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 |
|---|---|---|---|---|---|
The foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{b^2} = 1$ and the foci of the hyperbola
\[\frac{x^2}{169} - \frac{y^2}{121} = \frac{1}{49}\]coincide. Find $b^2.$ | tiger_math | [
0.016366668045520782,
0.02863563969731331,
-0.053385715931653976,
-0.04032444581389427,
0.03341134637594223,
-0.025899343192577362,
-0.008607367053627968,
0.04946507140994072,
0.05728274583816528,
-0.013751187361776829,
0.021730748936533928,
0.02734990231692791,
-0.01733679324388504,
-0.02... | 0.833879 | 0.821662 | 0.012217 |
Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=10$. What is the maximum value of \[A\cdot M\cdot C+A\cdot M+M\cdot
C+C\cdot A?\] | tiger_math | [
0.0415366031229496,
0.012669975869357586,
-0.044644661247730255,
-0.01876554824411869,
0.03922924026846886,
-0.01631261594593525,
-0.027719244360923767,
0.04722670838236809,
0.027761535719037056,
-0.0377914160490036,
0.009507301263511181,
0.04104417562484741,
-0.047769635915756226,
-0.0289... | 0.847045 | 0.843044 | 0.004001 |
In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC =1:2$. Point $F$ is on $AD$ such that $AF:FD=2:1$. If the area of $\triangle DEF$ is 12, determine the area of $\triangle ABC$. | tiger_math | [
0.05197451263666153,
0.007403113879263401,
-0.04903388395905495,
-0.0574381984770298,
0.025748971849679947,
-0.037663545459508896,
0.005284549202769995,
0.0499703586101532,
0.04195204749703407,
0.010108844377100468,
0.02021002769470215,
0.039394550025463104,
-0.03315693512558937,
-0.026649... | 0.833443 | 0.815109 | 0.018335 |
Suppose you need to plan a dinner for a book club that consists of 8 members, and the dinner budget is $120. Three items are on the menu: soup, salad, and pasta. Each soup costs $6, each salad costs $5, and each pasta dish costs $10. If the total number of soup, salad, and pasta dishes ordered equals twice the number o... | tiger_math | [
0.02710961364209652,
0.015588512644171715,
-0.03862718120217323,
-0.03613340109586716,
0.030689584091305733,
-0.01702628657221794,
-0.020982960239052773,
0.037433188408613205,
0.045751817524433136,
-0.020295752212405205,
0.01882316917181015,
0.0283688735216856,
-0.05606608837842941,
-0.028... | 0.834486 | 0.824087 | 0.010399 |
Positive integers $a$, $b$, and $c$ are chosen so that $a<b<c$, and the system of equations \[
2x + y = 2003\quad\text{and}\quad y = |x-a| + |x-b| + |x-c|
\]has exactly one solution. What is the minimum value of $c$? | tiger_math | [
0.030454743653535843,
0.02272235043346882,
-0.07453365623950958,
-0.030435634776949883,
0.046259213238954544,
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-0.030645884573459625,
0.0709277093410492,
0.04939042031764984,
-0.017879363149404526,
0.0038535960484296083,
0.04766872152686119,
-0.01803726702928543,
-0.0... | 0.846233 | 0.832084 | 0.014149 |
Which of the following could NOT be the lengths of the external diagonals of a right regular prism [a "box"]? (An $\textit{external diagonal}$ is a diagonal of one of the rectangular faces of the box.)
$\text{(A) }\{4,5,6\} \quad \text{(B) } X \quad \text{(C) } \{4,6,7\} \quad \text{(D) } \{5,6,7\} \quad \text{(E) } \... | tiger_math | [
0.020959319546818733,
0.027049193158745766,
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0.02556847222149372,
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0.06006500497460365,
0.04157618433237076,
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0.01879565790295601,
0.039516668766736984,
-0.023667000234127045,
-0.01... | 0.830789 | 0.823667 | 0.007122 |
For $-1<r<1$, let $S(r)$ denote the sum of the geometric series \[12+12r+12r^2+12r^3+\cdots .\]Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2019$. Find $S(a)+S(-a)$. | tiger_math | [
0.009612697176635265,
0.027898181229829788,
-0.04870113730430603,
-0.03479482978582382,
0.03224918991327286,
-0.030455684289336205,
-0.02963527850806713,
0.07141163945198059,
0.04167764261364937,
0.0002182530879508704,
0.013594661839306355,
0.043375417590141296,
-0.04641028121113777,
-0.02... | 0.837945 | 0.831749 | 0.006197 |
In a garden, there are two types of flowers, roses and tulips. Initially, you have a garden that consists of 120 roses and 80 tulips. You decide to add an equal number of both kinds of flowers to maintain the proportion. You have space to add up to 60 flowers in total. If you decide to add an even number of each type o... | tiger_math | [
0.020177701488137245,
0.010889818891882896,
-0.0391143262386322,
-0.033521562814712524,
0.03945016488432884,
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0.0671943873167038,
0.0492996871471405,
-0.015493083745241165,
-0.000466803991002962,
0.04661152511835098,
-0.05165805295109749,
-0.00657... | 0.827845 | 0.817232 | 0.010614 |
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to three times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | tiger_math | [
0.018408888950943947,
0.009198447689414024,
-0.057010263204574585,
-0.041117314249277115,
0.017354529350996017,
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0.018456412479281425,
0.032486237585544586,
-0.019917475059628487,
0.01586497202515602,
0.04165908694267273,
-0.04339682310819626,
-0... | 0.835947 | 0.829258 | 0.006689 |
Mia and her brother Leo are collecting sea shells on the beach. Mia collects twice as many shells per day as Leo. One morning, they decide to have a competition over 6 days to see who collects more shells. Each day, Mia has a \( \frac{4}{5} \) probability of collecting 10 shells (20 for herself as she collects twice as... | tiger_math | [
-0.006022181361913681,
0.034363094717264175,
-0.05810001119971275,
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0.03562391176819801,
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0.07296612858772278,
0.01923207752406597,
-0.007124770898371935,
-0.007045265287160873,
0.023996448144316673,
-0.02623888850212097,
-... | 0.822436 | 0.814599 | 0.007837 |
The eight horizontal and eight vertical lines on a $7\times7$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ | tiger_math | [
0.0254527535289526,
0.02149534411728382,
-0.044323474168777466,
-0.03969421610236168,
0.03025432676076889,
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-0.03239769488573074,
0.05980275943875313,
0.05063662678003311,
-0.01946599781513214,
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0.04393886774778366,
-0.030043601989746094,
-0.03465... | 0.841066 | 0.826579 | 0.014487 |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a^2 b^2 c^2 = 8.$ Find the minimum value of
\[(a + b)(b + c)(ac + 1).\] | tiger_math | [
0.022838477045297623,
0.0234398003667593,
-0.04691410809755325,
-0.04112383723258972,
0.04500347003340721,
-0.028568385168910027,
-0.01058664545416832,
0.06081351265311241,
0.04068432003259659,
-0.03110557049512863,
-0.0009520771563984454,
0.020230842754244804,
-0.04438330605626106,
-0.039... | 0.853016 | 0.845287 | 0.007729 |
A gardener is organizing pots in a unique way for a large floral exhibition. He visualizes a sequence and plans the placement of pots in different rows based on the sequence. The pots are to contain roses, with each row containing a fixed number of pots, each pot containing roses as per the number of the row. The seque... | tiger_math | [
0.022103143855929375,
0.026686426252126694,
-0.040783241391181946,
-0.02948044054210186,
0.025886740535497665,
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0.062090206891298294,
0.046024713665246964,
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-0.003632386913523078,
0.031448397785425186,
-0.02842462621629238,
-... | 0.838359 | 0.828704 | 0.009654 |
Square each integer $n$ in the range $1\le n\le 10$ and find the remainders when the squares are divided by $11$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $11$? | tiger_math | [
0.04506022483110428,
0.012264487333595753,
-0.02518720179796219,
-0.053166765719652176,
0.038539547473192215,
-0.04364224523305893,
-0.011936347000300884,
0.05075475573539734,
0.03640635684132576,
-0.002904670313000679,
0.018309151753783226,
0.052411340177059174,
-0.06799100339412689,
-0.0... | 0.846131 | 0.837708 | 0.008424 |
The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)
What is the sum of the $x$-coordinates of all points where $f(x) = x+1$? | tiger_math | [
0.006118023302406073,
0.019633224233984947,
-0.06317228823900223,
-0.058334581553936005,
0.03842524439096451,
-0.035693854093551636,
-0.020417477935552597,
0.05270873382687569,
0.04337351769208908,
-0.007116867229342461,
0.003913205116987228,
0.033238645642995834,
-0.031063398346304893,
-0... | 0.827353 | 0.823112 | 0.004241 |
The average weight of a, b, and c is 70 kg. If d, with a weight ratio of 2 to 1 compared to a, joins the group, the average weight of the group (a, b, c, and d) becomes 72 kg. Now, let f join the group, such that f's weight is 7 kg more than b, and the average weight of the group (a, b, c, d, and f) becomes 74 kg. Next... | tiger_math | [
0.036899566650390625,
0.03495338559150696,
-0.05253645405173302,
-0.050833795219659805,
0.03263029456138611,
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-0.025424087420105934,
0.057175468653440475,
0.0389973521232605,
-0.010977747850120068,
0.008414125069975853,
0.016315588727593422,
-0.05218568444252014,
-0.02... | 0.826938 | 0.818428 | 0.00851 |
Let $O$ and $H$ be the circumcenter and orthocenter of triangle $ABC$, respectively. Let $K$ denote the area of the triangle, and let $R$ denote the circumradius. Find $OH^2$ if $R = 5$ and $K = 10$. | tiger_math | [
0.023575812578201294,
0.036816779524087906,
-0.05013873055577278,
-0.0386556014418602,
0.035752084106206894,
-0.02177039533853531,
-0.016096720471978188,
0.046506498008966446,
0.04254794120788574,
-0.007873610593378544,
0.029374539852142334,
0.02423599734902382,
-0.031233256682753563,
-0.0... | 0.835909 | 0.817667 | 0.018242 |
How many non- empty subsets $S$ of $\{1,2,3,\ldots ,20\}$ have the following two properties?
$(1)$ No two consecutive integers belong to $S$.
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$. | tiger_math | [
0.03123868629336357,
0.012278277426958084,
-0.06322788447141647,
-0.031906839460134506,
0.03237787261605263,
-0.02687200903892517,
-0.010944494046270847,
0.03758108988404274,
0.035391177982091904,
-0.02069515734910965,
0.04722793027758598,
0.031569693237543106,
-0.04098116233944893,
-0.005... | 0.833025 | 0.823168 | 0.009856 |
We have that $3 \cdot f(x) + 4 \cdot g(x) = h(x)$ where $f(x),$ $g(x),$ and $h(x)$ are all polynomials in $x.$ If the degree of $f(x)$ is $8$ and the degree of $h(x)$ is X$, The minimum possible degree of $g(x)$ is 9. What is the value of unknown variable X? | tiger_math | [
0.020678002387285233,
0.013354908674955368,
-0.05929086357355118,
-0.030686775222420692,
0.05243389308452606,
-0.02944524772465229,
-0.021933771669864655,
0.06169978529214859,
0.04224782809615135,
0.00606047036126256,
0.008005339652299881,
0.03564062714576721,
-0.020557276904582977,
-0.025... | 0.83486 | 0.831199 | 0.003661 |
In a magical land, there are three types of coins: gold, silver, and bronze. The value of a gold coin is $g(x) = x^2 + 2x + 1$. The value of a silver coin is $s(x) = 3x + 4$. A wizard combines the values of a gold coin and a silver coin, creating a new spell valued at $h(x) = g(x) - s(x)$. Calculate the value of the sp... | tiger_math | [
0.022369423881173134,
0.05333612114191055,
-0.052391208708286285,
-0.04964173212647438,
0.035483863204717636,
-0.04742299020290375,
-0.024512555450201035,
0.06164538860321045,
0.04550354927778244,
-0.02737739495933056,
0.020884612575173378,
0.04463900625705719,
-0.024517925456166267,
-0.03... | 0.841412 | 0.828719 | 0.012693 |
Find the values of $A$, $B$, and $C$ in the partial fraction decomposition:
\[\frac{2x^2 + 5x - 11}{(x - 2)(x + 1)(x - 1)} = \frac{A}{x - 2} + \frac{B}{x + 1} + \frac{C}{x - 1}.\] | tiger_math | [
0.039058879017829895,
0.03554169833660126,
-0.04316484555602074,
-0.04468328133225441,
0.047439251095056534,
-0.045310452580451965,
0.002017148770391941,
0.08231861144304276,
0.03263634815812111,
0.0004464171070139855,
0.0033984945621341467,
0.030021723359823227,
-0.03234194591641426,
-0.0... | 0.824048 | 0.817186 | 0.006862 |
Zan has created this iterative rule for generating sequences of whole numbers:
1) If a number is 25 or less, double the number.
2) If a number is greater than 25, subtract 12 from it.
Let $F$ be the first number in a sequence generated by the rule above. $F$ is a "sweet number" if X is not a term in the sequence tha... | tiger_math | [
0.01552861463278532,
0.02006920799612999,
-0.03947731852531433,
-0.026819301769137383,
0.03580755740404129,
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-0.009064185433089733,
0.06124470382928848,
0.013864386826753616,
-0.03364524617791176,
0.009641396813094616,
0.039833832532167435,
-0.04860584810376167,
-0.00... | 0.840254 | 0.833847 | 0.006407 |
Two congruent cylinders each have radius 10 inches. The radius of one cylinder is increased by x inches, while the height of the other cylinder is decreased by x inches. If the resulting volumes are equal, what is the value of x? Express your answer as a decimal rounded to the nearest hundredth. | tiger_math | [
0.020832184702157974,
0.03062286600470543,
-0.03735436871647835,
-0.018237605690956116,
0.030575713142752647,
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-0.023543931543827057,
0.0708126351237297,
0.04252060502767563,
-0.025225205346941948,
0.01822708733379841,
0.02339031547307968,
-0.05224344879388809,
-0.0080... | 0.824207 | 0.820898 | 0.003309 |
In trapezoid $ABCD$, sides $\overline{AB}$ and $\overline{CD}$ are parallel, $\angle A = 2\angle D$, and $\angle C = 2\angle B$. Find $\angle D$. | tiger_math | [
0.016796696931123734,
0.011730540543794632,
-0.03213408961892128,
-0.047591619193553925,
0.0076474640518426895,
-0.003663628362119198,
-0.010050807148218155,
0.04385945945978165,
0.02479388751089573,
-0.013474442064762115,
0.0347655713558197,
0.025571729987859726,
-0.03532842919230461,
-0.... | 0.831131 | 0.812971 | 0.01816 |
In a candy store, the owner has 3 types of candies in piles: Type A, Type B, and Type C. Type A candies come in packages of 15, Type B in packages of 20, and Type C in packages of 25. The owner initially has 6 candies of Type A, 8 candies of Type B, and 10 candies of Type C on the display pile on the desk. The owner wa... | tiger_math | [
0.007288849446922541,
0.017710117623209953,
-0.03638178110122681,
-0.040298979729413986,
0.04195833206176758,
-0.019079266116023064,
-0.0382571779191494,
0.08555405586957932,
0.007829025387763977,
-0.005880619864910841,
0.01929323375225067,
0.03192993625998497,
-0.058230023831129074,
-0.01... | 0.820989 | 0.813863 | 0.007126 |
The foot of the perpendicular from the origin to a plane is $(p,q,r).$ Find the equation of the plane in the form $Ax + By + Cz + D = 0,$ where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$ What is the value of $A$? | tiger_math | [
0.02951805852353573,
0.025669673457741737,
-0.05425834655761719,
-0.03418190777301788,
0.034884680062532425,
-0.0015355810755863786,
-0.0011261618928983808,
0.05120976269245148,
0.04414493963122368,
-0.019229762256145477,
0.0166900884360075,
0.03496242314577103,
-0.02672157622873783,
-0.01... | 0.842992 | 0.831788 | 0.011204 |
Define the function $f(x)=\frac{c}{3x-2}$. If $f(1)=f^{-1}(c-2)$, find the product of all possible values of $c$. | tiger_math | [
0.03529402241110802,
0.039459019899368286,
-0.048432428389787674,
-0.017378894612193108,
0.033319659531116486,
-0.03777815029025078,
-0.0025694561190903187,
0.09054147452116013,
0.029621673747897148,
-0.013049027882516384,
0.012204701080918312,
0.04728247970342636,
-0.0158681720495224,
-0.... | 0.835537 | 0.832623 | 0.002914 |
Ray is climbing up the stairs $m$ at a time, as defined previously. Ray climbs up a flight of stairs of $n$ steps in two ways. When he does it $3$ steps at a time, there are $2$ steps left at the top. When he does it $5$ steps at a time, there are $4$ steps left at the top. What is the smallest possible value of $n$ th... | tiger_math | [
0.03248576074838638,
0.03780324384570122,
-0.026838010177016258,
-0.029484618455171585,
0.046025414019823074,
-0.005662629846483469,
-0.014965527690947056,
0.07006281614303589,
0.027522139251232147,
-0.01585528813302517,
0.018894720822572708,
0.038480959832668304,
-0.062540702521801,
-0.02... | 0.848859 | 0.84179 | 0.007068 |
Lucy's art studio is making custom triangular tabletops designed with a specific color pattern. The triangular tabletops have sides measuring 10 units, 15 units, and 20 units. Lucy wants to calculate the area of each tabletop to know how much paint is needed for the colored area. What is the area of each triangle? | tiger_math | [
0.025189140811562538,
0.02688576467335224,
-0.05560772866010666,
-0.03261909633874893,
0.04150491952896118,
-0.025004835799336433,
-0.045457031577825546,
0.04023134708404541,
0.037870343774557114,
-0.019816085696220398,
0.04396449774503708,
0.03837018832564354,
-0.02033817581832409,
-0.014... | 0.803075 | 0.795685 | 0.00739 |
A hiking group plans to travel across a rectangular field that measures 300 meters by 700 meters. They decide to walk diagonally from one corner to the opposite corner. Calculate the distance they will walk, taking into account only the straight diagonal path across the field. | tiger_math | [
0.0004802235634997487,
0.0235221516340971,
-0.06132712587714195,
-0.05903578922152519,
0.009069419465959072,
-0.04169014096260071,
-0.0205587986856699,
0.04735950380563736,
0.034705162048339844,
-0.02298983559012413,
0.011143798008561134,
0.032915037125349045,
-0.024659909307956696,
-0.023... | 0.819626 | 0.815718 | 0.003908 |
Chef Emily is preparing chocolate boxes as gifts for a culinary event. For a pair of unique ingredients, she needs to prepare three different types of cookies. If she has six unique ingredients available, how many different pairs of ingredients can she choose, and how many different types of cooking sets can she prepar... | tiger_math | [
0.016254598274827003,
0.0054122149012982845,
-0.021525882184505463,
-0.035044021904468536,
0.026427356526255608,
-0.022231072187423706,
-0.0395040288567543,
0.059103526175022125,
0.035957977175712585,
-0.03074173629283905,
0.030685443431138992,
0.03506148234009743,
-0.021319830790162086,
-... | 0.799657 | 0.79495 | 0.004707 |
For the rational function $f(x) = \frac{x^3-3x+1}{x^2-9}$, determine the equations of all vertical asymptotes. | tiger_math | [
0.007663734722882509,
0.00881256815046072,
-0.0414922796189785,
-0.028973275795578957,
0.01748160645365715,
-0.053134504705667496,
-0.004689144901931286,
0.05534321069717407,
0.03522508591413498,
-0.020461447536945343,
0.03329262137413025,
0.00845212023705244,
-0.03387787938117981,
-0.0189... | 0.82996 | 0.823413 | 0.006548 |
The equations of the asymptotes of a hyperbola are $y = 2x+5$ and $y = -2x+1.$ The hyperbola passes through the point $(0, 7)$ and its center is $(h, k).$ Find the value of $k - h.$ | tiger_math | [
0.029288876801729202,
0.02390659600496292,
-0.044249411672353745,
-0.038393281400203705,
0.03999378904700279,
-0.031211428344249725,
-0.024411413818597794,
0.05990993604063988,
0.042689431458711624,
-0.03722279146313667,
0.023203259333968163,
0.01772109605371952,
-0.03173932433128357,
-0.0... | 0.829297 | 0.824608 | 0.004688 |
Let $\mathbf{a}$ and $\mathbf{b}$ be unit vectors. If the angle between $\mathbf{a}$ and $\mathbf{b}$ is 60 degrees and $\mathbf{a} + 2 \mathbf{b}$ and $5 \mathbf{a} - 4 \mathbf{b}$ are orthogonal, find the dot product of $\mathbf{a}$ and $\mathbf{b}$. | tiger_math | [
0.03002467006444931,
0.02433222346007824,
-0.04382611811161041,
-0.028456078842282295,
0.042410459369421005,
-0.0037966156378388405,
-0.014880661852657795,
0.07824652642011642,
0.03832100331783295,
-0.013680404052138329,
0.008161368779838085,
0.02050168253481388,
-0.023504309356212616,
-0.... | 0.828674 | 0.819297 | 0.009377 |
Find a monic cubic polynomial $P(x)$ with integer coefficients such that
\[P(\sqrt[3]{2} + 1) = 0.\](A polynomial is monic if its leading coefficient is 1.) | tiger_math | [
0.018526002764701843,
0.049960654228925705,
-0.044742152094841,
-0.03135528415441513,
0.025298530235886574,
-0.03917977213859558,
-0.013461180031299591,
0.05265342444181442,
0.03329199552536011,
-0.0029057117644697428,
0.010670345276594162,
0.023151099681854248,
-0.0037351285573095083,
-0.... | 0.844845 | 0.830248 | 0.014597 |
A regular octagon $ABCDEFGH$ has sides of length two. If the area of $\bigtriangleup ADG$ is $4+3\sqrt{2}$, find the length of $OG$. | tiger_math | [
0.017684735357761383,
0.011028565466403961,
-0.054676201194524765,
-0.04707849398255348,
0.02883804216980934,
-0.036897651851177216,
-0.03875645250082016,
0.038786180317401886,
0.0425325371325016,
-0.01734870672225952,
0.017112763598561287,
0.02383175492286682,
-0.01729937456548214,
-0.028... | 0.841894 | 0.831903 | 0.009991 |
Find $a$ if $a$ and $b$ are integers such that $x^2 - 2x - 1$ is a factor of $ax^{17} + bx^{16} + 1$. | tiger_math | [
0.03291918337345123,
0.039636459201574326,
-0.037386052310466766,
-0.049500465393066406,
0.029172252863645554,
-0.021132459864020348,
0.005456990096718073,
0.08938297629356384,
0.046356115490198135,
-0.03009023703634739,
0.02174694463610649,
0.050966594368219376,
-0.032074060291051865,
-0.... | 0.85041 | 0.842547 | 0.007863 |
A curious case of fruit baskets involves apples and oranges. Mrs. Jonas goes to a market and decides to buy fruit baskets for her three children. Each basket contains only apples or oranges or a combination of both. If Mrs. Jonas buys 3 baskets that contain a total of $30$ pieces of fruit (apples and oranges combined),... | tiger_math | [
0.013989287428557873,
0.017665492370724678,
-0.053295549005270004,
-0.007139378692954779,
0.02949528768658638,
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0.06387951225042343,
0.02509470283985138,
-0.012682919390499592,
-0.0023150499910116196,
0.032566312700510025,
-0.04939090088009834,
-... | 0.831475 | 0.81565 | 0.015825 |
A recently constructed space bridge connects three planets in a triangular route. Each side of the triangle represents the distance between two planets. The distances are recorded as follows: the distance from Planet A to Planet B is \( X + 8 \) light-years, the distance from Planet B to Planet C is \( 2X \) light-year... | tiger_math | [
0.011793228797614574,
0.03355919197201729,
-0.03692066669464111,
-0.032950907945632935,
0.023189395666122437,
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0.04449521377682686,
0.04612717404961586,
-0.01746257022023201,
0.023513687774538994,
0.04532954841852188,
-0.004730138927698135,
-0.0... | 0.806912 | 0.804051 | 0.002861 |
Line $l_1$ has equation $3x - 2y = 1$ and goes through $A = (-1, -2)$. Line $l_2$ has equation $y = 1$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and meets $l_2$ at point $C$. The distance from point $A$ to line $l_2$ is equal to the length of $BC$. What is the slope of $l... | tiger_math | [
0.004867623560130596,
0.008372392505407333,
-0.0648353099822998,
-0.06304126232862473,
0.054294321686029434,
-0.020423738285899162,
-0.03455941006541252,
0.04475794732570648,
0.03582080826163292,
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0.010676689445972443,
0.04833359643816948,
-0.02730054035782814,
-0.008... | 0.835988 | 0.82736 | 0.008628 |
A circular spinner for a game has a radius of X cm. The probability of winning on one spin of this spinner is $\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\pi$.
[asy]import graph;
draw(Circle((0,0),25),black);
draw((0,0)--(7,18),Arrow);
draw((0,0)--(0,25));
draw((0... | tiger_math | [
-0.027496840804815292,
0.04167557135224342,
-0.052689604461193085,
-0.0411752387881279,
0.0381348691880703,
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0.043096110224723816,
0.038067400455474854,
0.010452868416905403,
0.025707950815558434,
0.023499149829149246,
-0.01376130897551775,
-0.00... | 0.827284 | 0.82126 | 0.006025 |
Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B=(0,0), C=(223,0), D=(680,380),$ and $E=(689,389).$ The distance from $A$ to line $DE$ is $18.$ What is the sum of all possible $x$-coordinates of $A$? | tiger_math | [
0.02139088325202465,
0.015979642048478127,
-0.05081348866224289,
-0.03885004296898842,
0.019550753757357597,
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0.043083831667900085,
0.04679778218269348,
-0.001950953621417284,
0.009400870651006699,
0.0453493706882,
-0.04484931379556656,
-0.01973... | 0.8363 | 0.832723 | 0.003577 |
An equilateral triangle T2 is formed by joining the mid points of the sides of another equilateral triangle T1. A third equilateral triangle T3 is formed by joining the mid-points of T2 and this process is continued indefinitely. Each side of T1 is 30 cm. What is the sum of the perimeters of all the triangles? | tiger_math | [
0.020502684637904167,
0.017061898484826088,
-0.029515709728002548,
-0.05441771820187569,
0.024543531239032745,
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0.04589901119470596,
0.030994538217782974,
-0.022899659350514412,
0.023340415209531784,
0.05115434154868126,
-0.04121539369225502,
-0... | 0.828401 | 0.813917 | 0.014484 |
Alice is planning an afternoon garden party for her friend Bob. She decides to set up round tables and, initially, wants each table to have exactly 4 guests. She invited a total of 48 friends. Later, Bob asks her to rearrange the seating so that each table sits exactly 6 guests to make the setting more intimate and to ... | tiger_math | [
0.04399692267179489,
0.03084646724164486,
-0.024143248796463013,
-0.018508896231651306,
0.024584105238318443,
-0.03739316761493683,
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0.04005477577447891,
0.046933338046073914,
0.006386166904121637,
-0.0070436373353004456,
0.02616037428379059,
-0.035167235881090164,
-0... | 0.822175 | 0.814935 | 0.007241 |
Find the equation of the plane passing through $(-1,1,1)$ and $(1,-1,1),$ and which is perpendicular to the plane $x + 2y + 3z = 5.$ Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = X
If we know the answer to the above question ... | tiger_math | [
0.013410449959337711,
0.0274571031332016,
-0.051211025565862656,
-0.04682621732354164,
0.05900876969099045,
0.013347486034035683,
-0.014288892038166523,
0.058566395193338394,
0.03529692068696022,
-0.004848194774240255,
0.020282408222556114,
0.04031515493988991,
-0.01976575143635273,
-0.020... | 0.833813 | 0.823705 | 0.010108 |
The function $g(x)$ satisfies
\[xg(y) = yg(x)\]for all non-zero real numbers $x$ and $y.$ If $g(5) = 8,$ find $g(2).$ | tiger_math | [
0.025378422811627388,
0.029470279812812805,
-0.05926556512713432,
-0.039256501942873,
0.0361936129629612,
-0.028024613857269287,
-0.010826466605067253,
0.0555107519030571,
0.034082960337400436,
-0.01931873895227909,
0.014902198687195778,
0.027972744777798653,
-0.03324924036860466,
-0.01489... | 0.836832 | 0.824099 | 0.012733 |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 1,$ $\|\mathbf{b}\| = 5,$ $\|\mathbf{c}\| = X and
\[\mathbf{a} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{c}.\]If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b},$ then find $\sin \theta.$
If we know the answer to the ab... | tiger_math | [
0.018618352711200714,
0.039073847234249115,
-0.08031310886144638,
-0.03144918754696846,
0.044844262301921844,
0.018986796960234642,
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0.07720615714788437,
0.028150707483291626,
-0.004955480806529522,
0.019390907138586044,
0.013252068310976028,
-0.028822368010878563,
-0.... | 0.836292 | 0.820769 | 0.015522 |
In the figure shown, the ratio of $BD$ to $DC$ is $2$ to $1$. The area of $\triangle ADC$ is $15$ square centimeters. What is the area of $\triangle ABD$? [asy] size(85); defaultpen(linewidth(1)+fontsize(10));
pair A = (0,5.5), B=(0,0), D = (2,0), C = (3,0);
draw(A--B--C--A--D); label("A",A,N); label("B",B,S); label(... | tiger_math | [
0.008545235730707645,
0.029925081878900528,
-0.046821821480989456,
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0.033052075654268265,
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0.05703139677643776,
0.04958552494645119,
0.010136319324374199,
0.022039715200662613,
0.03122808039188385,
-0.014681809581816196,
-0.... | 0.826701 | 0.820255 | 0.006447 |
Let $
f(n) =
\begin{cases}
n^2+1 & \text{if }n\text{ is odd} \\
\dfrac{n}{2} & \text{if }n\text{ is even}
\end{cases}.
$
For how many integers $n$ from 0 to 100, inclusive, does $f ( f (\dotsb f (n) \dotsb )) = 0$ for some number of applications of $f$? | tiger_math | [
0.022022292017936707,
0.019264450296759605,
-0.05131883919239044,
-0.025698937475681305,
0.029556460678577423,
-0.01634870655834675,
-0.009693578816950321,
0.05100294575095177,
0.023249058052897453,
-0.014517609030008316,
0.0030200111214071512,
0.020226076245307922,
-0.021437274292111397,
... | 0.844933 | 0.831809 | 0.013124 |
Dr. Peterson owns a small botanical garden that hosts various types of trees. In one section, there are eight rows of apple trees. Each row contains exactly seven apple trees. In another section, Dr. Peterson planted sixteen pear trees. Being a fan of mathematical patterns, Dr. Peterson arranged the number of cherry tr... | tiger_math | [
0.023911405354738235,
0.017158374190330505,
-0.04248110204935074,
-0.05212734267115593,
0.014035893604159355,
-0.02860897034406662,
-0.022030489519238472,
0.052807487547397614,
0.039220910519361496,
-0.023951759561896324,
-0.017422527074813843,
0.04237719625234604,
-0.04713232442736626,
-0... | 0.829884 | 0.817157 | 0.012728 |
Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-a$, the remainder is $b$, and when $P(x)$ is divided by $x-c$, the remainder is $d$. If $(x-a)(x-c)$ divides $P(x)$ with a remainder of $2x-20$, find the values of $a$, $b$, $c$, and $d$. | tiger_math | [
0.034904610365629196,
0.03325903043150902,
-0.05376097559928894,
-0.03958593308925629,
0.030193598940968513,
-0.0445370189845562,
-0.006064736749976873,
0.07205774635076523,
0.045157335698604584,
-0.014816632494330406,
-0.00004125144550926052,
0.048655394464731216,
-0.05113879218697548,
-0... | 0.842 | 0.829013 | 0.012987 |
If $e^{i \alpha} + e^{i \beta} = \frac{2}{5} + \frac{1}{3} i,$ then find $e^{-i \alpha} + e^{-i \beta}.$ | tiger_math | [
0.024096114560961723,
0.019300494343042374,
-0.03826777637004852,
-0.0330912321805954,
0.043494440615177155,
-0.013273775577545166,
-0.019582167267799377,
0.07529875636100769,
0.05302051827311516,
-0.0037221135571599007,
0.022255556657910347,
0.01324675977230072,
-0.05124007910490036,
-0.0... | 0.83722 | 0.83323 | 0.00399 |
Find the largest value of $t$ for which the graph of $r = \sin \theta$ covers only the semicircle above the polar axis when plotted for $0 \le \theta \le t$. | tiger_math | [
-0.007123321760445833,
0.034093227237463,
-0.03697936236858368,
-0.03245570883154869,
0.024989167228341103,
-0.019456954672932625,
-0.039107903838157654,
0.0642198696732521,
0.03500346094369888,
-0.019726477563381195,
0.0339152067899704,
0.020041082054376602,
-0.026092765852808952,
0.00252... | 0.837078 | 0.82449 | 0.012588 |
Let $a$ and $b$ be positive real numbers such that the equation $x^2 + ax + b = 0$ has real roots. If the roots of this equation are $r_1$ and $r_2$, find the minimum value of $(r_1 - r_2)^2$. | tiger_math | [
0.01811285875737667,
0.019162015989422798,
-0.04492485150694847,
-0.038303714245557785,
0.0345270037651062,
-0.025167476385831833,
0.0101087037473917,
0.06667616963386536,
0.04982408136129379,
-0.031516995280981064,
0.000541651388630271,
0.01799723505973816,
-0.023630520328879356,
-0.02676... | 0.842388 | 0.825482 | 0.016906 |
In triangle $ABC$, $\angle C=90^\circ$, $AC=6$ and $BC=8$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED=90^\circ$. If $BD=5$, then what is the length of $DE$? | tiger_math | [
0.00477774441242218,
-0.003884679637849331,
-0.05271542817354202,
-0.050662074238061905,
0.017659366130828857,
-0.02204812690615654,
-0.018247121945023537,
0.03214984014630318,
0.044584065675735474,
-0.020426571369171143,
0.011072084307670593,
0.029353754594922066,
-0.04460902512073517,
-0... | 0.835758 | 0.830152 | 0.005606 |
Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. If Dick is currently less than 50 years ... | tiger_math | [
0.018811970949172974,
0.01175722572952509,
-0.054613273590803146,
-0.0317099466919899,
0.010696664452552795,
-0.0032116039656102657,
-0.02317848987877369,
0.05890365689992905,
0.03219068795442581,
-0.02057081274688244,
0.018159564584493637,
0.02389228157699108,
-0.04704083129763603,
-0.004... | 0.855478 | 0.843428 | 0.012051 |
Max likes to collect model trains. He asks for one for every birthday of his, and asks for two each Christmas. Max always gets the gifts he asks for, and asks for these same gifts every year for 5 years. At the end of the 5 years, his parents give him double the number of trains he already has. How many trains does... | tiger_math | [
0.015199019573628902,
0.0041596656665205956,
-0.04503048211336136,
-0.0385739766061306,
0.020589644089341164,
-0.03146609291434288,
-0.03930917754769325,
0.05162736400961876,
0.025086592882871628,
-0.02915504015982151,
0.01808265969157219,
0.02896450273692608,
-0.03569486737251282,
-0.0209... | 0.819911 | 0.815057 | 0.004854 |
Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 8$, $BC = 4$, $\overline{PA}\perp \overline{AD}$, $\overline{PA}\perp \overline{AB}$, and $PB = 17$, then what is the volume of $PABCD$? | tiger_math | [
0.011460191570222378,
0.014865395613014698,
-0.03817812725901604,
-0.04172134026885033,
0.019852010533213615,
-0.04173628240823746,
-0.006628143135458231,
0.038210660219192505,
0.03546275943517685,
0.0074005210772156715,
0.012993848882615566,
0.03568585589528084,
-0.025439413264393806,
-0.... | 0.82588 | 0.821778 | 0.004102 |
Squares of side length 1 are arranged to form the figure shown. What is the perimeter of the outer rectangle? [asy]
size(6cm);
path sqtop = (0, 0)--(0, 1)--(1, 1)--(1, 0);
path sqright = (0, 1)--(1, 1)--(1, 0)--(0, 0);
path horiz = (0, 0)--(1, 0); path vert = (0, 0)--(0, 1);
picture pic;
draw(pic, shift(-4, -2) *... | tiger_math | [
0.01070378627628088,
0.04006693884730339,
-0.030353650450706482,
-0.03764228895306587,
0.03337208926677704,
-0.014057237654924393,
-0.024303901940584183,
0.056071776896715164,
0.03813818097114563,
0.0001483094965806231,
0.024295948445796967,
0.04435626044869423,
-0.019482744857668877,
-0.0... | 0.829754 | 0.822504 | 0.00725 |
Let $a$ and $b$ be nonzero complex numbers such that $a^2 + ab + b^2 = 0$ and $a^3 \neq b^3.$ Evaluate
\[\frac{a^9 + b^9}{(a + b)^9}.\] | tiger_math | [
0.022772038355469704,
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0.019316373392939568,
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-0.015... | 0.843227 | 0.828399 | 0.014828 |
Let $A$ be the vertex of the graph of the equation $y=x^2 - 2x + X $. Let $B$ be the vertex of the graph of the equation $y=x^2 + 4x + 10 $. What is the distance between $A$ and $B$?
If we know the answer to the above question is 5, what is the value of unknown variable X? | tiger_math | [
0.021798105910420418,
0.017271190881729126,
-0.0577826164662838,
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0.025932541117072105,
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0.05179291591048241,
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0.011270025745034218,
0.039399001747369766,
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0.0029... | 0.84112 | 0.833344 | 0.007776 |
A scout has 12 campsites to inspect, each situated at unique coordinates on a map. Three of the camps are located at the points (4, 5), (7, -2), and (0, 3). The scout decides to check the camps in a circular route, starting and ending at the camp which is geographically in the middle among these three. Find the coordin... | tiger_math | [
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0.022122010588645935,
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0.02072937786579132,
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0.03162803873419762,
0.01564418524503708,
-0.026894614100456238,
-0.029... | 0.822261 | 0.813005 | 0.009256 |
The numbers X, 2, 3, 4, 5, 6, 7,$ and $8$ are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where $8$ and $1$ are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are ... | tiger_math | [
0.007569367531687021,
0.0133933424949646,
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0.03620683401823044,
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0.06285997480154037,
0.039392948150634766,
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0.0047479732893407345,
0.01973014883697033,
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Amara is planning a garden which she decides will be in the shape of a right triangle. She wants to construct a fence around the garden, using the hypotenuse as one side of a decorative wooden fence, while the other two sides will be simple wire. She determines that the total cost for fencing the garden will be determi... | tiger_math | [
0.008683772757649422,
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0.0006784654106013477,
0.007440537214279175,
0.05508960410952568,
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In the magical land of Figureshire, the queen has a perfectly shaped garden in the form of a rectangle. The queen wants to plant two separate flower beds shaped as right triangles at opposite corners of the garden. Each flower bed will have legs that coincide with the garden's length and width. The garden's dimensions ... | tiger_math | [
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0.04326271638274193,
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Let $\mathbf{a} = \begin{pmatrix} 7 \\ -4 \\ -4 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} -2 \\ -1 \\ X \end{pmatrix}.$ Find the vector $\mathbf{b}$ such that $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are collinear, and $\mathbf{b}$ bisects the angle between $\mathbf{a}$ and $\mathbf{c}.$
[asy]
unitsize(0.5... | tiger_math | [
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Convert the point $(2 \sqrt{3}, 6, -4)$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$ | tiger_math | [
0.014677423052489758,
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0.042866989970207214,
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Let
\[\mathbf{M} = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}.\]If $\mathbf{M} \mathbf{M}^T = 16 \mathbf{I},$ then find the value of $a^2+b^2$.
Note: For a matrix $\mathbf{A},$ $\mathbf{A}^T$ is the transpose of $\mathbf{A},$ which is generated by reflecting the matrix $\mathbf{A}$ over the mai... | tiger_math | [
0.051477935165166855,
0.03727155923843384,
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0.030080925673246384,
0.030446847900748253,
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In the diagram, $AOB$ is a sector of a circle with $\angle AOB=60^\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ If the radius of the circle is 10 cm, what is the length of $XY
?$ | tiger_math | [
0.0054169450886547565,
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Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a}\| = X $\|\mathbf{b}\| = 5,$ and $\|\mathbf{a} \times \mathbf{b}\| = 8.$ Find $|\mathbf{a} \cdot \mathbf{b}|. 6. What is the value of unknown variable X? | tiger_math | [
0.028357520699501038,
0.03589004650712013,
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0.024065181612968445,
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-... | 0.825073 | 0.820517 | 0.004556 |
A chef is preparing meals for a large banquet. The banquet facilities have 5 different types of vegetables, and he must select three types to create a vegetable platter. Each platter is to include exactly 10 kilograms of vegetables, consisting of 4 kilograms of the first vegetable, 3 kilograms of the second vegetable, ... | tiger_math | [
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0.055684417486190796,
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0.... | 0.830768 | 0.826502 | 0.004265 |
Let $a,$ $b,$ $c,$ and $d$ be the roots of the equation $x^4 + 8x^3 + 9x^2 + 5x + X = 0$. If $\frac{1}{ab} + \frac{1}{ac} + \frac{1}{ad} + \frac{1}{bc} + \frac{1}{bd} + \frac{1}{cd} = \frac{5}{2}$, what is the value of $X$? | tiger_math | [
0.019052131101489067,
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0.03324830159544945,
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0.007149691227823496,
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-0.037... | 0.850618 | 0.845212 | 0.005406 |
In triangle $ABC$, $AB = 10$, $BC = 14$, and $CA = 16$. Let $D$ be a point on $\overline{BC}$. Let points $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD$, respectively. The circumcircles of triangles $BI_BD$ and $CI_CD$ meet at distinct points $P$ and $D$. If the maximum possible area of $\triangle BP... | tiger_math | [
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0.028064588084816933,
0.040262218564748764,
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-0... | 0.849283 | 0.817674 | 0.031609 |
Wanda is trying to locate the Fermat point $P$ of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,0)$, and $C$ is at $(3,5)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point is at $P = (4,2)$, and computes the sum of ... | tiger_math | [
0.026640530675649643,
0.03201338276267052,
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0.03561194986104965,
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0.04012836888432503,
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0.018237993121147156,
0.03803436830639839,
-0.007025990169495344,
-0.031... | 0.845159 | 0.830016 | 0.015144 |
An equilateral triangle has an altitude of $\sqrt6$ units. Determine the area of the triangle, in square units, expressing your answer in simplest radical form. | tiger_math | [
0.015255258418619633,
0.030193477869033813,
-0.04227568209171295,
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0.030217446386814117,
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0.04057711362838745,
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0.028250785544514656,
0.02956608682870865,
-0.03017452545464039,
-0.... | 0.820261 | 0.815871 | 0.00439 |
The polynomial $x^{2023} + Ax + B$ is divisible by $x^2 + x + 1$ for some real numbers $A$ and $B.$ Find $A + B.$ | tiger_math | [
0.03044860251247883,
0.029995866119861603,
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0.022384045645594597,
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0.01688203029334545,
0.03861080855131149,
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-0.01... | 0.840597 | 0.828561 | 0.012036 |
A massive number of placards are placed in a huge basket at the entrance of a stadium. There are two different sizes of placards: small placards and large placards. Every person entering the stadium collects one small placard for their left hand and one large placard for their right hand. People started entering the st... | tiger_math | [
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In a group of nine people each person shakes hands with exactly three of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other ... | tiger_math | [
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0.02568840980529785,
0.03674124553799629,
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-0.0... | 0.85346 | 0.8387 | 0.01476 |
At the local farm, Sally noticed that there are several rows of apple trees and pineapple trees. If she counted 45 apple trees and 120 pineapple trees, and wants to arrange them in a layout where each row has the same number of trees, being only of one type of tree, what is the maximum number of trees she can place in ... | tiger_math | [
0.014108244329690933,
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-0.0... | 0.825277 | 0.815354 | 0.009924 |
There is an angle $\theta$ in the range $0^\circ < \theta < 45^\circ$ which satisfies
\[\tan \theta + \tan 2 \theta + \tan X \theta = 0.\]Calculate $\tan \theta$ for this angle. The answer is 2. What is the value of unknown variable X? | tiger_math | [
0.016551610082387924,
0.0391344353556633,
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0.03889447823166847,
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0.027463901787996292,
0.020261691883206367,
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-0.0... | 0.831571 | 0.82714 | 0.004431 |
A rectangular sheet of paper with a length of 12 cm and a width of 8 cm is cut into four congruent right triangles. What is the area of the square that can be created by arranging the triangles such that their right angles form the corners of the square? | tiger_math | [
0.03630630671977997,
0.03927222639322281,
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0.028670435771346092,
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0.004490979481488466,
0.036057230085134506,
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-0.0... | 0.825549 | 0.820279 | 0.005271 |
Three mathematicians, Alice, Bob, and Charlie, meet for coffee and decide to each buy a novel math puzzle book. The coffee shop is running a special promotion where every third book is 50% off. If Alice buys a book that costs $12, Bob buys a book that costs $15, and Charlie buys a book that costs $18, how much will the... | tiger_math | [
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-... | 0.835757 | 0.83015 | 0.005607 |
The perimeter of an equilateral triangle is some value. If one of the sides of the equilateral triangle is the side of an isosceles triangle of perimeter 45, then the base of the isosceles triangle is 5. What is the perimeter of the equilateral triangle? | tiger_math | [
0.03872818127274513,
0.008054928854107857,
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0.03471243754029274,
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0.04698460176587105,
0.050993144512176514,
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-0.... | 0.79043 | 0.786889 | 0.003542 |
Determine the value of $m$ in the equation $\frac{y^2}{9} - \frac{x^2}{4} = 1$ where the asymptotes of the hyperbola are $y = \pm mx$, and $m$ is a positive number. | tiger_math | [
0.004900703206658363,
0.008750383742153645,
-0.05965777486562729,
-0.05073825269937515,
0.05012337863445282,
-0.04707909747958183,
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0.0619824193418026,
0.062190283089876175,
-0.02897714450955391,
0.01318434253334999,
0.019842689856886864,
-0.01990458555519581,
-0.0126... | 0.826762 | 0.819134 | 0.007628 |
The perpendicular bisector of the line segment with endpoints $A = (24, 7)$ and $B = (3, 4)$ intersects $AB$ at point $C = (x, y)$. Find the value of $x + 2y$. | tiger_math | [
0.042517777532339096,
0.004811648279428482,
-0.04988724738359451,
-0.03640042617917061,
0.045594438910484314,
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0.05293336510658264,
0.05111885070800781,
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0.024482205510139465,
0.03762838616967201,
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-0.0... | 0.834875 | 0.830764 | 0.004111 |
Let $\mathbf{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -5 \\ 2 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}.$ | tiger_math | [
0.04250337556004524,
0.03176790103316307,
-0.06879023462533951,
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0.03980005159974098,
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0.06330418586730957,
0.03488212451338768,
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0.008880502544343472,
0.026468897238373756,
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-0.0... | 0.828858 | 0.817778 | 0.011079 |
The figure shown consists of a right triangle and two squares. If the figure's total area equals 850 square inches, what is the length of the hypotenuse of the triangle in inches?
[asy]
unitsize(5mm);
defaultpen(linewidth(.7pt)+fontsize(10pt));
draw((0,5)--(0,-2)--(-2,-2)--(-2,0)--(5,0)--(5,5)--cycle--(-2,0));
draw(sc... | tiger_math | [
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0.016712427139282227,
0.03468132019042969,
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-0.... | 0.827764 | 0.824648 | 0.003116 |
In the diagram, $\triangle PQR$ is right-angled at $P$ and has $PQ=2$ and $PR=2\sqrt{3}$. Altitude $PL$ intersects median $RM$ at $F$. What is the area of $\triangle PFL$?
[asy]
draw((0,0)--(10,0)--(0,10*sqrt(3))--cycle);
draw((0,0)--(7.5,4.33)); draw((0,10*sqrt(3))--(5,0));
draw((6.68,3.86)--(7.17,3.01)--(7.99,3.49)... | tiger_math | [
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0.02481016330420971,
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-0.02... | 0.823045 | 0.81194 | 0.011106 |
During Everlee's extended family gathering, there are four adults, six children, and three cookie jars with different quantities of cookies. Each cookie jar has 240, 360, and 480 cookies, respectively. If the adults eat 'a' fraction of the cookies from the first jar, 'b' fraction from the second jar, and 'c' fraction f... | tiger_math | [
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-0... | 0.823662 | 0.818205 | 0.005457 |
In the Mathland fair, Lizzy and her little brother Max found a booth playing a number-challenge game. The game organizer told them that they should choose a number \( x \), and then they would get as many candies as the result of the expression \( |x-3| + |x+2| \). Lizzy wants to win as many candies as possible. For wh... | tiger_math | [
0.02155991829931736,
0.01159417163580656,
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0.02914213389158249,
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0.06618262827396393,
0.02711224928498268,
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0.0036254783626645803,
0.04643092304468155,
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-0.025867... | 0.83137 | 0.825162 | 0.006208 |
Dr. Wilson is organizing a treasure hunt for a group of adolescents. He has hidden the treasure behind three locked gates in succession, each requiring a separate key. The keys are labeled A, B, and C, and Dr. Wilson provides the teenagers with a list of clues:
1. The product of the key numbers in their natural order ... | tiger_math | [
0.05198275297880173,
0.027100957930088043,
-0.05216042324900627,
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0.0412171334028244,
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0.05872233957052231,
0.02916191704571247,
-0.011262801475822926,
-0.0007208954775705934,
0.03272430598735809,
-0.044102516025304794,
-0.00... | 0.82777 | 0.823146 | 0.004624 |
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are three male and two female professors in the mathematics department, two male and three female professors in the statistics department, and three male and two female prof... | tiger_math | [
0.012812166474759579,
0.009264950640499592,
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0.04472823813557625,
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0.022154059261083603,
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0.002868447918444872,
0.028234485536813736,
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-... | 0.82479 | 0.808003 | 0.016787 |
You have decided to buy two types of plants for your garden - daisies and tulips. The store offers a package deal of 4 daisies and 6 tulips for $44. Also, another package includes 3 daisies and 2 tulips for $19. You want at least one package of each type, how many of each plant will you get if you buy exactly one packa... | tiger_math | [
0.018887117505073547,
0.01402242574840784,
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0.040161050856113434,
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0.01721542701125145,
0.03232103958725929,
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-0.01... | 0.829751 | 0.826565 | 0.003186 |
A triangle has two sides of length 7 and 9 units. The length of the third side is $y$ units, where $y$ is a positive integer. What is the smallest possible perimeter of the triangle? | tiger_math | [
0.014118241146206856,
0.014078643172979355,
-0.046530164778232574,
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0.02521481364965439,
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0.04489125683903694,
0.04912212863564491,
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0.028127318248152733,
0.0431811548769474,
-0.024502195417881012,
-0.025... | 0.843343 | 0.837295 | 0.006048 |
A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. [asy]
draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));
fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,gray(.8));
fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,gray(.8));
fill((7,1)--(8,1)--(8,2)--(7,2)--cy... | tiger_math | [
0.0033319094218313694,
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0.009162751026451588,
0.0344189777970314,
0.03154221549630165,
-0.04394277557730675,
-0.0029... | 0.823589 | 0.819236 | 0.004352 |
Mr. Thompson owns 8 different colored houses arranged in a circle in his village - Red, Green, Blue, Yellow, Orange, Purple, Black, and White. He plans to paint each with a pattern such that no two adjacent houses have the same color, and he wants to know in how many ways he can do this without violating the adjacent c... | tiger_math | [
0.0012114833807572722,
0.0284182820469141,
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0.024963049218058586,
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0.043923430144786835,
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... | 0.822204 | 0.810586 | 0.011618 |
The polynomial $P(x)$ is a monic, quartic polynomial with real coefficients, and two of its roots are $\sin \theta + i \cos \theta$ and $\cos \theta - i \sin \theta,$ where $\frac{\pi}{4} < \theta < \frac{\pi}{2}.$ When the four roots of $P(x)$ are plotted in the complex plane, they form a quadrilateral whose area is ... | tiger_math | [
0.0172034353017807,
0.05300318822264671,
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0.01780354045331478,
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0.06394369900226593,
0.048084624111652374,
0.012643896043300629,
0.009009875357151031,
0.030368590727448463,
-0.020006701350212097,
-0.03878... | 0.827143 | 0.800457 | 0.026686 |
A portion of the graph of a quadratic function $f(x)$ is shown below.
Let $g(x)=-f(x)$ and $h(x)=f(-x)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $a+b$?
[asy]
size(150);
real ti... | tiger_math | [
0.01869438961148262,
0.012856346555054188,
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0.03253759816288948,
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0.08837515115737915,
0.04806206375360489,
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0.013364947400987148,
0.03245928883552551,
-0.022021068260073662,
-0.... | 0.834721 | 0.829484 | 0.005236 |
Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of
\[\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.\] | tiger_math | [
0.02367917262017727,
0.029621366411447525,
-0.04678738862276077,
-0.026368051767349243,
0.04165993630886078,
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0.05300837382674217,
0.052725061774253845,
-0.03399992734193802,
-0.0026451251469552517,
0.02174319326877594,
-0.022235678508877754,
-... | 0.841297 | 0.838357 | 0.00294 |
Find the maximum 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. The answer is 1. What is the value of unknown variable X? | tiger_math | [
0.03976414352655411,
0.025677787140011787,
-0.05800029635429382,
-0.029051227495074272,
0.026275357231497765,
-0.02508579008281231,
-0.006581112742424011,
0.04273423179984093,
0.04379607364535332,
-0.026858273893594742,
0.012569441460072994,
0.03869498893618584,
-0.03146019205451012,
-0.02... | 0.835442 | 0.829737 | 0.005705 |
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