instruction_seed stringlengths 54 2.22k | source stringclasses 1
value | embeddings listlengths 1.02k 1.02k | mean_positive_score float64 0.77 0.86 | mean_negative_score float64 0.76 0.85 | difference_score float64 0 0.04 |
|---|---|---|---|---|---|
Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[\begin{aligned} xy+x+y&=71 \\ x^2y+xy^2 &= 880.\end{aligned}\] | tiger_math | [
0.013836364261806011,
0.014582340605556965,
-0.04799028858542442,
-0.06699618697166443,
0.03109968826174736,
-0.017984507605433464,
-0.0123245595023036,
0.0563187301158905,
0.05640636384487152,
-0.02770242467522621,
0.01649092324078083,
0.03773735836148262,
-0.04874173179268837,
-0.0198649... | 0.834453 | 0.82583 | 0.008623 |
The function $g(x)$ satisfies
\[g(x - y) = \frac{g(x)}{g(y)}\]for all real numbers $x$ and $y.$ If $g(5) = 2,$ find $g(1)$. | tiger_math | [
0.031448863446712494,
0.040495987981557846,
-0.05534501373767853,
-0.043331556022167206,
0.05170286446809769,
-0.03746668994426727,
-0.02114127390086651,
0.05872778967022896,
0.03683193400502205,
-0.016708767041563988,
0.02281329222023487,
0.030950885266065598,
-0.029712166637182236,
-0.00... | 0.837081 | 0.829784 | 0.007297 |
For how many positive integers $n$ less than or equal to 2000 is $$(\cos t+i\sin t)^n=\cos nt+i\sin nt$$true for all real $t$? | tiger_math | [
0.02477237582206726,
0.03301502391695976,
-0.05828319117426872,
-0.01605508290231228,
0.018372492864727974,
-0.021586131304502487,
-0.020943619310855865,
0.03830103203654289,
0.03573286905884743,
-0.019457068294286728,
0.024977512657642365,
0.02839597873389721,
-0.025539137423038483,
-0.02... | 0.847478 | 0.844577 | 0.002901 |
Let $f(x) = (x+2)^2-5$. If the domain of $f$ is all real numbers, then $f$ does not have an inverse function. If we restrict the domain of $f$ to an interval $[c,\infty)$, then $f$ may have an inverse function. What is the smallest value of $c$ we can use here, so that $f$ has an inverse function? Explain why this valu... | tiger_math | [
0.00923876278102398,
0.0483267679810524,
-0.03517861291766167,
-0.03309235721826553,
0.03376404196023941,
-0.0193758774548769,
-0.010753512382507324,
0.0756300538778305,
0.02779463492333889,
-0.004693076014518738,
0.007824255153536797,
0.026707978919148445,
-0.016962843015789986,
-0.007583... | 0.834895 | 0.826371 | 0.008525 |
The sequence $a_1,$ $a_2,$ $a_3,$ $\dots$ satisfies $a_1 = 25,$ $a_7 = 85,$ and for all $n \ge 3,$ $a_n$ is the arithmetic mean of the first $n - 1$ terms. Find $a_2.$ | tiger_math | [
0.04457011818885803,
0.015575668774545193,
-0.03713371977210045,
-0.05186035856604576,
0.02906796708703041,
-0.014845849946141243,
-0.03213050961494446,
0.06245780736207962,
0.01816673018038273,
-0.025032103061676025,
0.010023168288171291,
0.013370247557759285,
-0.05280430242419243,
-0.012... | 0.846244 | 0.836401 | 0.009843 |
[asy]
draw((0,0)--(2,2)--(5/2,1/2)--(2,0)--cycle,dot);
MP("A",(0,0),W);
MP("B",(2,2),N);
MP("C",(5/2,1/2),SE);
MP("D",(2,0),S);
MP("a",(1,0),N);
MP("b",(17/8,1/8),N);
[/asy]
In the figure, $AB$ and $CD$ are parallel, the measure of angle $B$ is half that of angle $D$, and the measures of segments $AD$ and $CD$ are $a$... | tiger_math | [
0.02873869799077511,
0.01492228452116251,
-0.0473233237862587,
-0.04312858358025551,
0.03902546316385269,
-0.00903304759413004,
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0.050341296941041946,
0.04005615413188934,
0.008757520467042923,
0.007317350246012211,
0.020989960059523582,
-0.015421568416059017,
-0.0105... | 0.840496 | 0.824203 | 0.016294 |
On a certain planet, a round lake has marked underwater pillars for a race. Kyle starts traveling in a straight line from the northernmost pillar to the southernmost pillar across the middle of the lake, a distance of 6 km. From the eastern point due south to the nearest underwater pillar, he finds it to be 8 km. If we... | tiger_math | [
0.010838713496923447,
0.038079846650362015,
-0.03668017312884331,
-0.0557304322719574,
0.021371344104409218,
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0.03454756736755371,
0.030021894723176956,
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0.022324252873659134,
0.04325933754444122,
-0.02471141144633293,
-0.00... | 0.834582 | 0.825773 | 0.008809 |
[asy] draw((0,0)--(0,2)--(2,2)--(2,0)--cycle,dot); draw((2,2)--(0,0)--(0,1)--cycle,dot); draw((0,2)--(1,0),dot); MP("B",(0,0),SW);MP("A",(0,2),NW);MP("D",(2,2),NE);MP("C",(2,0),SE); MP("E",(0,1),W);MP("F",(1,0),S);MP("H",(2/3,2/3),E);MP("I",(2/5,6/5),N); dot((1,0));dot((0,1));dot((2/3,2/3));dot((2/5,6/5)); [/asy]
If $A... | tiger_math | [
0.03338276594877243,
0.010314620099961758,
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0.02513195388019085,
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0.0361262783408165,
0.0023635956458747387,
0.01673264242708683,
0.02094452455639839,
0.0005259480094537139,
-0.02... | 0.836317 | 0.821327 | 0.014989 |
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are positive integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find the value of $a+b+c$. | tiger_math | [
0.021597394719719887,
0.04995707795023918,
-0.05686326324939728,
-0.04090110585093498,
0.03725071996450424,
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0.0719589963555336,
0.053504299372434616,
-0.03275329992175102,
0.01529756560921669,
0.03435284644365311,
-0.042820606380701065,
-0.0318... | 0.831548 | 0.818948 | 0.0126 |
In the configuration below, $\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are line segments and $AB$ is tangent to the circle at $A$. The radius of the circle is 1.
[asy] defaultpen(fontsize(10pt)+linewidth(.8pt)); pair A=(0,-1), E=(0,1), C=(0,0), D=dir(10), F=dir(190), B=(-1/sin(10*... | tiger_math | [
0.0177198126912117,
0.04025019705295563,
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0.04562462866306305,
-0.05291135236620903,
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0.05891339108347893,
0.03833002597093582,
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0.021452393382787704,
0.010242586024105549,
-0.013388783670961857,
-0.00... | 0.828859 | 0.80523 | 0.023628 |
Let
\[x^8 - 98x^4 + 1 = p(x) q(x),\]where $p(x)$ and $q(x)$ are monic, non-constant polynomials with integer coefficients. Find $p(-1) + q(-1).$ | tiger_math | [
0.016259541735053062,
0.02453364059329033,
-0.038735438138246536,
-0.04672461003065109,
0.02306005172431469,
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0.06741416454315186,
0.052651118487119675,
-0.004799804650247097,
0.005847328342497349,
0.024639341980218887,
-0.038358598947525024,
-0.... | 0.846262 | 0.835807 | 0.010455 |
Find the sum of the cubes of the solutions to
\[\left| x^2 - x + \frac{1}{2008} \right| = \frac{1}{2008}.\] | tiger_math | [
-0.0015109980013221502,
0.041618745774030685,
-0.0567912794649601,
-0.028898559510707855,
0.0322902649641037,
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-0.011298783123493195,
0.0620390810072422,
0.053506217896938324,
-0.017852509394288063,
0.0035688169300556183,
0.006819036789238453,
-0.031091123819351196,
-... | 0.831416 | 0.827104 | 0.004312 |
Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than X$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. The answer is 0. What is the value of unknown variable X? | tiger_math | [
0.007814033888280392,
0.02354881912469864,
-0.0629245787858963,
-0.00962790660560131,
0.04695655778050423,
-0.010414534248411655,
-0.011382272467017174,
0.05730699747800827,
0.0470314621925354,
-0.031213363632559776,
0.011319663375616074,
0.054029591381549835,
-0.03575633838772774,
-0.0305... | 0.856546 | 0.852492 | 0.004054 |
Given $a \neq 0,$ solve for $x$ in
\[\begin{vmatrix} x - a & x & x \\ x & x - a & x \\ x & x & x - a \end{vmatrix} = 0.\]Give your answer in terms of $a.$ | tiger_math | [
0.035173896700143814,
0.029820505529642105,
-0.0381765142083168,
-0.031092569231987,
0.03864702209830284,
-0.027476975694298744,
-0.0018022313015535474,
0.083411306142807,
0.04803948849439621,
-0.023076049983501434,
0.008380694314837456,
0.0322834737598896,
-0.04090963676571846,
-0.0404708... | 0.831688 | 0.826614 | 0.005074 |
A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\ldots,2023$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it i... | tiger_math | [
0.014893599785864353,
0.01591714471578598,
-0.03899676352739334,
-0.04047781974077225,
0.021052580326795578,
-0.017135363072156906,
-0.04092201218008995,
0.06576228141784668,
0.02786436304450035,
0.000725198769941926,
0.020289726555347443,
0.02713623456656933,
-0.042866069823503494,
-0.009... | 0.852445 | 0.836547 | 0.015898 |
Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. If the dots on the boundary of the figure are required to be red, how many ways are there to color the remaining dots?
[asy]
draw((-75,0)--(-45,0)--(-60,26)--... | tiger_math | [
0.014272957108914852,
0.008640543557703495,
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0.007767754606902599,
-0.03100237250328064,
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0.0553482286632061,
0.03199183568358421,
-0.023024309426546097,
0.020808860659599304,
0.031537000089883804,
0.0010693364311009645,
-0.02... | 0.82858 | 0.824347 | 0.004233 |
The function $f$ is not defined for $x = 0,$ but for all non-zero real numbers $x,$
\[f(x) + 2f \left( \frac{1}{x} \right) = 3x.\]Find the real solutions to $f(2x) = f(-x).$ | tiger_math | [
0.03895970806479454,
0.034702420234680176,
-0.050632331520318985,
-0.02930402383208275,
0.03818061202764511,
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0.008649260737001896,
0.07278618961572647,
0.040837183594703674,
-0.0070599219761788845,
0.022482862696051598,
0.026831110939383507,
-0.02776968479156494,
-0.... | 0.820592 | 0.816755 | 0.003837 |
Let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that
\[\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}.\]Compute $\operatorname{proj}_{\mathbf{w}} (5 \mathbf{v}).$ | tiger_math | [
0.029719440266489983,
0.03649938479065895,
-0.05396578088402748,
-0.029926380142569542,
0.06005356088280678,
0.00819531362503767,
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0.07341595739126205,
0.03320183604955673,
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0.009034303016960621,
0.029259389266371727,
-0.029766689985990524,
-0.01... | 0.829693 | 0.821652 | 0.008041 |
Let $ f(x) = x^3 - 2x + 2$. Suppose $ g$ is a cubic polynomial such that $ g(1) = 2$, and the roots of $ g$ are the cubes of the roots of $ f$. Find $ g(8)$. | tiger_math | [
0.03377209231257439,
0.02974148653447628,
-0.060351643711328506,
-0.024579951539635658,
0.02715681679546833,
-0.05016947165131569,
-0.010765114799141884,
0.06762173771858215,
0.02378535084426403,
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0.011859949678182602,
0.023454630747437477,
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-0.0... | 0.838879 | 0.829664 | 0.009214 |
Given that $a$, $b$, and $c$ are complex numbers satisfying the equations $a + b + c = ab + ac + bc = abc = 1$, find the value of $(a-1)(b-1)(c-1)$. | tiger_math | [
0.046372752636671066,
0.041325367987155914,
-0.0429314486682415,
-0.05272892862558365,
0.046060800552368164,
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0.0011265981011092663,
0.05210747942328453,
0.026280848309397697,
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0.023011887446045876,
0.037987932562828064,
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-0.040... | 0.838882 | 0.833494 | 0.005388 |
Given that $\triangle PQR$ is isosceles with $\overline{PQ} \cong \overline{PR}$, and the perimeter of the triangle is 25, find the length of $\overline{QR}$ if the length of $\overline{PR}$ is 9. | tiger_math | [
0.009111134335398674,
0.01263429969549179,
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0.009053891524672508,
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0.057344529777765274,
0.014876714907586575,
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0.037794437259435654,
0.0420372411608696,
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-0.0... | 0.828487 | 0.818449 | 0.010039 |
In a galaxy far away, a space station orbits around a colossal planet in a circular path. Today, the orbiting station’s commander, upon analyzing the charted course, mathematical data, and celestial navigation system, noted an anomaly; the equation describing the current path was stated improperly as \(x^2 + y^2 - 12x ... | tiger_math | [
0.0254477858543396,
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0.025105910375714302,
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0.04473292827606201,
0.04268428310751915,
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0.006814824882894754,
0.008131211623549461,
-0.0063803791999816895,
-0.... | 0.829294 | 0.816647 | 0.012646 |
Find the ordered pair $(a,b)$ of real numbers such that the cubic polynomials $x^3 + ax^2 + 11x + 6 = 0$ and $x^3 + bx^2 + 14x + 8 = 0$ have a repeated root in common. | tiger_math | [
0.030402278527617455,
0.025876551866531372,
-0.05362913757562637,
-0.05157604068517685,
0.03301164135336876,
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0.0064034657552838326,
0.05869331955909729,
0.03498753160238266,
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0.01267775148153305,
0.007153470069169998,
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-0.0... | 0.851115 | 0.836167 | 0.014948 |
Determine the matrix $\mathbf{N}$ such that
\[\mathbf{N} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} -1 \\ 5 \end{pmatrix} \quad \text{and} \quad \mathbf{N} \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 2 \\ -7 \end{pmatrix}.\] | tiger_math | [
0.04492233693599701,
0.031014714390039444,
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0.05536499246954918,
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0.020433887839317322,
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0.006161906290799379,
0.01969338394701481,
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-0.... | 0.833048 | 0.828984 | 0.004064 |
Emily has decided to create a unique fence around her rectangular garden. The length of the garden is twice its width. She already has 30 meters of old fencing that she wants to use first. After using the old fencing, she purchases more fencing to complete the perimeter. Each meter of new fencing costs her $3. After co... | tiger_math | [
0.012755989097058773,
0.03266575187444687,
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0.0034137340262532234,
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0.029538778588175774,
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Paula the painter is working on a unique project. She has decided to create a hexagonal mural with a side length in each side of the hexagon of 3 feet. She began sketching circles within the hexagon, each with a diameter that equals the side length of the hexagon. Each of the circles' centers lies on each corner of the... | tiger_math | [
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0.061804261058568954,
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0.036147311329841614,
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... | 0.824018 | 0.802422 | 0.021596 |
Sarah sets out to ride her bicycle on a triangular route. First, she rides north for 12 kilometers, then turns and rides southeast for 17 kilometers. If Sarah is to return directly to her starting point, how far must she ride, and in which direction should she head? Assume all movements are perfectly linear and the sou... | tiger_math | [
0.03631466254591942,
0.0070325895212590694,
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0.020408296957612038,
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0.04707012325525284,
0.032159287482500076,
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0.03146401047706604,
0.03492911159992218,
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-0.04... | 0.816017 | 0.807453 | 0.008564 |
In a theme park, a magical prize machine dispenses a certain number of tickets based on a vector input. The machine calculates the tickets dispensed by the vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ with the following formula: $\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} 5 \\ 12 \end{... | tiger_math | [
0.03049297071993351,
0.03979059308767319,
-0.04216909408569336,
-0.046185992658138275,
0.03820652887225151,
-0.02138667181134224,
-0.04832681268453598,
0.06155066937208176,
0.033136382699012756,
-0.003130899043753743,
0.008015606552362442,
0.03719738498330116,
-0.012913999147713184,
-0.013... | 0.836416 | 0.814087 | 0.022329 |
In the figure shown, arc $ADB$ and arc $BEC$ are semicircles, each with a radius of one unit. Point $D$, point $E$ and point $F$ are the midpoints of arc $ADB$, arc $BEC$ and arc $DFE$, respectively. If arc $DFE$ is also a semicircle, what is the area of the region enclosed by the three arcs?
[asy]
unitsize(0.5inch);
... | tiger_math | [
0.02221161499619484,
0.013108095154166222,
-0.04398718848824501,
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0.023369690403342247,
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0.023650426417589188,
0.023986026644706726,
0.02627013809978962,
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-0.0... | 0.822164 | 0.808375 | 0.013789 |
A botanist is planting flowers in a circular pattern and each type of flower needs to be planted in a specific number of sections according to its species traits. There are 4 types of flowers: Roses, Tulips, Daisies, and Lilies. If Roses require 5 sections, Tulips need 7 sections, Daisies 4 sections, and Lilies 6 secti... | tiger_math | [
-0.0037506106309592724,
0.011316770687699318,
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0.030092278495430946,
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0.03819066658616066,
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0... | 0.830432 | 0.826059 | 0.004373 |
Nathan is stocking bathrooms at the hotel where he works. He has 18 rolls of toilet paper and 9 bars of soap. If he wants all bathrooms to be stocked identically, with the same combination of supplies in each one and nothing left over, what is the greatest combination of bathrooms Nathan can stock? | tiger_math | [
0.04819604381918907,
0.019333483651280403,
-0.05720002204179764,
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0.03425534442067146,
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0.056341856718063354,
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0.026896266266703606,
0.043821368366479874,
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-0.041... | 0.828339 | 0.821484 | 0.006855 |
A collection of cubes consists of one cube with edge-length $k$ for each positive integer $k$ up to a maximum value $n$. The rules for building a tower are the same as in the original question. Given that the remainder when the number of different towers that can be constructed is divided by 1000 is 729, find the value... | tiger_math | [
0.03130713850259781,
0.02371327579021454,
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0.004014287143945694,
0.04219966009259224,
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0.02296186052262783,
0.023848149925470352,
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0.004460... | 0.836316 | 0.823143 | 0.013173 |
Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=x$ while $HQ=y$, then calculate $(BP)(PC)-(AQ)(QC)$ in terms of $x$ and $y$. | tiger_math | [
0.03268272429704666,
0.005540091078728437,
-0.031069930642843246,
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0.03408422693610191,
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0.0534086599946022,
0.05373765155673027,
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0.02712758630514145,
0.033075541257858276,
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-0.01... | 0.831467 | 0.813926 | 0.017541 |
Rationalize the denominator of $\frac{\sqrt{32}}{\sqrt{16}-\sqrt{2}}$. The answer can be written as $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. Find the minimum possible value of $A+B+C+D$. | tiger_math | [
0.01163004245609045,
0.02851392701268196,
-0.047161877155303955,
-0.0217463206499815,
0.044673122465610504,
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0.06927856057882309,
0.0599721297621727,
-0.029692864045500755,
0.024884194135665894,
0.0368536151945591,
-0.01322126667946577,
-0.0354... | 0.840068 | 0.830321 | 0.009747 |
An $\textit{annulus}$ is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$, with $b>c$. Let $\overline{OX}$ be a radius of the larger circle, let $\overline{XZ}$ be tangent to the smaller circle at $Z$, and let $\overline{OY}$ be the radius of the larger circle that ... | tiger_math | [
0.003167476737871766,
0.047756824642419815,
-0.05541782081127167,
-0.02817631885409355,
0.016290923580527306,
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0.04971931874752045,
0.0406084842979908,
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0.006516582332551479,
0.04024571180343628,
-0.029797585681080818,
-0.... | 0.827685 | 0.801589 | 0.026097 |
In the figure, $m\angle A = 28^{\circ}$, $m\angle B = 74^\circ$ and $m\angle C = 26^{\circ}$. If $w$, $x$, $y$ and $z$ are the measures of the angles in which they are shown, what is the value of $w + x + y + z$? [asy]
size(150);
draw((0,5)--(0,0)--(15,0)--(15,5),linewidth(1));
draw((0,5)--(2,2)--(5,5)--(12,-2)--(15,5)... | tiger_math | [
0.007828792557120323,
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0.015682728961110115,
0.022970449179410934,
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-0.0... | 0.824352 | 0.818117 | 0.006235 |
In the land of Mathematica, dragons collect coins based on a special rule. If a dragon collects $x$ coins on day one, it collects $x^2 + 2$ coins on the next day. If a dragon started with 3 coins on the first day, how many coins does it have after three days? | tiger_math | [
0.018436651676893234,
0.010773372836411,
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0.0203426331281662,
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0.010450997389853,
0.03302166610956192,
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-0.03535298... | 0.841948 | 0.834338 | 0.00761 |
In the diagram, $O$ is the center of a circle with radii $OP=OQ=r$. What is the perimeter of the shaded region?
[asy]
size(100);
import graph;
label("$P$",(-1,0),W); label("$O$",(0,0),NE); label("$Q$",(0,-1),S);
fill(Arc((0,0),1,-90,180)--cycle,mediumgray);
draw(Arc((0,0),1,-90,180));
fill((0,0)--(-1,0)--(0,-1)--cyc... | tiger_math | [
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0.022977299988269806,
0.018253689631819725,
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-... | 0.823131 | 0.819471 | 0.00366 |
In a box with 40 cards numbered from 1 to 10 (4 cards per number), 4 cards are drawn randomly without replacement. Let r be the probability that at least three of the cards have the same number. Find the value of r. | tiger_math | [
0.012745372951030731,
0.023639952763915062,
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0.040501777082681656,
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0.002585139125585556,
0.021474961191415787,
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Lila is planning a garden party and wants to arrange the seating such that each table has an equal number of guests and each table offers a unique fruit plate. If Lila has 36 different fruits to distribute among the tables, and each table gets 3 unique fruits, what is the sum of all possible numbers of guests given tha... | tiger_math | [
0.006967192981392145,
0.032536379992961884,
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0.033917926251888275,
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0.011903943493962288,
0.04186609759926796,
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-0.... | 0.839411 | 0.82711 | 0.012302 |
The number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{12} \le 2010$ such that $a_i-i$ is odd for $1\le i \le 12$ can be expressed as ${m \choose n}$ for some positive integers $m > n$. Compute the remainder when $m$ is divided by 1000. | tiger_math | [
0.023668697103857994,
0.02108748070895672,
-0.038102008402347565,
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0.01690935716032982,
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0.0812886580824852,
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0.020112432539463043,
0.021911494433879852,
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-0.0... | 0.834267 | 0.819013 | 0.015253 |
There are $22n$ members in the Trumpington marching band, and when they line up in rows of 30, there are 4 band members left over. If $n$ is an integer and there are fewer than 1000 band members, what is the maximum number of people that could be in the Trumpington marching band? | tiger_math | [
0.0393030047416687,
0.021422989666461945,
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0.03411592170596123,
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0.03840465471148491,
0.021343659609556198,
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0.01325046457350254,
0.032757069915533066,
-0.03187907487154007,
-0.0260060... | 0.838328 | 0.834425 | 0.003903 |
Let $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 3 \\ 2 \end{pmatrix}$. Find the vector $\mathbf{c}$ such that $\mathbf{a} \times \mathbf{c} = \mathbf{b}$. | tiger_math | [
0.053153619170188904,
0.04894281551241875,
-0.05740225315093994,
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0.04542763531208038,
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0.08589348196983337,
0.02003335766494274,
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-0.003408880904316902,
0.017665911465883255,
-0.03610712289810181,
-0.... | 0.827544 | 0.819136 | 0.008408 |
Find the value of $X$ in the equation $16^{x^2 + y} + 16^{x + y^2} = X$ if there are 2 ordered pairs $(x,y)$ of real numbers that satisfy the equation. | tiger_math | [
0.011228923685848713,
0.013241555541753769,
-0.05931971222162247,
-0.04062642902135849,
0.050238918513059616,
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0.06562364101409912,
0.04898912459611893,
-0.02941020391881466,
0.030504802241921425,
0.0342843197286129,
-0.025846656411886215,
-0.01... | 0.836936 | 0.831312 | 0.005625 |
In the enchanted forest, there are magical trees that grow fruits in a very particular pattern. A particular magical tree doubles the number of fruits it bears every month starting from a single fruit. The fruits, once fully grown, can be turned into gold coins, each fruit converting into precisely 130 gold coins. Oliv... | tiger_math | [
0.016075316816568375,
0.033500559628009796,
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0.0015381858684122562,
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0.0590672492980957,
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0.007441374007612467,
0.02515258453786373,
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-0.... | 0.816515 | 0.812462 | 0.004053 |
For what value of $b$ do the graphs of $y = x^2 + bx - 2$ and $y = 2x + 2$ have exactly one point of intersection? | tiger_math | [
0.027234049513936043,
0.01421281322836876,
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0.028822140768170357,
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0.05129169300198555,
0.051653146743774414,
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0.019001828506588936,
0.03638915345072746,
-0.006044873967766762,
-0.0... | 0.831777 | 0.828697 | 0.00308 |
The hyperbola given by the equation \[\frac{(x-2)^2}{100} - \frac{(y+1)^2}{64} = 1\]has asymptotes $y = \pm mx + c,$ where $m$ is positive. Find $m$ and $c.$ | tiger_math | [
0.015959318727254868,
0.009931543841958046,
-0.03657715767621994,
-0.030095942318439484,
0.028856521472334862,
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-0.024568768218159676,
0.07786364108324051,
0.04343206807971001,
-0.033246636390686035,
0.003247076878324151,
0.026674993336200714,
-0.012675574980676174,
-... | 0.833115 | 0.819655 | 0.013461 |
In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\overline{AC}$ with $V$ on $\overline{AW}$, points $X$ and $Y$ are on $\overline{BC}$ with $X$ on $\overline{CY}$, and points $Z$ and $U$ are on $\overline{AB}$ with $Z$ on $\overline{BU}$. In addition, the points are positioned so that ... | tiger_math | [
0.02829776704311371,
0.036500535905361176,
-0.040420811623334885,
-0.0477752648293972,
0.026136310771107674,
-0.008244941011071205,
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0.04294510930776596,
0.03900941088795662,
-0.012210376560688019,
0.03386378288269043,
0.04729887843132019,
-0.019131436944007874,
-0.00... | 0.852229 | 0.824011 | 0.028218 |
Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 6.$ Find the maximum value of
\[\sqrt{3x + 1} + \sqrt{3y + 1} + \sqrt{3z + 1}.\] | tiger_math | [
0.03191627562046051,
0.03544801473617554,
-0.034897059202194214,
-0.03546889126300812,
0.044475849717855453,
0.0005230684764683247,
-0.02349722757935524,
0.0375959612429142,
0.02412363886833191,
-0.022992592304944992,
0.03128580003976822,
0.03987743705511093,
-0.03323058784008026,
-0.04305... | 0.846584 | 0.831123 | 0.015461 |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(y) - x) = g(x - y) + yg(x) + xy - y + 1\]for all real numbers $x$ and $y.$
Find the number of possible values of $g(-1).$ | tiger_math | [
0.01733129471540451,
0.031025627627968788,
-0.05204785615205765,
-0.047805383801460266,
0.0449073351919651,
-0.017898332327604294,
-0.010191388428211212,
0.06903060525655746,
0.03934573382139206,
-0.02459929883480072,
0.005699372384697199,
0.037258487194776535,
-0.0472567155957222,
-0.0233... | 0.849043 | 0.838261 | 0.010781 |
In a magical forest, three mythical creatures, a Dragon, a Unicorn, and a Griffin, each finds a certain number of golden apples. The relation $\oplus$ is defined by $X \oplus Y = X^3 + 3XY + Y^3$. If the Unicorn has 3 golden apples, and together with the Dragon's apples they relate as $D \oplus 3 = 351$, how many golde... | tiger_math | [
0.008638630621135235,
0.02605779655277729,
-0.05484393611550331,
-0.04731009900569916,
0.016094619408249855,
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-0.024347703903913498,
0.05174470692873001,
0.03878189995884895,
-0.01896590366959572,
0.002322223735973239,
0.032156143337488174,
-0.04395619034767151,
-0.01... | 0.848082 | 0.838948 | 0.009134 |
Suppose $\sqrt{1 + \sqrt{2y-3}} = \sqrt{6}$; find $y$. | tiger_math | [
0.020586566999554634,
0.04473889246582985,
-0.03142805024981499,
-0.04167293384671211,
0.0517343133687973,
-0.024761544540524483,
-0.01950513944029808,
0.06609485298395157,
0.039587508887052536,
-0.02173766866326332,
0.022585026919841766,
0.0330161489546299,
-0.04759762063622475,
-0.034878... | 0.837579 | 0.831602 | 0.005977 |
The circles whose equations are $x^2 + y^2 - 4x + 2y - 11 = 0$ and $x^2 + y^2 - 14x + 12y + 60 = 0$ intersect in the points $A$ and $B.$ Compute the distance between the points $A$ and $B.$ | tiger_math | [
0.026126502081751823,
0.015246913768351078,
-0.03723742812871933,
-0.0438288189470768,
0.05473896861076355,
-0.02721700258553028,
-0.030788376927375793,
0.045650385320186615,
0.04968228563666344,
-0.019846180453896523,
0.016388174146413803,
0.026930147781968117,
-0.013840455561876297,
-0.0... | 0.836689 | 0.82784 | 0.00885 |
In triangle $ABC$, we have that $E$ and $F$ are midpoints of sides $\overline{AC}$ and $\overline{AB}$, respectively. The area of $\triangle ABC$ is 24 square units. How many square units are in the area of $\triangle CEF$? | tiger_math | [
0.03532472997903824,
0.0073145427741110325,
-0.052607372403144836,
-0.049482718110084534,
0.028389358893036842,
-0.020718462765216827,
-0.015350375324487686,
0.042851708829402924,
0.04773727059364319,
-0.011346738785505295,
0.019546812400221825,
0.029985660687088966,
-0.03142799437046051,
... | 0.833055 | 0.82585 | 0.007205 |
Four points are chosen uniformly at random on a circle. What is the probability that no three of these points form an obtuse triangle with the circle's center? | tiger_math | [
0.0005775805911980569,
0.017560217529535294,
-0.03906766697764397,
-0.03669942170381546,
0.03439744934439659,
-0.009335032664239407,
-0.024500854313373566,
0.0197079349309206,
0.03760499507188797,
-0.01197450328618288,
0.020184993743896484,
0.023861723020672798,
-0.015474147163331509,
-0.0... | 0.831896 | 0.827669 | 0.004228 |
In an amusement park, there are 15 vehicles on a carousel with each displaying a unique number. These numbers are in arithmetic sequence. The first vehicle's number is 2 and the third vehicle's number is 10. A child wants to take a ride on each carousel vehicle whose number has an even tens digit. How many vehicles wil... | tiger_math | [
0.03278157114982605,
0.013304930180311203,
-0.03198942914605141,
-0.039963770657777786,
0.03597427159547806,
-0.019787456840276718,
-0.01721258834004402,
0.048598721623420715,
0.035744842141866684,
-0.023485293611884117,
0.014328462071716785,
0.025339443236589432,
-0.03715763986110687,
-0.... | 0.840233 | 0.835764 | 0.004469 |
If $a>0$ and $b>0,$ a new operation $\nabla$ is defined as follows: $$a \nabla b = \frac{a + b}{1 + ab}.$$For example, $$3 \nabla 6 = \frac{3 + 6}{1 + 3 \times 6} = \frac{9}{19}.$$Calculate $(1 \nabla 2) \nabla 3.$ | tiger_math | [
0.036777425557374954,
0.03343713656067848,
-0.04033639281988144,
-0.0036741646472364664,
0.04990292340517044,
-0.0069303493946790695,
-0.02809206023812294,
0.08249591290950775,
0.025662928819656372,
-0.0000918586811167188,
0.007262436207383871,
0.013755416497588158,
-0.042035412043333054,
... | 0.821539 | 0.816912 | 0.004628 |
The projection of $\begin{pmatrix} 1 \\ 2 \\ z \end{pmatrix}$ onto $\begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}$ is
\[\frac{4}{14} \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}.\]Find $z.$ | tiger_math | [
0.04455893859267235,
0.062054675072431564,
-0.045692749321460724,
-0.039920322597026825,
0.06935202330350876,
-0.0025005429051816463,
-0.0021008928306400776,
0.03545075282454491,
0.028267985209822655,
0.004850524011999369,
0.010723213665187359,
0.022137785330414772,
-0.021104250103235245,
... | 0.81923 | 0.816323 | 0.002907 |
In a magical forest, three wizard friends, Alice, Bob, and Charlie, each cast spells that double the number of mystical flowers in the garden every hour. Starting with X flowers, Alice casts her spell, followed by Bob, then Charlie in a consecutive manner. After the third hour, and three rounds of casting (one by each ... | tiger_math | [
0.018365712836384773,
0.03368154168128967,
-0.035643987357616425,
-0.039593037217855453,
0.014109606854617596,
-0.042213041335344315,
-0.033943045884370804,
0.05670464038848877,
0.04504948481917381,
-0.020729366689920425,
0.007310265675187111,
0.014807104133069515,
-0.0357217900454998,
-0.... | 0.823433 | 0.817254 | 0.006179 |
Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 8$, $BC = 4$, $\overline{PA}\perp \overline{AB}$, $\overline{PA}\perp\overline{AD}$, and $PA = 6$, then what is the volume of $PABCD$? | tiger_math | [
0.010006655007600784,
0.016683317720890045,
-0.036741238087415695,
-0.036058459430933,
0.02244718186557293,
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-0.00624138442799449,
0.03755738586187363,
0.03374041989445686,
0.0070044854655861855,
0.013783319853246212,
0.0356905460357666,
-0.023691147565841675,
-0.02319... | 0.827429 | 0.821731 | 0.005698 |
During a school science fair, a teacher is organizing booths in a large hall. The teacher wants to arrange 30 seats into rows such that each row contains the same number of seats. Initially, the teacher attempts to set 5 more seats per row than the total number of rows. How many seats will each row contain once they ar... | tiger_math | [
0.02159108594059944,
0.005662865471094847,
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0.019678587093949318,
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0.035222429782152176,
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0.010357815772294998,
0.03020760789513588,
-0.036072202026844025,
-0.0... | 0.838617 | 0.832198 | 0.006418 |
In the mystical land of Numeria, a castle has a collection of differently colored magic doors arranged in a row: 5 blue doors, 4 red doors, and 6 green doors, each leading to distinct magical realms. The Fairy Queen requires your help in determining how many distinct ways can she rearrange all the doors in her castle s... | tiger_math | [
0.02501320280134678,
0.009303819388151169,
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0.009976575151085854,
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0.03453141823410988,
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0.020037731155753136,
0.03787602484226227,
-0.032654136419296265,
-0.... | 0.810547 | 0.797981 | 0.012567 |
Iris, an event planner, is designing a triangular floral display for a wedding. She chooses three points in the garden where she will place large flower pots to mark the vertices of the triangular area. The pots are located at the coordinates (4, 3), (10, -2), and (1, 8). Calculate the area of the triangle formed by th... | tiger_math | [
0.02808619849383831,
0.021959509700536728,
-0.05450158938765526,
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0.03146360069513321,
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0.0492536835372448,
0.00871663074940443,
0.015566213056445122,
0.05767424777150154,
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-0.00664026... | 0.810386 | 0.800611 | 0.009776 |
In a triangular park, three garden paths connect each pair of entrance gates located at the vertices. Each path is straight and the distances between the gates are as follows: Gate A to Gate B is 300 meters, Gate B to Gate C is 500 meters, and Gate A to Gate C is 400 meters. A fountain is to be installed at a point wit... | tiger_math | [
0.04233775660395622,
0.023103920742869377,
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0.023152919486165047,
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0.023237569257616997,
0.045410677790641785,
-0.024515263736248016,
-... | 0.822487 | 0.805443 | 0.017044 |
A rectangular plot of private property is fenced in by a chain-link fence. The long sides of the plot are three times the length of the short sides. One short side of the fence is rusted from being hit by a sprinkler and needs to be replaced. If all the sides of the fence together are x feet long, how many feet of fenc... | tiger_math | [
0.002669280394911766,
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-0.01... | 0.8354 | 0.828349 | 0.007051 |
Let $\theta$ be the smallest acute angle for which $\sin \theta,$ $\sin 2 \theta,$ $\sin X \theta$ form an arithmetic progression, in some order. Find $\cos \theta. 4. What is the value of unknown variable X? | tiger_math | [
0.019167382270097733,
0.04304291680455208,
-0.055889759212732315,
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0.03554460406303406,
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0.025565151125192642,
0.0031712534837424755,
-0.025927647948265076,... | 0.847426 | 0.837115 | 0.010311 |
The equation $a^7xy-a^6y-a^5x=a^4(b^4-1)$ is equivalent to the equation $(a^mx-a^n)(a^py-a^q)=a^4b^4$ for some integers $m$, $n$, $p$, and $q$. Find $mn+pq$. | tiger_math | [
0.03817173093557358,
0.021812627092003822,
-0.035935088992118835,
-0.05823426693677902,
0.031803421676158905,
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-0.008028003387153149,
0.0694604367017746,
0.07017896324396133,
-0.028567146509885788,
0.011339719407260418,
0.0354858823120594,
-0.047615040093660355,
-0.032... | 0.839651 | 0.832542 | 0.007109 |
Find the minimum number of real roots to a polynomial of the form
\[x^n + x^{n - 1} + \dots + x + X = 0,\]where $n$ is a positive integer and $X$ is a positive real number. | tiger_math | [
0.027258388698101044,
0.021319009363651276,
-0.04091142117977142,
-0.044373467564582825,
0.024910349398851395,
-0.03359682485461235,
-0.009013297036290169,
0.04732280597090721,
0.04317081347107887,
-0.02408336102962494,
0.011884969659149647,
0.017087459564208984,
-0.03377361223101616,
-0.0... | 0.838306 | 0.83191 | 0.006396 |
Fido's leash is tied to a stake at the center of his yard, which is in the shape of a regular hexagon. His leash is exactly long enough to reach the midpoint of each side of his yard. If the perimeter of Fido's yard is $36$ units, and the fraction of the area of Fido's yard that he is able to reach while on his leash i... | tiger_math | [
0.010402288287878036,
0.043385811150074005,
-0.03957764059305191,
-0.03566592186689377,
0.03416375070810318,
-0.03221097216010094,
-0.02900456264615059,
0.052293308079242706,
0.043414488434791565,
-0.014360573142766953,
0.0016273879446089268,
0.04780718684196472,
-0.03892792761325836,
-0.0... | 0.840774 | 0.829269 | 0.011505 |
Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The products of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers? [asy]
path box=(0,0)--(1,0)--(1,1.5)--(0,... | tiger_math | [
0.04403042048215866,
0.026537345722317696,
-0.040423497557640076,
-0.029943669214844704,
0.053696684539318085,
-0.02888578735291958,
-0.013284897431731224,
0.0516938716173172,
0.02450520545244217,
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0.011288541369140148,
0.031063610687851906,
-0.04029728099703789,
-0.... | 0.835942 | 0.831234 | 0.004708 |
The total mask production was doubled every month by a company after the outbreak of Coronavirus due to increased demand. If the company produced 3000 masks in March, calculate the total mask production of July. | tiger_math | [
0.048890624195337296,
0.03843258321285248,
-0.07303177565336227,
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0.03618511185050011,
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0.07887715846300125,
0.008775126188993454,
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0.03197886422276497,
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-0.... | 0.799931 | 0.795861 | 0.004071 |
Margot is planning a themed birthday party and wants to serve fruit punch in a large cylindrical punch bowl. The bowl has a radius of 5 inches and a height of 10 inches. She plans to fill it exactly three-quarters full with her punch. If each party guest consumes about 8 ounces of punch, how many guests can Margot serv... | tiger_math | [
0.007385516073554754,
0.05698363855481148,
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0.018886998295783997,
0.04052312299609184,
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-0.020482... | 0.811088 | 0.807146 | 0.003943 |
In the magical land of Numeralia, three wizards - Astra, Balthazar, and Cyrus - each have gardens where they grow mystical sunflowers and moonflowers. Astra has 12 sunflowers and 8 moonflowers. Balthazar has 18 sunflowers and 12 moonflowers, while Cyrus has 24 sunflowers and 16 moonflowers. They decide to create a bouq... | tiger_math | [
0.017402300611138344,
0.031475163996219635,
-0.05085798725485802,
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0.02638305351138115,
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0.03559550642967224,
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0.05282722786068916,
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0.000... | 0.829869 | 0.823049 | 0.00682 |
In triangle $ABC,$ $\angle C = \frac{\pi}{2}.$ Find
\[\arctan \left( \frac{a}{b + c} \right) + \arctan \left( \frac{b}{a + c} \right).\] | tiger_math | [
0.04451616108417511,
0.04181036353111267,
-0.035307127982378006,
-0.025220828130841255,
0.0331471748650074,
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0.048195719718933105,
0.04066590964794159,
-0.0059029958210885525,
0.014083007350564003,
0.019296426326036453,
-0.035359762609004974,
-0... | 0.832627 | 0.814685 | 0.017942 |
In a magical forest, a treasure chest moves according to the mystical laws described by the parametric equations for its position: \(x = 3t^2 - 5t + 4\) and \(y = -2t^2 + t + 6\), where \(t\) is the time in minutes. If you can calculate the speed of the treasure chest at exactly \(t = 2\) minutes, you earn the right to... | tiger_math | [
0.04107215628027916,
0.03390529379248619,
-0.0386224240064621,
-0.029230503365397453,
0.0291635449975729,
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0.06189253553748131,
0.0309178177267313,
-0.02794044092297554,
0.014713047072291374,
0.01941571570932865,
-0.0335172601044178,
-0.00712601142... | 0.829939 | 0.826187 | 0.003751 |
In triangle $XYZ$, $XY = 12$ and $XZ = 15$. Let $E$ be the foot of the perpendicular from $X$ to $YZ$. If $YE:EZ = 3:4$, then find the area of triangle $XEY$. | tiger_math | [
0.03266172483563423,
0.028496356680989265,
-0.030032716691493988,
-0.04268660396337509,
0.03876234591007233,
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0.0441339947283268,
0.01939932070672512,
0.005770137533545494,
0.039116859436035156,
0.03794597089290619,
-0.041449256241321564,
-0.0044... | 0.832789 | 0.825855 | 0.006935 |
Altitudes $\overline{AX}$ and $\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\angle BAC = \alpha$ and $\angle ABC = \beta$, then what is $\angle CHX$ in terms of $\alpha$ and $\beta$? | tiger_math | [
0.02290612831711769,
-0.00287312432192266,
-0.044076379388570786,
-0.04238348826766014,
0.019506439566612244,
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0.046646974980831146,
0.048654891550540924,
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0.02566707134246826,
0.017487315461039543,
-0.04582011699676514,
-0.... | 0.819353 | 0.808422 | 0.010931 |
Compute the least positive value of $t$ such that
\[\arcsin (\sin \alpha), \ \arcsin (\sin 3 \alpha), \ \arcsin (\sin 5 \alpha), \ \arcsin (\sin t \alpha)\]is a geometric progression for some $\alpha$ with $0 < \alpha < \frac{\pi}{2}.$ | tiger_math | [
0.009001677855849266,
0.03646636754274368,
-0.05287761986255646,
-0.02929273247718811,
0.03888288512825966,
-0.002088577253744006,
-0.03799783065915108,
0.06380073726177216,
0.0295647494494915,
-0.006434020586311817,
0.030217144638299942,
0.005450519733130932,
-0.02798648178577423,
-0.0111... | 0.838478 | 0.823604 | 0.014874 |
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are X feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Ken... | tiger_math | [
0.011461140587925911,
0.05191338434815407,
-0.06400489807128906,
-0.026156364008784294,
0.039921462535858154,
0.005333546549081802,
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0.07884670794010162,
0.043002426624298096,
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0.020783795043826103,
0.020495083183050156,
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-0.... | 0.837934 | 0.834696 | 0.003238 |
In the magical world of Mathlandia, there are two cities: Algebria and Calculopolis. A train departs from Algebria to Calculopolis every 2 hours, and the journey time between the two cities is precisely 4 hours. If a train leaves Algebria at 8:00 am, what are the possible times that a traveler could arrive in Calculopo... | tiger_math | [
0.04281726852059364,
0.014363009482622147,
-0.03389083594083786,
-0.028440970927476883,
0.0292446780949831,
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0.05148561671376228,
0.03685778006911278,
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0.015587816014885902,
0.025071097537875175,
-0.011215440928936005,
-0.01... | 0.826932 | 0.818398 | 0.008535 |
Define $P(z) = z^8 + (4\sqrt{3} + 6)z^4 - (4\sqrt{3} + 7)$. Among all the $8$-sided polygons in the complex plane whose vertices are the zeros of $P(z)$, what is the smallest perimeter? | tiger_math | [
0.02198861539363861,
0.04507550969719887,
-0.027509795501828194,
-0.042533207684755325,
0.031828079372644424,
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0.042890600860118866,
0.04458998143672943,
-0.0058652544394135475,
0.004203618969768286,
0.03505411371588707,
-0.023600347340106964,
-... | 0.839063 | 0.829402 | 0.009661 |
Orlan had two ropes: Rope A with a length of x meters and Rope B with a length of y meters. He first gave 40% of Rope A to Allan, then one-third of the remaining Rope A to Jack. After that, he gave 25% of Rope B to Maria, and finally one-fourth of the remaining Rope B to Mike. If Orlan has p meters of Rope A and q mete... | tiger_math | [
0.012605537660419941,
0.023797687143087387,
-0.04808327555656433,
-0.04070742055773735,
0.016784805804491043,
-0.027086669579148293,
-0.00997897144407034,
0.05480874329805374,
0.04253605380654335,
0.004574131686240435,
0.015111078508198261,
0.03939139097929001,
-0.035750892013311386,
-0.02... | 0.825431 | 0.822284 | 0.003146 |
Let \[f(x) =
\begin{cases}
3x^2 + 2&\text{if } x\le 3, \\
ax - 1 &\text{if } x>3.
\end{cases}
\]Find $a$ if the graph of $y=f(x)$ is continuous (which means the graph can be drawn without lifting your pencil from the paper). | tiger_math | [
0.03261692449450493,
0.03195551037788391,
-0.03862041234970093,
-0.02842751517891884,
0.039321135729551315,
-0.03659842163324356,
-0.017842823639512062,
0.087153360247612,
0.03469420224428177,
-0.02162293531000614,
0.02449708990752697,
0.05101265385746956,
-0.012816859409213066,
-0.0247832... | 0.825662 | 0.810611 | 0.015051 |
the surface area of a sphere is same as the curved surface area of a right circular cylinder whose height and diameter are 16 cm each . the radius of the sphere is | tiger_math | [
0.004754094406962395,
0.021285131573677063,
-0.029822571203112602,
-0.03559762239456177,
0.040134772658348083,
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-0.041211411356925964,
0.04919552430510521,
0.03161998838186264,
-0.022368501871824265,
0.02322864718735218,
0.04012225195765495,
-0.026107899844646454,
-0.... | 0.789338 | 0.784187 | 0.005152 |
Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 1.$ Find the minimum value of
\[\frac{1}{a + 2b} + \frac{1}{b + 2c} + \frac{1}{c + 2a}.\] | tiger_math | [
0.02375788614153862,
0.03187498822808266,
-0.04924444481730461,
-0.034555889666080475,
0.046147629618644714,
-0.018360260874032974,
-0.01454673707485199,
0.05163417011499405,
0.030704190954566002,
-0.028332576155662537,
-0.001582643366418779,
0.017277196049690247,
-0.01816754974424839,
-0.... | 0.843982 | 0.83198 | 0.012002 |
Lilly is planning her birthday party and decided to place party favors in groups so that each favor box contains the same number of items and every group has the same color of balloons tied to it. Lilly has 120 balloons and 75 favor boxes. What is the greatest number of favor boxes she can have per group so that each g... | tiger_math | [
0.014136500656604767,
0.01959284581243992,
-0.04498368501663208,
-0.0464201457798481,
0.021437248215079308,
-0.03498295322060585,
-0.02675427682697773,
0.03744113817811012,
0.026087703183293343,
-0.01439624559134245,
0.012699958868324757,
0.049726031720638275,
-0.025901062414050102,
-0.001... | 0.827299 | 0.820697 | 0.006602 |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.5 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
... | tiger_math | [
0.03727820888161659,
0.032234661281108856,
-0.03612986207008362,
-0.043725110590457916,
0.034164898097515106,
-0.03855364769697189,
-0.0242566280066967,
0.10366503149271011,
0.05227048695087433,
-0.011991920880973339,
0.019321465864777565,
0.03239550441503525,
-0.00948173925280571,
-0.0036... | 0.818289 | 0.814166 | 0.004123 |
A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called `hearts' and `diamonds') are red, the other two ($\spadesuit$ and $\clubsuit$, called `spades' and `clubs') are black. The cards in the deck are placed in random order (usu... | tiger_math | [
0.0018529650988057256,
0.018873164430260658,
-0.05603794381022453,
-0.029006868600845337,
0.023584650829434395,
-0.010807116515934467,
-0.030421104282140732,
0.04912415146827698,
0.02900821901857853,
-0.012198044918477535,
0.014250013045966625,
0.010925615206360817,
-0.027706848457455635,
... | 0.821995 | 0.814266 | 0.007729 |
Given that the matrix
\[\mathbf{M} = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\]satisfies $\mathbf{M}^T \mathbf{M} = \mathbf{I},$ where $\mathbf{I}$ is the identity matrix, find the value of $a^2 + d^2 + g^2 + b^2 + e^2 + h^2 + c^2 + f^2 + i^2.$ | tiger_math | [
0.04363188147544861,
0.039028801023960114,
-0.04428994655609131,
-0.050320882350206375,
0.0539529025554657,
-0.048894595354795456,
-0.010873786173760891,
0.07331228256225586,
0.03453826531767845,
-0.023826364427804947,
0.007913989946246147,
0.012264734134078026,
-0.03615458309650421,
-0.03... | 0.828395 | 0.813169 | 0.015226 |
With all angles measured in degrees, the product $\prod_{k=1}^{22} \csc^2(4k-3)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. | tiger_math | [
0.025999993085861206,
0.03001430444419384,
-0.0382353775203228,
-0.04323659464716911,
0.03569531440734863,
-0.02492111176252365,
-0.036490585654973984,
0.05772709101438522,
0.03296928480267525,
-0.00008367103146156296,
0.018599212169647217,
0.043266717344522476,
-0.03003540448844433,
-0.02... | 0.846351 | 0.828817 | 0.017534 |
Real numbers $x$ and $y$ have an arithmetic mean of 18 and a geometric mean of $\sqrt{92}$. Find $x^2+y^2$. | tiger_math | [
0.021467948332428932,
0.03487816080451012,
-0.0553586408495903,
-0.06757838279008865,
0.028237052261829376,
-0.01860196143388748,
-0.030714862048625946,
0.05539128556847572,
0.05005593225359917,
-0.03378580883145332,
0.02899901382625103,
0.023811176419258118,
-0.035660337656736374,
-0.0081... | 0.835285 | 0.82608 | 0.009205 |
Find the equation of the plane that passes through the point $(-3,4,-2)$ and is parallel to the plane $2x - 3y + 4z = 5.$ Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$ | tiger_math | [
0.0257489662617445,
0.031620100140571594,
-0.0439382903277874,
-0.05350151285529137,
0.045008692890405655,
0.009262628853321075,
-0.010844822973012924,
0.04641636461019516,
0.03654007986187935,
0.006774757988750935,
0.01545832771807909,
0.04364033415913582,
-0.0038417307659983635,
-0.01884... | 0.836655 | 0.825092 | 0.011564 |
At a local cupcake competition, there are 8 different flavored cupcakes and each flavor has exactly 5 identical cupcakes. Contestants need to choose exactly 3 different flavors to create a unique flavor combination for the judges. The combination needs to include one vanilla cupcake, which is a flavor always available.... | tiger_math | [
-0.000778605870436877,
0.02790168486535549,
-0.00895305909216404,
-0.03919705003499985,
0.034426525235176086,
-0.01155376061797142,
-0.04046749323606491,
0.04833626747131348,
0.032327909022569656,
-0.017229216173291206,
0.016788074746727943,
0.027064112946391106,
-0.019455892965197563,
-0.... | 0.826058 | 0.819614 | 0.006444 |
Maria has decided to throw a party for her five closest friends and wants to serve them her newly invented drinks. She has created five different drinks for the occasion. However, each friend must try each drink exactly once in a unique sequential order so every friend has a unique arrangement. How many ways can Maria ... | tiger_math | [
0.02963007241487503,
0.011007977649569511,
-0.028092049062252045,
-0.031190747395157814,
0.01830039732158184,
-0.015934176743030548,
-0.0295918807387352,
0.05594340339303017,
0.04268709197640419,
-0.011244278401136398,
0.008562629111111164,
0.022293703630566597,
-0.022991221398115158,
-0.0... | 0.821656 | 0.813293 | 0.008363 |
In quadrilateral $ABCD$, sides $\overline{AB}$ and $\overline{BC}$ both have length $a$, sides $\overline{CD}$ and $\overline{DA}$ both have length $b$, and the measure of angle $ADC$ is $\theta$. What is the length of diagonal $\overline{AC}$ in terms of $a$, $b$, and $\theta$? | tiger_math | [
-0.00439224299043417,
0.0007808925583958626,
-0.06951192766427994,
-0.058018721640110016,
0.014101870357990265,
-0.002155638998374343,
-0.013941554352641106,
0.041928261518478394,
0.030366117134690285,
-0.017511216923594475,
0.005904164165258408,
0.018990421667695045,
-0.04221690446138382,
... | 0.831746 | 0.821953 | 0.009793 |
A rectangular garden's length is $(3n-1)$ meters and its width is $(n+4)$ meters. The area of the garden is Y square meters. If $n=3$, determine the value of Y. | tiger_math | [
0.02727317437529564,
0.028565075248479843,
-0.05792495980858803,
-0.05279030278325081,
0.03178904205560684,
-0.04221219941973686,
-0.03583696484565735,
0.04898913577198982,
0.046021901071071625,
0.0017357039032503963,
0.015029330737888813,
0.046458709985017776,
-0.060397159308195114,
-0.00... | 0.827631 | 0.823239 | 0.004391 |
Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=x$ while $HQ=y$, then calculate $(BP)(PC)-(AQ)(QC)$ in terms of $x$ and $y$. | tiger_math | [
0.03268272429704666,
0.005540091078728437,
-0.031069930642843246,
-0.039670780301094055,
0.03408422693610191,
-0.03563999757170677,
-0.014317024499177933,
0.0534086599946022,
0.05373765155673027,
-0.017233390361070633,
0.02712758630514145,
0.033075541257858276,
-0.034316904842853546,
-0.01... | 0.831467 | 0.813926 | 0.017541 |
Let $x$, $y$, and $z$ be positive real numbers such that $(x \cdot y) + 2z = (x + 2z) \cdot (y + 2z)$. What is the maximum possible value of $xyz$? | tiger_math | [
0.04291802644729614,
0.035687901079654694,
-0.04235855117440224,
-0.03749324753880501,
0.024128612130880356,
-0.00401151180267334,
-0.011464530602097511,
0.03873259201645851,
0.03263647109270096,
-0.034691471606492996,
0.01338327769190073,
0.04754217714071274,
-0.035617027431726456,
-0.030... | 0.846666 | 0.836856 | 0.00981 |
A binary operation $\star$ has the properties that $a\, \star\, (b\, \star \,c) = (a\, \star \,b) + c$ and that $a\, \star \,a=0$ for all real numbers $a, b,$ and $c$. Find the solution to the equation $2016 \, \star\, (6\, \star\, x)=100.$ | tiger_math | [
0.030140075832605362,
0.008166434243321419,
-0.04281681403517723,
-0.011082273907959461,
0.050784461200237274,
-0.02854522317647934,
-0.012712698429822922,
0.09134913235902786,
0.0642683357000351,
-0.02800891548395157,
0.024619299918413162,
0.0438375361263752,
-0.025415167212486267,
-0.036... | 0.837165 | 0.826571 | 0.010594 |
In three-dimensional space, find the number of lattice points that have a distance of X from the origin.
Note: A point is a lattice point if all its coordinates are integers.
If we know the answer to the above question is 30, what is the value of unknown variable X? | tiger_math | [
0.012496412731707096,
0.025460777804255486,
-0.05932888016104698,
-0.023244423791766167,
0.036559153348207474,
-0.014212916605174541,
-0.01814618520438671,
0.0437387079000473,
0.03977392986416817,
-0.021070506423711777,
0.02882181666791439,
0.04090653359889984,
-0.05172784626483917,
-0.025... | 0.826028 | 0.823096 | 0.002932 |
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