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 |
|---|---|---|---|---|---|
We define two sequences of vectors $(\mathbf{v}_n)$ and $(\mathbf{w}_n)$ as follows: First, $\mathbf{v}_0 = \begin{pmatrix} 1 \\ 3 \end{pmatrix},$ $\mathbf{w}_0 = \begin{pmatrix} X \\ 0 \end{pmatrix}.$ Then for all $n \ge 1,$ $\mathbf{v}_n$ is the projection of $\mathbf{w}_{n - 1}$ onto $\mathbf{v}_0,$ and $\mathbf{w}... | tiger_math | [
0.01856590248644352,
0.034966982901096344,
-0.07012028992176056,
-0.028606658801436424,
0.04377251863479614,
0.002942686667665839,
-0.005724216811358929,
0.08281808346509933,
0.04737568646669388,
-0.011103815399110317,
0.01801495999097824,
0.020852666348218918,
-0.02893710322678089,
-0.003... | 0.842468 | 0.829501 | 0.012968 |
An auto assembly plant performs 6 functions with each frame that arrives: add axles, add wheels to the axles, install the windshield to the frame, install the instrument panel, install the steering wheel, and install the interior seating. Once those 6 tasks are performed, each car goes to a separate building for finish... | tiger_math | [
0.028030244633555412,
0.010829451493918896,
-0.03184555843472481,
-0.012301674112677574,
0.02752023935317993,
-0.004601707682013512,
-0.053143762052059174,
0.05146849527955055,
0.03228451684117317,
-0.01224883459508419,
0.008472437039017677,
0.037078190594911575,
-0.019503911957144737,
-0.... | 0.809359 | 0.803725 | 0.005634 |
In the diagram below, suppose $\sin R = \frac{3}{8}$. What is the length of $RQ$?
[asy]
pair Q,R,S;
S = (0,0);
Q = (sqrt(55),0);
R = (sqrt(55),3);
draw(S--Q--R--S);
draw(rightanglemark(S,Q,R,13));
label("$S$",S,NW);
label("$Q$",Q,NE);
label("$R$",R,SE);
label("$8$",(R+S)/2,SW);
[/asy] | tiger_math | [
0.006357353646308184,
0.020813975483179092,
-0.052289027720689774,
-0.051880646497011185,
0.029998470097780228,
-0.009366795420646667,
-0.03091619908809662,
0.054023005068302155,
0.028932087123394012,
-0.0008992744842544198,
0.017229173332452774,
0.019792694598436356,
-0.025845054537057877,
... | 0.834631 | 0.831485 | 0.003146 |
A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(-i)$ are both real. What is the smallest possible value of $| \alpha | + |\gamma |$? | tiger_math | [
0.005596502218395472,
0.040798015892505646,
-0.0522555410861969,
-0.040238913148641586,
0.04232727736234665,
-0.009527609683573246,
0.004321873653680086,
0.060274429619312286,
0.048715684562921524,
-0.0014027878642082214,
0.014454914256930351,
0.02188742347061634,
-0.0261625275015831,
-0.0... | 0.848801 | 0.83464 | 0.014162 |
Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of
\[\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.\] | tiger_math | [
0.02367917262017727,
0.029621366411447525,
-0.04678738862276077,
-0.026368051767349243,
0.04165994003415108,
-0.014275837689638138,
-0.0008617579587735236,
0.05300837382674217,
0.052725061774253845,
-0.03399992734193802,
-0.0026451253797858953,
0.02174319326877594,
-0.022235678508877754,
-... | 0.841297 | 0.838357 | 0.00294 |
Let $m \ge 4$ be an integer and let $S = \{4,5,6,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$. | tiger_math | [
0.030549103394150734,
0.017101678997278214,
-0.0484183244407177,
-0.015927763655781746,
0.03476496413350105,
-0.018386974930763245,
-0.016998104751110077,
0.05489249899983406,
0.02925286442041397,
-0.018664512783288956,
0.029714088886976242,
0.023022236302495003,
-0.014256405644118786,
-0.... | 0.864788 | 0.840296 | 0.024492 |
Let $\mathbf{M}$ be a matrix, and let $\mathbf{v}$ and $\mathbf{w}$ be vectors, such that
\[\mathbf{M} \mathbf{v} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \quad \text{and} \quad \mathbf{M} \mathbf{w} = \begin{pmatrix} -2 \\ -5 \end{pmatrix}.\]Compute $\mathbf{M} (\mathbf{v} + 3 \mathbf{w}).$ | tiger_math | [
0.0489545501768589,
0.02789193205535412,
-0.04925578460097313,
-0.038364145904779434,
0.060463737696409225,
-0.005181546323001385,
-0.03232954069972038,
0.08034320175647736,
0.034291382879018784,
-0.004830093123018742,
0.016401780769228935,
0.034563373774290085,
-0.04438585042953491,
-0.03... | 0.826804 | 0.821702 | 0.005102 |
Positive real numbers $r,s$ satisfy the equations $r^2 + s^2 = X and $r^4 + s^4= \frac{7}{8}$. Find $rs$.
If we know the answer to the above question is \frac{1}{4}, what is the value of unknown variable X? | tiger_math | [
0.007092196494340897,
0.028592979535460472,
-0.06840772181749344,
-0.022044813260436058,
0.05311060696840286,
-0.005046047270298004,
0.00038474687607958913,
0.04912377521395683,
0.05913065746426582,
-0.013590997084975243,
0.0034652752801775932,
0.026890233159065247,
-0.020514313131570816,
... | 0.845137 | 0.832282 | 0.012855 |
If $\mathbf{v} \times \mathbf{w} = \begin{pmatrix} 5 \\ -2 \\ 4 \end{pmatrix}$ and $|\mathbf{v}| = 3,$ find $|(\mathbf{v} + \mathbf{w}) \times (\mathbf{v} + \mathbf{w})|.$ | tiger_math | [
0.03537185117602348,
0.03344404697418213,
-0.04868736118078232,
-0.028999626636505127,
0.05526495724916458,
0.004884648602455854,
-0.025064952671527863,
0.08999592065811157,
0.02552800439298153,
0.0043138680048286915,
0.007749210111796856,
0.030516378581523895,
-0.031018519774079323,
-0.03... | 0.824041 | 0.817533 | 0.006509 |
Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\mathbf{a}\| = 4,$ $\|\mathbf{b}\| = 6,$ and $\|\mathbf{a} - \mathbf{b}\| = 5.$ Find $\cos \theta,$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}.$ | tiger_math | [
0.04053494706749916,
0.038351766765117645,
-0.049213577061891556,
-0.037853728979825974,
0.03874319791793823,
0.0003312070039100945,
-0.019439827650785446,
0.059795111417770386,
0.036695655435323715,
-0.0028548231348395348,
0.023325525224208832,
0.01750938966870308,
-0.013117535971105099,
... | 0.822858 | 0.816821 | 0.006037 |
A town has a park shaped like a right triangle. The longer leg of the triangle is three times as long as the shorter leg, and the hypotenuse is 15 meters longer than the shorter leg. What are the lengths of the legs and the hypotenuse of the triangular park? | tiger_math | [
0.02741117961704731,
0.012917793355882168,
-0.05254875868558884,
-0.03783174976706505,
0.03176364302635193,
-0.031538352370262146,
-0.0348828062415123,
0.04683418199419975,
0.04424167051911354,
-0.02074367180466652,
0.02009524591267109,
0.04488655924797058,
-0.04002838954329491,
-0.0241961... | 0.827028 | 0.821639 | 0.005389 |
Compute
\[\begin{pmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{pmatrix} \begin{pmatrix} a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2 \end{pmatrix}.\] | tiger_math | [
0.03936440497636795,
0.05208444222807884,
-0.04570801556110382,
-0.03166177123785019,
0.05053161457180977,
-0.03356991335749626,
-0.006194248329848051,
0.05909371003508568,
0.0368838794529438,
-0.011077750474214554,
0.00594909256324172,
0.021540582180023193,
-0.04240154102444649,
-0.061995... | 0.815775 | 0.811971 | 0.003804 |
In triangle $XYZ$, we have $\angle X = 90^\circ$ and $\cos Y = \frac{5}{9}$. Find $\sin Z$. | tiger_math | [
0.016187481582164764,
0.02530614845454693,
-0.034786876291036606,
-0.04558427631855011,
0.025788486003875732,
-0.005621090531349182,
-0.013735366053879261,
0.03200109302997589,
0.038799818605184555,
-0.021941635757684708,
0.04640461504459381,
0.024040546268224716,
-0.026475533843040466,
0.... | 0.832769 | 0.826076 | 0.006693 |
In a bin, there are 5 green balls and an unknown positive integer k number of purple balls. A ball is randomly drawn from the bin. If a green ball is drawn, the player wins 2 dollars. If a purple ball is drawn, the player loses 2 dollars. If the expected amount won for playing the game is 50 cents, what is the value of... | tiger_math | [
-0.005211615469306707,
0.031065216287970543,
-0.06244770810008049,
-0.03850402683019638,
0.02893110178411007,
-0.015624814666807652,
-0.024920988827943802,
0.03620417043566704,
0.04679105058312416,
-0.015914281830191612,
0.024363627657294273,
0.03665778785943985,
-0.02790094166994095,
-0.0... | 0.833668 | 0.830516 | 0.003152 |
A point $Q$ is chosen at random in the interior of a square $ABCD$. What is the probability that $\triangle ABQ$ has a greater area than at least two of $\triangle BCQ$, $\triangle CDQ$, and $\triangle DAQ$? | tiger_math | [
0.006891897413879633,
0.017982713878154755,
-0.045239515602588654,
-0.04057019576430321,
0.027254337444901466,
-0.010729405097663403,
-0.021528318524360657,
0.0410604327917099,
0.0354803204536438,
-0.011317775584757328,
0.012778541073203087,
0.029896147549152374,
-0.044582586735486984,
-0.... | 0.842601 | 0.823676 | 0.018924 |
Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x \le a$ and $0\le y \le b$. Find $\rho^2.$ | tiger_math | [
0.034888964146375656,
0.023138733580708504,
-0.04386455938220024,
-0.027115877717733383,
0.027213215827941895,
-0.015159035101532936,
-0.011430264450609684,
0.05610530450940132,
0.04752959683537483,
-0.02962154895067215,
0.013573355972766876,
0.03903380408883095,
-0.032486964017152786,
-0.... | 0.854437 | 0.823645 | 0.030792 |
A mythical land called Mathemagica uses a unique currency system based on triangular coins. Every triangular coin has a distinct edge length which corresponds to the number of units it is worth. For example, a coin with an edge length of 3 units is worth 3 Mathemagica Dollars. Jake exchanges 15 such coins at the bank, ... | tiger_math | [
0.023243864998221397,
0.012044677510857582,
-0.0444401316344738,
-0.05269867554306984,
0.01862446404993534,
-0.02982836775481701,
-0.041369516402482986,
0.04487544670701027,
0.017504949122667313,
-0.027884462848305702,
0.011128378100693226,
0.026886774227023125,
-0.02628898434340954,
-0.02... | 0.836344 | 0.818006 | 0.018338 |
While practising for his upcoming math exams, Hayes realised that the area of a circle he had just solved was equal to the perimeter of a square he had solved in the previous problem. The area of the circle was 100. What is the length of one side of the square? | tiger_math | [
0.03647405281662941,
0.013161718845367432,
-0.04073246940970421,
-0.055959079414606094,
0.03801538795232773,
-0.023114169016480446,
-0.02398715168237686,
0.06072154641151428,
0.03261914849281311,
-0.016217779368162155,
-0.0022114517632871866,
0.026692846789956093,
-0.025979219004511833,
-0... | 0.819586 | 0.812323 | 0.007263 |
Let $p(x)$ be a monic quartic polynomial such that $p(-1) = -2,$ $p(0) = 1,$ $p(1) = 6,$ and $p(2) = 13.$ Find $p(3).$ | tiger_math | [
0.014050420373678207,
0.03170914947986603,
-0.049944888800382614,
-0.039054691791534424,
0.02634160779416561,
-0.021095622330904007,
-0.0012737121433019638,
0.043523166328668594,
0.025529637932777405,
-0.006750569678843021,
0.012585305608808994,
0.03133009374141693,
-0.039766278117895126,
... | 0.850129 | 0.84429 | 0.005839 |
Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 1.$ Find the set of all possible values of $xy + xz + yz.$ | tiger_math | [
0.028350356966257095,
0.035862892866134644,
-0.03640350326895714,
-0.045037128031253815,
0.04157780110836029,
-0.009102972224354744,
-0.0012220788048580289,
0.03913269564509392,
0.046983275562524796,
-0.03523833304643631,
0.028086859732866287,
0.033397823572158813,
-0.04302084818482399,
-0... | 0.845698 | 0.829546 | 0.016152 |
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 2 belongs to $\mathcal{S}$ and that 2023 is the largest element of $\mathcal{S},$ what is t... | tiger_math | [
0.05801503360271454,
0.03253697603940964,
-0.04320143535733223,
-0.048653073608875275,
0.01854642853140831,
-0.033829767256975174,
-0.041553884744644165,
0.04903636500239372,
0.04043811559677124,
-0.02876649983227253,
0.017723236232995987,
0.024309704080224037,
-0.027846744284033775,
0.002... | 0.840574 | 0.831262 | 0.009313 |
Let $X$ be a point not on line $YZ$ and $W$ a point on line $YZ$ such that $XW \perp YZ.$ Meanwhile, $V$ is a point on line $XZ$ such that $UV \perp XZ.$ If $UV = 12,$ $XW = 6,$ and $YZ = 10,$ then what is the length of $XZ?$ | tiger_math | [
-0.0018750008894130588,
0.01417961809784174,
-0.04040290787816048,
-0.05124948173761368,
0.03241204470396042,
-0.006954045034945011,
-0.02678675763309002,
0.04064934328198433,
0.02739466167986393,
-0.027844209223985672,
0.01752842776477337,
0.03736152499914169,
-0.011009246110916138,
-0.00... | 0.843714 | 0.835743 | 0.007972 |
In a rectangular array of points, with 6 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Six points, $P_1, P_2, P_3, P_4, P_5,$ and $P_6,$ are selected so... | tiger_math | [
0.011295676231384277,
0.050755683332681656,
-0.02893076464533806,
-0.03232051432132721,
0.0286197979003191,
-0.020562928169965744,
-0.017647191882133484,
0.0649321973323822,
0.029498552903532982,
-0.0072307582013309,
0.022464552894234657,
0.022120652720332146,
-0.03615950420498848,
-0.0025... | 0.844654 | 0.825973 | 0.01868 |
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{3}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer? | tiger_math | [
0.02001139149069786,
0.050469040870666504,
-0.04072243720293045,
-0.03698083013296127,
0.04006265103816986,
-0.011238561011850834,
-0.025944020599126816,
0.07682868093252182,
0.04577105864882469,
-0.01958698220551014,
0.011768968775868416,
0.0274548027664423,
-0.039407748728990555,
-0.0356... | 0.850676 | 0.83982 | 0.010856 |
Suppose that we are given 36 points equally spaced around the perimeter of a regular hexagon, so that six of them are located at the vertices and the remaining points divide each side into six congruent segments. If $P$, $Q$, and $R$ are chosen to be any three of these points which are not collinear, then how many dif... | tiger_math | [
0.021623950451612473,
0.02091791480779648,
-0.04083191230893135,
-0.03857395052909851,
0.021373672410845757,
-0.018397796899080276,
-0.030065739527344704,
0.02761014737188816,
0.03937610983848572,
-0.024192839860916138,
0.021578382700681686,
0.029299380257725716,
-0.04294493794441223,
-0.0... | 0.84534 | 0.830949 | 0.014391 |
A right square pyramid with base edges of length $8\sqrt{2}$ units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 4 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane? | tiger_math | [
0.03039846196770668,
0.03181076794862747,
-0.05183921381831169,
-0.025910578668117523,
0.02863049879670143,
-0.024661947041749954,
-0.0375886894762516,
0.051554206758737564,
0.043950408697128296,
-0.00211350223980844,
0.013698533177375793,
0.024694349616765976,
-0.028466472402215004,
-0.01... | 0.831531 | 0.825199 | 0.006332 |
$\triangle ABC$ is similar to $\triangle DEF$ . If the length of $\overline{BC}$ is 8 cm, the length of $\overline{DF}$ is 6 cm, and the length of $\overline{EF}$ is 12 cm, what is the length of $\overline{AB}$ in centimeters?
[asy]
draw((0,0)--(8,-2)--(5,4)--cycle);
label("8cm",(2.5,2),NW);
label("5cm",(6.1,1),NE);
d... | tiger_math | [
0.029682675376534462,
-0.006190941669046879,
-0.04711666330695152,
-0.05751235410571098,
0.03890887647867203,
-0.03520699217915535,
-0.026028279215097427,
0.05220125615596771,
0.03847896680235863,
0.005884621292352676,
0.015820205211639404,
0.029159123077988625,
-0.033018968999385834,
-0.0... | 0.824989 | 0.817182 | 0.007808 |
Math City has eight streets, all of which are straight. No street is parallel to another street, and no three streets intersect at the same point. One police officer is stationed at each intersection. What is the greatest number of police officers needed? | tiger_math | [
0.03927965462207794,
0.00906603317707777,
-0.04241282492876053,
-0.04151426628232002,
0.03240301460027695,
0.004654992371797562,
-0.047527916729450226,
0.024449434131383896,
0.04145841300487518,
-0.027085723355412483,
-0.00866981316357851,
0.032891903072595596,
-0.012043094262480736,
-0.05... | 0.812551 | 0.802819 | 0.009732 |
The number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{10} \le 2008$ such that $a_i-i$ is even for $1\le i \le 10$ can be expressed as ${m \choose n}$ for some positive integers $m > n$. Compute the remainder when $m$ is divided by 1000. | tiger_math | [
0.023076573386788368,
0.02122974582016468,
-0.033335085958242416,
-0.018481815233826637,
0.01713273860514164,
-0.02147149108350277,
-0.05219492316246033,
0.0816984549164772,
0.03978574275970459,
-0.017182860523462296,
0.019939301535487175,
0.03243175521492958,
-0.06341966986656189,
-0.0012... | 0.836997 | 0.819558 | 0.017439 |
Given vectors $\mathbf{a} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$, determine the area of the triangle formed by the points $\mathbf{0}$, $\mathbf{a}$, and $\mathbf{b}$. | tiger_math | [
0.039210256189107895,
0.040422022342681885,
-0.05938535928726196,
-0.04238296300172806,
0.04357552528381348,
-0.010324426926672459,
-0.013652973808348179,
0.07493428885936737,
0.041589170694351196,
0.008461009711027145,
0.00229633622802794,
0.020947694778442383,
-0.022118518128991127,
-0.0... | 0.823041 | 0.81395 | 0.009092 |
In triangle $ABC$, $BC = 18 \sqrt{3}$ and $\angle C = 30^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$. | tiger_math | [
0.020011233165860176,
0.006496078334748745,
-0.04701242223381996,
-0.04785964637994766,
0.03508936986327171,
-0.010434325784444809,
-0.029293781146407127,
0.04812651500105858,
0.042392533272504807,
-0.012000161223113537,
0.03085065633058548,
0.03002758137881756,
-0.04306581988930702,
-0.02... | 0.83766 | 0.825256 | 0.012404 |
Tiffany is constructing a fence around a rectangular tennis court. She must use exactly 300 feet of fencing. The fence must enclose all four sides of the court. Regulation states that the length of the fence enclosure must be at least 76 feet and the width must be at least 42 feet. Tiffany wants the area enclosed by th... | tiger_math | [
0.0008324346272274852,
0.03436814621090889,
-0.05269433930516243,
-0.04213031381368637,
0.022041792050004005,
-0.02251904085278511,
-0.03358254209160805,
0.05208619311451912,
0.035382550209760666,
-0.00005453580160974525,
0.005425248295068741,
0.044187285006046295,
-0.02890382707118988,
-0... | 0.794301 | 0.781616 | 0.012684 |
A teacher wants to arrange 3 copies of Introduction to Geometry and X copies of Introduction to Number Theory on a bookshelf. In how many ways can he do that?
If we know the answer to the above question is 35, what is the value of unknown variable X? | tiger_math | [
0.030847841873764992,
0.004945650696754456,
-0.05904780328273773,
-0.021994005888700485,
0.028965841978788376,
-0.0003713774785865098,
-0.022826362401247025,
0.0416877306997776,
0.032874707132577896,
-0.015879563987255096,
0.012407202273607254,
0.03228813782334328,
-0.04838120937347412,
-0... | 0.839843 | 0.834965 | 0.004878 |
Coach Grunt is selecting a 5-person starting lineup for his basketball team, the Grunters, from a roster of 12 players. Two of the players, Ace and Zeppo, are guaranteed to be in the starting lineup. How many different possible starting lineups can Coach Grunt create? (Note: The order of the players in the lineup does ... | tiger_math | [
0.03700634837150574,
0.006909522227942944,
-0.036422427743673325,
-0.025297582149505615,
0.04181120917201042,
-0.035686638206243515,
-0.0066269515082240105,
0.051183994859457016,
0.030880598351359367,
-0.0382695347070694,
0.0068931253626942635,
0.047395698726177216,
-0.035766828805208206,
... | 0.813895 | 0.809899 | 0.003996 |
Given vectors $\mathbf{u}$ and $\mathbf{v}$ such that $\|\mathbf{u}\| = 5,$ $\|\mathbf{v}\| = 10,$ and $\mathbf{u} \cdot \mathbf{v} = 25,$ then find $\|\operatorname{proj}_{\mathbf{v}} \mathbf{u}\|.$ | tiger_math | [
0.021235452964901924,
0.031150169670581818,
-0.04690384119749069,
-0.026766633614897728,
0.036981578916311264,
-0.002596741309389472,
-0.02085052989423275,
0.07438655197620392,
0.02455076202750206,
-0.009272011928260326,
0.011524112895131111,
0.03041253611445427,
-0.029582520946860313,
-0.... | 0.828333 | 0.818645 | 0.009688 |
We know that $X$ and $Y$ are integers with $100 < X < 500$ and $Y > 1$. If $X = Y \times Y^3$, what is the value of $Y$ for which $X$ is a perfect cube? | tiger_math | [
0.01744784042239189,
0.022482458502054214,
-0.057178497314453125,
-0.01794520765542984,
0.033050648868083954,
-0.017488086596131325,
-0.002412435133010149,
0.05275743082165718,
0.06285681575536728,
-0.03434399515390396,
0.017281463369727135,
0.030780136585235596,
-0.03184202313423157,
-0.0... | 0.843352 | 0.834709 | 0.008642 |
The parabola $y = x^2+2$ and the hyperbola $y^2 - mx^2 = 1$ intersect at a single point. Find $m.$ | tiger_math | [
0.03201338276267052,
0.030478624626994133,
-0.026835989207029343,
-0.0606495700776577,
0.04070122167468071,
-0.012970062904059887,
-0.02359631098806858,
0.07481905817985535,
0.04863246902823448,
-0.03481506183743477,
0.014532904140651226,
0.018012898042798042,
-0.005450983066111803,
-0.025... | 0.835562 | 0.822052 | 0.01351 |
Let $S$ be a subset of $\{1,2,3,\ldots,2023\}$ such that no two members of $S$ differ by $5$ or $9$. What is the largest number of elements $S$ can have? | tiger_math | [
0.03343479335308075,
0.016800900921225548,
-0.05177953466773033,
-0.040913283824920654,
0.021016797050833702,
-0.03152923285961151,
-0.022603660821914673,
0.046045370399951935,
0.029683107510209084,
-0.0301069226115942,
0.04715900495648384,
0.03508315235376358,
-0.03202119469642639,
0.0072... | 0.837794 | 0.834367 | 0.003426 |
In a conference room, 40 chairs with a capacity of 2 people each were arranged in rows in preparation for the board meeting of a company, whose number of members was the same as the chairs' capacity. If 2/5 of the chairs were not occupied, and the rest each had two people, calculate the number of board members who did ... | tiger_math | [
0.02750188298523426,
0.01713475026190281,
-0.050478994846343994,
-0.019376639276742935,
0.028164802119135857,
-0.02595287747681141,
-0.032318100333213806,
0.049190785735845566,
0.0349404402077198,
-0.0016951224533841014,
0.002339123049750924,
0.01711437851190567,
-0.04152647778391838,
-0.0... | 0.817136 | 0.811898 | 0.005238 |
A farmer is planting two types of crops, wheat and corn, on his farm. Together, they should cover 120 acres. He wants the area planted with wheat to be ten times that of the area planted with corn. Also, he plans to spend $90 on seeds for these crops. Each acre of wheat costs $1 for seeds, while each acre of corn costs... | tiger_math | [
0.01138310320675373,
0.023419607430696487,
-0.04586779326200485,
-0.036118071526288986,
0.02006174996495247,
-0.0238358061760664,
-0.027112090960144997,
0.07684556394815445,
0.03633911535143852,
-0.015674594789743423,
-0.014150010421872139,
0.02813701331615448,
-0.05064185708761215,
-0.022... | 0.80435 | 0.800397 | 0.003953 |
In a town, there is a circular park with a perimeter surrounded by 4 gates. One fine morning, two friends, Alice and Bob, started cycling from the same gate but in opposite directions. Alice cycles at a constant speed of 11 mph, and Bob cycles at 9 mph. If the circumference of the park is 5 miles, after how long will t... | tiger_math | [
0.027636589482426643,
0.022745316848158836,
-0.0558769591152668,
-0.037194449454545975,
0.014192555099725723,
-0.02728596143424511,
-0.04089410975575447,
0.031480997800827026,
0.04444116726517677,
-0.01111168134957552,
0.0087957214564085,
0.022691797465085983,
-0.017092248424887657,
-0.017... | 0.829512 | 0.825542 | 0.00397 |
In a magical land, a gardener is planting two types of magical trees in a large circular field. The first type, Moonlight Trees, are planted such that they form the exact circumference of the circle. The gardener places 20 Moonlight Trees evenly around the circle. The second type, Starlight Trees, are planted along two... | tiger_math | [
0.005188560113310814,
0.04681989923119545,
-0.04071209952235222,
-0.053277388215065,
0.030434206128120422,
-0.034825243055820465,
-0.02926715649664402,
0.04642412066459656,
0.03463756665587425,
-0.01873852126300335,
0.00460470374673605,
0.037819717079401016,
-0.02372606284916401,
-0.009943... | 0.827404 | 0.808763 | 0.018641 |
Let $z_1$ and $z_2$ be the complex roots of $z^2 + az + b = 0,$ where $a$ and $b$ are complex numbers. In the complex plane, 0, $z_1,$ and $z_2$ form the vertices of an equilateral triangle with side length $1$. Find $\frac{a^2}{b}.$ | tiger_math | [
0.03765658661723137,
0.040022511035203934,
-0.03929679095745087,
-0.04572943225502968,
0.027370762079954147,
-0.010933266952633858,
0.011712188832461834,
0.04530561342835426,
0.03706678748130798,
-0.017243418842554092,
0.015487064607441425,
0.03367768973112106,
-0.031054753810167313,
-0.02... | 0.840232 | 0.815311 | 0.024921 |
Alice is planning a garden and needs to purchase fencing. Her garden is in the shape of a right triangle. The two legs of the triangle measure 30 meters and 16 meters, respectively. However, she wants to include an additional 2 meters of space around her garden on all sides, requiring more fencing. Calculate the amount... | tiger_math | [
0.034889232367277145,
0.03676333278417587,
-0.046904727816581726,
-0.046046532690525055,
0.03529768064618111,
-0.044375475496053696,
-0.03399490565061569,
0.05364350229501724,
0.04028429463505745,
0.003343421034514904,
0.015522897243499756,
0.043097030371427536,
-0.028239266946911812,
-0.0... | 0.80753 | 0.801433 | 0.006096 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z}.\] | tiger_math | [
0.018419500440359116,
0.036973003298044205,
-0.040643543004989624,
-0.04620219022035599,
0.05159900337457657,
0.0005998157430440187,
-0.007146771531552076,
0.026960588991642,
0.041192714124917984,
-0.019391527399420738,
0.010689876973628998,
0.008722279220819473,
-0.020261486992239952,
-0.... | 0.841098 | 0.830023 | 0.011076 |
Express $\sqrt{x} \div\sqrt{y}$ as a common fraction, given:
$\frac{ {\left( \frac{1}{2} \right)}^2 - {\left( \frac{1}{3} \right)}^2 }{ {\left( \frac{1}{4} \right)}^2 - {\left( \frac{1}{5} \right)}^2} = \frac{5x}{11y} $ | tiger_math | [
0.015477782115340233,
0.043230727314949036,
-0.04502728208899498,
-0.06606504321098328,
0.041894350200891495,
-0.001351725310087204,
-0.027052998542785645,
0.06657110899686813,
0.05121445283293724,
-0.01631012000143528,
0.02494465932250023,
0.03147421032190323,
-0.037716444581747055,
-0.00... | 0.828817 | 0.814123 | 0.014695 |
Determine the maximum possible value of the expression
\[\frac{2b}{a} + \frac{4c}{b} + \frac{8a}{c},\]where $a,$ $b,$ and $c$ are positive real numbers. | tiger_math | [
0.04031487554311752,
0.03777847811579704,
-0.045806847512722015,
-0.02183823473751545,
0.03788906708359718,
-0.012542118318378925,
-0.01706635393202305,
0.06730733066797256,
0.05035719648003578,
-0.02949799969792366,
0.00016419855819549412,
0.03168611600995064,
-0.018862679600715637,
-0.03... | 0.829233 | 0.825862 | 0.003371 |
A circle with center $A$ and radius three inches is tangent at $C$ to a circle with center $B$, as shown. If point $B$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
[asy]
filldraw(circle((0,0),6),gray,linewidth(2));
filldraw(circle(3dir(-30),3),white,linewidth(2)... | tiger_math | [
0.0034656256902962923,
0.0452730767428875,
-0.041443295776844025,
-0.03570869565010071,
0.03995668143033981,
-0.04032671079039574,
-0.02303200401365757,
0.051311034709215164,
0.03850339725613594,
0.0024465464521199465,
0.02658950537443161,
0.03664354607462883,
-0.015285085886716843,
-0.019... | 0.830762 | 0.821976 | 0.008786 |
Let $f(x) = \frac{3}{9^x + 3}.$ Find
\[f \left( \frac{1}{1001} \right) + f \left( \frac{2}{1001} \right) + f \left( \frac{3}{1001} \right) + \dots + f \left( \frac{1000}{1001} \right).\] | tiger_math | [
0.018201610073447227,
0.0280496496707201,
-0.04957185685634613,
-0.03186613321304321,
0.020883122459053993,
-0.027288153767585754,
-0.027731817215681076,
0.07677273452281952,
0.03589944168925285,
-0.005707495380192995,
0.03063482604920864,
0.02799239009618759,
-0.0339062437415123,
-0.01920... | 0.840137 | 0.835627 | 0.00451 |
At a zoo, the animal keeper arranges a math-themed scavenger hunt. The first clue leads to the pandas, where Greta spots 4 groups of pandas. Each group has 5 pandas. The second clue takes the group to the flamingos, where they see 3 sets of flamingo pairs. Calculate the total number of animals Greta spotted on her scav... | tiger_math | [
0.013257247395813465,
0.011498578824102879,
-0.05783563479781151,
-0.028077257797122,
0.023336347192525864,
-0.02613900788128376,
-0.05590444803237915,
0.035931333899497986,
0.032221805304288864,
0.0004962445236742496,
-0.008644022978842258,
0.024728301912546158,
-0.040747951716184616,
-0.... | 0.815796 | 0.812146 | 0.00365 |
A store sells four different items, each with multiple costs of production and fluctuating sale prices over a three-month period.
Item A:
- Sales prices: $44 in month 1, $47 in month 2, $50 in month 3
- Costs of production: $20 in month 1, $22 in month 2, $25 in month 3
- Gross profit: 120% of cost
Item B:
- Sales ... | tiger_math | [
0.04442395642399788,
0.016224229708313942,
-0.05167248845100403,
-0.023900743573904037,
0.024637538939714432,
-0.04201393574476242,
-0.056403324007987976,
0.082036592066288,
0.033128153532743454,
0.009986952878534794,
0.031963929533958435,
0.03573686257004738,
-0.04893023520708084,
-0.0150... | 0.806151 | 0.80147 | 0.004682 |
Convert the point $\left( 6, \frac{5\pi}{6}, 2 \right)$ in cylindrical coordinates to rectangular coordinates. | tiger_math | [
0.0003277140494901687,
0.05734917148947716,
-0.023243216797709465,
-0.034323837608098984,
0.0272271279245615,
-0.020949233323335648,
-0.03043937496840954,
0.059983380138874054,
0.04478350281715393,
-0.015289859846234322,
0.00390700064599514,
0.039757777005434036,
-0.018478242680430412,
-0.... | 0.839053 | 0.834389 | 0.004664 |
Given a sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ let $S_n$ denote the sum of the first $n$ terms of the sequence.
If $a_1 = X and
\[a_n = \frac{2S_n^2}{2S_n - 1}\]for all $n \ge 2,$ then find $a_{100}. The answer is 39203. What is the value of unknown variable X? | tiger_math | [
0.0004000298504251987,
0.02803024649620056,
-0.05283196642994881,
-0.021639598533511162,
0.025526225566864014,
-0.01127098873257637,
-0.021913331001996994,
0.0687861442565918,
0.031580399721860886,
-0.021151160821318626,
0.018168650567531586,
0.02958010509610176,
-0.03663676232099533,
0.00... | 0.844128 | 0.835073 | 0.009055 |
Let a sequence $b_1, b_2, \ldots$ be defined as $b_1 = 1$, $b_2 = X$, and $b_{n+1} = b_n b_{n-1}$. Calculate $b_{20}$
If we know the answer to the above question is 2^{4181}, what is the value of unknown variable X? | tiger_math | [
0.0023120874539017677,
0.036842864006757736,
-0.059785448014736176,
-0.02198183536529541,
0.027228515595197678,
-0.018428213894367218,
-0.015328727662563324,
0.07006555795669556,
0.04719538614153862,
-0.010946162976324558,
0.0036484217271208763,
0.028195686638355255,
-0.04651128873229027,
... | 0.843633 | 0.838089 | 0.005544 |
In a magical land, a magician offers to create either a circular, a square, or a triangular cookie for each guest at a party. Knowing there are 15 guests, and each shape must be used at least once, how many ways can the magician distribute these shapes if no guest may receive more than one cookie? | tiger_math | [
-0.014436905272305012,
0.027861058712005615,
-0.024153685197234154,
-0.044726114720106125,
0.024508444592356682,
-0.009179879911243916,
-0.033547114580869675,
0.0482843779027462,
0.020611081272363663,
-0.02404305525124073,
0.02012486569583416,
0.03909463435411453,
-0.018373459577560425,
-0... | 0.831806 | 0.823028 | 0.008778 |
Amanda and her family are going to re-paint all the walls inside their house. Before they get started they want to divide up the work. Since all the rooms in the house have different numbers and sizes of walls in them, they figure the fairest way to divide up the work is to count all the walls in the house and assign a... | tiger_math | [
0.013457811437547207,
0.028586873784661293,
-0.038096003234386444,
-0.02945798635482788,
0.03859619423747063,
-0.02354452759027481,
-0.028674645349383354,
0.04691028967499733,
0.03342596814036369,
-0.015818055719137192,
0.021607378497719765,
0.04175557196140289,
-0.04938409850001335,
-0.02... | 0.823755 | 0.819562 | 0.004193 |
Emily is planning to organize a space-themed math treasure hunt for her little brother's birthday party. She decides to hide clues in the form of math puzzles at various stations. At the first station, the hosts must calculate the area of a square-shaped moon map where each side is represented by \(x + 4\). At the seco... | tiger_math | [
0.046425823122262955,
0.04573999345302582,
-0.041253652423620224,
-0.051937323063611984,
0.034072794020175934,
-0.0057831103913486,
-0.035645850002765656,
0.04435235261917114,
0.04957868903875351,
-0.012191454879939556,
-0.001707873190753162,
0.039154719561338425,
-0.014074218459427357,
-0... | 0.831065 | 0.815248 | 0.015816 |
The sequence $\{a_n\}$ satisfies $a_1 = 1$ and $5^{a_{n + 1} - a_n} - 1 = \frac {1}{n + \frac {2}{3}}$ for $n \geq X$. Find the least integer $k$ greater than $1$ for which $a_k$ is an integer.
If we know the answer to the above question is 41, what is the value of unknown variable X? | tiger_math | [
0.01379449013620615,
0.039484526962041855,
-0.04216978698968887,
-0.0359467975795269,
0.04366379231214523,
-0.007182524539530277,
-0.023611152544617653,
0.08631631731987,
0.04440226033329964,
-0.010868929326534271,
0.021189309656620026,
0.011871250346302986,
-0.03776004910469055,
0.0101432... | 0.844208 | 0.830414 | 0.013793 |
Let $\triangle PQR$ be an isosceles triangle such that $P$ is the vertex angle. A circle with diameter of $QR$ meets side $PQ$ at $S$. If the perimeter of $\triangle PQR$ is $50$ and $PQ = 15$, then what is $QS$? | tiger_math | [
0.011480451561510563,
0.025016261264681816,
-0.031588032841682434,
-0.04473577067255974,
0.021590163931250572,
-0.006505852565169334,
-0.016116291284561157,
0.04327210411429405,
0.02773056924343109,
-0.008139068260788918,
0.046883441507816315,
0.026889190077781677,
-0.0347026102244854,
-0.... | 0.839552 | 0.824192 | 0.01536 |
Let $\omega$ be a nonreal root of $z^3 = 1.$ Let $a_1,$ $a_2,$ $\dots,$ $a_n$ be real numbers such that
\[\frac{1}{a_1 + \omega} + \frac{1}{a_2 + \omega} + \dots + \frac{1}{a_n + \omega} = 2 + 5i.\]If
\[\frac{2a_1 - 1}{a_1^2 - a_1 + 1} + \frac{2a_2 - 1}{a_2^2 - a_2 + 1 + \dots + \frac{2a_n - 1}{a_n^2 - a_n + 1} = 5,\]... | tiger_math | [
0.031110730022192,
0.04124315828084946,
-0.047229476273059845,
-0.05119900405406952,
0.02435402385890484,
-0.005972876213490963,
-0.008135179057717323,
0.05954958125948906,
0.010754032991826534,
-0.0038541448302567005,
0.01784989796578884,
0.018827388063073158,
-0.04831825569272041,
-0.026... | 0.849387 | 0.830034 | 0.019353 |
Find the smallest positive integer $n$ such that the sum of $20$ consecutive positive integers starting from $n$ is a perfect cube. | tiger_math | [
-0.0029681629966944456,
0.03611672297120094,
-0.04711635783314705,
-0.04342101141810417,
0.04032423719763756,
-0.018118547275662422,
-0.03479021415114403,
0.043064218014478683,
0.0491124652326107,
-0.03432190418243408,
0.02233852632343769,
0.024682149291038513,
-0.039089277386665344,
-0.00... | 0.848773 | 0.844252 | 0.004522 |
In triangle $ABC,$ $AB = 5,$ $AC = 7,$ and $\cos \angle A = \frac{1}{10}.$ Find the length of angle bisector $\overline{AD}.$ | tiger_math | [
0.030977590009570122,
0.004444546066224575,
-0.043327800929546356,
-0.04786651208996773,
0.03639829158782959,
-0.012790592387318611,
-0.021362610161304474,
0.044540885835886,
0.03965083509683609,
-0.008863844908773899,
0.01860656961798668,
0.018137404695153236,
-0.03848521038889885,
-0.001... | 0.832572 | 0.82372 | 0.008852 |
If $a>0$ and $b>0,$ a new operation $\nabla$ is defined as follows: $$a \nabla b = \frac{a + b}{1 + ab}.$$For example, $$3 \nabla 6 = \frac{3 + X + 3 \times 6} = \frac{9}{19}.$$Calculate $(1 \nabla 2) \nabla 3.$
If we know the answer to the above question is 1, what is the value of unknown variable X? | tiger_math | [
0.014645729213953018,
0.031122656539082527,
-0.061497125774621964,
-0.004374054726213217,
0.04921989142894745,
-0.0022293447982519865,
-0.016739176586270332,
0.08676307648420334,
0.03134780749678612,
0.0011483478592708707,
0.0034095074515789747,
0.0174137894064188,
-0.037795599550008774,
-... | 0.828053 | 0.822746 | 0.005308 |
$\triangle ABC$ is similar to $\triangle DEF$. If the length of $\overline{AB}$ is 10 cm, the length of $\overline{BC}$ is 16 cm, and the length of $\overline{DE}$ is 5 cm, what is the length of $\overline{EF}$ in centimeters? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(10,-2.5)--(7.5,5)-... | tiger_math | [
0.028314964845776558,
0.008592556230723858,
-0.0432269386947155,
-0.05720572918653488,
0.041227709501981735,
-0.04030213877558708,
-0.02777063101530075,
0.052880316972732544,
0.033289991319179535,
-0.0029079169034957886,
0.029606062918901443,
0.02479308284819126,
-0.02772475965321064,
-0.0... | 0.839472 | 0.832036 | 0.007436 |
Let $a,$ $b,$ $c$ be non-zero real numbers such that $a + b + c = 1.$ Find all possible values of
\[\frac{a^3 + b^3 + c^3 - 3abc}{abc}.\]Enter all the possible values, separated by commas. | tiger_math | [
0.038632627576589584,
0.029900122433900833,
-0.034183405339717865,
-0.027257831767201424,
0.035789284855127335,
-0.026377471163868904,
0.001884142984636128,
0.058590151369571686,
0.037913329899311066,
-0.030971847474575043,
0.019988972693681717,
0.03326839581131935,
-0.046268586069345474,
... | 0.836028 | 0.82444 | 0.011588 |
In a math competition, a display tower made of stacking cubes is designed so that each level has a square number of white cubes nested in a corner formation around a central black cube. If there are 5 levels in total, and the pattern of the number of white cubes in each level starts from 3 at the base and increases by ... | tiger_math | [
0.02608293853700161,
0.014757703058421612,
-0.055975280702114105,
-0.03034023940563202,
0.05032936856150627,
-0.012344872578978539,
-0.05668044090270996,
0.059161316603422165,
0.03310338780283928,
-0.006096992641687393,
-0.004109127912670374,
0.02665712870657444,
-0.0374566912651062,
0.004... | 0.838437 | 0.82367 | 0.014767 |
Find the matrix $\mathbf{P}$ such that for any vector $\mathbf{v},$ $\mathbf{P} \mathbf{v}$ is the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 2 \\ -2 \\ -1 \end{pmatrix}.$ | tiger_math | [
0.02996744215488434,
0.03889453411102295,
-0.049924254417419434,
-0.035664480179548264,
0.04324236884713173,
-0.018792996183037758,
-0.010038677603006363,
0.07554584741592407,
0.026567138731479645,
-0.018389416858553886,
0.008319901302456856,
0.03210332244634628,
-0.023971103131771088,
-0.... | 0.831323 | 0.820821 | 0.010502 |
Three vessels A, B, and C contain mixtures of milk and water in different ratios. Vessel A contains x liters of the mixture with milk and water in the ratio of 4 : 2, Vessel B contains y liters of the mixture with milk and water in the ratio of 5 : 1, and Vessel C contains z liters of the mixture with milk and water in... | tiger_math | [
0.047264762222766876,
0.02844894863665104,
-0.02144368179142475,
-0.02887416072189808,
0.0337192676961422,
-0.009667793288826942,
-0.017168298363685608,
0.059472255408763885,
0.03226045146584511,
0.0018754656193777919,
0.012037822976708412,
0.01888873614370823,
-0.026651082560420036,
-0.01... | 0.815391 | 0.812457 | 0.002934 |
Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF=3$ and $EF=8$. Then $BE$ can be expressed as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$... | tiger_math | [
0.037305284291505814,
0.036487601697444916,
-0.033358972519636154,
-0.042103394865989685,
0.031017854809761047,
-0.015072704292833805,
-0.010165045037865639,
0.0551321767270565,
0.05268041044473648,
0.0041399491019546986,
0.023594385012984276,
0.04403774440288544,
-0.027127444744110107,
-0... | 0.854696 | 0.822561 | 0.032136 |
In triangle $XYZ,$ $\cos X = \frac{4}{5}$ and $\sin Y = \frac{12}{13}.$ Find $\sin Z.$ | tiger_math | [
0.03111409954726696,
0.04214972257614136,
-0.0279338788241148,
-0.04190748929977417,
0.04088825359940529,
0.0004073386371601373,
-0.012023324146866798,
0.028424009680747986,
0.033661436289548874,
-0.011132242158055305,
0.04442819952964783,
0.022180620580911636,
-0.012165017426013947,
-0.01... | 0.838065 | 0.832286 | 0.00578 |
Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate
\[\begin{vmatrix} \cos^2 A & \tan A & 1 \\ \cos^2 B & \tan B & 1 \\ \cos^2 C & \tan C & 1 \end{vmatrix}.\] | tiger_math | [
0.04534740000963211,
0.0323927104473114,
-0.03852235898375511,
-0.04583342373371124,
0.04616229236125946,
-0.026784054934978485,
-0.016924725845456123,
0.05365386977791786,
0.044100236147642136,
-0.0012471535010263324,
0.02959173545241356,
0.010027960874140263,
-0.03855617716908455,
-0.026... | 0.830059 | 0.812707 | 0.017353 |
Let $x_1,$ $x_2,$ $\dots,$ $x_n$ be nonnegative real numbers such that $x_1 + x_2 + \dots + x_n = 1$ and
\[x_1^2 + x_2^2 + \dots + x_n^2 \le \frac{1}{25}.\]Find the smallest possible value of $n.$ | tiger_math | [
0.007333079352974892,
0.0312892310321331,
-0.041292112320661545,
-0.04575803503394127,
0.035307373851537704,
-0.007056899834424257,
-0.019737932831048965,
0.06548704206943512,
0.0273366067558527,
-0.016790496185421944,
0.0004899187479168177,
0.01595582440495491,
-0.0416734479367733,
-0.029... | 0.85953 | 0.840291 | 0.019238 |
In a mystical land, Alice, Bob, and Cindy share magical coins that cast spells when their totals satisfy a specific mathematical property. Alice starts with `a` coins, Bob starts with `b` coins, and Cindy starts with `c` coins. They noticed that whenever the product of two individuals' coin totals equals the third indi... | tiger_math | [
0.01759982295334339,
0.05621912702918053,
-0.045936644077301025,
-0.04477633908390999,
0.02726227045059204,
-0.02908828854560852,
-0.027785740792751312,
0.05298491194844246,
0.036116208881139755,
-0.02354852296411991,
0.012289832346141338,
0.02052350528538227,
-0.03299054875969887,
-0.0385... | 0.843836 | 0.829335 | 0.0145 |
A space traveler, Mina, collected crystals from three different asteroids. On the first asteroid, she collected 48 red crystals, the second she found 120 blue crystals, and on the third, she picked up 180 green crystals. Mina wants to evenly distribute each type of crystal into identical kits without mixing the colors ... | tiger_math | [
0.031245313584804535,
0.022033797577023506,
-0.04677958786487579,
-0.039653487503528595,
0.031516555696725845,
-0.009029058739542961,
-0.04019945487380028,
0.06828952580690384,
0.04180442914366722,
-0.028899675235152245,
0.020916922017931938,
0.03622809424996376,
-0.020836737006902695,
-0.... | 0.816829 | 0.811785 | 0.005043 |
Let $R$ be a unit square region and $n \geq 3$ an odd integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $2023$-ray partitional but not $2021$-ray partitional? | tiger_math | [
0.023183872923254967,
0.030371271073818207,
-0.060138095170259476,
-0.03647949919104576,
0.0136193772777915,
-0.0016806137282401323,
-0.03633818402886391,
0.05801381543278694,
0.026467761024832726,
-0.0027895369566977024,
0.014140522107481956,
0.03248167783021927,
-0.041017815470695496,
-0... | 0.844361 | 0.828126 | 0.016235 |
The vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 4 \\ 1 \\ -5 \end{pmatrix}$ are mutually orthogonal. There exist scalars $p,$ $q,$ and $r$ such that
\[\begin{pmatrix} 4 \\ -3 \\ -5 \end{pmatrix} = p \math... | tiger_math | [
0.041216325014829636,
0.04120153188705444,
-0.05508094280958176,
-0.0372166708111763,
0.04873024672269821,
-0.002118104137480259,
-0.00722901988774538,
0.07188011705875397,
0.025783123448491096,
0.00018686082330532372,
-0.0028910678811371326,
0.023959806188941002,
-0.031877174973487854,
-0... | 0.832223 | 0.813282 | 0.01894 |
Triangle $ABC$ has side lengths $AB=100,BC=180$, and $AC=160$. Lines $\ell_A,\ell_B$, and $\ell_C$ are drawn parallel to $\overline{BC},\overline{AC}$, and $\overline{AB}$, respectively, such that the intersections of $\ell_A,\ell_B$, and $\ell_C$ with the interior of $\triangle ABC$ are segments of lengths $45,35$, an... | tiger_math | [
0.024732783436775208,
0.01790020801126957,
-0.02901029959321022,
-0.043960392475128174,
0.01748746819794178,
-0.010992946103215218,
-0.0305478572845459,
0.06398522853851318,
0.03792130947113037,
-0.00750774797052145,
0.014605501666665077,
0.0507938489317894,
-0.026332492008805275,
-0.01532... | 0.835727 | 0.807267 | 0.02846 |
Given any two positive real numbers $x$ and $y$, then $x \, \Diamond \, y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x \, \Diamond \, y$ satisfies the equations $(xy) \, \Diamond \, y=x(y \, \Diamond \, y)$ and $(x \, \Diamond \, 1) \, \Diamond \, x = x \, \Di... | tiger_math | [
0.013268821872770786,
0.02210141159594059,
-0.06717858463525772,
-0.02650456503033638,
0.028016258031129837,
0.00030176914879120886,
-0.024633217602968216,
0.07452796399593353,
0.05171291530132294,
-0.016644319519400597,
0.016244232654571533,
0.039265409111976624,
-0.031729310750961304,
-0... | 0.833183 | 0.821599 | 0.011584 |
a perfect square is defined as the square of an integer and a perfect cube is defined as the cube of an integer . how many positive integers n are there such that n is less than 50,000 and at the same time n is a perfect square and a perfect cube ? | tiger_math | [
0.03360831364989281,
0.02766423672437668,
-0.050556596368551254,
-0.020411405712366104,
0.04898317530751228,
-0.03660183399915695,
-0.02762962132692337,
0.0329652838408947,
0.04398050531744957,
-0.011848336085677147,
0.031550243496894836,
0.015031703747808933,
-0.03247668594121933,
-0.0412... | 0.814708 | 0.808607 | 0.006101 |
The line $y - x \sqrt{3} + 3 = 0$ intersects the parabola $2y^2 = 2x + 3$ at points $A$ and $B.$ Let $P = (\sqrt{3},0).$ Find $|AP + BP|.$ | tiger_math | [
0.03171089291572571,
0.024918224662542343,
-0.045463405549526215,
-0.03806725889444351,
0.039133910089731216,
-0.012975260615348816,
-0.01615198515355587,
0.07901029288768768,
0.04525772109627724,
-0.023450259119272232,
0.016907496377825737,
0.05010990425944328,
-0.013095556758344173,
-0.0... | 0.839088 | 0.81854 | 0.020548 |
Find the product of all real values of $x$ for which the expression $\frac{x^2+2x+1}{x^2-x-2}$ is undefined. | tiger_math | [
0.016727572306990623,
0.03271063044667244,
-0.03827763348817825,
-0.033321596682071686,
0.026083681732416153,
-0.02570178359746933,
0.006121389102190733,
0.0837843269109726,
0.04043098911643028,
-0.04630003869533539,
0.008778845891356468,
0.016733460128307343,
-0.018856607377529144,
-0.028... | 0.83977 | 0.836098 | 0.003672 |
The vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ X \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 4 \\ 1 \\ -5 \end{pmatrix}$ are mutually orthogonal. There exist scalars $p,$ $q,$ and $r$ such that
\[\begin{pmatrix} -4 \\ 7 \\ 3 \end{pmatrix} = p \mathb... | tiger_math | [
0.02776847407221794,
0.04242024943232536,
-0.07456635683774948,
-0.030842414125800133,
0.05612746998667717,
0.0064770011231303215,
0.0022120336070656776,
0.07584566622972488,
0.037173427641391754,
-0.008139390498399734,
-0.00032342693884857,
0.02638922818005085,
-0.03319204971194267,
-0.01... | 0.830322 | 0.823317 | 0.007005 |
While practising for his upcoming math exams, Hayes realised that the area of a circle he had just solved was equal to the perimeter of a square he had solved in the previous problem. If the area of the circle was 100, what's the length of one side of the square? | tiger_math | [
0.03426702693104744,
0.010747038759291172,
-0.037024855613708496,
-0.055017728358507156,
0.03802807256579399,
-0.022263910621404648,
-0.024371346458792686,
0.05917690694332123,
0.034715425223112106,
-0.015665998682379723,
-0.00014790533168707043,
0.025017058476805687,
-0.030792536213994026,
... | 0.824982 | 0.814755 | 0.010227 |
Convert the point $(\rho,\theta,\phi) = \left( 3, \frac{5 \pi}{12}, 0 \right)$ in spherical coordinates to rectangular coordinates. | tiger_math | [
0.012289069592952728,
0.04141395166516304,
-0.02764444425702095,
-0.04941234365105629,
0.018171897158026695,
-0.021967129781842232,
-0.027037113904953003,
0.05749160423874855,
0.05385559797286987,
0.0010326422052457929,
0.006608528085052967,
0.03817463666200638,
-0.026768697425723076,
-0.0... | 0.83058 | 0.824961 | 0.005619 |
Let $x_1,$ $x_2,$ $\dots,$ $x_n$ be nonnegative real numbers such that $x_1 + x_2 + \dots + x_n = 1$ and
\[x_1^2 + x_2^2 + \dots + x_n^2 \le \frac{1}{100}.\]Find the smallest possible value of $n.$ | tiger_math | [
0.006028824485838413,
0.024967342615127563,
-0.03714139014482498,
-0.04279368370771408,
0.03554711490869522,
-0.005666053853929043,
-0.020383618772029877,
0.06277415156364441,
0.03754589334130287,
-0.01756208762526512,
0.0017714777495712042,
0.020438095554709435,
-0.036482881754636765,
-0.... | 0.858844 | 0.838212 | 0.020632 |
Imagine you're a pirate captain, and you've gathered 240 gold coins and 360 pearls from your adventures. You wish to distribute these treasures among your loyal crew in such a way that each crew member gets an equal share of gold coins and pearls, with no remainder. What is the maximum number of crew members you can ha... | tiger_math | [
0.02425864338874817,
0.012612811289727688,
-0.03676297515630722,
-0.033496685326099396,
0.012524737045168877,
-0.018992533907294273,
-0.02815469354391098,
0.028788136318325996,
0.026122543960809708,
-0.03367974981665611,
0.02220403403043747,
0.027385445311665535,
-0.04470491781830788,
-0.0... | 0.838236 | 0.832561 | 0.005675 |
In $\triangle RED$, $\measuredangle DRE=75^{\circ}$ and $\measuredangle RED=45^{\circ}$. $RD=1$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC}\perp\overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA=AR$. If $AE=2$, f... | tiger_math | [
0.03461400792002678,
0.008731055073440075,
-0.028963303193449974,
-0.0649731457233429,
0.002088027074933052,
-0.008948011323809624,
-0.01984047144651413,
0.06416837126016617,
0.015259482897818089,
-0.012785580940544605,
0.02157357893884182,
0.02328634262084961,
-0.04720485582947731,
-0.012... | 0.839487 | 0.81374 | 0.025747 |
A small lemonade stand is operated by three friends: Alice, Bob, and Charlie. They decided to pool their savings to buy ingredients. Alice contributes $15, Bob contributes twice as much as Alice, and Charlie contributes $5 less than Bob. At the end of the day, they calculate their total earnings, only to realize they'v... | tiger_math | [
0.01772439293563366,
0.03186218813061714,
-0.044155679643154144,
-0.03404208645224571,
0.034281037747859955,
-0.020538533106446266,
-0.01715364307165146,
0.06650882214307785,
0.02889948897063732,
-0.022666120901703835,
0.02397281490266323,
0.010406022891402245,
-0.047651030123233795,
-0.03... | 0.832323 | 0.820636 | 0.011687 |
Bronson is an apple dealer who buys apples from farmers and then transports them to grocery stores to sell them for profit. He buys individual apples from farmers at a cost of $12 per bushel and sells them to stores at a price of $0.40 per apple. Each bushel contains 48 apples. Bronson needs to deliver apples to 3 diff... | tiger_math | [
0.027056580409407616,
0.025487801060080528,
-0.020344780758023262,
-0.013418943621218204,
0.015472792088985443,
-0.005706207826733589,
-0.048282794654369354,
0.05761842429637909,
0.026534998789429665,
-0.015310497023165226,
0.0030932072550058365,
0.04350162297487259,
-0.035937923938035965,
... | 0.806145 | 0.799663 | 0.006481 |
A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a self-centered polynomial of degree $n$, what is the maximum number of integer solutions $k$ to the equation $p(k) = k^n$? | tiger_math | [
0.030973831191658974,
0.016447417438030243,
-0.046309370547533035,
-0.04005604609847069,
0.005631661973893642,
-0.010760591365396976,
-0.004131515976041555,
0.02712778002023697,
0.047984927892684937,
-0.008709464222192764,
0.021789630874991417,
0.02554638683795929,
-0.03667853772640228,
-0... | 0.827515 | 0.817649 | 0.009866 |
Two distinct vectors $\mathbf{c}$ and $\mathbf{d}$ lie on a line. If the vector $\frac{2}{3} \mathbf{c} + m \mathbf{d}$ also lies on this line, determine the value of $m$. | tiger_math | [
0.02132018283009529,
0.0345904715359211,
-0.06831100583076477,
-0.04304936155676842,
0.0533628948032856,
-0.0034057674929499626,
-0.003131334902718663,
0.07505033165216446,
0.054165519773960114,
-0.0239395909011364,
-0.001565039623528719,
0.02626092918217182,
-0.024955961853265762,
-0.0186... | 0.827676 | 0.823364 | 0.004313 |
In the figure below, all corner angles are right angles and each number represents the unit-length of the segment which is nearest to it. What is the area of the shaded region?
[asy]
draw((0,0)--(12,0)--(12,5)--(8,5)--(8,4)--(5,4)
--(5,6)--(0,6)--(0,0));
label("6",(0,3),W);
label("5",(2.5,6),N);
label("2",(5,5),W);
... | tiger_math | [
0.02230668254196644,
0.0256489384919405,
-0.04434345290064812,
-0.05224738270044327,
0.025543684139847755,
-0.04915023595094681,
-0.018212715163826942,
0.042682237923145294,
0.03439642861485481,
0.012554215267300606,
0.027152227237820625,
0.03172655031085014,
-0.020474622026085854,
-0.0113... | 0.823518 | 0.816711 | 0.006807 |
George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8, and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations of each seating arrangement are considered different but refle... | tiger_math | [
0.010052712634205818,
0.012046034447848797,
-0.04708677530288696,
-0.020445453003048897,
0.03407610207796097,
-0.010087031871080399,
-0.014894519932568073,
0.05181393399834633,
0.03702538087964058,
-0.00835895910859108,
0.020735396072268486,
0.025033114477992058,
-0.03598308190703392,
-0.0... | 0.827375 | 0.821746 | 0.005629 |
Find the sum of the positive roots of the equations $\sqrt{x+2+2\sqrt{x+1}}+\sqrt{x+2-2\sqrt{x+1}}=2$ and $\sqrt{x+6+4\sqrt{x+2}}+\sqrt{x+6-4\sqrt{x+2}}=4$. | tiger_math | [
0.019183631986379623,
0.022181207314133644,
-0.04910881444811821,
-0.042201440781354904,
0.028490284457802773,
-0.03826787695288658,
-0.031263094395399094,
0.049381107091903687,
0.03928714618086815,
-0.012967975810170174,
0.015161655843257904,
0.012379390187561512,
-0.04160352796316147,
-0... | 0.833796 | 0.828817 | 0.004979 |
Two interior angles $A$ and $B$ of pentagon $ABCDE$ are $75^{\circ}$ and $90^{\circ}$. Two of the remaining angles, $C$ and $D$, are equal and the fifth angle $E$ is $20^{\circ}$ less than twice $C$. Find the measure of the largest angle. | tiger_math | [
0.028832433745265007,
0.013028483837842941,
-0.039364952594041824,
-0.06199932470917702,
0.003352597588673234,
-0.019480282440781593,
-0.012652328237891197,
0.04531046375632286,
0.033774234354496,
-0.012532477267086506,
0.022173892706632614,
0.027469320222735405,
-0.051336128264665604,
0.0... | 0.830923 | 0.816288 | 0.014635 |
Point $P$ is located inside triangle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=14, BC=16,$ and $CA=18.$ Find $\tan \angle PAB.$ | tiger_math | [
0.04004836082458496,
0.04116062819957733,
-0.039304692298173904,
-0.035565074533224106,
0.03757081180810928,
-0.019762489944696426,
-0.0032620399724692106,
0.04216884821653366,
0.0361003652215004,
0.020922821015119553,
0.014891771599650383,
0.025791926309466362,
-0.01860704831779003,
-0.00... | 0.838873 | 0.822256 | 0.016618 |
Find the largest value of $n$ such that $2x^2 +nx + 60$ can be factored as the product of two linear factors with integer coefficients. | tiger_math | [
0.02452436275780201,
0.039788585156202316,
-0.037112221121788025,
-0.04374154284596443,
0.03128725290298462,
-0.022956984117627144,
-0.019755158573389053,
0.06654751300811768,
0.03803792595863342,
-0.02911038137972355,
0.018154703080654144,
0.04776648432016373,
-0.03877050429582596,
-0.029... | 0.844708 | 0.840513 | 0.004196 |
In a charm bracelet store, Emma has combined unique charms to sell in special packs. Every charm pack consists of a star charm, a heart charm, and a number of round charms that equals the sum of the first two types of charms in each pack. If Emma decides to price the star charm at 3 dollars, the heart charm at 5 dollar... | tiger_math | [
0.029594335705041885,
0.021923741325736046,
-0.054498281329870224,
-0.03152095526456833,
0.04523194208741188,
-0.030699914321303368,
-0.03498683124780655,
0.03715848550200462,
0.023685017600655556,
-0.0210515595972538,
0.010366334579885006,
0.031439900398254395,
-0.06059112399816513,
-0.03... | 0.805459 | 0.800821 | 0.004638 |
A standard six-sided die is rolled, and $P$ is the product of the four numbers that are visible. What is the largest number that is certain to divide $P$? | tiger_math | [
0.03233540058135986,
0.020784573629498482,
-0.0309266597032547,
-0.036856960505247116,
0.03862319886684418,
-0.017934421077370644,
-0.015573776327073574,
0.054856251925230026,
0.035794008523225784,
-0.032016340643167496,
0.03106105513870716,
0.035918936133384705,
-0.017949368804693222,
-0.... | 0.841568 | 0.837523 | 0.004045 |
You have $5$ red shirts, $5$ green shirts, $6$ pairs of pants, $8$ green hats and $8$ red hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats if you are not allowed to wear red pants? | tiger_math | [
0.010600817389786243,
0.019418874755501747,
-0.04158176854252815,
-0.017179271206259727,
0.011708763428032398,
-0.027507206425070763,
-0.03782785311341286,
0.04980328679084778,
0.0331161804497242,
-0.03225860744714737,
0.006370132323354483,
0.022953368723392487,
-0.02887537330389023,
-0.02... | 0.847636 | 0.841374 | 0.006262 |
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