0.026952028274536133,
-0.03150109201669693,
0.04536880552768707,
0.036195721477270126,
-0.007406085263937712,
0.017505204305052757,
0.021685095503926277,
-0.06057627871632576,
-0.0... | 0.847363 | 0.828432 | 0.018931 |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{c}\| = 2,$ and
\[\mathbf{a} \times (\mathbf{a} \times \mathbf{c}) + \mathbf{b} = \mathbf{0}.\]Find the smallest possible angle between $\mathbf{a}$ and $\mathbf{c},$ in degrees. | nvidia_math | [
0.034461893141269684,
0.03733188658952713,
-0.03834378346800804,
-0.0358419194817543,
0.02516653575003147,
0.013919164426624775,
-0.018006298691034317,
0.05930584669113159,
0.031766705214977264,
-0.010240999050438404,
0.007059402298182249,
0.019047366455197334,
-0.02605803869664669,
-0.025... | 0.840687 | 0.823554 | 0.017133 |
Let $g(x)$ be a function that satisfies
\[g(x^2 + y^2) = g(x)g(y) + x^2 y^2\]for all real numbers $x$ and $y.$ Find $g(x).$ | nvidia_math | [
0.0138371167704463,
0.03539000079035759,
-0.04785797372460365,
-0.04738478362560272,
0.04197007045149803,
-0.030129866674542427,
-0.01137655321508646,
0.05040156841278076,
0.0373409129679203,
-0.0234367772936821,
0.014562299475073814,
0.032923806458711624,
-0.02561555989086628,
-0.01224456... | 0.846571 | 0.820895 | 0.025676 |
Let $a,$ $b,$ $c$ be integers such that
\[\mathbf{A} = \frac{1}{3} \begin{pmatrix} 2 & a \\ b & c \end{pmatrix}\]and $\mathbf{A}^2 = \mathbf{I}.$ Find the largest possible value of $a + b + c.$ | nvidia_math | [
0.06127214804291725,
0.0418056845664978,
-0.051742009818553925,
-0.032199226319789886,
0.035328708589076996,
-0.01304525975137949,
-0.005106722004711628,
0.065400131046772,
0.02108817733824253,
-0.027594152837991714,
0.016689904034137726,
0.04008900374174118,
-0.041355688124895096,
-0.0285... | 0.849064 | 0.828518 | 0.020546 |
The diagonals of a parallelogram intersect at the point $(2, 3)$. One vertex of the parallelogram is located at the point $(5, 7)$. Find the equation of the circle passing through all four vertices of the parallelogram. | nvidia_math | [
0.023300237953662872,
0.02995915897190571,
-0.041981615126132965,
-0.0508568212389946,
0.030439602211117744,
-0.003976775798946619,
-0.02149537205696106,
0.0468192957341671,
0.03368007391691208,
-0.01196823175996542,
0.023801784962415695,
0.025768399238586426,
-0.015944235026836395,
-0.016... | 0.841356 | 0.821895 | 0.019461 |
Let's consider a similar problem where we also need to find the probability of a specific outcome. The problem can be stated as follows:
There are three distinct positive integers from 1 to 10 inclusive that are chosen. Let their sum be T and their product be M. What is the probability that M - T is a multiple of 3? | nvidia_math | [
0.030516421422362328,
0.010563279502093792,
-0.04508022218942642,
-0.011676624417304993,
0.06306441128253937,
-0.008966142311692238,
-0.03342849016189575,
0.06139093264937401,
0.026972219347953796,
0.006047938950359821,
0.007559613790363073,
0.0259111225605011,
-0.03816371038556099,
-0.040... | 0.828625 | 0.815238 | 0.013387 |
Problem:
Points $A$, $B$, $C$, and $T$ are in space such that each of $\overline{TA}$, $\overline{TB}$, and $\overline{TC}$ is perpendicular to the other two. If $TA = TB = 10$ and $TC = 9$, then what is the volume of pyramid $TABC$?
Write another problem inspired by this one:
In a three-dimensional coordinate syste... | nvidia_math | [
0.028930561617016792,
0.02645949460566044,
-0.038504041731357574,
-0.021665338426828384,
0.047804154455661774,
-0.024758532643318176,
-0.03287249058485031,
0.05828242748975754,
0.035216547548770905,
0.018085181713104248,
0.01473522000014782,
0.026071414351463318,
-0.03236747905611992,
-0.0... | 0.832401 | 0.8073 | 0.0251 |
Let's consider the polynomial $P(x) = x^5 + x^4 + x^3 + x^2 + x + 1$. It is known that the roots of $P(x)$ are the complex sixth roots of unity excluding 1.
Find the minimum value of $|x^3 - 1|$ among all the roots of $P(x)$. | nvidia_math | [
0.025746328756213188,
0.028493009507656097,
-0.043121516704559326,
-0.03828239440917969,
0.03495919704437256,
-0.05011352524161339,
0.0007649099570699036,
0.049401625990867615,
0.037309546023607254,
-0.017073718830943108,
0.012257272377610207,
0.015565290115773678,
-0.020077230408787727,
-... | 0.845011 | 0.830024 | 0.014987 |
A regular hexagon $ABCDEF$ is inscribed in a circle of radius $10$ units. The arc $\widehat{AB}$ of the circle has a length of $8$ units. What is the radius of the circle that circumscribes the triangle $ACE$?
(Note: I tried to create a problem that requires a different approach to solve, involving the properties of r... | nvidia_math | [
0.02634563110768795,
0.02671862207353115,
-0.04239659383893013,
-0.03494737669825554,
0.032524473965168,
-0.028852595016360283,
-0.02489016018807888,
0.04483447223901749,
0.025275202468037605,
-0.022233709692955017,
0.007375573273748159,
0.016589509323239326,
-0.03427828103303909,
-0.02515... | 0.840262 | 0.826019 | 0.014243 |
Given that $x^2 + y^2 = 14x + 6y + 6,$ find the largest possible value of $3x + 4y.$
Another problem inspired by this one:
A point P moves along the curve $x^2 + y^2 = 14x + 6y + 6$. At what point on the curve does the distance from P to the line $3x + 4y = 0$ become maximum? | nvidia_math | [
0.022513141855597496,
0.041242655366659164,
-0.03288434445858002,
-0.037152789533138275,
0.054090917110443115,
-0.018174972385168076,
-0.01257812138646841,
0.05725482478737831,
0.040162067860364914,
-0.014958991669118404,
0.02871750481426716,
0.03373803198337555,
-0.021904047578573227,
-0.... | 0.849534 | 0.835891 | 0.013643 |
In quadrilateral $ABCD$, the diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $O$. Let $E$ and $F$ be the projections of $O$ onto $\overline{AB}$ and $\overline{CD}$, respectively. If $AE = 1$, $EB = 2$, $CF = 3$, and $FD = 4$, compute the ratio of the areas of triangles $AOB$ and $COD$. | nvidia_math | [
0.021322259679436684,
0.013643631711602211,
-0.057925526052713394,
-0.05204544588923454,
0.02855711430311203,
-0.005380974151194096,
-0.006006152834743261,
0.05182850733399391,
0.04309435188770294,
0.006773467641323805,
0.01123473048210144,
0.01977013796567917,
-0.03179154917597771,
-0.012... | 0.843967 | 0.818082 | 0.025885 |
Let $x_1$ and $x_2$ be the roots of the cubic equation $2x^3+3x^2-7x-12=0$. Determine $x_1^2+x_2^2+x_3^2$, where $x_3$ is the third root of the cubic. | nvidia_math | [
0.04869663715362549,
0.03272301331162453,
-0.06727679818868637,
-0.035713471472263336,
0.03481440991163254,
-0.043641503900289536,
0.008817030116915703,
0.06512736529111862,
0.029492830857634544,
-0.017142821103334427,
0.0076827178709208965,
0.01417203713208437,
-0.03503800928592682,
-0.03... | 0.846887 | 0.828605 | 0.018282 |
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