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 |
|---|---|---|---|---|---|
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 8$ and $\angle O_{1}PO_{2} = 150^{\circ}$, then $AP = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are positive integers. Find $a + b$. | tiger_math | [
0.012361614964902401,
0.0235295370221138,
-0.04475835710763931,
-0.045708026736974716,
0.025984086096286774,
-0.0011414564214646816,
0.00038457888877019286,
0.07461794465780258,
0.055658288300037384,
0.0025481022894382477,
0.0066033536568284035,
0.029118001461029053,
-0.03802909702062607,
... | 0.841065 | 0.817644 | 0.023421 |
Find the values of $c_3$ and $c_4$ such that the following equation holds true:
$$c_3 \begin{pmatrix} 4 \\ -1 \end{pmatrix} + c_4 \begin{pmatrix} -3 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \end{pmatrix}$$ | tiger_math | [
0.033278629183769226,
0.04414950683712959,
-0.05576392635703087,
-0.0435636043548584,
0.0651063621044159,
-0.025256279855966568,
-0.006799254100769758,
0.06567893177270889,
0.039711203426122665,
-0.009963024407625198,
-0.0016402134206146002,
0.027328358963131905,
-0.031011631712317467,
-0.... | 0.844006 | 0.836264 | 0.007742 |
Let $u$ and $v$ be positive integers satisfying $0 < v < u$. Let $A = (u,v)$, let $B$ be the reflection of $A$ across the line $y = x$, let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $AB... | tiger_math | [
0.014221809804439545,
0.035013724118471146,
-0.05842497944831848,
-0.04196930676698685,
0.036423683166503906,
-0.029116163030266762,
-0.0094853974878788,
0.05179208889603615,
0.06142018362879753,
-0.01032746210694313,
0.020892782136797905,
0.03875325620174408,
-0.031286146491765976,
-0.025... | 0.850359 | 0.829023 | 0.021336 |
Quadrilateral $ABCD$ is a square, and segment $AE$ is perpendicular to segment $ED$. If $AE = 8$ units and $DE = 6$ units, what is the area of pentagon $AEDCB$, in square units? [asy]
size(150);
pair A, B, C, D, E;
A=(0,10);
B=(0,0);
C=(10,0);
D=(10,10);
E=(6.4,5.2);
draw(A--B--C--D--A);
draw(A--E--D);
label("A", A, NW... | tiger_math | [
0.025799209251999855,
0.010997449047863483,
-0.05336267501115799,
-0.05478238686919212,
0.029156457632780075,
-0.02372375875711441,
-0.014731809496879578,
0.060450319200754166,
0.04063091427087784,
0.004238244146108627,
0.01061604917049408,
0.027559053152799606,
-0.035578303039073944,
-0.0... | 0.833538 | 0.821838 | 0.0117 |
Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} - \begin{pmatrix} 2 \\ 1 \end{pmatrix} \right\| = 5.\]Find the smallest possible value of $\|\mathbf{v}\|.$ | tiger_math | [
0.020631197839975357,
0.03801928833127022,
-0.044878553599119186,
-0.03629925101995468,
0.04911188781261444,
0.000592254102230072,
-0.02757558599114418,
0.0757082924246788,
0.039615534245967865,
-0.011186517775058746,
0.004587564617395401,
0.010368282906711102,
-0.022594662383198738,
-0.00... | 0.833593 | 0.826967 | 0.006626 |
A rectangular playground measures 30 meters in length and 20 meters in width. On a sunny day, Sarah decides to play hopscotch by drawing a straight path diagonal across the playground. She starts at one corner and aims to hop diagonally to the opposite corner. Calculate the square of the length of Sarah's hopscotch pat... | tiger_math | [
0.017362475395202637,
0.023970596492290497,
-0.061627347022295,
-0.0550769604742527,
0.02611885592341423,
-0.027116650715470314,
-0.03185367211699486,
0.04792698845267296,
0.04222200810909271,
-0.03730910271406174,
0.01698482036590576,
0.046365633606910706,
-0.03194998577237129,
-0.0507552... | 0.813158 | 0.809389 | 0.00377 |
A farmer is planning to plant trees in his large, rectangular field. The field measures 120 meters by 180 meters. He intends to plant one tree every 10 meters along the length and every 12 meters along the width. If each tree requires exactly 4 square meters of space around it and no two trees can overlap this space re... | tiger_math | [
0.010303223505616188,
0.027272943407297134,
-0.04850083217024803,
-0.046401601284742355,
0.022481421008706093,
-0.02632066234946251,
-0.0062852539122104645,
0.0644894689321518,
0.027761563658714294,
-0.00796873401850462,
-0.00581571739166975,
0.04312174767255783,
-0.038189612329006195,
-0.... | 0.814245 | 0.808082 | 0.006162 |
In a triangle with sides of lengths $a$, $b$, and $c$, the following equation holds:
$$(a + b + c)(b + c - a) = 3bc.$$
Find the angle opposite the side of length $a$, in degrees. | tiger_math | [
0.026252422481775284,
0.010318612679839134,
-0.03728745877742767,
-0.03928253427147865,
0.021387679502367973,
-0.028902249410748482,
-0.028215983882546425,
0.056015755981206894,
0.03837508335709572,
-0.008860288187861443,
0.02299327775835991,
0.04676975682377815,
-0.0417405441403389,
-0.02... | 0.826486 | 0.8221 | 0.004386 |
Donna has $n$ boxes of doughnuts. Each box contains $12$ doughnuts.
After eating two doughnuts, Donna is able to rearrange the remaining doughnuts into bags so that each bag contains $10$ doughnuts, and none are left over. What is the smallest possible value of $n$? | tiger_math | [
0.012037977576255798,
0.025535468012094498,
-0.016737094148993492,
-0.033200424164533615,
0.014332792721688747,
0.007175510283559561,
-0.017659850418567657,
0.06683649122714996,
0.024865930899977684,
-0.023448020219802856,
0.015435176901519299,
0.03437308594584465,
-0.03268180787563324,
-0... | 0.848745 | 0.838636 | 0.010109 |
In the land of Mathemagica, three wizards share a garden where they grow magical sunflowers, moonlilies, and starblossoms. Wizard A plants 15 sunflowers, each emitting 7 sparkles. Wizard B plants 10 moonlilies, each producing 12 glows. Wizard C cultivates 8 starblossoms, which together emit 14 twinkles. Calculate the t... | tiger_math | [
0.025106506422162056,
0.03624895215034485,
-0.03941749781370163,
-0.07159307599067688,
0.03726276010274887,
-0.04827238991856575,
-0.042924925684928894,
0.04339071735739708,
0.04570388048887253,
-0.023670701310038567,
0.005380208604037762,
0.029665851965546608,
-0.02529274672269821,
-0.031... | 0.819847 | 0.814203 | 0.005643 |
For a nonnegative integer $n$, let $r_8(n)$ stand for the remainder left when $n$ is divided by $8.$ For example, $r_8(25)=1.$
What is the $18^{\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_8(3n)\le 4~?$$(Note that the first entry in this list is $0$.) | tiger_math | [
0.02763458527624607,
0.012125125154852867,
-0.030233114957809448,
-0.037005648016929626,
0.024601856246590614,
-0.01816604845225811,
-0.026011187583208084,
0.04393910616636276,
0.029502250254154205,
-0.011001422069966793,
0.006376436445862055,
0.03205455467104912,
-0.04790116474032402,
-0.... | 0.85086 | 0.843389 | 0.007471 |
In triangle $ABC,$ $AB = 10,$ $BC = 12,$ $AC = 14,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area of the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$ | tiger_math | [
0.027692105621099472,
0.021987907588481903,
-0.04371694102883339,
-0.04022442549467087,
0.021678557619452477,
-0.00767779815942049,
-0.025836383923888206,
0.04607202112674713,
0.03488728776574135,
-0.0035899977665394545,
0.009252303279936314,
0.02834203466773033,
-0.007634276989847422,
-0.... | 0.828636 | 0.819123 | 0.009513 |
The ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]has its foci at $(0,\pm 5),$ and the hyperbola \[\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1\]has its foci at $(\pm 8,0).$ Find the value of $ab.$ | tiger_math | [
0.02011936344206333,
0.04357314109802246,
-0.04850256070494652,
-0.029253143817186356,
0.04226956516504288,
-0.014236467890441418,
-0.016994450241327286,
0.05024382099509239,
0.05571004003286362,
-0.02204437367618084,
0.011536717414855957,
0.022839777171611786,
-0.011037101969122887,
-0.02... | 0.828646 | 0.81434 | 0.014306 |
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 $f(1 + i)$ and $f(1 - i)$ are both real. What is the smallest possible value of $| \alpha | + |\gamma |$? | tiger_math | [
0.010562360286712646,
0.04278620332479477,
-0.05240488797426224,
-0.03889138996601105,
0.041854649782180786,
-0.010766050778329372,
0.0016216217773035169,
0.06342093646526337,
0.04772990196943283,
-0.0011839140206575394,
0.015715181827545166,
0.022870410233736038,
-0.026173489168286324,
-0... | 0.850001 | 0.833529 | 0.016472 |
An isosceles triangle has an area of 120 square units. A line segment parallel to its base divides it into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 80 square units. If the altitude of the original triangle from the vertex opposite the base is 15 units, find the length of the... | tiger_math | [
0.03859145566821098,
0.003343238029628992,
-0.04284152761101723,
-0.049227140843868256,
0.04272359237074852,
-0.022441746667027473,
-0.01415029726922512,
0.059147074818611145,
0.031204653903841972,
-0.008031930774450302,
0.03740557283163071,
0.0471062958240509,
-0.030546288937330246,
-0.02... | 0.805483 | 0.797697 | 0.007786 |
Real numbers $x$ and $y$ have an arithmetic mean of 18 and a geometric mean of $\sqrt{92}$. Find $x^2+y^2$. | tiger_math | [
0.021467948332428932,
0.03487816080451012,
-0.0553586408495903,
-0.06757838279008865,
0.028237052261829376,
-0.01860196143388748,
-0.030714862048625946,
0.05539128556847572,
0.05005593225359917,
-0.03378580883145332,
0.02899901382625103,
0.023811176419258118,
-0.035660337656736374,
-0.0081... | 0.835285 | 0.82608 | 0.009205 |
Find all positive real numbers $x$ that satisfy
\[x \sqrt{10 - x} + \sqrt{10x - x^3} \ge 10.\]Enter all solutions, separated by commas. | tiger_math | [
0.019247809424996376,
0.02143106237053871,
-0.030592981725931168,
-0.028290562331676483,
0.028164906427264214,
-0.034592434763908386,
-0.020864252001047134,
0.062362827360630035,
0.05354820191860199,
-0.032528046518564224,
0.018176991492509842,
0.039993081241846085,
-0.057557377964258194,
... | 0.832009 | 0.818998 | 0.013011 |
For a positive integer $n,$ let $H_n$ be defined as in the original question. Find the value of
\[\sum_{n = 1}^\infty \frac{H_{n+2}-H_n}{(n + 1)(n + 2) H_{n+1}}.\] | tiger_math | [
0.010235792025923729,
0.030185185372829437,
-0.03391900286078453,
-0.04749434441328049,
0.030496319755911827,
-0.025495348498225212,
-0.035438865423202515,
0.05246533080935478,
0.03390543907880783,
-0.025327475741505623,
0.018309803679585457,
0.020842010155320168,
-0.022837383672595024,
-0... | 0.847491 | 0.838921 | 0.00857 |
The legs of a right triangle have lengths $\log_4 27$ and $\log_2 9.$ If the length of the hypotenuse is $h,$ compute $4^h.$ | tiger_math | [
0.019432030618190765,
0.032563671469688416,
-0.0325937420129776,
-0.028877627104520798,
0.021199408918619156,
-0.03668436408042908,
-0.02267860434949398,
0.0449058972299099,
0.05365247651934624,
-0.010386927984654903,
0.029818953946232796,
0.0389799140393734,
-0.027971303090453148,
-0.0316... | 0.852653 | 0.845509 | 0.007144 |
Square $EFGH$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 12$ and the $x$-axis, as shown below. The vertices $E$ and $F$ lie on the $x$-axis. Find the area of square $EFGH.$
[asy]
unitsize(0.8 cm);
real parab (real x) {
return(x^2 - 8*x + 12);
}
pair E, F, G, H;
real x = -1 + sqrt(5);
E = (... | tiger_math | [
0.043627552688121796,
0.017906449735164642,
-0.027603235095739365,
-0.054146166890859604,
0.046704404056072235,
-0.04483328014612198,
-0.023041702806949615,
0.08385570347309113,
0.044690947979688644,
0.00016615336062386632,
0.01276095025241375,
0.044420257210731506,
-0.027978673577308655,
... | 0.82401 | 0.810857 | 0.013153 |
Find the number of integer solutions to the inequality $\sqrt{x+1} > 2$ and $\sqrt{x+1} < 5$. | tiger_math | [
0.012771019712090492,
0.018416957929730415,
-0.034989677369594574,
-0.04617986083030701,
0.0352318175137043,
-0.0011580820428207517,
-0.03044968470931053,
0.06617176532745361,
0.054644905030727386,
-0.0282144732773304,
0.029334135353565216,
0.014063259586691856,
-0.04077944532036781,
-0.03... | 0.834605 | 0.83128 | 0.003325 |
The figure shows two concentric circles. If the length of chord AB is 100 units and chord AB is tangent to the smaller circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
[asy]
defaultpen(linewidth(.8pt));
dotfactor=4;
filldraw(circle((0,0),60),gray);
filldraw(circle((0,0),40),white);... | tiger_math | [
0.002240607049316168,
0.02132694609463215,
-0.03714662790298462,
-0.03630780801177025,
0.021525627002120018,
-0.05223832279443741,
-0.022331705316901207,
0.052179887890815735,
0.049142781645059586,
0.008173792622983456,
0.020497813820838928,
0.022480927407741547,
-0.019616473466157913,
-0.... | 0.83132 | 0.823307 | 0.008013 |
Let $u_n$ be the $n^\text{th}$ term of the sequence
\[1,\,\,\,\,\,\,2,\,\,\,\,\,\,5,\,\,\,\,\,\,6,\,\,\,\,\,\,9,\,\,\,\,\,\,12,\,\,\,\,\,\,13,\,\,\,\,\,\,16,\,\,\,\,\,\,19,\,\,\,\,\,\,22,\,\,\,\,\,\,23,\ldots,\]
where the first term is the smallest positive integer that is $1$ more than a multiple of $3$, the next two ... | tiger_math | [
0.008196917362511158,
0.04767642542719841,
-0.03263528645038605,
-0.020859435200691223,
0.034480877220630646,
-0.022321781143546104,
-0.02825644612312317,
0.04669824615120888,
0.01892247051000595,
0.004528833553195,
0.026583176106214523,
0.047760192304849625,
-0.030999865382909775,
-0.0054... | 0.836434 | 0.824312 | 0.012122 |
Jenny and Kenny are walking in the same direction, Kenny at 2 feet per second and Jenny at 1 foot per second, on parallel paths that are 100 feet apart. A tall circular building 50 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Ke... | tiger_math | [
0.015698475763201714,
0.05375213921070099,
-0.05599093809723854,
-0.03296441584825516,
0.03346522897481918,
-0.0002184330951422453,
-0.03766249492764473,
0.07344009727239609,
0.0412575900554657,
-0.014289134182035923,
0.006438943557441235,
0.015076734125614166,
-0.02897028438746929,
-0.007... | 0.845786 | 0.84138 | 0.004406 |
Seven points are evenly spaced out on a circle and connected as shown below to form a 7-pointed star. What is the measure of each angle at the tips of the star, in degrees? One such angle is marked as $\alpha$ below.
[asy]
dotfactor=4;
draw(Circle((0,0),1));
real x = 2*pi/7;
pair A,B,C,D,E,F,G;
A=(cos(4*x), sin(4*x)... | tiger_math | [
0.02356152981519699,
0.025694558396935463,
-0.031561195850372314,
-0.04599679633975029,
0.022443486377596855,
-0.015518500469624996,
-0.021031180396676064,
0.02934654802083969,
0.035350143909454346,
-0.005997758358716965,
0.00016460113693028688,
0.04579145833849907,
-0.03430875763297081,
-... | 0.834087 | 0.822163 | 0.011924 |
Vijay owns a furniture store where he sells various items. He sold a cupboard at a discount of 16% on the cost price to a customer. Later, the same customer bought another similar cupboard but this time Vijay gave an additional 10% discount on the already discounted price. On this discounted price, sales tax at 9% was ... | tiger_math | [
0.005521082319319248,
0.019898077473044395,
-0.05499276891350746,
-0.02014818787574768,
0.046722475439310074,
0.0036215162836015224,
-0.023953847587108612,
0.07550567388534546,
0.041632600128650665,
0.010417690500617027,
0.030965493991971016,
0.03494400531053543,
-0.05186593160033226,
-0.0... | 0.78877 | 0.785525 | 0.003244 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x+y+z = 1.$ Find the minimum value of
\[(x + 2y)(y + 2z)(xz + 1).\] | tiger_math | [
0.023961598053574562,
0.030795641243457794,
-0.04318198561668396,
-0.049721427261829376,
0.04815579205751419,
-0.010872774757444859,
-0.01583399809896946,
0.04212348908185959,
0.03555380553007126,
-0.02353927493095398,
0.019935982301831245,
0.018849758431315422,
-0.03079364076256752,
-0.02... | 0.855305 | 0.844617 | 0.010688 |
Triangle $ABC$ has sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of length 35, 15, and 40, respectively. Let $\omega$ be the circle circumscribed around $\triangle ABC$ and let $D$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{AC}$ that is not on the same side of $\overline{... | tiger_math | [
0.0275900736451149,
0.03290624916553497,
-0.030425729230046272,
-0.0425974540412426,
0.025637632235884666,
-0.007563503924757242,
-0.02852381393313408,
0.05346331000328064,
0.03254535794258118,
0.004105894360691309,
0.03747804835438728,
0.03035101294517517,
-0.03760053962469101,
-0.0234874... | 0.855303 | 0.825488 | 0.029814 |
A curve is described parametrically by
\[(x,y) = (2 \cos t - \sin t, 4 \sin t).\]The graph of the curve can be expressed in the form
\[ax^2 + bxy + cy^2 + dx + ey = 1.\]Enter the ordered quintuple $(a,b,c,d,e).$ | tiger_math | [
0.021662142127752304,
0.02776656486093998,
-0.04151975363492966,
-0.03901495784521103,
0.04469780623912811,
-0.03305145353078842,
-0.03393441438674927,
0.05968363955616951,
0.04754426330327988,
-0.002592758508399129,
0.012929784134030342,
0.0403122641146183,
-0.03222227469086647,
-0.015688... | 0.826039 | 0.812256 | 0.013784 |
For what values of the constant $c$ does the graph of $f(x) = \frac{x^2+2x+c}{x^2+x-12}$ have exactly one vertical asymptote?
Enter all possible values, separated by commas. | tiger_math | [
0.011961227282881737,
0.022394699975848198,
-0.04093432053923607,
-0.019587958231568336,
0.022590020671486855,
-0.0591290257871151,
-0.01312489528208971,
0.07536140829324722,
0.02767036482691765,
-0.03463711589574814,
0.0181389469653368,
0.026396464556455612,
-0.003136872313916683,
-0.0231... | 0.829257 | 0.82058 | 0.008677 |
The sum of the first $n$ terms in the infinite geometric sequence $\left\{2,\frac{2}{3},\frac{2}{9},\frac{2}{27},\dots \right\}$ is $\frac{122}{81}$. Find $n$. | tiger_math | [
0.013598456978797913,
0.020061934366822243,
-0.028854331001639366,
-0.0530347116291523,
0.02409646287560463,
-0.012547897174954414,
-0.017582843080163002,
0.04887555539608002,
0.03748887777328491,
-0.013004115782678127,
0.020652757957577705,
0.01709301955997944,
-0.03639430180191994,
-0.01... | 0.838021 | 0.829424 | 0.008597 |
Emily and her brother Tom are planning an interesting board game evening. They decide to play a custom game where they must roll a single 10-sided die, which lands on any number between 1 and 10 with equal probability. The game rules say that Emily wins if she rolls any number greater than 7. What is the probability th... | tiger_math | [
0.03054346702992916,
0.014694556593894958,
-0.028351420536637306,
-0.03558224439620972,
0.011600720696151257,
-0.03978952765464783,
-0.022882748395204544,
0.05502629280090332,
0.043501853942871094,
-0.022237395867705345,
0.020065918564796448,
0.023616386577486992,
-0.03144054114818573,
-0.... | 0.799243 | 0.791536 | 0.007707 |
****
[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); ... | tiger_math | [
0.024297384545207024,
0.012780651450157166,
-0.037665367126464844,
-0.04039150848984718,
0.044063497334718704,
-0.03686394914984703,
-0.03326144441962242,
0.05376993864774704,
0.034079816192388535,
0.006013899575918913,
0.02799966186285019,
0.03173447027802467,
-0.01733471266925335,
-0.029... | 0.846904 | 0.827419 | 0.019485 |
Find all real numbers $k$ such that
\[\left\| k \begin{pmatrix} 2 \\ -3 \end{pmatrix} - \begin{pmatrix} 4 \\ 7 \end{pmatrix} \right\| = 2 \sqrt{13}.\]Enter all the solutions, separated by commas. | tiger_math | [
0.04664236679673195,
0.04561631754040718,
-0.0494738444685936,
-0.04743088409304619,
0.051236800849437714,
-0.013917362317442894,
-0.005402009934186935,
0.04334720969200134,
0.04328247159719467,
-0.009979638271033764,
0.018330827355384827,
0.04215526953339577,
-0.02977570705115795,
-0.0339... | 0.843383 | 0.837671 | 0.005713 |
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 |
Alice is an avid collector of stamps and coins. She has decided to organize her stamp and coin collections in special boxes. If each box is meant to hold the same number of items, whether they are stamps or coins, and she has a total of 200 stamps and coins combined, how many boxes does she need if there are 8 items pe... | tiger_math | [
0.018753452226519585,
0.0217402596026659,
-0.05432659760117531,
-0.03332186117768288,
0.039174702018499374,
-0.029566477984189987,
-0.03795482963323593,
0.0486663356423378,
0.02838655188679695,
-0.0022970866411924362,
-0.0008906784350983799,
0.027058623731136322,
-0.031151030212640762,
-0.... | 0.814676 | 0.805048 | 0.009628 |
A circle with radius X cm is tangent to three sides of a rectangle, as shown. The area of the rectangle is twice the area of the circle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\pi$.
[asy]
import graph;
draw((0,0)--(30,0)--(30,20)--(0,20)--cycle);
draw(C... | tiger_math | [
-0.01538319606333971,
0.03904063627123833,
-0.05031907558441162,
-0.027892835438251495,
0.03435451164841652,
-0.018395962193608284,
-0.023112138733267784,
0.043883346021175385,
0.035117194056510925,
0.0037491051480174065,
0.02019897848367691,
0.033502183854579926,
-0.02665984258055687,
0.0... | 0.832643 | 0.82542 | 0.007223 |
In right triangle $ABC$ with $\angle A = 90^\circ$, we have $AB = 12$ and $BC = 20$. Find $\cos C$. | tiger_math | [
0.024851350113749504,
0.032555948942899704,
-0.05013185366988182,
-0.026532374322414398,
0.025076381862163544,
-0.02900017239153385,
-0.01612132415175438,
0.03926336020231247,
0.04756145924329758,
-0.017911314964294434,
0.02284933440387249,
0.03914997726678848,
-0.028737209737300873,
-0.01... | 0.84039 | 0.833753 | 0.006637 |
Captain Emma supervises a fleet of toy ships. Each ship requires a crew, with the number of crew members determined by the equation $C(n) = 3n + 4$, where $n$ represents the number of sails on the ship. If the total number of crew members available is 64, how many sails should a ship have to exactly fit the number of a... | tiger_math | [
0.02249874360859394,
0.0189334936439991,
-0.04193662106990814,
-0.03979752957820892,
0.03713522478938103,
-0.03281562775373459,
-0.03602060303092003,
0.04256761819124222,
0.01323565375059843,
-0.017777174711227417,
0.03534446656703949,
0.0338093526661396,
-0.026478247717022896,
-0.02045670... | 0.830705 | 0.826895 | 0.003811 |
In a magical forest, there are three different types of trees. Each type of tree grows triangular fruits. The areas of the triangle-shaped fruits from the three types of trees are 50 cm², Y cm², and 200 cm², respectively. What is the average (mean) height of the fruits if the height from their bases to their vertices i... | tiger_math | [
0.014669704250991344,
0.018814222887158394,
-0.06251007318496704,
-0.06593472510576248,
0.02084176428616047,
-0.03691750019788742,
-0.02071787416934967,
0.05104966089129448,
0.02722376212477684,
-0.029911700636148453,
0.014102714136242867,
0.03446657210588455,
-0.04992165043950081,
0.00306... | 0.827588 | 0.817863 | 0.009725 |
The equation $a^7xy-a^6y-a^5x-a^4=a^4(b^4-1)$ is equivalent to the equation $(a^mx-a^n)(a^py-a^2)=a^4b^4$ for some integers $m$, $n$, and $p$. Find $m+n+p$. | tiger_math | [
0.03763362765312195,
0.01926969550549984,
-0.03515223413705826,
-0.0572577528655529,
0.03425640985369682,
-0.037620801478624344,
-0.011161692440509796,
0.06847362220287323,
0.06706329435110092,
-0.025138244032859802,
0.009188243187963963,
0.03766943886876106,
-0.04800480231642723,
-0.03705... | 0.842681 | 0.834098 | 0.008583 |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be unit vectors such that
\[\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \frac{\mathbf{b} - \mathbf{c}}{\sqrt{2}},\]and such that $\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$ is a linearly independent set.
Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees. | tiger_math | [
0.0361785814166069,
0.04130008444190025,
-0.04907596483826637,
-0.03863655775785446,
0.03039437159895897,
-0.0021526652853935957,
-0.0157032310962677,
0.05242731794714928,
0.03256971761584282,
-0.01594223640859127,
0.015166503377258778,
0.02190784551203251,
-0.03791451081633568,
-0.0148450... | 0.842972 | 0.817626 | 0.025346 |
Find the scalar $c$ such that
\[\mathbf{i} \times (\mathbf{v} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{v} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{v} \times \mathbf{k}) = c \mathbf{v}\]holds true for any arbitrary vector $\mathbf{v}.$ | tiger_math | [
0.025602808222174644,
0.04803873226046562,
-0.045824673026800156,
-0.03014470264315605,
0.049676861613988876,
-0.007985862903296947,
0.0011293546995148063,
0.0735667496919632,
0.02922583743929863,
-0.02715320698916912,
0.006537761073559523,
0.028280416503548622,
-0.026594050228595734,
-0.0... | 0.836991 | 0.819431 | 0.01756 |
A square is inscribed in the ellipse
\[\frac{x^2}{3} + \frac{y^2}{6} = 1,\]so that its sides are parallel to the coordinate axes. Find the area of the square.
[asy]
unitsize(1 cm);
draw(xscale(sqrt(3))*yscale(sqrt(6))*Circle((0,0),1));
draw((sqrt(2),sqrt(2))--(-sqrt(2),sqrt(2))--(-sqrt(2),-sqrt(2))--(sqrt(2),-sqrt(2... | tiger_math | [
0.003326829755678773,
0.061416737735271454,
-0.04529521241784096,
-0.04277293384075165,
0.05210179463028908,
-0.0072189560160040855,
-0.019871847704052925,
0.0584724023938179,
0.04665710777044296,
-0.011539988219738007,
0.02095782943069935,
0.030392451211810112,
-0.013156333938241005,
-0.0... | 0.823784 | 0.812274 | 0.01151 |
A natural number greater than 1 is called "super-nice" if it is equal to the product of its distinct proper divisors plus 1. Find the sum of the first five super-nice numbers. | tiger_math | [
0.014688974246382713,
0.00786677561700344,
-0.013628924265503883,
-0.025104204192757607,
0.04318743199110031,
-0.010783049277961254,
-0.03426900878548622,
0.03665192425251007,
0.03078017197549343,
-0.009820064529776573,
0.03342361003160477,
0.025313042104244232,
-0.06366045773029327,
-0.02... | 0.83399 | 0.828072 | 0.005918 |
Five identical rectangles are arranged to form a larger rectangle $ABCD$, as shown. The area of $ABCD$ is $3000$. If the width of each rectangle is half of its length, what is the length of each rectangle, rounded to the nearest integer? | tiger_math | [
0.01063164696097374,
0.032938990741968155,
-0.05606964975595474,
-0.03137076273560524,
0.013940609060227871,
-0.020952608436346054,
-0.01095140166580677,
0.04538784921169281,
0.051809731870889664,
-0.0128836864605546,
0.009686712175607681,
0.032748784869909286,
-0.03956451267004013,
-0.026... | 0.846401 | 0.840303 | 0.006097 |
A plane is expressed parametrically by
\[\mathbf{v} = \begin{pmatrix} 1 + s - t \\ 2 - s \\ 3 - 2s + 2t \end{pmatrix}.\]Find the equation of the plane in the form
\[Ax + By + Cz = D,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|) = 1.$ | tiger_math | [
0.04474890977144241,
0.037330321967601776,
-0.05329475179314613,
-0.04138755798339844,
0.04285799339413643,
-0.004398982971906662,
-0.015768056735396385,
0.062447234988212585,
0.0395677275955677,
-0.006533127743750811,
0.015022900886833668,
0.04311194270849228,
-0.02653549797832966,
-0.019... | 0.840243 | 0.82011 | 0.020133 |
Rectangle $ABCD$ has sides $\overline {AB}$ of length 6 and $\overline {CB}$ of length 4. Divide $\overline {AB}$ into 252 congruent segments and $\overline {CB}$ into 168 congruent segments. Draw segments connecting the corresponding points on $\overline {AB}$ and $\overline {CB}$, and also on $\overline {AD}$ and $\o... | tiger_math | [
-0.0040615014731884,
0.022470321506261826,
-0.04451754689216614,
-0.035790059715509415,
0.013036460615694523,
-0.012699081562459469,
-0.013285604305565357,
0.04393396154046059,
0.034041840583086014,
0.011034754104912281,
0.01149765308946371,
0.0438276082277298,
-0.013360564596951008,
0.004... | 0.839376 | 0.819883 | 0.019493 |
Let $x,$ $y,$ $z$ be positive real numbers such that $x + 2y + 3z = 10.$ Find the minimum value of
\[\frac{1}{x} + \frac{2}{y} + \frac{3}{z}.\] | tiger_math | [
0.021887121722102165,
0.03656797111034393,
-0.04388951137661934,
-0.028570611029863358,
0.04257550090551376,
-0.008158509619534016,
-0.01617981120944023,
0.033920176327228546,
0.03299130126833916,
-0.023405632004141808,
0.015642086043953896,
0.024127710610628128,
-0.0351262167096138,
-0.01... | 0.836567 | 0.832338 | 0.004229 |
Below is the graph of $y = a \sin bx$ for some constants $a < 0$ and $b > 0.$ Find $a.$
[asy]import TrigMacros;
size(400);
real g(real x)
{
return (-2*sin(x/3));
}
draw(graph(g,-3*pi,3*pi,n=700,join=operator ..),red);
trig_axes(-3*pi,3*pi,-3,3,pi/2,1);
layer();
rm_trig_labels(-5, 5, 2);
label("$1$", (0,1), E);
l... | tiger_math | [
0.020403563976287842,
0.040858522057533264,
-0.03706559166312218,
-0.034018624573946,
0.029521791264414787,
-0.020292850211262703,
-0.020663350820541382,
0.07672730088233948,
0.035107724368572235,
0.012205406092107296,
0.0318288654088974,
0.03269367665052414,
-0.02161465212702751,
-0.00669... | 0.833529 | 0.825296 | 0.008233 |
If $x$ satisfies the equation $x^2 + 3x + \frac{3}{x} + \frac{1}{x^2} = 26$, and $x$ can be written as $a + \sqrt{b}$ where $a$ and $b$ are positive integers, what is the value of $a + b$? | tiger_math | [
0.011968319304287434,
0.013579421676695347,
-0.05232793092727661,
-0.03738927096128464,
0.04071751981973648,
-0.027346370741724968,
-0.004172968212515116,
0.061110712587833405,
0.049518268555402756,
-0.02695736475288868,
0.028073476627469063,
0.03770287334918976,
-0.03728141263127327,
-0.0... | 0.842977 | 0.840089 | 0.002889 |
A candy shop receives weekly shipments of various candies. This week, the number of jelly beans ($J$) they received has a mathematical relationship to the number of lollipops ($L$) and the number of chocolate bars ($C$) described by the equation $J = 2L + 3C - 5$.
If the shop received exactly 100 jelly beans and 25 m... | tiger_math | [
0.010249720886349678,
0.030510954558849335,
-0.03876551240682602,
-0.04776737093925476,
0.04089692607522011,
-0.03314204886555672,
-0.03988385945558548,
0.05676911398768425,
0.02312881126999855,
-0.020815087482333183,
0.013271104544401169,
0.024184782058000565,
-0.03995151445269585,
-0.016... | 0.838029 | 0.832865 | 0.005164 |
A circle has a radius of $\sqrt{50}$ cm. A chord of length 6 cm is drawn in the circle, forming a right triangle with the radius. If the distance from the midpoint of the chord to the center of the circle is y cm, find the value of y^2. | tiger_math | [
0.020586080849170685,
0.0347541868686676,
-0.03812355175614357,
-0.03265642374753952,
0.034789275377988815,
-0.035120464861392975,
-0.02523547224700451,
0.04525983706116676,
0.04882519692182541,
-0.013863583095371723,
0.029637103900313377,
0.029019761830568314,
-0.02230517566204071,
-0.011... | 0.823033 | 0.807284 | 0.015749 |
Isosceles triangle $ABE$ of area 100 square inches is cut by $\overline{CD}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 75 square inches. If the altitude of triangle $ABE$ from $A$ is 20 inches, what is the number of inches in the length of $\overline{CD}$?
[asy]
draw((-... | tiger_math | [
0.033541012555360794,
0.01903774030506611,
-0.04096175730228424,
-0.042007990181446075,
0.024161923676729202,
-0.02443065121769905,
-0.02046000398695469,
0.06077491492033005,
0.020999271422624588,
0.01119908131659031,
0.03803037479519844,
0.032539963722229004,
-0.039463549852371216,
-0.014... | 0.832402 | 0.820739 | 0.011663 |
Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the "base" $-n+i$ using the integers $1,2,\ldots,n^2$ as "digits." That is, the equation\[r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0\]is true for a uni... | tiger_math | [
0.037335656583309174,
0.02668408676981926,
-0.04408048093318939,
-0.03820623829960823,
0.037972256541252136,
-0.028073202818632126,
-0.001420739572495222,
0.05298776179552078,
0.04609445482492447,
-0.005962933413684368,
0.027840595692396164,
0.02356819622218609,
-0.04806032031774521,
-0.03... | 0.840802 | 0.817888 | 0.022914 |
A mail carrier delivers mail to the twenty-one houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible? | tiger_math | [
0.00010112501331605017,
0.01684681698679924,
-0.022141866385936737,
-0.026959389448165894,
0.025301095098257065,
-0.02428717538714409,
-0.03763285651803017,
0.04236246645450592,
0.0396113395690918,
-0.03171786293387413,
0.02646355703473091,
0.047991760075092316,
-0.02194088324904442,
-0.01... | 0.820223 | 0.810903 | 0.00932 |
A parallelogram has adjacent sides of lengths $x$ units and $2x$ units forming a 45-degree angle. The area of the parallelogram is $16\sqrt Y$ square units. What is the value of $Y$? Express your answer in simplest radical form. | tiger_math | [
0.037093956023454666,
0.028294578194618225,
-0.0466739684343338,
-0.05528531223535538,
0.02975604310631752,
-0.002766477642580867,
-0.026681993156671524,
0.07936511188745499,
0.047399748116731644,
-0.01808333583176136,
0.02641020342707634,
0.03413638472557068,
-0.04063177481293678,
-0.0251... | 0.831459 | 0.825937 | 0.005522 |
Solve
\[\sqrt{1 + \sqrt{2 + \sqrt{x+1}}} = \sqrt[3]{1 + \sqrt{x+1}}.\] | tiger_math | [
0.04143163189291954,
0.044011689722537994,
-0.022254059091210365,
-0.06010652706027031,
0.034220896661281586,
-0.02721414528787136,
-0.01814473234117031,
0.06387501955032349,
0.05499817430973053,
-0.018482903018593788,
0.018567057326436043,
0.01636900007724762,
-0.0370447151362896,
-0.0429... | 0.827593 | 0.818167 | 0.009426 |
An equilateral triangle and a square are inscribed in a circle as shown. $ABC$ is isosceles. The triangle and square share a common vertex. If the radius of the circle is 1, what is the area of the shaded region?
[asy]
import markers; defaultpen(linewidth(0.8));
pair A,B,C,D,E,F,G;
draw(unitcircle);
A=(-1,0);
B=(0... | tiger_math | [
0.01820974051952362,
0.022307144477963448,
-0.02368529513478279,
-0.04867388308048248,
0.03864111751317978,
-0.030441034585237503,
-0.008350957185029984,
0.04778312146663666,
0.03467494249343872,
-0.0031215890776365995,
0.030004827305674553,
0.022194044664502144,
-0.01043562963604927,
-0.0... | 0.826415 | 0.813284 | 0.01313 |
Given that the equation $px^3 + qx^2 + rx + s = 0$ has roots of 1 and 2, and that $p = 1$, find $\frac{q-r}{p}$. | tiger_math | [
0.02137322910130024,
0.02894606627523899,
-0.05009712278842926,
-0.042070481926202774,
0.03400074318051338,
-0.029842423275113106,
0.0031715582590550184,
0.05493245646357536,
0.03505720570683479,
-0.023662671446800232,
0.00992309395223856,
0.025244291871786118,
-0.04079823940992355,
-0.045... | 0.837972 | 0.831877 | 0.006095 |
Given the twelve-sided regular polygon $A_1A_2A_3\cdots A_{12}$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $\{A_1, A_2, \ldots A_{12}\}$? | tiger_math | [
0.019879434257745743,
0.01282785926014185,
-0.04081323370337486,
-0.036695513874292374,
0.0021400756668299437,
-0.023379962891340256,
-0.029345912858843803,
0.04194048047065735,
0.02435200847685337,
-0.012348717078566551,
0.0005922724376432598,
0.051559556275606155,
-0.01481806393712759,
-... | 0.833453 | 0.824227 | 0.009226 |
Let $a$ and $b$ be positive real numbers such that $a^2 + b^2 = 25.$ Find the maximum value of $\frac{1}{a} + \frac{1}{b}.$ | tiger_math | [
0.024769602343440056,
0.022557273507118225,
-0.03953249007463455,
-0.03545580804347992,
0.028229685500264168,
-0.012163704261183739,
0.0014867187710478902,
0.0529167577624321,
0.03304829075932503,
-0.032398197799921036,
0.01339087076485157,
0.03457212820649147,
-0.03313001990318298,
-0.005... | 0.840259 | 0.827691 | 0.012568 |
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} 2 \\ -4 \\ 4 \end{pmatrix} = p \mathb... | tiger_math | [
0.04370318353176117,
0.04050738736987114,
-0.05432450771331787,
-0.03502163290977478,
0.05197367072105408,
-0.0021450105123221874,
-0.003858886193484068,
0.0719001293182373,
0.030760152265429497,
-0.004013060592114925,
-0.005820762366056442,
0.024619946256279945,
-0.0296921469271183,
-0.03... | 0.833096 | 0.811829 | 0.021266 |
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 |
If $A$, $B$, and $C$ are all distinct non-zero digits greater than $5$ and the following is true, find the difference between the three values $A$, $B$, and $C$, expressing your answer in base $7$.
$$\begin{array}{c@{}c@{}c@{}c} &A&B&C_7\\ &-&B&C_7\\ \cline{2-4} &A&C&B_7\\ \end{array}$$ | tiger_math | [
0.04220990836620331,
0.02910769172012806,
-0.04591277986764908,
-0.027556676417589188,
0.04450830817222595,
-0.0278619434684515,
-0.021322747692465782,
0.047545358538627625,
0.04279555380344391,
-0.02702716924250126,
0.02979991026222706,
0.038741335272789,
-0.05349205061793327,
-0.00520362... | 0.842518 | 0.838463 | 0.004055 |
Find the equation of the plane passing through $(2,0,-1)$ and $(0,1,1),$ and which is perpendicular to the plane $2x - y + 4z = 8.$ Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = X$. What is the value of $X$? | tiger_math | [
0.014112756587564945,
0.031721096485853195,
-0.04248717799782753,
-0.042151786386966705,
0.03919780254364014,
0.010774007067084312,
-0.013694542460143566,
0.0542057566344738,
0.04047967493534088,
-0.00773137342184782,
0.009713904932141304,
0.04339894279837608,
-0.0194251649081707,
-0.02454... | 0.834733 | 0.825315 | 0.009418 |
A group of 12 students includes 5 math majors. A committee of 3 students is to be selected from this group, and the committee must include at least one math major. In how many ways can this committee be formed? | tiger_math | [
0.035688694566488266,
0.01690674014389515,
-0.050854701548814774,
-0.036056797951459885,
0.044195063412189484,
-0.02742285095155239,
-0.041216276586055756,
0.03282945603132248,
0.012370514683425426,
-0.011288885027170181,
0.007092510350048542,
0.0421457402408123,
-0.02712690457701683,
-0.0... | 0.820692 | 0.815786 | 0.004906 |
Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 3$ and $\frac{\cos x}{\cos y} = \frac12$. If $\sin y = \frac{1}{5}$, find the value of
\[\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}.\] | tiger_math | [
0.01916508935391903,
0.04668363556265831,
-0.05093924328684807,
-0.04068119823932648,
0.0433819405734539,
0.010091941803693771,
0.0005352788139134645,
0.06357382982969284,
0.04122380167245865,
-0.013782544992864132,
0.024167079478502274,
0.023582877591252327,
-0.02636146731674671,
-0.01271... | 0.837195 | 0.820795 | 0.0164 |
A math teacher is organizing a math fair with 4 different activities: Algebra, Geometry, Statistics, and Number Theory. She wants each student to participate in exactly 2 of these activities. How many different combinations of 2 activities can each student choose from? | tiger_math | [
0.037380751222372055,
-0.0018629609839990735,
-0.055149611085653305,
-0.034027326852083206,
0.024690749123692513,
0.0030397309456020594,
-0.05487866327166557,
0.033597879111766815,
0.05182061716914177,
-0.009820339269936085,
-0.0021889854688197374,
0.02489253506064415,
-0.011022970080375671,... | 0.83095 | 0.823853 | 0.007097 |
Four siblings took the GMAT exam, with their respective scores being u, v, w, and x such that u > v > w > x. The difference between the highest and lowest scores (u - x) is equal to the average of all four scores. Also, the difference between the highest and second highest scores (u - v) is twice the difference between... | tiger_math | [
0.0033733367454260588,
0.011524475179612637,
-0.0386066772043705,
-0.03971908986568451,
0.05032991245388985,
-0.016914432868361473,
-0.03844763711094856,
0.05617675185203552,
0.048028357326984406,
-0.019367113709449768,
0.007781235035508871,
0.011785192415118217,
-0.02035108581185341,
-0.0... | 0.828653 | 0.824884 | 0.003769 |
Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{PS}$, $\overline{PQ}$, and $\overline{PR}$ is perpendicular to the other two. If $PS = 8$, $PQ = 15$, and the distance from $S$ to face $PQR$ is 10, find the length of $\overline{SR}$. | tiger_math | [
-0.000060553142247954383,
0.020714567974209785,
-0.05318576097488403,
-0.04538630321621895,
0.0388304628431797,
0.0025313717778772116,
-0.03148750215768814,
0.057846274226903915,
0.03579341247677803,
-0.011102396994829178,
0.017858313396573067,
0.032588306814432144,
-0.038223493844270706,
... | 0.842554 | 0.829239 | 0.013316 |
In triangle $PQR,$ $\angle Q = 32^\circ,$ $\angle R = 104^\circ,$ and $PR = 4 \sqrt{2}.$ Find $QR.$ | tiger_math | [
0.006840715184807777,
0.037266843020915985,
-0.03517334163188934,
-0.04622370004653931,
0.023185783997178078,
-0.0015322137624025345,
-0.03072291798889637,
0.0479198694229126,
0.040035128593444824,
-0.007746739778667688,
0.03608807176351547,
0.03200247511267662,
-0.020980529487133026,
-0.0... | 0.839228 | 0.832538 | 0.006691 |
Grandma Molly created statues of turtles for her front lawn. The first year, she created x statues and placed them on her lawn. The second year, she quadrupled the number of statues on her front lawn. In the third year, she added another 12 statues to the front lawn, but a hail storm broke 3 of the statues, which sh... | tiger_math | [
-0.011447899974882603,
0.024912117049098015,
-0.04693177342414856,
-0.023616796359419823,
0.04274367913603783,
-0.026607943698763847,
-0.033568620681762695,
0.07838162779808044,
0.0354977548122406,
-0.002992746653035283,
0.030987102538347244,
0.036999575793743134,
-0.041407205164432526,
-0... | 0.834728 | 0.830461 | 0.004267 |
[asy] draw(circle((0,6sqrt(2)),2sqrt(2)),black+linewidth(.75)); draw(circle((0,3sqrt(2)),sqrt(2)),black+linewidth(.75)); draw((-8/3,16sqrt(2)/3)--(-4/3,8sqrt(2)/3)--(0,0)--(4/3,8sqrt(2)/3)--(8/3,16sqrt(2)/3),dot); MP("B",(-8/3,16*sqrt(2)/3),W);MP("B'",(8/3,16*sqrt(2)/3),E); MP("A",(-4/3,8*sqrt(2)/3),W);MP("A'",(4/3,8*s... | tiger_math | [
0.007642980664968491,
0.0431768000125885,
-0.03576024994254112,
-0.034435078501701355,
0.05157880857586861,
-0.020016441121697426,
-0.028977394104003906,
0.05488239973783493,
0.04443592578172684,
0.007625181693583727,
0.015447534620761871,
0.032290760427713394,
-0.00553516298532486,
-0.013... | 0.833087 | 0.818373 | 0.014714 |
Find the $3 \times 3$ matrix $\mathbf{M}$ such that for a $3 \times 3$ matrix $\mathbf{N},$ $\mathbf{M} \mathbf{N}$ is the result of swapping the first row and second row of $\mathbf{N},$ and doubling the third row of $\mathbf{N}.$ In other words,
\[\mathbf{M} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{p... | tiger_math | [
0.04140709340572357,
0.02641509473323822,
-0.042984332889318466,
-0.04449305683374405,
0.052065666764974594,
-0.023554546758532524,
-0.028552554547786713,
0.07228463888168335,
0.01758904755115509,
-0.007778520695865154,
0.0068426961079239845,
0.0071028415113687515,
-0.061581190675497055,
-... | 0.815947 | 0.801294 | 0.014653 |
Let $a,$ $b,$ $c,$ and $d$ be the roots of \[x^4 + 8x^3 + 9x^2 + 5x + 4 = 0.\]Find the value of \[\frac{1}{a^2b^2} + \frac{1}{a^2c^2} + \frac{1}{a^2d^2} + \frac{1}{b^2c^2} + \frac{1}{b^2d^2} + \frac{1}{c^2d^2}.\] | tiger_math | [
0.025272848084568977,
0.04526791721582413,
-0.06350531429052353,
-0.0527694933116436,
0.04351930320262909,
-0.031073374673724174,
0.008895473554730415,
0.06054539605975151,
0.05597212165594101,
-0.01357466634362936,
-0.003410585690289736,
0.009450986981391907,
-0.03498315066099167,
-0.0377... | 0.843157 | 0.833621 | 0.009536 |
The function $g(x)$ satisfies
\[g(x+y) = xg(y)\]for all real numbers $x$ and $y.$ If $g(0) = 25,$ find $g(79).$ | tiger_math | [
0.028603140264749527,
0.004213630687445402,
-0.05851641297340393,
-0.04871402308344841,
0.03545543551445007,
-0.03202781826257706,
-0.008521795272827148,
0.05740184336900711,
0.03505905717611313,
-0.020117001608014107,
0.022447431460022926,
0.04368411377072334,
-0.05184325948357582,
-0.019... | 0.842461 | 0.83444 | 0.008021 |
Compute the least positive value of $t$ such that
\[\arctan (\tan \alpha), \ \arctan (\tan 2 \alpha), \ \arctan (\tan 7 \alpha), \ \arctan (\tan t \alpha)\]is a geometric progression for some $\alpha$ with $0 < \alpha < \frac{\pi}{4}.$ | tiger_math | [
0.026862189173698425,
0.047023575752973557,
-0.052541229873895645,
-0.03485357016324997,
0.032928649336099625,
-0.00980236753821373,
-0.03319640830159187,
0.06483010947704315,
0.038844022899866104,
-0.011059032753109932,
0.010953362099826336,
0.005930624902248383,
-0.03578949719667435,
-0.... | 0.835746 | 0.823218 | 0.012527 |
Suppose $g(x) = \frac{1}{px+q}$ for some nonzero constants $p$ and $q$. If $g^{-1}(2) = -1$, find the value of $q$. | tiger_math | [
0.010062644258141518,
0.03398781642317772,
-0.04518115520477295,
-0.041018977761268616,
0.03609069064259529,
-0.02464144304394722,
0.0018983311019837856,
0.06834103167057037,
0.029459679499268532,
-0.006473756395280361,
0.01741308905184269,
0.02438093163073063,
-0.04372386261820793,
-0.021... | 0.845397 | 0.836966 | 0.008431 |
A square sheet of paper has area $8 \text{ cm}^2$. The front is white and the back is black. When the sheet is folded so that point $A$ rests on the diagonal as shown, the visible black area is equal to the visible white area. How many centimeters is $A$ from its original position? Express your answer in simplest radic... | tiger_math | [
0.024213867262005806,
0.024165969341993332,
-0.03367715701460838,
-0.055821336805820465,
0.03510400652885437,
-0.007494333200156689,
-0.013767797499895096,
0.06962183117866516,
0.04937327280640602,
-0.013266663998365402,
0.012268590740859509,
0.01748337782919407,
-0.03442240133881569,
-0.0... | 0.846188 | 0.83953 | 0.006658 |
Find the quadratic polynomial, with real coefficients, which has $1 - 2i$ as a root, and where the coefficient of $x^2$ is 3. | tiger_math | [
0.030069636180996895,
0.031607095152139664,
-0.055137839168310165,
-0.049294229596853256,
0.020817451179027557,
-0.024833831936120987,
-0.0017955177463591099,
0.06921481341123581,
0.01679355837404728,
-0.024897271767258644,
0.03161467984318733,
0.03324061259627342,
-0.03912845253944397,
-0... | 0.837064 | 0.828853 | 0.008211 |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.3 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -8; i <= 8; ++i) {
draw((i,-8)--(i,8),gray(0.7));
... | tiger_math | [
0.027128150686621666,
0.03694828599691391,
-0.03569071367383003,
-0.04415702819824219,
0.03908061236143112,
-0.044761307537555695,
-0.03859560936689377,
0.09154412150382996,
0.04715368524193764,
-0.005052168387919664,
0.024745065718889236,
0.033935800194740295,
-0.017586877569556236,
-0.00... | 0.818571 | 0.815379 | 0.003192 |
Given that $a$, $b$, and $c$ are real numbers, with $a$ and $b$ nonzero, find all possible values of the expression
\[\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{ab}{|ab|}.\]Enter all possible values, separated by commas. | tiger_math | [
0.03617842122912407,
0.030948325991630554,
-0.049167707562446594,
-0.025710713118314743,
0.033462729305028915,
-0.013227040879428387,
-0.009161291643977165,
0.06246456503868103,
0.04906243830919266,
-0.024767925962805748,
0.015277473255991936,
0.03180979937314987,
-0.029014775529503822,
-0... | 0.829586 | 0.820303 | 0.009283 |
Find $b$ such that $4x^2+bx-3$ is the square of a binomial. | tiger_math | [
0.023466091603040695,
0.032749537378549576,
-0.04143685847520828,
-0.0520746186375618,
0.03686176612973213,
-0.03758883848786354,
-0.015347102656960487,
0.06716884672641754,
0.039648741483688354,
-0.019983960315585136,
0.02539808116853237,
0.05360481142997742,
-0.02853175811469555,
-0.0345... | 0.842161 | 0.839076 | 0.003085 |
The roots of the equation $2x^2 - 5x - 4 = 0$ can be written in the form $x = \frac{m \pm \sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers with a greatest common divisor of 1. What is the value of $n$? | tiger_math | [
0.04545722156763077,
0.03269009664654732,
-0.059503354132175446,
-0.05107172578573227,
0.03942633420228958,
-0.0402207151055336,
0.001575224450789392,
0.07104579359292984,
0.05770101025700569,
-0.016584886237978935,
0.01354258880019188,
0.02498347871005535,
-0.019304003566503525,
-0.041846... | 0.829714 | 0.826431 | 0.003283 |
The Rangers are playing the Yankees in the American League Championship Series. To win the ALCS, a team must win 4 games before the other team does. If the Rangers win each game with probability $\dfrac{2}{3}$ and there are no ties, what is the probability that the Rangers will win the ALCS? Express your answer as a pe... | tiger_math | [
0.016740815714001656,
0.02864365093410015,
-0.04888861998915672,
-0.013735450804233551,
0.019424069672822952,
-0.015821412205696106,
-0.014077571220695972,
0.040823161602020264,
0.021573634818196297,
-0.012332452461123466,
-0.0011453611077740788,
0.0380149781703949,
-0.010185341350734234,
... | 0.819044 | 0.81232 | 0.006724 |
Let $ABCDE$ be a convex pentagon with $AB \parallel CE, BC \parallel AD, AC \parallel DE, \angle ABC=120^\circ, AB=3, BC=5,$ and $AE = 15.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ | tiger_math | [
0.018895460292696953,
0.023639362305402756,
-0.044985152781009674,
-0.052949208766222,
0.02269485965371132,
-0.021237315610051155,
-0.01668146811425686,
0.057567015290260315,
0.03653935715556145,
0.004320830572396517,
0.018744822591543198,
0.040597233921289444,
-0.048328571021556854,
-0.00... | 0.848037 | 0.82176 | 0.026277 |
Find the $3 \times 3$ matrix $\mathbf{M}$ such that for a X \times 3$ matrix $\mathbf{N},$ $\mathbf{M} \mathbf{N}$ is the result of swapping the first row and second row of $\mathbf{N},$ and doubling the third row of $\mathbf{N}.$ In other words,
\[\mathbf{M} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pm... | tiger_math | [
0.021680284291505814,
0.02605832740664482,
-0.06123439595103264,
-0.030884502455592155,
0.055013325065374374,
-0.013756207190454006,
-0.019886314868927002,
0.08030608296394348,
0.026910342276096344,
-0.011403925716876984,
0.007322144694626331,
0.013760844245553017,
-0.060694605112075806,
-... | 0.815757 | 0.80927 | 0.006487 |
In the town of Mathville, there is an annual festival where participants engage in a unique game of distribution. The game involves distributing apples and bananas to three participants: Alice, Bob, and Carlos. The organizer has a total of 120 fruits, consisting of apples and bananas in the ratio of 3:5. Alice receives... | tiger_math | [
0.020608631893992424,
0.005585678853094578,
-0.04603920876979828,
-0.0376749224960804,
0.025686293840408325,
-0.017423061653971672,
-0.024844659492373466,
0.06643708795309067,
0.023638902232050896,
-0.023554136976599693,
0.0028460111934691668,
0.036673638969659805,
-0.04223479703068733,
-0... | 0.81726 | 0.807503 | 0.009756 |
A stationery store sells large and small ink cartridges. A large ink cartridge costs $25 and a small one costs $10. However, the store offers a discount: you can buy a pack of 2 large ink cartridges for $45 or a pack of 3 small ink cartridges for $27. With a budget of $182, what is the maximum number of ink cartridges ... | tiger_math | [
0.008995997719466686,
0.03255721181631088,
-0.05491539090871811,
-0.027488764375448227,
0.03819119185209274,
-0.018322952091693878,
-0.03334788233041763,
0.050192300230264664,
0.036042071878910065,
-0.02466469444334507,
0.024861173704266548,
0.023428767919540405,
-0.055941153317689896,
-0.... | 0.813532 | 0.806112 | 0.00742 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x) f(y) - f(xy) = x - y\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | tiger_math | [
0.02922794036567211,
0.033929578959941864,
-0.06879737973213196,
-0.03487313166260719,
0.047715380787849426,
0.0012413267977535725,
-0.02225235290825367,
0.06471554934978485,
0.029784122481942177,
-0.011793876066803932,
0.011511941440403461,
0.05371056869626045,
-0.05974046140909195,
-0.01... | 0.851726 | 0.833271 | 0.018455 |
Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive integers that satisfy $\sum_{i = 1}^4 x_i = 99.$ Find $\frac n{100}.$ | tiger_math | [
0.011457225307822227,
-0.0020064108539372683,
-0.04807666689157486,
-0.028711237013339996,
0.01478702574968338,
-0.010691463015973568,
-0.02102307789027691,
0.027076588943600655,
0.04041964188218117,
-0.007267327513545752,
0.017899788916110992,
0.02693014033138752,
-0.030665239319205284,
-... | 0.850099 | 0.838444 | 0.011655 |
Consider the polynomials \[f(x)=1-12x+3x^2-4x^3+5x^4\] and \[g(x)=3-2x-6x^3+9x^4.\] Find all values of $c$ such that the polynomial $f(x)+cg(x)$ has degree less than or equal to 3. | tiger_math | [
0.02417137660086155,
0.018305139616131783,
-0.05432302877306938,
-0.047328196465969086,
0.04073433205485344,
-0.05034623667597771,
-0.012051339261233807,
0.06643063575029373,
0.026855336502194405,
0.008771655149757862,
0.014579584822058678,
0.03429762274026871,
-0.03134878724813461,
-0.025... | 0.842435 | 0.832096 | 0.01034 |
Find the $2 \times 2$ matrix $\mathbf{M}$ such that $\mathbf{M} \begin{pmatrix} 3 \\ 0 \end{pmatrix} = \begin{pmatrix} 6 \\ 21 \end{pmatrix}$ and $\mathbf{M} \begin{pmatrix} -1 \\ 5 \end{pmatrix} = \begin{pmatrix} 3 \\ -17 \end{pmatrix}.$ | tiger_math | [
0.04586510360240936,
0.02012324519455433,
-0.0575898140668869,
-0.04886602610349655,
0.04842210188508034,
-0.01620064303278923,
-0.023400429636240005,
0.07697463780641556,
0.021992754191160202,
-0.013714269734919071,
0.013136973604559898,
0.029448341578245163,
-0.054349128156900406,
-0.026... | 0.838092 | 0.835053 | 0.003039 |
Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of thre... | tiger_math | [
0.00940885953605175,
0.021943191066384315,
-0.058234695345163345,
-0.030550144612789154,
0.03710382059216499,
-0.02402018941938877,
-0.03556709736585617,
0.031820736825466156,
0.03299573063850403,
-0.024072526022791862,
0.02696058712899685,
0.03517984226346016,
-0.027712883427739143,
-0.01... | 0.82391 | 0.806392 | 0.017517 |
Rationalize the denominator of $\frac{3}{2\sqrt{3} - 3\sqrt{7}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$? | tiger_math | [
0.029422368854284286,
0.03637322410941124,
-0.03321473300457001,
-0.0366382971405983,
0.04358544945716858,
-0.023724498227238655,
-0.002348720794543624,
0.060836274176836014,
0.043709661811590195,
-0.026472004130482674,
0.010855286382138729,
0.03565705940127373,
-0.03572303056716919,
-0.04... | 0.829633 | 0.824721 | 0.004912 |
In a small village, there are seven households. Each household decides to plant trees in their backyard. Each household can choose between planting one, two, or three apple trees. What is the probability that exactly five of the seven households decide to plant three apple trees? | tiger_math | [
0.01711236871778965,
0.018492894247174263,
-0.045127592980861664,
-0.032576821744441986,
0.04274589940905571,
-0.025338394567370415,
0.00020594480156432837,
0.048961423337459564,
0.02642604522407055,
-0.02049334906041622,
0.0029548078309744596,
0.031129101291298866,
-0.027778292074799538,
... | 0.826586 | 0.822522 | 0.004064 |
There are 8 stores in town that had a total of 21 visitors on a particular day. However, only 12 people went shopping that day; some people visited more than one store. If 8 people visited exactly two stores each, and everyone visited at least one store, what is the largest number of stores anyone could have visited? | tiger_math | [
0.04149378836154938,
0.008468913845717907,
-0.04114336520433426,
-0.03436930477619171,
0.0477968268096447,
-0.018521925434470177,
-0.048419054597616196,
0.031147552654147148,
0.018512330949306488,
-0.0035199294798076153,
0.025997553020715714,
0.03929996117949486,
-0.05821312218904495,
-0.0... | 0.819803 | 0.816462 | 0.003341 |
Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ and $c,$
\[\sqrt{\frac{a}{b + c}} + \sqrt{\frac{b}{a + c}} + \sqrt{\frac{c}{a + b}} > m.\] | tiger_math | [
0.027602603659033775,
0.04325096309185028,
-0.03296499326825142,
-0.0375862680375576,
0.03292730823159218,
-0.015269417315721512,
-0.02893202193081379,
0.05774080380797386,
0.0421779602766037,
-0.017799226567149162,
-0.002891461830586195,
0.03970762714743614,
-0.03753514215350151,
-0.01772... | 0.849526 | 0.83083 | 0.018696 |
Let $x$ be a real number, $|x| > 1.$ Compute
\[\sum_{n = 0}^\infty \frac{1}{x^{2^n} - x^{-2^n}}.\] | tiger_math | [
0.019054437056183815,
0.045558396726846695,
-0.03464335575699806,
-0.03488833084702492,
0.026889506727457047,
-0.025222567841410637,
-0.016393186524510384,
0.05346217751502991,
0.05488409474492073,
-0.020039832219481468,
0.005156948231160641,
0.013330536894500256,
-0.04325468838214874,
-0.... | 0.848926 | 0.840043 | 0.008884 |
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