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 the number of positive integers $n \le 2000$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor = n\]for some real number $x.$ | tiger_math | [
0.016158422455191612,
0.02437783032655716,
-0.06288246065378189,
-0.022333437576889992,
0.012013237923383713,
-0.013131172396242619,
-0.030795229598879814,
0.03647230193018913,
0.05163923278450966,
-0.005351729691028595,
0.030258914455771446,
0.056869350373744965,
-0.04359188303351402,
-0.... | 0.844847 | 0.834653 | 0.010194 |
A woman labels the squares of a very large chessboard $1$ through $64$. On each square $k$, the woman puts $2^k$ grains of rice. How many more grains of rice are placed on the $10^{th}$ square than on the first $8$ squares combined? | tiger_math | [
0.011661291122436523,
0.017859430983662605,
-0.04082288220524788,
-0.0248273853212595,
0.01927601546049118,
-0.008974838070571423,
-0.029554717242717743,
0.06706138700246811,
0.04254473000764847,
-0.022143792361021042,
-0.003243476152420044,
0.023470666259527206,
-0.04236225783824921,
-0.0... | 0.844848 | 0.838115 | 0.006733 |
Let $f(x)=2x+1$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x+2)=f(x^{-1}+2)$. | tiger_math | [
0.042497456073760986,
0.03198770061135292,
-0.049468886107206345,
-0.05272863060235977,
0.030445029959082603,
-0.03091956488788128,
-0.0067084068432450294,
0.07357736676931381,
0.02392278052866459,
-0.01458736602216959,
0.022456742823123932,
0.023695021867752075,
-0.03531105816364288,
-0.0... | 0.846866 | 0.836003 | 0.010863 |
At point P, two concentric circles are present. The sides of a 45 degree angle at P create an arc on the smaller circle that is equivalent in length to an arc on the larger circle formed by the sides of a 36 degree angle at P. What is the ratio of the area of the smaller circle to the area of the larger circle, express... | tiger_math | [
0.018242966383695602,
0.03443804755806923,
-0.04799789562821388,
-0.04825447499752045,
0.011906762607395649,
-0.005709152203053236,
-0.018009942024946213,
0.05447804927825928,
0.03599175065755844,
-0.008725776337087154,
0.029203355312347412,
0.02360442467033863,
-0.039662983268499374,
-0.0... | 0.825262 | 0.81579 | 0.009473 |
Let $a,$ $b,$ $c$ be positive real numbers such that $1 \le a \le b \le c \le 4.$ Find the minimum value of
\[(a - 1)^2 + \left( \frac{b}{a} - 1 \right)^2 + \left( \frac{c}{b} - 1 \right)^2 + \left( \frac{4}{c} - 1 \right)^2.\] | tiger_math | [
0.029226237908005714,
0.03241414204239845,
-0.04941580444574356,
-0.03181634843349457,
0.051190488040447235,
-0.01138810534030199,
-0.01487708743661642,
0.04825986921787262,
0.038317590951919556,
-0.020192239433526993,
-0.004101040307432413,
0.02523917518556118,
-0.024217024445533752,
-0.0... | 0.854379 | 0.831703 | 0.022676 |
In the figure, the visible gray area within the larger circle is equal to three times the area of the white circular region. What is the ratio of the radius of the small circle to the radius of the large circle? Express your answer as a common fraction.
[asy]size(101);
filldraw(Circle((0,0),2)^^Circle((.8,-.3),1),gray... | tiger_math | [
0.00621351832523942,
0.029709218069911003,
-0.036934249103069305,
-0.056806016713380814,
0.030671222135424614,
-0.01881735771894455,
-0.01669224724173546,
0.06559482961893082,
0.036796171218156815,
-0.0037051872350275517,
0.02365894801914692,
0.030450941994786263,
-0.023193655535578728,
-0... | 0.832637 | 0.829428 | 0.003209 |
Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown. If $BA' = 2$ and $A'C = 3,$ then find the length of crease $\overline{PQ}.$ | tiger_math | [
0.039299312978982925,
0.005216559860855341,
-0.032115448266267776,
-0.04975039139389992,
0.01857287436723709,
-0.02895723283290863,
-0.01088915579020977,
0.0641457661986351,
0.03238246962428093,
-0.011086972430348396,
0.0177807305008173,
0.04480587691068649,
-0.017251865938305855,
-0.01833... | 0.833373 | 0.816006 | 0.017366 |
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 |
Let $f(x) = \sin{x} + 2\cos{x} + 3\tan{x}$, using radian measure for the variable $x$. Let $r$ be the smallest positive value of $x$ for which $f(x) = 0$. Find $\lfloor r \rfloor.$ | tiger_math | [
0.012172091752290726,
0.041163068264722824,
-0.0580013208091259,
-0.030598729848861694,
0.027179032564163208,
-0.012468368746340275,
-0.02221846953034401,
0.05516507104039192,
0.026876820251345634,
0.006496209651231766,
0.021315382793545723,
0.015728505328297615,
-0.034948043525218964,
-0.... | 0.841229 | 0.829946 | 0.011283 |
The graphs of $y=2-x^2+x^3$ and $y=1+x^2+x^3$ intersect in multiple points. Find the maximum difference between the $y$-coordinates of these intersection points. | tiger_math | [
0.03601778671145439,
0.017803579568862915,
-0.04909166693687439,
-0.03976704925298691,
0.03987918794155121,
-0.029442591592669487,
-0.039291247725486755,
0.06297411769628525,
0.05418485403060913,
-0.02451171912252903,
0.018040725961327553,
0.02603861317038536,
-0.03388053551316261,
-0.0058... | 0.839266 | 0.828568 | 0.010699 |
Dean is playing a game with calculators. The 42 participants (including Dean) sit in a circle, and Dean holds 3 calculators. One calculator reads 1, another 0, and the last one -1. Dean starts by pressing the cube button on the calculator that shows 1, pressing the square button on the one that shows X, and on the calc... | tiger_math | [
0.02400006726384163,
0.04458541050553322,
-0.058969851583242416,
-0.032085876911878586,
0.03389870002865791,
-0.01428100187331438,
-0.008704149164259434,
0.06042623519897461,
0.033404506742954254,
-0.019613660871982574,
0.01772483065724373,
0.03710583224892616,
-0.02780119702219963,
-0.028... | 0.82111 | 0.815268 | 0.005842 |
In an isosceles triangle $PQR$ with $PQ=PR$, the altitude $PS$ intersects the base $QR$ at $T$. If the area of $\triangle PQR$ is 200 and $QT=8$, find the length of $PS$. | tiger_math | [
0.02363579533994198,
0.016930360347032547,
-0.04153621569275856,
-0.05094693973660469,
0.01840132288634777,
-0.017592931166291237,
-0.022502075880765915,
0.05378806218504906,
0.03664650395512581,
-0.0019403700716793537,
0.023723267018795013,
0.03388373181223869,
-0.025251323357224464,
-0.0... | 0.827348 | 0.816028 | 0.01132 |
Joy is winding balls of yarn for her cat to play with. The first ball, B1, has an unknown size in feet. The second ball, B2, is twice as large as the first ball. The third ball, B3, is three times as large as the first ball. The fourth ball, B4, is twice as large as the second ball, and the fifth ball, B5, is the sum o... | tiger_math | [
0.022540627047419548,
0.03802470490336418,
-0.05969826504588127,
-0.03322095051407814,
0.02938247099518776,
-0.02993100881576538,
-0.04225087910890579,
0.05481971800327301,
0.027095871046185493,
0.0010438230820000172,
0.017020123079419136,
0.02899368852376938,
-0.036869876086711884,
-0.023... | 0.820471 | 0.814979 | 0.005492 |
In a garden, there are 10 trees planted in a row. Each tree has a distinct number of apples, from 1 apple on the first tree to 10 apples on the last tree. A child is allowed to pick apples from only three consecutive trees. What is the maximum number of apples the child can collect, and from which trees? | tiger_math | [
0.0382661446928978,
0.008457939140498638,
-0.03570304811000824,
-0.02304754965007305,
0.014234041795134544,
-0.037370674312114716,
-0.01669098436832428,
0.05834454670548439,
0.02619548700749874,
-0.036817874759435654,
0.0043345107696950436,
0.03678330034017563,
-0.05784548446536064,
0.0023... | 0.828362 | 0.814864 | 0.013498 |
The students in Mrs. Reed's English class are reading the same $760$-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in $20$ seconds, Bob reads a page in $45$ seconds and Chandra reads a page in X seconds. Chandra and Bob, who each have a copy of the book, decide that they can ... | tiger_math | [
0.014123521745204926,
0.024523401632905006,
-0.04966909438371658,
-0.04982355237007141,
0.027293985709547997,
-0.023492956534028053,
-0.019820740446448326,
0.06442280113697052,
0.02981426566839218,
-0.013767370954155922,
0.011766215786337852,
0.011523518711328506,
-0.041666027158498764,
0.... | 0.82867 | 0.817825 | 0.010845 |
For real numbers $x > 0,$ find the maximum value of
\[\frac{\sqrt{x}}{\sqrt{x + 1}}.\] | tiger_math | [
0.019620873034000397,
0.03215065598487854,
-0.026642579585313797,
-0.02712131477892399,
0.016861477866768837,
-0.02506376802921295,
-0.01273496262729168,
0.05879238247871399,
0.052498187869787216,
-0.03417208790779114,
0.021925585344433784,
0.028910068795084953,
-0.034822218120098114,
-0.0... | 0.839749 | 0.829373 | 0.010376 |
Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that
\[z + \frac{1}{z} = r.\]Find the real part of $z$. | tiger_math | [
0.023255299776792526,
0.043447498232126236,
-0.026636404916644096,
-0.043730080127716064,
0.034020133316516876,
0.0015254570171236992,
0.007534780539572239,
0.04416204243898392,
0.0327894352376461,
-0.0228241179138422,
0.012730239890515804,
0.02805754914879799,
-0.04093954339623451,
-0.014... | 0.833627 | 0.825394 | 0.008233 |
At an annual town festival, three friends, Alice, Bob, and Charlie, won money in the trivia competition. They decided to split their total winnings of $150 based on fractions that reflect the number of questions each answered correctly. Alice answered $\frac{1}{6}$ of the questions, Bob answered $\frac{1}{3}$, and Char... | tiger_math | [
0.03245691955089569,
0.023142771795392036,
-0.04390936717391014,
-0.0451807864010334,
0.030424010008573532,
-0.03476085513830185,
-0.01294674538075924,
0.04686289280653,
0.020881997421383858,
-0.015219496563076973,
0.007389879785478115,
0.015025310218334198,
-0.023512976244091988,
-0.01641... | 0.833751 | 0.820905 | 0.012846 |
Let $r$ and $s$ denote the two real roots of $x^2 - 2x \sqrt{2} + 2 = 0.$ Then determine $r^4 + s^4.$ | tiger_math | [
0.018594976514577866,
0.0339551717042923,
-0.042719945311546326,
-0.03161519765853882,
0.037651900202035904,
-0.028967877849936485,
-0.002252490958198905,
0.0768754780292511,
0.052388373762369156,
-0.021603122353553772,
0.01791607216000557,
0.02537413313984871,
-0.02048094943165779,
-0.035... | 0.832742 | 0.819892 | 0.01285 |
An elementary school teacher is making Halloween goodie bags for her class. She wants the bags to be personalized, so she surveys her students asking whether they'd like a vampire-themed bag or a pumpkin-themed bag. Of her 25 students, 11 indicate they want the vampire-themed bag and 14 indicate they want the pumpkin-t... | tiger_math | [
0.015429725870490074,
0.005658556707203388,
-0.03539392724633217,
-0.04171835258603096,
0.023124581202864647,
-0.021452536806464195,
-0.039705790579319,
0.02963803894817829,
0.02057754620909691,
-0.012214647606015205,
0.02907566912472248,
0.02879634127020836,
-0.030665714293718338,
-0.0180... | 0.82934 | 0.825375 | 0.003964 |
Let $G$ be the centroid of triangle $ABC,$ and let $P$ be an arbitrary point on the plane of the triangle. Then there exists a constant $k$ so that
\[PA^2 + PB^2 + PC^2 = k \cdot PG^2 + GA^2 + GB^2 + GC^2.\]Find $k.$ | tiger_math | [
0.04180531203746796,
0.022650953382253647,
-0.04881027340888977,
-0.04877781122922897,
0.032471489161252975,
-0.034958869218826294,
-0.010495309717953205,
0.036908335983753204,
0.03571658581495285,
-0.017732953652739525,
0.025706816464662552,
0.03794454038143158,
-0.03166656568646431,
-0.0... | 0.854841 | 0.829287 | 0.025554 |
Let $P$ be the product of the first $200$ positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k .$ | tiger_math | [
0.022468741983175278,
0.02957603521645069,
-0.04897226393222809,
-0.044198472052812576,
0.023595191538333893,
-0.019538631662726402,
-0.02289627306163311,
0.04964436590671539,
0.04137546941637993,
-0.013241404667496681,
0.01800946705043316,
0.03474779054522514,
-0.03037356585264206,
-0.021... | 0.859004 | 0.846536 | 0.012468 |
If $a>0$ and $b>0$, the operation $\nabla$ is defined as $a \nabla b = \frac{a + b}{1 + ab}$. Let $x = 2 \nabla 3$. Calculate $(x \nabla 4) \nabla 5$. | tiger_math | [
0.0302607212215662,
0.030010465532541275,
-0.04206739366054535,
-0.004958318080753088,
0.04194863140583038,
-0.00847594439983368,
-0.027997542172670364,
0.08722846210002899,
0.030924778431653976,
0.006129808258265257,
0.008043579757213593,
0.017345109954476357,
-0.03686157986521721,
-0.033... | 0.82103 | 0.816649 | 0.004381 |
a, b, and c travel from their respective starting points to place x, y, and z. a leaves for x with a speed of Va km/h, b leaves for y with a speed of Vb km/h, and c leaves for z with a speed of Vc km/h. After traveling for a certain amount of time, a has covered 180 km more than b and 100 km more than c. At this point,... | tiger_math | [
0.028040189296007156,
0.03688362240791321,
-0.031563181430101395,
-0.038469552993774414,
0.02706410363316536,
-0.006594357546418905,
-0.03458334133028984,
0.04749048128724098,
0.03442048653960228,
-0.0076063405722379684,
0.017984358593821526,
0.016003115102648735,
-0.013423907570540905,
-0... | 0.815652 | 0.812208 | 0.003444 |
Triangle $ABC$ with right angle at $C$, $\angle BAC < 45^\circ$ and $AB = 6$. Point $P$ on $\overline{AB}$ is chosen such that $\angle APC = 3\angle ACP$ and $CP = 2$. The ratio $\frac{AP}{BP}$ can be represented in the form $p + q\sqrt{r}$, where $p$, $q$, $r$ are positive integers and $r$ is not divisible by the squa... | tiger_math | [
0.038423407822847366,
0.027936913073062897,
-0.03284405916929245,
-0.0343698188662529,
0.01763751544058323,
-0.006823485251516104,
-0.0002698981552384794,
0.057543713599443436,
0.033353291451931,
0.00220668432302773,
0.02798176184296608,
0.0413174107670784,
-0.03110893815755844,
-0.0164679... | 0.842527 | 0.814426 | 0.028102 |
Ms. Lopez is preparing a special batch of homemade chocolates to distribute among her students as a treat during their mathematics class. She plans to use dark and white chocolate in her recipe. Initially, she has 8 dark and 12 white chocolate bars. If each batch requires 2 dark and 3 white chocolate bars, how many com... | tiger_math | [
0.03833608329296112,
0.024656670168042183,
-0.041119713336229324,
-0.035973064601421356,
0.028928937390446663,
-0.00702414195984602,
-0.030251333490014076,
0.07261896133422852,
0.021991655230522156,
-0.03341126814484596,
0.004823021590709686,
0.03343155235052109,
-0.044618815183639526,
-0.... | 0.816483 | 0.813328 | 0.003155 |
The polynomial $P(x) = x^3 + ax^2 + bx - 1$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. What is $b$? | tiger_math | [
0.020426634699106216,
0.009897792711853981,
-0.06384774297475815,
-0.04629554972052574,
0.040182508528232574,
-0.032705556601285934,
-0.0067889089696109295,
0.05676162242889404,
0.04359220340847969,
-0.01834636926651001,
0.004389026202261448,
0.032901059836149216,
-0.034629009664058685,
-0... | 0.845153 | 0.834701 | 0.010452 |
Let $\mathbf{a} = \begin{pmatrix} 3 \\ p \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ X \\ q \end{pmatrix}$ be vectors that are equal in magnitude, and orthogonal. Enter the ordered pair $(p,q).$
If we know the answer to the above question is (-\frac{31}{12},\frac{41}{12}), what is the value of unknown ... | tiger_math | [
0.021608620882034302,
0.02099003829061985,
-0.08098553121089935,
-0.035173092037439346,
0.043652430176734924,
0.011451099067926407,
0.010113066993653774,
0.0671660527586937,
0.041221898049116135,
-0.005172200966626406,
0.00348537671379745,
0.031244643032550812,
-0.042631056159734726,
-0.02... | 0.831612 | 0.825721 | 0.005891 |
How many interior diagonals does an icosahedron have? (An $\emph{icosahedron}$ is a 3-dimensional figure with 20 triangular faces and 12 vertices, with 5 faces meeting at each vertex. An $\emph{interior}$ diagonal is a segment connecting two vertices which do not lie on a common face.) | tiger_math | [
0.024273358285427094,
0.008421001024544239,
-0.03363260254263878,
-0.03459469601511955,
0.00279243104159832,
-0.015458404086530209,
-0.01743653602898121,
0.04290592297911644,
0.01777900569140911,
0.0034639749210327864,
0.013895883224904537,
0.05916588008403778,
-0.032461922615766525,
-0.02... | 0.790336 | 0.78734 | 0.002997 |
A convex pentagon has interior angles with measures $x+1$, $2x$, $3x$, $4x$, and $5x-1$ degrees. What is the measure of the largest angle? | tiger_math | [
0.02044699899852276,
0.023580577224493027,
-0.052767276763916016,
-0.05375882610678673,
0.012341645546257496,
-0.0002433591289445758,
-0.024426601827144623,
0.032515060156583786,
0.03342941775918007,
-0.012109796516597271,
0.031208833679556847,
0.037093207240104675,
-0.04588445648550987,
-... | 0.833002 | 0.829079 | 0.003923 |
In an isosceles triangle, the measure of the largest angle is $120^{\circ}$. Determine the measure of each of the other two angles in the triangle. | tiger_math | [
0.02152053453028202,
0.0118433628231287,
-0.034780941903591156,
-0.059772856533527374,
0.028925389051437378,
-0.011986330151557922,
-0.018166813999414444,
0.03810539096593857,
0.03177015110850334,
-0.010721241123974323,
0.03456718847155571,
0.02855548821389675,
-0.03911935165524483,
-0.004... | 0.828886 | 0.815668 | 0.013218 |
The sequences of positive integers $1,a_2, a_3, \dots$ and $1,b_2, b_3, \dots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k}=100$ and $c_{k+2}=1000$. Find $c_{k+1}$. | tiger_math | [
0.03789959475398064,
0.038289379328489304,
-0.040124475955963135,
-0.02871919982135296,
0.01842803880572319,
-0.0026509349700063467,
-0.027744967490434647,
0.08046918362379074,
0.04190448299050331,
-0.016334783285856247,
0.02182723954319954,
0.013675777241587639,
-0.048995498567819595,
-0.... | 0.855359 | 0.822696 | 0.032663 |
In square ABCD with side length of 1 unit, points E and F are located on sides AB and CB respectively, with AE = CF. When the square is folded along lines DE and DF, sides AD and CD align and lie on diagonal BD. The length of segment AE can be written as √k - m units. What is the integer value of k + m? | tiger_math | [
0.04005426540970802,
0.00008555491513106972,
-0.05727485567331314,
-0.07019497454166412,
0.03853177651762962,
-0.014208287000656128,
0.00022099919442553073,
0.050681859254837036,
0.04815768450498581,
-0.0017654125113040209,
0.02214818261563778,
0.02558206394314766,
-0.041428327560424805,
0... | 0.825352 | 0.811466 | 0.013886 |
Positive real numbers $x,y$ satisfy the equations $x^2 + y^2 = 4$ and $x^4 + y^4= 24$. Find $x^2y^2$. | tiger_math | [
0.033980753272771835,
0.03071293607354164,
-0.04812885448336601,
-0.04730788618326187,
0.04320145025849342,
-0.02993335761129856,
-0.00975507777184248,
0.044486384838819504,
0.0640401616692543,
-0.026808707043528557,
0.017745839431881905,
0.03139116242527962,
-0.02229594625532627,
-0.02818... | 0.832018 | 0.823174 | 0.008844 |
If $\det \mathbf{A} = 2$ and $\det \mathbf{B} = 12,$ then find $\det (2\mathbf{A} \cdot 3\mathbf{B}).$ | tiger_math | [
0.07443530857563019,
0.01881798729300499,
-0.05586087331175804,
-0.028460673987865448,
0.03680357336997986,
-0.02119738608598709,
-0.02685610018670559,
0.06633202731609344,
0.029752369970083237,
-0.005074172280728817,
0.012758536264300346,
0.028913220390677452,
-0.043163832277059555,
-0.01... | 0.821033 | 0.81635 | 0.004683 |
Given vectors $\mathbf{a}$ and $\mathbf{b},$ let $\mathbf{p}$ be a vector such that
\[\|\mathbf{p} - \mathbf{b}\| = k \|\mathbf{p} - \mathbf{a}\|,\]where $k$ is a positive constant. Among all such vectors $\mathbf{p},$ there exists constants $t$ and $u$ such that $\mathbf{p}$ is at a fixed distance from $t \mathbf{a} ... | tiger_math | [
0.046822357922792435,
0.03034866414964199,
-0.04710511863231659,
-0.029746370390057564,
0.0353567972779274,
-0.008147364482283592,
-0.024634050205349922,
0.056763291358947754,
0.054432302713394165,
-0.0044962032698094845,
0.013827544637024403,
0.030088497325778008,
-0.024228287860751152,
0... | 0.84427 | 0.81614 | 0.02813 |
Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate
\[\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & X \\ \sin^2 C & \cot C & 1 \end{vmatrix}. The answer is 0. What is the value of unknown variable X? | tiger_math | [
0.011768629774451256,
0.031921159476041794,
-0.05885095149278641,
-0.02900598756968975,
0.04052778333425522,
-0.01086821872740984,
-0.007504092529416084,
0.07024271041154861,
0.042071737349033356,
-0.006057892460376024,
0.028738494962453842,
0.024965260177850723,
-0.030426116660237312,
-0.... | 0.833243 | 0.822301 | 0.010941 |
If the measure of angle $C$ is 3 times the measure of angle $B$, what is the measure of angle $A$ in triangle $ABC$?
[asy]
pair A,B,C;
A=(0,0);
B=(5,0);
C=(-0.8,2.5);
draw(A--B--C--A);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
draw((4,0.6)..(3.8,0.4)..(3.9,0.1),ArcArrow);
label("$21^\circ$",(4,0.6)... | tiger_math | [
0.029852358624339104,
0.004819788970053196,
-0.045336734503507614,
-0.034397583454847336,
0.03967781364917755,
-0.017494022846221924,
-0.030558042228221893,
0.04038906469941139,
0.030112847685813904,
0.005480851978063583,
0.019003018736839294,
0.024540480226278305,
-0.03364235907793045,
-0... | 0.822153 | 0.818085 | 0.004068 |
Alice is trying to bake cookies for a school fundraiser. She can bake 3 batches of cookies, with each batch containing $b$ cookies. She wants to give out the cookies to 45 students such that each student gets exactly the same number of cookies with no leftovers. Find the smallest number of cookies per batch $b$ to sati... | tiger_math | [
0.01799003593623638,
0.01704024337232113,
-0.02935025282204151,
-0.051976583898067474,
0.02859182097017765,
-0.009739506058394909,
-0.008557225577533245,
0.07675819098949432,
0.027149593457579613,
-0.017303025349974632,
0.03137845918536186,
0.01641971245408058,
-0.028718680143356323,
-0.02... | 0.836854 | 0.825474 | 0.01138 |
Find the minimum number of real roots to a polynomial of the form
\[x^n - x^{n - X + \dots - x - 1 = 0,\]where $n$ is a positive integer. If the maximum number of real roots is 1, what is the value of unknown variable X? | tiger_math | [
0.02875637821853161,
0.021420584991574287,
-0.05455086752772331,
-0.02845710888504982,
0.02601318247616291,
-0.026786580681800842,
-0.002863297937437892,
0.04692379757761955,
0.03693892061710358,
-0.02456642873585224,
0.00980109628289938,
0.03228738531470299,
-0.032598040997982025,
-0.0256... | 0.836331 | 0.830712 | 0.005619 |
a canoe has two oars , left and right . each oar either works or breaks . the failure or non - failure of each oar is independent of the failure or non - failure of the other . you can still row the canoe with one oar . the probability that the left oar works is 3 / 5 . the probability that the right oar works is also ... | tiger_math | [
0.018019771203398705,
0.020400557667016983,
-0.05239326134324074,
-0.026467518880963326,
0.035112414509058,
-0.01821013353765011,
-0.014500647783279419,
0.07516902685165405,
0.016585439443588257,
0.0055161286145448685,
0.025509297847747803,
0.02367321401834488,
-0.006348897237330675,
-0.01... | 0.818279 | 0.81488 | 0.003399 |
The sum\[\sum_{x=3}^{45} 2\sin{x}\sin{1}[1 + \sec (x-2) \sec (x)]\]can be written in the form $\sum_{n=1}^{4} (-1)^n \frac{\Phi(\theta_n)}{\Psi(\theta_n)}$, where $\Phi,\, \Psi$ are trigonometric functions and $\theta_1,\, \theta_2, \, \theta_3, \, \theta_4$ are degrees $\in [0,45]$. Find $\theta_1 + \theta_2 + \theta_... | tiger_math | [
-0.0019509810954332352,
0.021793492138385773,
-0.04744016006588936,
-0.04288817197084427,
0.02493227832019329,
0.00018011171778198332,
-0.028046805411577225,
0.0736297145485878,
0.03278401866555214,
0.015133725479245186,
0.02860843762755394,
0.026612484827637672,
-0.04800055921077728,
-0.0... | 0.828079 | 0.813792 | 0.014287 |
Two concentric circles are centered at point P. The sides of a 45 degree angle at P form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 36 degree angle at P. What is the ratio of the area of the smaller circle to the area of the larger circle? Express your ... | tiger_math | [
0.020866762846708298,
0.03820196911692619,
-0.0470886155962944,
-0.048998065292835236,
0.014216665178537369,
-0.007646473590284586,
-0.012963649816811085,
0.04686363413929939,
0.03528060391545296,
-0.013696670532226562,
0.031740665435791016,
0.018953543156385422,
-0.04302896559238434,
-0.0... | 0.824888 | 0.816442 | 0.008446 |
For any positive integer $n,$ let $\langle n \rangle$ denote the closest integer to $\sqrt{n}.$ Evaluate
\[\sum_{n = 2}^\infty \frac{2^{\langle n \rangle} + 2^{-\langle n \rangle}}{2^{n+1}}.\] | tiger_math | [
0.02181035280227661,
0.05319754406809807,
-0.02182328887283802,
-0.041268497705459595,
0.028266705572605133,
-0.019976258277893066,
-0.013501424342393875,
0.04455096647143364,
0.04908810928463936,
-0.01068704854696989,
0.019551364704966545,
0.016696086153388023,
-0.03001810610294342,
-0.00... | 0.850404 | 0.828252 | 0.022153 |
Rational Man and Irrational Man are driving on the same racetrack. Rational Man's path is parameterized by $x = \cos t$ and $y = \sin t$, while Irrational Man's path is parameterized by $x = 1 + 4 \cos \frac{t}{\sqrt{2}}$ and $y = 2 \sin \frac{t}{\sqrt{2}}$. If Rational Man starts driving at time $t = 0$, at what time ... | tiger_math | [
0.015526351518929005,
0.03382973000407219,
-0.023927459493279457,
-0.020044298842549324,
0.03407890349626541,
-0.004363001324236393,
-0.023313304409384727,
0.0506959892809391,
0.047279249876737595,
-0.02361069992184639,
0.005056270398199558,
0.023706400766968727,
-0.011685248464345932,
-0.... | 0.838 | 0.825298 | 0.012702 |
Let $p$ and $q$ be positive integers such that\[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\]and $p$ is as large as possible. What is $p+q$?
$\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$ | tiger_math | [
0.03114725835621357,
0.02738839015364647,
-0.04779776185750961,
-0.04421750456094742,
0.019627101719379425,
-0.006553548388183117,
-0.01025399286299944,
0.04325346648693085,
0.03521016240119934,
-0.023168964311480522,
0.017123937606811523,
0.04556221142411232,
-0.045300375670194626,
-0.006... | 0.859846 | 0.848239 | 0.011608 |
Let
\[f(x) = \frac{ax+b}{x + 1}.\]Find the constants $a$ and $b$ so that $f(f(x)) = x$ for all $x \neq -1.$ | tiger_math | [
0.034978125244379044,
0.023797810077667236,
-0.04745945334434509,
-0.046536967158317566,
0.038065772503614426,
-0.03449804335832596,
0.015248158946633339,
0.08571397513151169,
0.033834781497716904,
-0.014694178476929665,
0.012120453640818596,
0.011374176479876041,
-0.021982837468385696,
-0... | 0.836699 | 0.825462 | 0.011238 |
Lisa, an event planner, has four types of table centerpieces to decorate tables for a party: tulips, roses, daisies, and lilies. Each centerpiece type can use either a gold or silver vase. If Lisa wants to ensure each table at the party has a unique centerpiece, assuming she decorates 120 tables, what is the maximum nu... | tiger_math | [
0.01948920451104641,
0.014284622855484486,
-0.04379117488861084,
-0.03185528516769409,
0.02307831682264805,
-0.036471445113420486,
-0.02798503264784813,
0.06527138501405716,
0.04945284128189087,
-0.012240041047334671,
0.018304022029042244,
0.05010579153895378,
-0.019968288019299507,
0.0046... | 0.80057 | 0.793972 | 0.006598 |
The sides of rectangle $ABCD$ have lengths $a$ and $b$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle in terms of $a$ and $b$. | tiger_math | [
0.0256564449518919,
0.014608585275709629,
-0.04517040774226189,
-0.03099067322909832,
0.008922652341425419,
-0.028868574649095535,
-0.00045446312287822366,
0.048304419964551926,
0.058017224073410034,
-0.014370165765285492,
0.00977295357733965,
0.0573115199804306,
-0.023927059024572372,
-0.... | 0.845182 | 0.81891 | 0.026272 |
For real numbers $t,$ the point $(x,y)$ is plotted such that $x = \sin^2 t$ and $y = \cos^2 t$. All the plotted points lie on what type of curve? | tiger_math | [
0.010843136347830296,
0.03584843501448631,
-0.06731850653886795,
-0.047289397567510605,
0.03294498845934868,
-0.026270991191267967,
-0.013767106458544731,
0.052776116877794266,
0.0465119294822216,
-0.03584804758429527,
0.015638092532753944,
0.037476081401109695,
-0.023992087692022324,
0.00... | 0.829824 | 0.821248 | 0.008576 |
Let $r$, $s$, and $t$ be distinct solutions of the equation $x^3-5x^2+6x=9$.
Compute $\frac{rs}t + \frac{st}r + \frac{tr}s$. | tiger_math | [
0.030747149139642715,
0.032373495399951935,
-0.05319200083613396,
-0.028144877403974533,
0.04310867190361023,
-0.03494537994265556,
-0.0005902705015614629,
0.0815134048461914,
0.05924810469150543,
-0.015784503892064095,
0.01713140308856964,
0.019317295402288437,
-0.04265078902244568,
-0.01... | 0.836305 | 0.829432 | 0.006873 |
A group of friends went apple picking. They collected apples and decided to distribute them evenly among their 5 baskets. After filling the baskets, they had 3 apples leftover. When they tried to evenly distribute the same apples among 3 baskets, they had 2 apples remaining. What is the smallest number of apples they c... | tiger_math | [
0.027482176199555397,
0.009838791564106941,
-0.028407394886016846,
-0.032802682369947433,
0.029178231954574585,
-0.02149137668311596,
-0.016388053074479103,
0.049727119505405426,
0.021500857546925545,
-0.016821719706058502,
0.008572131395339966,
0.015642058104276657,
-0.044346679002046585,
... | 0.843002 | 0.839457 | 0.003545 |
Lucy has a garden divided into a rectangular grid, having a length of 12 meters and a width of 10 meters. She plans to plant flowers in a circular patch such that the circle just fits within the garden. Calculate the maximum radius of the circular garden patch and the area it will occupy in the garden. | tiger_math | [
0.007174442522227764,
0.05017068609595299,
-0.04506086930632591,
-0.046473558992147446,
0.023055966943502426,
-0.05089603364467621,
-0.049435488879680634,
0.04094219580292702,
0.03647368773818016,
0.006061977706849575,
0.00045547555782832205,
0.047567568719387054,
-0.04276416078209877,
-0.... | 0.802889 | 0.796066 | 0.006823 |
Find the product of all positive integer values of $c$ such that $8x^2+15x+c=0$ has two real roots. | tiger_math | [
0.02083640918135643,
0.02846270240843296,
-0.04367366060614586,
-0.04379968345165253,
0.03074096515774727,
-0.038108546286821365,
0.0011484710266813636,
0.07943031191825867,
0.03709445148706436,
-0.03978155925869942,
0.0033612255938351154,
0.027412569150328636,
-0.026917122304439545,
-0.04... | 0.846916 | 0.841868 | 0.005048 |
A cylindrical water tank has a radius of 5 units and a height of 10 units. A smaller cylindrical tank with a radius of 3 units and a height of 6 units is placed inside the larger tank. Find the volume of water that can be filled in the larger tank without overflowing the smaller tank, in terms of pi. | tiger_math | [
0.015446222387254238,
0.04779103398323059,
-0.03576371818780899,
-0.02298586256802082,
0.048879507929086685,
-0.043128758668899536,
-0.04081517085433006,
0.05058193951845169,
0.015733523294329643,
-0.01678275875747204,
0.01160768885165453,
0.02547094039618969,
-0.04595266655087471,
-0.0117... | 0.802613 | 0.796776 | 0.005837 |
In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have lengths in the ratio $2:3$. Find the square of the length of $QP$. | tiger_math | [
0.02284606732428074,
0.023182684555649757,
-0.042988188564777374,
-0.037243444472551346,
0.035186734050512314,
-0.01659710705280304,
-0.04049999266862869,
0.051042646169662476,
0.03848663344979286,
0.010213013738393784,
0.02107222005724907,
0.026573896408081055,
-0.025469163432717323,
-0.0... | 0.841911 | 0.825305 | 0.016606 |
Triangle $ABC$ is an obtuse, isosceles triangle. Angle $A$ measures 20 degrees. What is number of degrees in the measure of the largest interior angle of triangle $ABC$?
[asy]
draw((-20,0)--(0,8)--(20,0)--cycle);
label("$20^{\circ}$",(-13,-0.7),NE);
label("$A$",(-20,0),W);
label("$B$",(0,8),N);
label("$C$",(20,0),E);
... | tiger_math | [
0.02082388661801815,
0.01818760111927986,
-0.042561162263154984,
-0.04269855469465256,
0.02314099296927452,
-0.021405501291155815,
-0.035077061504125595,
0.033882107585668564,
0.02663920447230339,
0.005700232461094856,
0.025009121745824814,
0.041230734437704086,
-0.03969540446996689,
-0.01... | 0.831207 | 0.824052 | 0.007154 |
Find the equation of the locus of all points P such that the distance between P and the point (-3, 1) is 1 unit greater than the distance between P and the line x = -3. | tiger_math | [
0.02103174664080143,
0.00405401736497879,
-0.05554945021867752,
-0.05256667360663414,
0.034520670771598816,
-0.039475664496421814,
-0.01853954792022705,
0.036088086664676666,
0.03099607117474079,
-0.01138526201248169,
0.02916448377072811,
0.054808326065540314,
-0.031504787504673004,
-0.012... | 0.84421 | 0.841101 | 0.003109 |
An ice cream shop offers X kinds of ice cream. The greatest number of two scoop sundaes that can be made such that each sundae contains two types of ice cream and no two sundaes are the same combination is 15. What is the value of unknown variable X? | tiger_math | [
0.0062025198712944984,
0.02157367579638958,
-0.04146326705813408,
-0.02168763428926468,
0.03444034233689308,
0.0007129809237085283,
-0.03630714491009712,
0.06406670063734055,
0.04117688536643982,
-0.03072216920554638,
0.02854476124048233,
0.036187317222356796,
-0.021610062569379807,
-0.010... | 0.838555 | 0.832673 | 0.005882 |
Given that $a$, $b$, and $c$ are real numbers such that at least one of them is nonzero, find all possible values of the expression
\[\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{abc}{|abc|}.\]Enter all possible values, separated by commas. | tiger_math | [
0.03437373414635658,
0.029087821021676064,
-0.04464040696620941,
-0.029903046786785126,
0.03506571054458618,
-0.012308478355407715,
-0.007754223421216011,
0.05892359837889671,
0.03869007155299187,
-0.023546241223812103,
0.018871493637561798,
0.02594839781522751,
-0.024505117908120155,
-0.0... | 0.834216 | 0.821427 | 0.012788 |
During the annual school sports day, Mia is in charge of buying refreshments for the athletes. There are three varieties of sports drinks she plans to buy: Energy Booster, Hydration Plus, and Quick Recharge. Each drink type has a different size bottle:
- Energy Booster comes in 250 ml bottles.
- Hydration Plus comes i... | tiger_math | [
0.03403976559638977,
0.02067379839718342,
-0.04070006683468819,
-0.030093401670455933,
0.015544590540230274,
-0.04505832493305206,
-0.03952676057815552,
0.0604838989675045,
0.030838005244731903,
-0.030418187379837036,
0.008403473533689976,
0.03522950038313866,
-0.061559420078992844,
-0.019... | 0.807811 | 0.803764 | 0.004047 |
The function $f(x)$ satisfies
\[a^3 f(b) = b^3 f(a)\]for all real numbers $a$ and $b.$ If $f(3) \neq 0,$ find
\[\frac{f(8) - f(2)}{f(3)}.\] | tiger_math | [
0.043712813407182693,
0.02911558747291565,
-0.06163507699966431,
-0.03423517197370529,
0.02758558839559555,
-0.021558189764618874,
0.0005649930099025369,
0.0731782391667366,
0.0429999902844429,
-0.007607355248183012,
0.012761484831571579,
0.02327675186097622,
-0.03766840696334839,
-0.03364... | 0.833029 | 0.822521 | 0.010508 |
Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, and let $\mathbf{w}$ be a vector such that $\mathbf{u} \times \mathbf{v} + \mathbf{v} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{v} = \mathbf{u}.$ Compute $\mathbf{v} \cdot (\mathbf{u} \times \mathbf{w}).$ | tiger_math | [
0.04062872380018234,
0.04590515419840813,
-0.03994816914200783,
-0.03867720440030098,
0.05322793871164322,
-0.0071123456582427025,
-0.01008167490363121,
0.0762535110116005,
0.03522120416164398,
-0.006210261955857277,
0.008459044620394707,
0.029170677065849304,
-0.033916834741830826,
-0.031... | 0.826251 | 0.814031 | 0.01222 |
In an orchard, Charlie found an ancient treasure chest under an apple tree. The password to unlock it was stated in an old riddle: "Find the number 'x' that satisfies the equation $4x - 9 = 3x + 6$. What must 'x' be for you to unlock the treasure?" | tiger_math | [
0.03879893571138382,
0.01732436940073967,
-0.035705383867025375,
-0.03557693585753441,
0.03627690672874451,
-0.04141310602426529,
-0.014066245406866074,
0.05361669510602951,
0.028435884043574333,
-0.024165501818060875,
0.02136564999818802,
0.030478904023766518,
-0.042197879403829575,
-0.01... | 0.832908 | 0.829144 | 0.003764 |
The terms of the sequence $(a_i)$ defined by $a_{n + 2} = \frac {2009 + a_n} {1 + a_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $a_1 + a_2$. | tiger_math | [
0.011342013254761696,
0.013985943049192429,
-0.05286949500441551,
-0.031239446252584457,
0.030671164393424988,
-0.026379769667983055,
-0.028134731575846672,
0.07243778556585312,
0.04794524237513542,
-0.004964152816683054,
0.014599312096834183,
0.014106858521699905,
-0.03279224410653114,
0.... | 0.837141 | 0.830022 | 0.007119 |
A pirate is searching for buried treasure on 6 islands. On each island, there is a $\frac{1}{4}$ chance that the island has buried treasure and no traps, a $\frac{1}{12}$ chance that the island has traps but no treasure, and a $\frac{2}{3}$ chance that the island has neither traps nor treasure. What is the probability ... | tiger_math | [
0.015310694463551044,
0.028991028666496277,
-0.037738535553216934,
-0.033675145357847214,
0.0392080619931221,
-0.004548129625618458,
-0.019912850111722946,
0.02588675357401371,
0.006360411178320646,
-0.02527228556573391,
0.022748971357941628,
0.013411485590040684,
-0.019837869331240654,
-0... | 0.835048 | 0.826893 | 0.008155 |
A series of equilateral triangles is formed by the following process: An equilateral triangle T1 has side length 40 cm. A second equilateral triangle T2 is formed by joining the mid-points of the sides of T1. A third equilateral triangle T3 is formed by joining the mid-points of the sides of T2. A fourth equilateral tr... | tiger_math | [
0.021239684894680977,
0.024338627234101295,
-0.03098367527127266,
-0.05120797082781792,
0.027305733412504196,
-0.01974032074213028,
-0.037027448415756226,
0.0698033794760704,
0.03410429134964943,
-0.012841287069022655,
0.01935334876179695,
0.04296838119626045,
-0.045869130641222,
-0.027960... | 0.823627 | 0.79725 | 0.026377 |
If $a$, $b$, and $c$ are distinct real numbers such that $\frac{a}{b - c} + \frac{b}{c - a} + \frac{c}{a - b} = 0$, what are all the possible values of $\frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2}$? Please enter all the possible values, separated by commas. | tiger_math | [
0.034861449152231216,
0.039928436279296875,
-0.039396703243255615,
-0.026652999222278595,
0.04147506132721901,
-0.02519259788095951,
0.005060652270913124,
0.06362905353307724,
0.06153079494833946,
-0.04564180597662926,
0.0037557475734502077,
0.02948009967803955,
-0.03683348745107651,
-0.03... | 0.836193 | 0.821974 | 0.01422 |
Let $\mathbf{a} = \begin{pmatrix} X \\ 1 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}.$ Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}. What is the value of u... | tiger_math | [
0.0340607687830925,
0.046118106693029404,
-0.06499089300632477,
-0.030238106846809387,
0.044281646609306335,
0.010330368764698505,
0.002320805098861456,
0.0915893092751503,
0.03152620419859886,
-0.011672371067106724,
0.006177341099828482,
0.03011353500187397,
-0.0305128563195467,
-0.021913... | 0.822576 | 0.815074 | 0.007502 |
Amelia painted all the outer faces of a structure formed by 12 interconnected cuboids, arranged in a particular pattern. Some faces are shared between neighboring cuboids. Calculate the total surface area of the painted faces. | tiger_math | [
0.024243008345365524,
0.041400473564863205,
-0.04439602047204971,
-0.03193414583802223,
0.0241222083568573,
-0.04973648488521576,
-0.04116004705429077,
0.05922701954841614,
0.025863902643322945,
-0.008602943271398544,
0.02404770255088806,
0.043092552572488785,
-0.01822950318455696,
-0.0105... | 0.809941 | 0.805136 | 0.004805 |
The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | tiger_math | [
0.013698618859052658,
0.03230563551187515,
-0.048355016857385635,
-0.04948735609650612,
0.03148321434855461,
-0.007586097810417414,
-0.01272005494683981,
0.04859806224703789,
0.03856457397341728,
-0.018092280253767967,
0.026934491470456123,
0.024782516062259674,
-0.04087359085679054,
-0.00... | 0.835849 | 0.826842 | 0.009007 |
In a theme park, there are four different zones: Adventureland, Fantasyland, Tomorrowland, and Frontierland. A group of friends decides to visit each zone exactly once during their day at the park. Each transition between zones involves traveling on a monorail, train, or walking. They start at Adventureland and are equ... | tiger_math | [
0.014785023406147957,
0.025038715451955795,
-0.04750451073050499,
-0.04113982990384102,
0.03467830643057823,
-0.02171110175549984,
-0.03169378265738487,
0.053911540657281876,
0.02380865067243576,
-0.008342019282281399,
0.011277708224952221,
0.021872011944651604,
-0.00805064756423235,
-0.00... | 0.809885 | 0.802073 | 0.007812 |
The vector $\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ is rotated $120^\circ$ about the origin. During the rotation, it passes through the $xy$-plane. Find the resulting vector. | tiger_math | [
0.011137035675346851,
0.037735965102910995,
-0.035965245217084885,
-0.041551392525434494,
0.03959198668599129,
-0.02222178317606449,
-0.02836141362786293,
0.06478247791528702,
0.03996683284640312,
-0.0067983996123075485,
0.0032047440763562918,
0.022574661299586296,
-0.04347215220332146,
-0... | 0.820381 | 0.817419 | 0.002962 |
At point P, two circles are concentric. The sides of a 30 degree angle at P create an arc on the larger circle that has the same length as an arc on the smaller circle formed by the sides of an angle at P. If the radius of the larger circle is 10, what is the radius of the smaller circle? | tiger_math | [
0.020879140123724937,
0.03554047271609306,
-0.04905840381979942,
-0.024901753291487694,
0.017423123121261597,
-0.018643897026777267,
-0.023803718388080597,
0.04004386067390442,
0.03287219628691673,
-0.021318307146430016,
0.023120228201150894,
0.03022041730582714,
-0.053404390811920166,
-0.... | 0.814176 | 0.803361 | 0.010815 |
A point has rectangular coordinates $(x,y,z)$ and spherical coordinates $\left(2, \frac{8 \pi}{7}, \frac{2 \pi}{9} \right).$ Find the spherical coordinates of the point with rectangular coordinates $(-x,y,-z).$ Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \p... | tiger_math | [
0.00017624169413466007,
0.03967591002583504,
-0.03278058022260666,
-0.041019268333911896,
0.01529859658330679,
-0.002801573369652033,
-0.011735256761312485,
0.042827315628528595,
0.057165250182151794,
0.014161298982799053,
0.005089608021080494,
0.04614127427339554,
-0.033681582659482956,
-... | 0.818001 | 0.806024 | 0.011978 |
If $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix},$ then its transpose is given by
\[\mathbf{A}^T = \begin{pmatrix} a & c \\ b & d \end{pmatrix}.\]Given that $\mathbf{A}^T = \mathbf{A}^{-1},$ find $a^2 + b^2 + c^2 + d^2.$ | tiger_math | [
0.06154649704694748,
0.038555581122636795,
-0.04180014505982399,
-0.044172365218400955,
0.05882047489285469,
-0.023132624104619026,
0.0072215828113257885,
0.07485514134168625,
0.03540697321295738,
0.00006851266516605392,
0.010909209959208965,
0.016009444370865822,
-0.04492612183094025,
-0.... | 0.809227 | 0.795751 | 0.013476 |
Lucy has five unique flowers in her garden: a rose, a tulip, a daisy, a sunflower, and a lily. She decides to plant them in a row in such a way that the tulip is never next to the sunflower. How many different ways can she arrange the flowers under this condition? | tiger_math | [
0.002254193415865302,
0.015772586688399315,
-0.04086325690150261,
-0.04385729506611824,
0.020862875506281853,
-0.02656007930636406,
-0.04941781982779503,
0.05542320758104324,
0.046273451298475266,
-0.011809495277702808,
0.0020705442875623703,
0.026975097134709358,
-0.02251237817108631,
-0.... | 0.812517 | 0.807093 | 0.005424 |
The Mathletes have $Y$ members. Three members, Alice, Ben, and Carol, cannot be on the same team. How many ways can the coach create two teams of 3 members each, ensuring that Alice, Ben, and Carol are not on the same team? | tiger_math | [
0.022763706743717194,
0.004970578011125326,
-0.05023924261331558,
-0.030788222327828407,
0.03758508339524269,
-0.020557735115289688,
-0.02502293698489666,
0.03901824727654457,
0.030248234048485756,
-0.023963619023561478,
0.0018161297775804996,
0.03992820531129837,
-0.02529301308095455,
-0.... | 0.828309 | 0.823858 | 0.00445 |
In a large neighborhood, there is a popular walking competition. Joe and four of his friends decide to participate. They can walk through any of the 6 different parks each day for a week (Monday to Friday). Joe likes variety and wants to make sure he walks in a different park each day. How many different combinations o... | tiger_math | [
0.035288434475660324,
0.02783314883708954,
-0.06058891862630844,
-0.025544945150613785,
0.0319172665476799,
-0.042786117643117905,
-0.031223254278302193,
0.04204085096716881,
0.02926204726099968,
-0.02134695276618004,
0.007986856624484062,
0.03970613330602646,
-0.04104387015104294,
-0.0199... | 0.824248 | 0.813832 | 0.010415 |
If there are $n$ distinct points on the circumference of a circle, how many different triangles can be formed if each vertex must be one of these $n$ points? | tiger_math | [
-0.0025195851922035217,
0.02783951722085476,
-0.028553076088428497,
-0.025514677166938782,
0.023020610213279724,
-0.023074518889188766,
-0.015548430383205414,
0.026453670114278793,
0.034039318561553955,
-0.01908586360514164,
0.012188936583697796,
0.036417681723833084,
-0.027697574347257614,
... | 0.83601 | 0.829165 | 0.006845 |
Convert the point $\left( 8, \frac{\pi}{4}, \sqrt{3} \right)$ in cylindrical coordinates to rectangular coordinates. | tiger_math | [
-0.0013607846340164542,
0.04970835894346237,
-0.0248197503387928,
-0.04350970685482025,
0.027325280010700226,
-0.013732538558542728,
-0.03423767909407616,
0.06401780992746353,
0.055295661091804504,
-0.004496936220675707,
-0.002372222486883402,
0.03786887228488922,
-0.029725102707743645,
-0... | 0.839815 | 0.833432 | 0.006383 |
In the diagram below, the rectangle has a width of $x$, a length of $12$, and an area of $48$. What is the ratio of its length to its width?
[asy]
pair a = (0, 0); pair b = (12, 0); pair c = (12, 4); pair d = (0, 4);
draw(a--b--c--d--cycle);
label("$x$", midpoint(a--d), W); label("$12$", midpoint(c--d), N);
[/asy]
Wr... | tiger_math | [
0.02535383403301239,
0.014311796985566616,
-0.04957784712314606,
-0.044318776577711105,
0.023116331547498703,
-0.03590267896652222,
-0.03450055420398712,
0.06623604148626328,
0.04033634066581726,
0.00974472425878048,
0.024936029687523842,
0.035461634397506714,
-0.02277478761970997,
-0.0165... | 0.83492 | 0.827692 | 0.007228 |
[asy] unitsize(27); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E,F,X,Y,Z; A=(3,3); B=(0,0); C=(6,0); D=(4,0); E=(4,2); F=(1,1); draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(C--F); X=intersectionpoint(A--D,C--F); Y=intersectionpoint(B--E,A--D); Z=intersectionpoint(B--E,C--F); label("$A$",A,N); label(... | tiger_math | [
0.024400219321250916,
0.01220406498759985,
-0.03914168104529381,
-0.042059775441884995,
0.04960653558373451,
-0.03713078796863556,
-0.030212631449103355,
0.052720312029123306,
0.03789098560810089,
0.0007572300382889807,
0.03456903621554375,
0.03213071450591087,
-0.01677405834197998,
-0.026... | 0.841379 | 0.822956 | 0.018423 |
The isosceles triangle and the square shown here have the same perimeter in units. What is the height of the triangle, $h$, in terms of the side length of the square, $s$?
[asy]
draw((0,0)--(0,10)--(10,10)--(10,0)--cycle);
fill((0,0)--(17,5)--(0,10)--cycle,white);
draw((0,0)--(17,5)--(0,10)--cycle);
label("$s$",(5,10)... | tiger_math | [
0.02069249376654625,
0.02946195937693119,
-0.034924495965242386,
-0.04369150847196579,
0.04573723301291466,
-0.02256814017891884,
-0.0193271916359663,
0.06226027011871338,
0.02439059130847454,
0.000513831153512001,
0.03846941515803337,
0.0354180634021759,
-0.02872404456138611,
-0.029087970... | 0.834252 | 0.826429 | 0.007823 |
There are two distinct unit vectors $\mathbf{v}$ such that the angle between $\mathbf{v}$ and $\begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}$ is $45^\circ,$ and the angle between $\mathbf{v}$ and $\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$ is $60^\circ.$ Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be these vectors. Find $\|\... | tiger_math | [
0.03248991444706917,
0.024802038446068764,
-0.03800719976425171,
-0.041134562343358994,
0.02423640713095665,
0.004141727928072214,
-0.013872593641281128,
0.0695391371846199,
0.02734380215406418,
-0.0209148321300745,
0.031316015869379044,
0.02080303616821766,
-0.032569605857133865,
-0.01396... | 0.830879 | 0.812036 | 0.018843 |
Let the ordered triples $(x,y,z)$ of complex numbers that satisfy the system of equations:
\begin{align*}
x + yz &= X, \\
y + xz &= 10, \\
z + xy &= 10
\end{align*}
be $(x_1,y_1,z_1),$ $(x_2,y_2,z_2),$ $\dots,$ $(x_n,y_n,z_n).$ If the sum of the values of $z_1,z_2,\dots,z_n$ is 10, find the value of $X$. | tiger_math | [
0.02888385020196438,
0.018624190241098404,
-0.04888244345784187,
-0.03802824020385742,
0.03593456372618675,
-0.012114267796278,
-0.012501050718128681,
0.038649946451187134,
0.02237531542778015,
-0.0023925791028887033,
0.02242940478026867,
0.04103574901819229,
-0.05761079117655754,
-0.02457... | 0.838122 | 0.817593 | 0.020529 |
In a small park, there are three types of trees: oak, pine, and maple. There are 11 oaks, each producing 120 acorns. Each pine tree produces twice as many pinecones as there are pine trees, and there are 7 pines. Each maple produces 30 fewer seeds than ten times the number of maples, with a total of 5 maples in the par... | tiger_math | [
0.016303375363349915,
0.033983975648880005,
-0.053721971809864044,
-0.04186635464429855,
0.03417683765292168,
-0.03457237407565117,
-0.02223799377679825,
0.08670788258314133,
0.05224534496665001,
-0.008136524818837643,
0.001252337358891964,
0.026254432275891304,
-0.0514577217400074,
-0.012... | 0.836579 | 0.830847 | 0.005732 |
Points $A$, $B$, $C$, and $D$ are located on $\overline{AB}$ such that $AB = 3AD = 9BC$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $B$ and $D$? Express your answer as a common fraction.
[asy]
draw((0,0)--(12,.0000000001));
dot((0,0)); dot((4,0)); dot((11,0)); dot(... | tiger_math | [
0.019076958298683167,
0.007452350575476885,
-0.05398279428482056,
-0.04175969958305359,
0.02973085269331932,
-0.03773767873644829,
-0.008497079834342003,
0.042218293994665146,
0.04497842490673065,
0.013417214155197144,
0.02422676794230938,
0.02951919287443161,
-0.004745247308164835,
-0.020... | 0.840572 | 0.829479 | 0.011093 |
Find the equation of the plane which passes through the point $(1,4,-2),$ and which is parallel to the plane $-2x + y - 3z = 7.$ 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|) = X
If we know the answer to the above question is 2... | tiger_math | [
0.027899213135242462,
0.029740117490291595,
-0.058771055191755295,
-0.0525335930287838,
0.04097866266965866,
0.01022179052233696,
-0.013049010187387466,
0.056269366294145584,
0.0394480936229229,
-0.006711078342050314,
0.012988491915166378,
0.04820915311574936,
-0.011403735727071762,
-0.022... | 0.827333 | 0.82173 | 0.005603 |
Define $a \Delta b = a^2 -b^2 $. What is the value of $ (2^{4 \Delta13})\Delta(3^{3\Delta5})$ | tiger_math | [
0.03659427911043167,
0.04352466017007828,
-0.037029609084129333,
-0.042535293847322464,
0.04186802729964256,
-0.02004234306514263,
-0.009921660646796227,
0.06545741856098175,
0.04782514274120331,
-0.01874525099992752,
0.022492581978440285,
0.02896231599152088,
-0.032358262687921524,
-0.040... | 0.850023 | 0.846377 | 0.003645 |
The graphs of $2y + x + 3 = 0$ and $3y + ax + 2 = 0$ are perpendicular. Solve for $a.$ | tiger_math | [
0.021225236356258392,
0.030352164059877396,
-0.033823367208242416,
-0.03139135241508484,
0.036587562412023544,
-0.010223736055195332,
-0.007933896034955978,
0.07784034311771393,
0.055434707552194595,
-0.023070259019732475,
0.011629065498709679,
0.03383508324623108,
-0.0264374278485775,
-0.... | 0.831756 | 0.826673 | 0.005083 |
Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate
\[\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}.\] | tiger_math | [
0.029245059937238693,
0.037025198340415955,
-0.03358388692140579,
-0.04395626485347748,
0.04238622263073921,
-0.02661193162202835,
-0.0198217760771513,
0.05968732386827469,
0.04071112722158432,
0.003157197032123804,
0.041256342083215714,
0.01921871118247509,
-0.04012337327003479,
-0.028764... | 0.83178 | 0.810224 | 0.021556 |
For a positive integer $n,$ let
\[a_n = \sum_{k = 0}^n \frac{1}{\binom{n+1}{k}} \quad \text{and} \quad b_n = \sum_{k = 0}^n \frac{k}{\binom{n}{k}}.\]Simplify $\frac{a_n}{b_n}.$ | tiger_math | [
0.028890321031212807,
0.03162652626633644,
-0.052033454179763794,
-0.02792872115969658,
0.04242163151502609,
-0.014607703313231468,
-0.014646693132817745,
0.052383903414011,
0.041142016649246216,
-0.009161467663943768,
0.015525512397289276,
0.0131918266415596,
-0.03265564143657684,
-0.0128... | 0.841872 | 0.819641 | 0.022231 |
Ed and Sue bike, jog, and swim at equal and constant rates. Ed covers $74$ kilometers after biking for $2$ hours, jogging for $3$ hours, and swimming for $4$ hours, while Sue covers $91$ kilometers after jogging for $2$ hours, swimming for $3$ hours, and biking for $4$ hours. The sum of their biking, jogging, and swimm... | tiger_math | [
0.027967721223831177,
0.022837433964014053,
-0.06451334804296494,
-0.046751927584409714,
0.0325516052544117,
-0.024685369804501534,
-0.026192007586359978,
0.047629158943891525,
0.044802431017160416,
-0.02136611007153988,
0.02595050260424614,
0.017434781417250633,
-0.042653411626815796,
-0.... | 0.838621 | 0.835378 | 0.003244 |
A 2 by 2004 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fraction. [asy]
size(7cm);
defaultpen(linewidt... | tiger_math | [
0.007717180997133255,
0.04441248998045921,
-0.04478885978460312,
-0.03495173156261444,
0.024214522913098335,
-0.027059292420744896,
-0.024600131437182426,
0.07228101789951324,
0.01562599465250969,
0.010184889659285545,
0.030219359323382378,
0.02846796251833439,
0.0009897822747007012,
-0.01... | 0.83184 | 0.824222 | 0.007617 |
Three of the four vertices of a rectangle are $(5, 11)$, $(16, 11)$ and $(16, -2)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 5)^2 + (y + 2)^2 = X$? Express your answer in terms of $\pi$.
If we know the answer to the above question is \frac{9}{... | tiger_math | [
0.014698359183967113,
0.0259324312210083,
-0.0644233301281929,
-0.03482786566019058,
0.034312885254621506,
-0.009889342822134495,
-0.009490556083619595,
0.06249530240893364,
0.048956431448459625,
0.008893284946680069,
0.025076262652873993,
0.04165305197238922,
-0.01667466014623642,
-0.0154... | 0.842282 | 0.830293 | 0.011989 |
In $\triangle ABC$, the sum of $\angle A$ and $\angle B$ is half the measure of $\angle C$. Find the measure of $\angle C$. | tiger_math | [
0.019857851788401604,
0.0152904586866498,
-0.038843825459480286,
-0.04115087166428566,
0.03191688656806946,
-0.011703355237841606,
-0.016868699342012405,
0.04323052987456322,
0.03756151348352432,
-0.0207875594496727,
0.016530415043234825,
0.030364379286766052,
-0.039242979139089584,
-0.014... | 0.8408 | 0.826135 | 0.014665 |
The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$? | tiger_math | [
0.03977540507912636,
0.017770081758499146,
-0.06111579388380051,
-0.034252021461725235,
0.026018615812063217,
-0.012244034558534622,
-0.020249484106898308,
0.026681920513510704,
0.04034138098359108,
-0.029123185202479362,
0.015987718477845192,
0.023842807859182358,
-0.030457619577646255,
-... | 0.836349 | 0.830203 | 0.006146 |
A shopkeeper has three items with varying cost prices. For the first item, she fixes the marked price 40% above its cost price and aims for a 12% profit. For the second item, she fixes the marked price 60% above its cost price and targets a 20% profit. For the third item, she fixes the marked price 35% above its cost p... | tiger_math | [
0.03631476312875748,
0.0047971284948289394,
-0.05267716944217682,
-0.03361313045024872,
0.05221128091216087,
-0.016752254217863083,
-0.02761039137840271,
0.057755451649427414,
0.050481460988521576,
0.004420885816216469,
0.021984906867146492,
0.030228201299905777,
-0.03810787945985794,
-0.0... | 0.81874 | 0.815324 | 0.003416 |
Find the ordered pair of prime numbers $(p,q)$ that satisfy the system of equations:
\begin{align*}
p^q+1&=q^p,\\
2p^q&=q^p+14.
\end{align*} | tiger_math | [
0.05023161321878433,
0.02787286601960659,
-0.03448813036084175,
-0.05894653499126434,
0.04992036893963814,
-0.026306768879294395,
0.004224721807986498,
0.05358009412884712,
0.033950336277484894,
-0.017665855586528778,
0.028361845761537552,
0.006591638084501028,
-0.037255920469760895,
-0.03... | 0.846585 | 0.833544 | 0.013041 |
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