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 |
|---|---|---|---|---|---|
At a fantasy-themed party, three types of magical drink potions—Elixir, Brew, and Charm—are being served in equal ratios. The total number of potions served is 120. Each Elixir increases happiness by 8 points, each Brew by 5 points, and each Charm by 3 points. A mischievous witch casts a spell that instantly doubles th... | tiger_math | [
0.036058999598026276,
0.022770915180444717,
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0.016311854124069214,
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0.04799642786383629,
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0.00892183929681778,
0.026410682126879692,
-0.04784327372908592,
-0.022... | 0.814263 | 0.810846 | 0.003417 |
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 | [
0.028532683849334717,
0.035270918160676956,
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0.04034910351037979,
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-0.0... | 0.829681 | 0.821834 | 0.007847 |
Let $a,$ $b,$ and $c$ be the roots of $x^3 - 7x^2 + 5x + 2 = 0.$ Find
\[\frac{a}{bc + 2} + \frac{b}{ac + 2} + \frac{c}{ab + 2}.\] | tiger_math | [
0.045788709074258804,
0.0378105454146862,
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0.029589056968688965,
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0.009117485024034977,
0.07128487527370453,
0.04031897708773613,
-0.032852903008461,
-0.001969114178791642,
0.014297757297754288,
-0.03357473388314247,
-0.04009... | 0.834038 | 0.828405 | 0.005633 |
Find all values of $k$ so that the domain of
\[b(x) = \frac{kx^2 - 5x + 2}{-2x^2 + 5x + k}\]is the set of all real numbers. | tiger_math | [
0.027669360861182213,
0.03842393681406975,
-0.04906325787305832,
-0.04554789885878563,
0.03780835494399071,
-0.04223087802529335,
0.023531368002295494,
0.05257034674286842,
0.054383374750614166,
-0.019514033570885658,
0.011101619340479374,
0.04517175629734993,
-0.013543713837862015,
0.0002... | 0.84552 | 0.831766 | 0.013754 |
Olga purchases a rectangular mirror (the shaded region) that fits exactly inside a frame. The outer perimeter of the frame measures 60 cm by X cm. The width of each side of the frame is 10 cm. What is the area of the mirror?
[asy]
unitsize(0.15inch);
defaultpen(black);
draw(((0,0)--(8,0)--(8,6)--(0,6)--cycle));
draw((... | tiger_math | [
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0.017629319801926613,
0.02835233137011528,
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-0.00534... | 0.821853 | 0.818533 | 0.00332 |
Let $f(n)$ be the integer closest to $\sqrt[3]{n}.$ Find $\sum_{k=1}^{2023}\frac 1{f(k)}.$ | tiger_math | [
0.01155713852494955,
0.05383152514696121,
-0.04685145244002342,
-0.03936481103301048,
0.024881569668650627,
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0.03971991315484047,
0.02714812010526657,
0.007531630340963602,
0.023153964430093765,
0.02549343928694725,
-0.027324436232447624,
-0.004... | 0.859182 | 0.848757 | 0.010424 |
A regular polygon has an interior angle of $150^\circ$. Find the sum of all its exterior angles. | tiger_math | [
0.001090696663595736,
0.017205480486154556,
-0.027292998507618904,
-0.05562365800142288,
0.020394500344991684,
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0.038992177695035934,
0.029582906514406204,
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0.0146988769993186,
0.03781023249030113,
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-0.... | 0.831681 | 0.827993 | 0.003688 |
A cube with an edge length of X units has the same volume as a square-based pyramid with base edge lengths of 8 units and a height of $h$ units. What is the value of $h$?
If we know the answer to the above question is 3, what is the value of unknown variable X? | tiger_math | [
0.01819741353392601,
0.016473202034831047,
-0.0626433789730072,
-0.015013806521892548,
0.04203634336590767,
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0.054300885647535324,
0.04065686836838722,
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0.020103469491004944,
0.030521413311362267,
-0.03328695893287659,
-0.0... | 0.833478 | 0.826833 | 0.006644 |
If we write $\sqrt{5}+\frac{1}{\sqrt{5}} + \sqrt{7} + \frac{1}{\sqrt{7}}$ in the form $\dfrac{a\sqrt{5} + b\sqrt{7}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$? | tiger_math | [
0.005441218614578247,
0.059678636491298676,
-0.046664007008075714,
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0.04165783151984215,
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0.0657903254032135,
0.05322451889514923,
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0.025362230837345123,
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-0.... | 0.849352 | 0.84015 | 0.009202 |
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of X points attempted on the second day. Beta, who did not attem... | tiger_math | [
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0.07292429357767105,
0.042684413492679596,
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0.012434769421815872,
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... | 0.821451 | 0.812641 | 0.00881 |
In a magical garden, there are mystical flowers and magical bees. Each flower has a spell that doubles the number of visiting bees every hour. If initially there are 3 bees visiting a flower, and you noticed that after some hours there are 384 bees on the single flower, find out how many hours have lapsed? | tiger_math | [
0.01928732357919216,
0.01624065823853016,
-0.03925935551524162,
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0.019299587234854698,
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0.037841152399778366,
0.04647720977663994,
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0.008619804866611958,
0.02213497832417488,
-0.036286965012550354,
-0.0... | 0.819018 | 0.814765 | 0.004253 |
Cutting equilateral triangle $BFC$ out of square $ABCD$ and translating it to the left of the square creates the shaded figure $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. What is the perimeter, in inches, of $ABFCDE$?
[asy]
filldraw((0,0)--(sqrt(3),1)--(sqrt(3)+2,1)--(2,0)--(sqrt(3)+2,-1)--(sqrt(3),-1)--(0... | tiger_math | [
0.03772563859820366,
0.01572885550558567,
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0.02870120480656624,
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0.05485732480883598,
0.051594145596027374,
0.00087424908997491,
0.02250644378364086,
0.04519324004650116,
-0.0373578779399395,
-0.03617390... | 0.824443 | 0.815968 | 0.008475 |
In a magical land, there are three different types of magical orbs: red, blue, and green. Each red orb grants 3 power points, each blue orb grants 5 power points, and each green orb grants 8 power points. A young mage, Zara, collects a total of 100 magical orbs consisting of twice as many blue orbs as green orbs and 10... | tiger_math | [
0.02416578307747841,
0.03022262640297413,
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0.009901042096316814,
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0.04316072165966034,
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0.011070874519646168,
0.0364210344851017,
-0.03515905886888504,
-0.0387... | 0.816469 | 0.812672 | 0.003797 |
Mr. Thompson has bought some candy bars for his three children, Alice, Bob, and Chris, to divide amongst them equally. On his way home, Mr. Thompson couldn’t resist and ate 2 of the candy bars. When he reached home, he divided the remaining candy bars equally among his three children, and each child received 4 candy ba... | tiger_math | [
0.026554886251688004,
0.018598061054944992,
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0.02876930683851242,
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0.00036416860530152917,
0.05978052318096161,
0.021848123520612717,
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0.008129040710628033,
0.02394351176917553,
-0.05267105624079704,
-0.0... | 0.824658 | 0.818103 | 0.006555 |
Suppose that $a$ is a nonzero constant for which the equation $ax^2+bx+7=0$ has only one solution. If $b=20$, find this solution. | tiger_math | [
0.015067039988934994,
0.021890264004468918,
-0.050523269921541214,
-0.04873194172978401,
0.028989898040890694,
-0.015458340756595135,
0.008864487521350384,
0.08717004209756851,
0.051959194242954254,
-0.030309034511446953,
0.006991779897361994,
0.021032089367508888,
-0.034598179161548615,
-... | 0.846974 | 0.836094 | 0.01088 |
In triangle $ABC,$ $D$ lies on $\overline{BC}$ extended past $C$ such that $BD:DC = 3:1,$ and $E$ lies on $\overline{AC}$ such that $AE:EC = 5:3.$ Let $P$ be the intersection of lines $BE$ and $AD.$
[asy]
unitsize(0.8 cm);
pair A, B, C, D, E, F, P;
A = (1,4);
B = (0,0);
C = (6,0);
D = interp(B,C,3/2);
E = interp(A,... | tiger_math | [
0.036608241498470306,
0.009716516360640526,
-0.04877137392759323,
-0.044158127158880234,
0.02997721917927265,
-0.01825401559472084,
-0.018169604241847992,
0.05672627314925194,
0.04621155560016632,
0.002835019724443555,
0.04688243567943573,
0.029906565323472023,
-0.019913095980882645,
-0.01... | 0.844017 | 0.821675 | 0.022342 |
Allison, Brian and Noah each have a 6-sided cube. All of the faces on Allison's cube have a 5. The faces on Brian's cube are numbered 1, 2, X, 4, 5 and 6. Three of the faces on Noah's cube have a 2 and three of the faces have a 6. All three cubes are rolled. What is the probability that Allison's roll is greater than e... | tiger_math | [
0.02423262782394886,
0.046889785677194595,
-0.06798968464136124,
-0.018618952482938766,
0.03691937401890755,
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0.055073417723178864,
0.034972600638866425,
-0.010909227654337883,
0.013315467163920403,
0.01586722955107689,
-0.024709226563572884,
-0... | 0.833447 | 0.823043 | 0.010404 |
Given that $a$, $b$, and $c$ are non-zero real numbers with the sum of $a + b + c = 0$, determine all possible values of $\frac{a^3 + b^3 + c^3}{abc}$. Enter all the possible values, separated by commas. | tiger_math | [
0.029050510376691818,
0.02417393960058689,
-0.034527432173490524,
-0.024157483130693436,
0.03564973175525665,
-0.022411081939935684,
0.0052831778302788734,
0.05351484566926956,
0.04690421372652054,
-0.034195903688669205,
0.014582650735974312,
0.026572929695248604,
-0.043898921459913254,
-0... | 0.828745 | 0.81841 | 0.010336 |
Points $P$ and $Q$ are midpoints of two sides of the square. The area of the shaded region is 7 units squared. What is the area of the square?
[asy]
filldraw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,1)--(1,2)--(2,2)--(0,1)--cycle,white,linewidth(1));
label("P",(0,1),W);
label("Q",(1,2),... | tiger_math | [
0.02758682891726494,
0.028508659452199936,
-0.052673038095235825,
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0.035809293389320374,
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0.0576116107404232,
0.04503657668828964,
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0.031015241518616676,
0.027775244787335396,
-0.028288908302783966,
-0.... | 0.82869 | 0.820571 | 0.00812 |
When the vectors $\mathbf{a} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 0 \\ 3 \\ X \end{pmatrix}$ are both projected onto the same vector $\mathbf{v},$ the result is $\mathbf{p}$ in both cases. Furthermore, the vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{p}$ are collinear. F... | tiger_math | [
0.018626300618052483,
0.04254573956131935,
-0.06658487021923065,
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0.05715370923280716,
0.0034055982250720263,
0.0035487033892422915,
0.08570171147584915,
0.04239373654127121,
-0.0046927062794566154,
0.000575013633351773,
0.016288140788674355,
-0.03156988322734833,
-0.... | 0.812119 | 0.803329 | 0.008789 |
In the same figure as the original question, let $F$ be the intersection of $\overline{BD}$ and $\overline{CE}$. If $EC = 8 \sqrt{3}$ units, what is the length of $\overline{DF}$ in the form $p\sqrt{q}$, where $q$ has no perfect-square factors? | tiger_math | [
0.011799221858382225,
0.01800455152988434,
-0.050292007625103,
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0.01110252644866705,
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0.04849601164460182,
0.04392768815159798,
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0.014872987754642963,
0.03254743292927742,
-0.034669436514377594,
-0.0419... | 0.842399 | 0.828333 | 0.014067 |
In a magical kingdom, three wizards use their magic staffs to control water, fire, and air. Each staff can generate an elemental blast in a vector direction. Wizard A's water staff releases a force represented by vector $\mathbf{a} = (2, 3, 1)$ units, Wizard B's fire staff by vector $\mathbf{b} = (-1, 2, 4)$ units, and... | tiger_math | [
0.03913767263293266,
0.05881509184837341,
-0.04698289558291435,
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0.03775453567504883,
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0.07472553104162216,
0.03618102893233299,
-0.019391559064388275,
0.020087750628590584,
0.029241353273391724,
-0.02954617142677307,
-0.038... | 0.827304 | 0.816738 | 0.010566 |
A right cylinder with a base radius of X units is inscribed in a sphere of radius 5 units. The total volume, in cubic units, of the space inside the sphere and outside the cylinder is $W\pi$. Find $W$, as a common fraction. The answer is 3. What is the value of unknown variable X? | tiger_math | [
0.008464229293167591,
0.031424783170223236,
-0.059120647609233856,
-0.027569690719246864,
0.03622616454958916,
-0.0212544072419405,
-0.02450720965862274,
0.05577931925654411,
0.04637046530842781,
-0.0034999696072191,
0.01611802540719509,
0.0341746024787426,
-0.004959081765264273,
-0.018941... | 0.832504 | 0.825351 | 0.007153 |
If a point P is randomly chosen from a square region with vertices at (±2, ±2), what is the probability that P is within one unit of the origin? Express your answer as a common fraction in terms of π. | tiger_math | [
0.014758912846446037,
0.01577337644994259,
-0.04414587840437889,
-0.03473182022571564,
0.022061707451939583,
-0.02211236022412777,
-0.02314036525785923,
0.03411402180790901,
0.027031978592276573,
0.010447302833199501,
0.01353489700704813,
0.021433982998132706,
-0.014046004973351955,
-0.015... | 0.828423 | 0.822994 | 0.005429 |
Let $\theta$ be the smallest acute angle for which $\sin \theta,$ $\sin 2 \theta,$ $\sin X \theta$ form an arithmetic progression, in some order. Find $\cos \theta.$
If we know the answer to the above question is \frac{3}{4}, what is the value of unknown variable X? | tiger_math | [
0.016378385946154594,
0.039926927536726,
-0.06135639175772667,
-0.031421300023794174,
0.032476652413606644,
0.0050145164132118225,
-0.0058346823789179325,
0.06553378701210022,
0.040715448558330536,
-0.004244411364197731,
0.024893375113606453,
0.0019235826330259442,
-0.024284537881612778,
-... | 0.851405 | 0.836572 | 0.014833 |
Evaluate the expression:
$$\left( X + \frac{1}{\cos 45^\circ} \right) \left( 1 - \frac{1}{\sin 45^\circ} \right) \left( 1 + \frac{1}{\sin 60^\circ} \right) \left( 1 - \frac{1}{\cos 60^\circ} \right)$$ | tiger_math | [
0.0002913645002990961,
0.03174624964594841,
-0.03457728400826454,
-0.04380282759666443,
0.03896122798323631,
-0.01349328551441431,
-0.017434611916542053,
0.07608747482299805,
0.04861986264586449,
-0.010710850358009338,
0.037636686116456985,
0.018829552456736565,
-0.04092210903763771,
-0.02... | 0.840664 | 0.835888 | 0.004777 |
For some integers that are not palindromes, like 91, a person can create a palindrome by repeatedly reversing the number and adding the original number to its reverse. For example, $91 + 19 = 110$. Then $110+011 = 121$, which is a palindrome, so 91 takes two steps to become a palindrome. Of all positive integers betwee... | tiger_math | [
0.023550286889076233,
-0.007476150523871183,
-0.023297850042581558,
-0.047916848212480545,
0.02348250523209572,
0.0003319520619697869,
-0.018694229423999786,
0.05728078633546829,
0.02828141488134861,
-0.011137256398797035,
0.018314290791749954,
0.032650671899318695,
-0.05426830053329468,
-... | 0.834292 | 0.829846 | 0.004446 |
Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 2,$ and $x \ge -\frac{1}{3},$ $y \ge -1,$ and $z \ge -\frac{5}{3}.$ Find the maximum value of
\[\sqrt{3x + 1} + \sqrt{3y + 3} + \sqrt{3z + 5}.\] | tiger_math | [
0.03722141683101654,
0.04781098663806915,
-0.03913058340549469,
-0.045063842087984085,
0.040992338210344315,
0.008197442628443241,
-0.027093203738331795,
0.04577283188700676,
0.028610624372959137,
-0.00428253598511219,
0.021614309400320053,
0.040389664471149445,
-0.03129749000072479,
-0.03... | 0.841658 | 0.822446 | 0.019211 |
Elena is organizing a bake sale where she offers cookies, cupcakes, and pies. Someone can mix and match their order from these items, and the total cost is determined as follows: $6 for a set of 6 cookies, $3 for each cupcake, and $12 for a pie. On a particular day, Elena realizes she sold twice as many cupcakes as pie... | tiger_math | [
0.013184149749577045,
0.012291999533772469,
-0.04852181673049927,
-0.016100704669952393,
0.018196256831288338,
-0.014275399968028069,
-0.038559626787900925,
0.057356543838977814,
0.0302880872040987,
-0.01187090203166008,
0.029985591769218445,
0.03776639327406883,
-0.04667101427912712,
-0.0... | 0.811797 | 0.807812 | 0.003986 |
For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$, and $g(2020) = \text{the digit sum of }12_{\text{8}} = 3$. Let... | tiger_math | [
0.018250582739710808,
0.022904856130480766,
-0.03984901309013367,
-0.021795153617858887,
0.03667587786912918,
-0.02910693734884262,
-0.021128831431269646,
0.05665954574942589,
0.03766883537173271,
0.000013496526662493125,
0.01959981396794319,
0.03596113622188568,
-0.03386076167225838,
-0.0... | 0.851359 | 0.838108 | 0.013251 |
Find the determinant of the matrix
$\begin{vmatrix} \cos^2 A & \tan A & 1 \\ \cos^2 B & \tan B & 1 \\ \cos^2 C & \tan C & 1 \end{vmatrix}$
where $A$, $B$, and $C$ are the angles of a triangle. | tiger_math | [
0.025521939620375633,
0.03735416755080223,
-0.03969245031476021,
-0.04797203838825226,
0.04354320093989372,
-0.023024365305900574,
-0.02537515014410019,
0.052577000111341476,
0.041559092700481415,
-0.00490145618095994,
0.02278788574039936,
0.009112697094678879,
-0.032616570591926575,
-0.02... | 0.821692 | 0.809718 | 0.011974 |
In the magical land of Numerica, a jester has three types of juggling balls: red, blue, and green. He discovers that if he uses 6 red balls, 15 blue balls, or 10 green balls, he performs his act perfectly without dropping any. Unfortunately, one day all three bags of balls get mixed up. If he randomly picks balls from ... | tiger_math | [
-0.0015331351896747947,
0.015095471404492855,
-0.046859778463840485,
-0.03591643646359444,
0.027882715687155724,
0.00016998119826894253,
-0.019162852317094803,
0.046941883862018585,
0.014521834440529346,
-0.02175297401845455,
0.009372848086059093,
0.04493897408246994,
-0.02098918706178665,
... | 0.843897 | 0.832528 | 0.011369 |
Let $x_1,$ $x_2,$ $\dots,$ $x_n$ be nonnegative real numbers such that $x_1 + x_2 + \dots + x_n = 1$ and
\[x_1^2 + x_2^2 + \dots + x_n^2 \le \frac{1}{25}.\]Find the smallest possible value of $n$. | tiger_math | [
0.00687140179798007,
0.02610989660024643,
-0.04168230667710304,
-0.045965537428855896,
0.03744981437921524,
-0.008953312411904335,
-0.020498063415288925,
0.06395840644836426,
0.029399504885077477,
-0.017135759815573692,
0.0022239964455366135,
0.017306938767433167,
-0.041532695293426514,
-0... | 0.855376 | 0.838778 | 0.016598 |
Given a point $P$ on a triangular piece of paper $ABC,\,$ consider the creases that are formed in the paper when $A, B,\,$ and $C\,$ are folded onto $P.\,$ Let us call $P$ a fold point of $\triangle ABC\,$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=30, AC=... | tiger_math | [
0.030267443507909775,
0.03939434140920639,
-0.04442066699266434,
-0.03861116245388985,
0.033729106187820435,
-0.02029474638402462,
-0.00805653166025877,
0.045885954052209854,
0.04232164844870567,
-0.006515943445265293,
0.02771833911538124,
0.04478295147418976,
-0.020604848861694336,
-0.036... | 0.842248 | 0.817235 | 0.025012 |
Let $a$ and $b$ be the roots of the equation $x^2-mx+2=0.$ Suppose that $a-(1/b)$ and $b-(1/a)$ are the roots of the equation $x^2-rx+s=0.$ What is $s$? | tiger_math | [
0.034246984869241714,
0.03264317288994789,
-0.0650351345539093,
-0.03719476982951164,
0.042712558060884476,
-0.03696276992559433,
0.011160684749484062,
0.07162472605705261,
0.04346983879804611,
-0.025233807042241096,
0.00480948481708765,
0.021640421822667122,
-0.03121284209191799,
-0.03143... | 0.837573 | 0.832457 | 0.005116 |
Let $\mathbf{D}$ be the $2 \times 2$ matrix corresponding to the dilation, centered at the origin, with scale factor 7. Find $\det \mathbf{D}.$ | tiger_math | [
0.035887930542230606,
0.019881505519151688,
-0.039628226310014725,
-0.05399983748793602,
0.02879287488758564,
-0.0026579455006867647,
-0.028613699600100517,
0.06656483560800552,
0.03850056976079941,
-0.02385345846414566,
0.00047152308980003,
0.03434857726097107,
-0.04265565052628517,
-0.00... | 0.828113 | 0.823399 | 0.004714 |
Let $x$ and $y$ be complex numbers such that
\[\frac{x + y}{x - y} + \frac{x - y}{x + y} = 2.\]Find
\[\frac{x^4 + y^4}{x^4 - y^4} + \frac{x^4 - y^4}{x^4 + y^4}.\] | tiger_math | [
0.026552030816674232,
0.035235729068517685,
-0.04130113124847412,
-0.0517859123647213,
0.04242466762661934,
-0.010067636147141457,
-0.015345108695328236,
0.055794768035411835,
0.06364180892705917,
-0.012542632408440113,
0.011338360607624054,
0.026889588683843613,
-0.041231147944927216,
-0.... | 0.845025 | 0.833522 | 0.011504 |
Emily loves to collect stamps and decides to arrange her stamp album during a long weekend. On the first day, she realizes she can arrange her stamps on pages such that each page has 12 stamps without any leftover. On the second day, she notices if she uses pages with 10 stamps each, this time she has exactly 4 stamps ... | tiger_math | [
0.020144671201705933,
0.014742547646164894,
-0.01943616010248661,
-0.038871098309755325,
0.03826560452580452,
-0.034975241869688034,
-0.03011314384639263,
0.07155442237854004,
0.024893809109926224,
-0.013241146691143513,
0.00654015876352787,
0.028110163286328316,
-0.030587997287511826,
-0.... | 0.846703 | 0.836052 | 0.010651 |
In a certain alley, Mary notices five birdhouses aligned in a row, each a different color. She observes that the green birdhouse is exactly in the middle, and the blue birdhouse is not next to the green one. The red birdhouse is to the immediate left of the yellow one, and the blue birdhouse is to the immediate right o... | tiger_math | [
0.03188808262348175,
0.036724820733070374,
-0.037803877145051956,
-0.04629184678196907,
0.0011268939124420285,
0.0004857658932451159,
-0.035857636481523514,
0.06026221066713333,
0.05292784795165062,
-0.01913987472653389,
0.010464937426149845,
0.0329589881002903,
-0.02388325333595276,
-0.00... | 0.803959 | 0.794293 | 0.009666 |
Suppose that $n$ is a positive integer such that in base $5$, then $n$ can be expressed as $\overline{ABCD}_5$, and in base $9$, then $n$ can be expressed as $\overline{DCBA}_9$. Find the largest possible value of $n$ in base X$. What is the value of unknown variable X? | tiger_math | [
0.03045082278549671,
0.012825895100831985,
-0.0754447653889656,
-0.02520742267370224,
0.034472543746232986,
-0.014877814799547195,
0.0005256071453914046,
0.04692035913467407,
0.04209588095545769,
-0.011633635498583317,
0.0307275652885437,
0.04695654660463333,
-0.04568641632795334,
-0.00755... | 0.854893 | 0.843295 | 0.011598 |
Convert the point $(-\sqrt{3},\sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | tiger_math | [
0.013237481005489826,
0.03095172718167305,
-0.028134653344750404,
-0.036551445722579956,
0.025809092447161674,
-0.013727870769798756,
-0.020023616030812263,
0.04818198084831238,
0.049884263426065445,
-0.009435993619263172,
0.014847103506326675,
0.03511209785938263,
-0.04538869485259056,
-0... | 0.828698 | 0.821173 | 0.007525 |
Alice is organizing a science competition with 5 different stations. If 24 students attend the competition and are divided evenly among the stations, each student will be asked 14 math questions. However, if 3 more students join and the number is distributed equally among the stations, how many questions does each stud... | tiger_math | [
0.026076722890138626,
-0.006760294083505869,
-0.05697309970855713,
-0.05704972520470619,
0.035207685083150864,
-0.03265408053994179,
-0.03642906993627548,
0.03712771460413933,
0.02965504489839077,
-0.00555457966402173,
-0.006277834996581078,
0.0224973876029253,
-0.022978665307164192,
-0.02... | 0.821714 | 0.81681 | 0.004904 |
Let $\{b_k\}$ be a sequence such that $b_1=2$ and $b_{m+n}=b_m+b_n+2mn,$ for all positive integers $m$ and $n.$ Calculate $b_{10}.$ | tiger_math | [
0.024945532903075218,
0.03501914069056511,
-0.034269656985998154,
-0.03161679208278656,
0.023827824741601944,
-0.03017641417682171,
-0.033290453255176544,
0.05934130772948265,
0.026543907821178436,
-0.017029348760843277,
0.010447773151099682,
0.04306887090206146,
-0.05716323107481003,
-0.0... | 0.847533 | 0.833589 | 0.013944 |
A soccer field is being designed in a rectangular shape where both diagonals are equal in length. The designer plans to place a water cooler at the exact center of the field so that it is an equal distance from each corner of the rectangle. If each diagonal measures 100 meters, how far will the water cooler be from any... | tiger_math | [
0.010539907030761242,
0.006355131510645151,
-0.0676744282245636,
-0.048921871930360794,
0.012431529350578785,
-0.016306133940815926,
-0.02778307907283306,
0.0544838048517704,
0.04138994216918945,
-0.03181963413953781,
0.01731199398636818,
0.0433928519487381,
-0.041251178830862045,
-0.01590... | 0.823923 | 0.818 | 0.005923 |
Find the quadratic polynomial, with real coefficients, which has $3 + i$ as a root, and where the coefficient of $x^2$ is 2. | tiger_math | [
0.027155034244060516,
0.029111335054039955,
-0.058424338698387146,
-0.04597194492816925,
0.014867941848933697,
-0.028719160705804825,
-0.005305408034473658,
0.06629455089569092,
0.02517121285200119,
-0.020085250958800316,
0.030735794454813004,
0.030711328610777855,
-0.03659611940383911,
-0... | 0.839141 | 0.828709 | 0.010432 |
Kenny plans to mow lawns all summer, and then use the profits to buy video-games and books. He initially charges $15 per lawn, but as the summer progresses and his skills improve, he increases his rate by $2 per lawn for every additional 5 lawns he mows. The video-games he wants have different prices: $40, $45, $50, $5... | tiger_math | [
0.0072039649821817875,
0.01965039223432541,
-0.04723449423909187,
-0.02374628558754921,
0.026717403903603554,
-0.00316047971136868,
-0.03571192920207977,
0.05672140046954155,
0.030810469761490822,
-0.013881790451705456,
0.007582951802760363,
0.01656978577375412,
-0.07327339798212051,
-0.01... | 0.802707 | 0.799531 | 0.003176 |
A group of 5 friends visited a carnival that has 8 different games. Each friend plays exactly 3 games, and each game is played by at least one of the friends. If each game prohibits more than one player at a time, how many possible combinations of games can the friends choose, assuming no two friends can play the exact... | tiger_math | [
0.025525959208607674,
0.017504340037703514,
-0.05052667856216431,
-0.0329749658703804,
0.029866350814700127,
-0.024817919358611107,
-0.02739226631820202,
0.042867910116910934,
0.031024277210235596,
-0.02561790868639946,
0.014381667599081993,
0.035287465900182724,
-0.03457232564687729,
-0.0... | 0.834069 | 0.830598 | 0.003471 |
Let $a,$ $b,$ $c$ be distinct, nonzero real numbers such that
\[a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a} = 2023.\]Find $|abc|.$ | tiger_math | [
0.029638085514307022,
0.049880847334861755,
-0.04163229092955589,
-0.03802410140633583,
0.022076131775975227,
-0.011379715986549854,
-0.017401866614818573,
0.059186872094869614,
0.027545148506760597,
-0.017619436606764793,
0.002641210565343499,
0.01745270937681198,
-0.0264703668653965,
-0.... | 0.852267 | 0.836082 | 0.016185 |
A troop of 15 scouts is planning to distribute exactly three types of native tree saplings—oak, maple, and pine—among themselves for a reforestation project. They aim to distribute these saplings in such a way that each scout gets an equal total number of saplings, but with the condition that no scout can have more tha... | tiger_math | [
0.02544541284441948,
0.01212798710912466,
-0.050520606338977814,
-0.04633106291294098,
0.026624275371432304,
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-0.007648012600839138,
0.07353239506483078,
0.02069975808262825,
-0.00869428925216198,
0.02983710914850235,
0.02068001590669155,
-0.042045336216688156,
-0.0272... | 0.814937 | 0.807938 | 0.006999 |
In a mythical forest, three types of magical trees grow: Ember Trees, Breeze Trees, and Crystal Trees. They grow in a perfect triangular pattern forming isosceles triangles in which Ember Trees form the two equal sides, and a Breeze Tree forms the base. The length from an Ember Tree to the Breeze Tree is always 10 mete... | tiger_math | [
0.031324602663517,
0.025358201935887337,
-0.03868812695145607,
-0.06271374970674515,
0.012864838354289532,
-0.047642625868320465,
-0.03409761190414429,
0.051912397146224976,
0.028181329369544983,
-0.018704179674386978,
0.022441348060965538,
0.030246024951338768,
-0.04301466420292854,
-0.00... | 0.817013 | 0.801687 | 0.015326 |
The medians of a triangle with area $T$ are joined to form a triangle with area $N$. What is the ratio of $N$ to $T$? Express your answer as a common fraction. | tiger_math | [
0.02681880258023739,
0.020772026851773262,
-0.05051594600081444,
-0.04154542088508606,
0.029028939083218575,
-0.00891812238842249,
-0.017070388421416283,
0.05204595997929573,
0.03473040461540222,
-0.0050299037247896194,
0.03727918118238449,
0.029354659840464592,
-0.03449976444244385,
-0.00... | 0.829558 | 0.82599 | 0.003568 |
Let the operation $\star$ be defined as $\star(x, y, z) = y^3 + 8xz$, for all real numbers $x, y$ and $z$. Find the value of $\star(2, -1, 1)$. | tiger_math | [
0.034978948533535004,
0.028473718091845512,
-0.04256144538521767,
-0.03566198796033859,
0.030774088576436043,
-0.002551246667280793,
-0.015885939821600914,
0.07437410205602646,
0.04844866320490837,
-0.017656126990914345,
0.0107717951759696,
0.037340521812438965,
-0.05071297660470009,
-0.04... | 0.846029 | 0.840613 | 0.005416 |
Alicia is hosting a dinner party and needs to prepare tables. She has decided to organize the seating such that each table has an equal number of guests and all guests are at a table with either one more or one less than triple the number of tables. The total number of guests she can invite given these restrictions is ... | tiger_math | [
0.052695006132125854,
0.026616450399160385,
-0.05009802430868149,
-0.02500005252659321,
0.04184852913022041,
-0.03147371485829353,
-0.013476432301104069,
0.06704045087099075,
0.048258449882268906,
-0.00786919891834259,
0.011020779609680176,
0.04987214505672455,
-0.015295451506972313,
-0.02... | 0.843801 | 0.832742 | 0.011059 |
In the town of Mathville, the math club hosts an annual competition where participants travel along roads connecting 5 different checkpoints labeled A, B, C, D, and E. The town council reports that the total length of the roads AB, AC, and AD are 7 kilometers, 4 kilometers, and 5 kilometers respectively. Given that AC ... | tiger_math | [
0.02653331868350506,
0.001265264581888914,
-0.030097680166363716,
-0.05063839256763458,
0.03178615868091583,
-0.03424603119492531,
-0.0417175218462944,
0.027411073446273804,
0.0560808926820755,
-0.02761990949511528,
-0.019889097660779953,
0.03615861013531685,
-0.015047864057123661,
-0.0363... | 0.825041 | 0.814166 | 0.010875 |
Let $x,$ $y,$ $z$ be distinct nonzero real numbers such that $x + y + z = 0,$ and $xy + xz + yz \neq 0.$ Find all possible values of
\[\frac{x^4 + y^4 + z^4}{x^2y^2 + x^2z^2 + y^2z^2}.\]Enter all possible values, separated by commas. | tiger_math | [
0.02594655007123947,
0.039771951735019684,
-0.024118250235915184,
-0.041327767074108124,
0.03968305513262749,
-0.009473429061472416,
-0.0030911483336240053,
0.04349450394511223,
0.055378299206495285,
-0.017874591052532196,
0.018959974870085716,
0.021590426564216614,
-0.02771596983075142,
-... | 0.841588 | 0.824688 | 0.0169 |
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. If the vector $\mathbf{d} = \begin{pmatrix} -4 \\ 7 \\ 3 \end{pmatrix}$ can be written as a linear combi... | tiger_math | [
0.043896857649087906,
0.040531616657972336,
-0.05900227278470993,
-0.041015155613422394,
0.04750746488571167,
-0.004715010058134794,
0.009586971253156662,
0.07647274434566498,
0.037647198885679245,
-0.01705170050263405,
-0.011614780873060226,
0.030056947842240334,
-0.03858145326375961,
-0.... | 0.831052 | 0.816659 | 0.014393 |
Consider the region $A^{}_{}$ in the complex plane that consists of all points $z^{}_{}$ such that both $\frac{z^{}_{}}{30}$ and $\frac{30^{}_{}}{\overline{z}}$ have real and imaginary parts between $0^{}_{}$ and $1^{}_{}$, inclusive. Find the area of $A.$ | tiger_math | [
0.016904009506106377,
0.03675607591867447,
-0.05059114471077919,
-0.04105183109641075,
0.022654609754681587,
-0.009453820995986462,
0.00850878469645977,
0.054126717150211334,
0.02754657156765461,
-0.018869612365961075,
0.020759958773851395,
0.044176407158374786,
-0.03398948162794113,
-0.01... | 0.836394 | 0.813664 | 0.02273 |
In a local comic book store, the manager decides to organize a special sale where customers can guess a number puzzle to win a rare comic. The customers are told to summatively multiply their guess by 5, then add 24 to the result, and finally, the ultimate result should be divisible by 7. If Paul guesses the number 36,... | tiger_math | [
0.014713739976286888,
0.022426048293709755,
-0.043495967984199524,
-0.027952732518315315,
0.04388285055756569,
0.003685293486341834,
-0.03800442814826965,
0.039961982518434525,
0.04983198642730713,
-0.006145161110907793,
-0.004704797640442848,
0.033742066472768784,
-0.04966040700674057,
-0... | 0.824889 | 0.818498 | 0.00639 |
An isosceles right triangle is removed from each corner of a square piece of paper, as shown, to create a rectangle. If the length of each leg of the removed triangles is 6 units, what is the combined area of the four removed triangles, in square units? [asy]
unitsize(5mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
p... | tiger_math | [
0.02857070229947567,
0.04133934900164604,
-0.04853548854589462,
-0.055576615035533905,
0.041200824081897736,
-0.03571927174925804,
-0.020356232300400734,
0.05154765024781227,
0.037144679576158524,
0.003061952767893672,
0.031221220269799232,
0.03310102969408035,
-0.012843099422752857,
-0.03... | 0.832604 | 0.827591 | 0.005013 |
[asy] draw((0,0)--(0,3)--(4,0)--cycle,dot); draw((4,0)--(7,0)--(7,10)--cycle,dot); draw((0,3)--(7,10),dot); MP("C",(0,0),SW);MP("A",(0,3),NW);MP("B",(4,0),S);MP("E",(7,0),SE);MP("D",(7,10),NE); [/asy]
Triangle $ABC$ has a right angle at $C, AC=3$ and $BC=4$. Triangle $ABD$ has a right angle at $A$ and $AD=12$. Points $... | tiger_math | [
0.03509874641895294,
0.02137690596282482,
-0.033126767724752426,
-0.041453927755355835,
0.03718944266438484,
-0.0262470543384552,
-0.02293967455625534,
0.04613577201962471,
0.033715296536684036,
0.01194421760737896,
0.024830570444464684,
0.02522633969783783,
-0.004304532893002033,
-0.01844... | 0.852563 | 0.83025 | 0.022313 |
Let $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}.$ Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$ | tiger_math | [
0.0561983548104763,
0.05002228543162346,
-0.04754649102687836,
-0.042067769914865494,
0.04508465155959129,
0.003487068461254239,
-0.004724991042166948,
0.08313541114330292,
0.02250034734606743,
-0.010673679411411285,
0.006642178166657686,
0.028319789096713066,
-0.031090684235095978,
-0.026... | 0.822037 | 0.808284 | 0.013753 |
Given any two positive real numbers $a$ and $b$, then $a \, \star \, b$ is a positive real number defined in terms of $a$ and $b$ by some fixed rule. Suppose the operation $a \, \star \, b$ satisfies the equations $(ab) \, \star \, b=a(b \, \star \, b)$ and $(a \, \star \, 2) \, \star \, a = a \, \star \, 2$ for all $... | tiger_math | [
0.046014152467250824,
0.025328313931822777,
-0.04222444072365761,
-0.028414957225322723,
0.029292846098542213,
0.00019047757086809725,
-0.02648324705660343,
0.07650574296712875,
0.049379464238882065,
-0.012488744221627712,
-0.0016057378379628062,
0.03745133802294731,
-0.04331282526254654,
... | 0.843141 | 0.822216 | 0.020925 |
$ABCD$, a rectangle with $AB = 12$ and $BC = 16$, is the base of pyramid $P$, which has a height of $24$. A plane parallel to $ABCD$ is passed through $P$, dividing $P$ into a frustum $F$ and a smaller pyramid $P'$. Let $X$ denote the center of the circumsphere of $F$, and let $T$ denote the apex of $P$. If the volume ... | tiger_math | [
0.027458084747195244,
0.0347343385219574,
-0.03644086793065071,
-0.03817511722445488,
0.023286323994398117,
-0.031937651336193085,
-0.007134862709790468,
0.04965807870030403,
0.04173379763960838,
0.004511752165853977,
0.020924003794789314,
0.04737008735537529,
-0.025083178654313087,
-0.002... | 0.84349 | 0.81486 | 0.02863 |
In the figure, the visible gray area within the larger circle is equal to three times the area of the white circular region. What is the ratio of the radius of the small circle to the radius of the large circle? Express your answer as a common fraction.
[asy]size(101);
filldraw(Circle((0,0),2)^^Circle((.8,-.3),1),gray... | tiger_math | [
0.00621351832523942,
0.029709218069911003,
-0.036934249103069305,
-0.056806016713380814,
0.030671222135424614,
-0.01881735771894455,
-0.01669224724173546,
0.06559482961893082,
0.036796171218156815,
-0.0037051872350275517,
0.02365894801914692,
0.030450941994786263,
-0.023193655535578728,
-0... | 0.832637 | 0.829428 | 0.003209 |
The greatest integer function of $x$ is expressed as $[x]$ and is defined to be the greatest integer less than or equal to $x$. Find $[\pi - 4]$ | tiger_math | [
0.03107437863945961,
0.03500300273299217,
-0.028880342841148376,
-0.039964500814676285,
0.04363969713449478,
-0.034666527062654495,
-0.028295671567320824,
0.03132424131035805,
0.04649914801120758,
0.008900758810341358,
0.04570489376783371,
0.03324175253510475,
-0.02334236353635788,
-0.0065... | 0.840702 | 0.837442 | 0.00326 |
Compute $A(1, A(2, 0))$ using the recursive definition of the function $A(m, n)$. | tiger_math | [
0.03254697844386101,
0.03960151597857475,
-0.014668595977127552,
-0.026098055765032768,
0.030660925433039665,
-0.023285577073693275,
-0.026218410581350327,
0.06143864244222641,
0.029395170509815216,
-0.017644284293055534,
0.019046226516366005,
0.022191377356648445,
-0.031357862055301666,
-... | 0.837637 | 0.829283 | 0.008354 |
If
\[\frac{\sin x}{\cos y} + \frac{\sin y}{\cos x} = 2 \quad \text{and} \quad \frac{\cos x}{\sin y} + \frac{\cos y}{\sin x} = 8,\]then find $\frac{\tan x}{\tan y} + \frac{\tan y}{\tan x}.$ | tiger_math | [
0.028236379846930504,
0.040069352835416794,
-0.03893290460109711,
-0.030925458297133446,
0.03638552874326706,
0.00789041817188263,
-0.017290839925408363,
0.06393551081418991,
0.04891524836421013,
-0.004396098665893078,
0.0174282044172287,
0.005308005027472973,
-0.030981656163930893,
-0.010... | 0.819622 | 0.806503 | 0.013119 |
The parabolas defined by $y=x^2+2x+2$ and $y=-2x^2+4x+2$ intersect at points $(p,q)$ and $(r,s)$, with $p\le r$. Find the value of $r-p$ as a simplified fraction. | tiger_math | [
0.03671878203749657,
0.024944307282567024,
-0.03392500802874565,
-0.034963157027959824,
0.04434003308415413,
-0.017796305939555168,
-0.01055917888879776,
0.08731507509946823,
0.041912756860256195,
-0.012261344119906425,
0.01756867952644825,
0.019183604046702385,
-0.01970888487994671,
-0.01... | 0.835057 | 0.828335 | 0.006722 |
Let $p(x)$ be a polynomial of degree 6 such that
\[p(2^n) = \frac{1}{2^n}\]for $n = 0,$ 1, 2, $\dots,$ X. Find $p(0). The answer is 64. What is the value of unknown variable X? | tiger_math | [
0.009750587865710258,
0.032613255083560944,
-0.04846729710698128,
-0.012872974388301373,
0.030113890767097473,
-0.01890687271952629,
-0.0076732030138373375,
0.05147545784711838,
0.037320107221603394,
-0.019228095188736916,
0.012047016061842442,
0.0330248698592186,
-0.021022280678153038,
-0... | 0.84068 | 0.837448 | 0.003232 |
Find the sum of the infinite geometric series $\sum_{k = 0}^\infty \frac{(k+1)^2}{2^{k+1}}.$ | tiger_math | [
-0.0017304051434621215,
0.050729312002658844,
-0.025055941194295883,
-0.056163862347602844,
0.028511017560958862,
-0.01793418638408184,
-0.005433593410998583,
0.06111499294638634,
0.04906528815627098,
-0.03510791435837746,
0.01611952856183052,
0.0009530876995995641,
-0.018565349280834198,
... | 0.843017 | 0.833404 | 0.009614 |
Alex runs a boutique that specializes in making luxury dresses. He currently works with three types of fabric: silk, satin, and chiffon. To make a dress, he needs 5 meters of silk, 3 meters of satin, and 2 meters of chiffon. In his storage, he has 600 meters of silk, 400 meters of satin, and 350 meters of chiffon. Alex... | tiger_math | [
0.02373572438955307,
0.0104371327906847,
-0.046848930418491364,
-0.02813858352601528,
0.014992727898061275,
-0.012111160904169083,
-0.05663088336586952,
0.06277972459793091,
0.030586285516619682,
-0.009060034528374672,
-0.011606527492403984,
0.052693311125040054,
-0.03447593376040459,
-0.0... | 0.813291 | 0.805004 | 0.008287 |
Farmer Ellie has a peculiarly shaped field which consists of a perfect square and four identical right-angled triangle pieces at each corner of the square. Each side of the square is 100 meters long, and each triangle's legs are equal to half the length of the square's sides. Farmer Ellie wants to calculate the total a... | tiger_math | [
0.008510936051607132,
0.025090230628848076,
-0.044151317328214645,
-0.051852304488420486,
0.025636952370405197,
-0.0387498214840889,
-0.029798397794365883,
0.04237816482782364,
0.042875923216342926,
-0.0025427856016904116,
-0.0018619404872879386,
0.03283478319644928,
-0.025732973590493202,
... | 0.81614 | 0.805071 | 0.01107 |
The circles whose equations are $x^2 + y^2 - 4x + 2y - 11 = 0$ and $x^2 + y^2 - 14x + 12y + 60 = 0$ intersect in the points $A$ and $B.$ Find the equation of the line containing $\overline{AB}.$ | tiger_math | [
0.017826266586780548,
0.004236239939928055,
-0.036698371171951294,
-0.05001886561512947,
0.05369920656085014,
-0.034939851611852646,
-0.013331263326108456,
0.059280913323163986,
0.04357166588306427,
-0.01677480712532997,
0.018344629555940628,
0.028922326862812042,
-0.016519801691174507,
-0... | 0.833482 | 0.820643 | 0.01284 |
Compute the ordered pair of positive integers $(x,y)$ such that
\begin{align*}
x^y+2&=y^x,\\
3x^y&=y^x+14.
\end{align*} | tiger_math | [
0.044660814106464386,
0.0434105284512043,
-0.03328010067343712,
-0.03592722490429878,
0.0348689928650856,
-0.021221555769443512,
-0.02286113053560257,
0.05363595485687256,
0.048745132982730865,
-0.020431341603398323,
0.03383179381489754,
0.018824761733412743,
-0.034270357340574265,
-0.0277... | 0.8442 | 0.837106 | 0.007094 |
Let $0 \le a,$ $b,$ $c \le 2.$ Find the maximum value of
\[\sqrt{abc} + \sqrt{(2 - a)(2 - b)(2 - c)}.\] | tiger_math | [
0.037888962775468826,
0.04337558150291443,
-0.04337018355727196,
-0.027897467836737633,
0.03018440119922161,
-0.006491918582469225,
-0.03010573238134384,
0.05439275503158569,
0.026814796030521393,
-0.025842955335974693,
0.01608855463564396,
0.04793214052915573,
-0.03769751265645027,
-0.030... | 0.848907 | 0.83433 | 0.014577 |
Find the value of $Y$ in the matrix $\mathbf{B}$ given that:
\[\mathbf{B} = \begin{pmatrix} 0 & 0 & Y \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}\]
and
\[\mathbf{B}^{99} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}\] | tiger_math | [
0.0220346637070179,
0.02006976306438446,
-0.059537746012210846,
-0.04105742648243904,
0.04190845414996147,
-0.030719583854079247,
-0.00706403749063611,
0.06817785650491714,
0.05981145799160004,
-0.020884038880467415,
0.01215003989636898,
0.0311734601855278,
-0.04788407310843468,
-0.0010001... | 0.826341 | 0.82268 | 0.003661 |
Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1.$ If $g(64) = 16,$ then compute $[g(4)]^3.$ | tiger_math | [
0.035753749310970306,
0.03736002743244171,
-0.045784518122673035,
-0.02200492098927498,
0.05240287259221077,
-0.026613734662532806,
-0.016671789810061455,
0.060845132917165756,
0.032316673547029495,
-0.010677049867808819,
0.029454786330461502,
0.02977130375802517,
-0.03433925285935402,
-0.... | 0.842781 | 0.833141 | 0.00964 |
A rectangular garden has side lengths of $(2x+5)$ and $(x+3)$. There is a rectangular path inside the garden with side lengths of $(x-1)$ and $(x-2)$. What is the area of the garden (excluding the area of the path)? Express your answer as a polynomial in $x$. | tiger_math | [
0.015322273597121239,
0.028494957834482193,
-0.05689596012234688,
-0.03696753457188606,
0.012859372422099113,
-0.03682126849889755,
-0.019675686955451965,
0.06469622999429703,
0.04511703550815582,
-0.0028682139236479998,
0.022078169509768486,
0.046622250229120255,
-0.03793299198150635,
-0.... | 0.835978 | 0.827618 | 0.00836 |
In a small village, a baker is preparing for the annual fair. He decides to bake a number of cupcake packs, each containing either 5 or 7 cupcakes. He aims to prepare exactly 100 cupcakes for the day. If the baker wants as few packs as possible, how should he organize the pack sizes? | tiger_math | [
0.01732574962079525,
0.02515340968966484,
-0.03259972482919693,
-0.03279544785618782,
0.030587518587708473,
-0.01548044104129076,
-0.03202196955680847,
0.046852804720401764,
0.03204794600605965,
-0.014641514047980309,
-0.00722266873344779,
0.035112399607896805,
-0.027591152116656303,
-0.00... | 0.82883 | 0.817451 | 0.011379 |
Let $\mathrm {P}$ be the product of the roots of $z^8-z^4+z^2-1=0$ that have a positive imaginary part, and suppose that $\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})$, where $0<r$ and $0\leq \theta <360$. Find $\theta$. | tiger_math | [
0.014313875697553158,
0.024996347725391388,
-0.02892555296421051,
-0.041535139083862305,
0.02576838620007038,
-0.01673256605863571,
-0.004476436413824558,
0.06217590346932411,
0.03439111262559891,
0.003935556858778,
0.000041289018554380164,
0.018927065655589104,
-0.04125986620783806,
-0.03... | 0.841546 | 0.824761 | 0.016784 |
Determine the value of $a+b+c$ in the expression $a+b\sqrt{2}+c\sqrt{5}$, where the perimeter of a hexagon is obtained by joining the points $(0,1)$, $(1,2)$, $(2,2)$, $(2,1)$, $(3,1)$, $(2,0)$, and $(0,1)$. All variables are whole numbers. | tiger_math | [
0.023688942193984985,
0.05103347450494766,
-0.05220622196793556,
-0.03448612615466118,
0.04469897970557213,
-0.025608250871300697,
-0.02631951868534088,
0.07601814717054367,
0.03652220591902733,
-0.01251300610601902,
0.012840203940868378,
0.045219529420137405,
-0.023383455350995064,
-0.043... | 0.833627 | 0.8187 | 0.014927 |
Let $A=(0,0),$ $B=(3,4),$ $C=(6,1),$ and $D=(8,-1).$ A figure is created by connecting $A$ to $B,$ $B$ to $C,$ $C$ to $D,$ and $D$ to $A.$ The perimeter of $ABCD$ can be expressed in the form $a\sqrt2+b\sqrt{10}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$? | tiger_math | [
0.01121436432003975,
0.03637513518333435,
-0.04432280361652374,
-0.028364861384034157,
0.03265191614627838,
-0.009691761806607246,
-0.02487870864570141,
0.05027172714471817,
0.05317246541380882,
-0.017095036804676056,
0.005334728863090277,
0.05187481641769409,
-0.026761887595057487,
-0.033... | 0.858262 | 0.836771 | 0.021491 |
The Enchanted Forest holds a treasure hunt every summer where treasure chests are hidden, each containing different numbers of enchanted gems. This year, three treasure chests were hidden. The first chest contains 50 gems, the second chest contains 70 gems, and the third contains 90 gems. To access these chests, a part... | tiger_math | [
0.014011738821864128,
0.02927485667169094,
-0.03552735224366188,
-0.05289919301867485,
0.019672367721796036,
-0.018360182642936707,
-0.029259560629725456,
0.03798295930027962,
0.006299341097474098,
-0.05061564967036247,
0.02196027897298336,
0.021364979445934296,
-0.04007072001695633,
-0.01... | 0.811814 | 0.806511 | 0.005303 |
Suppose $z$ and $w$ are complex numbers such that
\[|z| = |w| = 1 \text{ and } z \overline{w} + \overline{z} w= 2.\]Find the largest possible value of the real part of $z + w.$ | tiger_math | [
0.03722240775823593,
0.03848831728100777,
-0.03661782294511795,
-0.04642213508486748,
0.035305801779031754,
0.009959450922906399,
-0.0005800092476420105,
0.04447852820158005,
0.031740374863147736,
-0.008905479684472084,
0.0338604561984539,
0.038199543952941895,
-0.041933100670576096,
-0.01... | 0.838035 | 0.823104 | 0.014931 |
Determine the area, in square units, of the triangle with vertices at $A(1, 1)$, $B(6, 1)$, and $C(3, 7)$. | tiger_math | [
0.037024714052677155,
0.0493454709649086,
-0.054420195519924164,
-0.03995183855295181,
0.02829088643193245,
-0.018910475075244904,
-0.02506783977150917,
0.03777404874563217,
0.03039126843214035,
-0.02716156467795372,
0.027163758873939514,
0.0307006798684597,
-0.03127752244472504,
-0.026747... | 0.82718 | 0.82255 | 0.00463 |
Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 6.$ Find the maximum value of
\[\sqrt{2x + 3} + \sqrt{2y + 3} + \sqrt{2z + 3}.\] | tiger_math | [
0.0265138391405344,
0.04001648724079132,
-0.039723191410303116,
-0.029529890045523643,
0.042773690074682236,
0.001376363798044622,
-0.024853641167283058,
0.03694620355963707,
0.022157050669193268,
-0.02008737623691559,
0.025388281792402267,
0.04328695312142372,
-0.03694276511669159,
-0.040... | 0.844048 | 0.830639 | 0.013409 |
Convert the point $\left( 5, \frac{3 \pi}{2}, 4 \right)$ in cylindrical coordinates to rectangular coordinates. | tiger_math | [
-0.0012120162136852741,
0.051543161273002625,
-0.027795512229204178,
-0.040694158524274826,
0.02894807793200016,
-0.014541776850819588,
-0.03147616982460022,
0.06300000846385956,
0.052559994161129,
-0.002077536890283227,
0.0012463752645999193,
0.03636879473924637,
-0.021784653887152672,
-0... | 0.841056 | 0.836629 | 0.004427 |
If $\tan \alpha = \frac{1}{2}$ and $\tan \beta = \frac{1}{3},$ find $\tan (2\alpha + \beta).$ | tiger_math | [
0.03380086272954941,
0.04433874040842056,
-0.042173031717538834,
-0.02345145307481289,
0.03452618420124054,
-0.012221510522067547,
-0.0175241120159626,
0.05064253881573677,
0.033827777951955795,
-0.019339971244335175,
0.01691609062254429,
0.016941508278250694,
-0.04207152873277664,
-0.0195... | 0.826547 | 0.816922 | 0.009624 |
A given sequence $r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the thir... | tiger_math | [
0.007319006137549877,
0.027452245354652405,
-0.026271210983395576,
-0.03176060691475868,
0.009985196404159069,
-0.010239249095320702,
-0.027236316353082657,
0.07586725801229477,
0.006478733848780394,
-0.0061506181955337524,
0.013047980144619942,
0.02134762704372406,
-0.051842667162418365,
... | 0.834721 | 0.815854 | 0.018868 |
In $\triangle{ABC}$ with $AB = 10$, $BC = 17$, and $AC = 21$, let $M$ be a point on $\overline{AC}$ such that the incircles of $\triangle{ABM}$ and $\triangle{BCM}$ have equal radii. Then $\frac{AM}{CM} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | tiger_math | [
0.03470979258418083,
0.03403262421488762,
-0.034098051488399506,
-0.042396944016218185,
0.02812931500375271,
-0.019870679825544357,
-0.0077890329994261265,
0.0498950220644474,
0.042393799871206284,
-0.015362469479441643,
0.018859580159187317,
0.035759337246418,
-0.03674010932445526,
-0.005... | 0.858606 | 0.831433 | 0.027173 |
Karlanna places 600 marbles into $m$ total boxes such that each box contains an equal number of marbles. There is more than one box, and each box contains more than one marble. For how many values of $m$ can this be done? | tiger_math | [
0.008671097457408905,
0.02731919102370739,
-0.036661017686128616,
-0.025098543614149094,
0.047544896602630615,
-0.014766690321266651,
-0.024194439873099327,
0.06474864482879639,
0.024468662217259407,
-0.03024310991168022,
0.020728105679154396,
0.0441225990653038,
-0.023254791274666786,
-0.... | 0.831916 | 0.821902 | 0.010014 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x - y)) = f(x) f(y) - f(x) + f(y) - xy\]for all $x,$ $y.$ If $f(0) = 0$, find the sum of all possible values of $f(2)$. | tiger_math | [
0.02124873735010624,
0.03126777336001396,
-0.055306173861026764,
-0.03776424378156662,
0.035285480320453644,
-0.006318717263638973,
-0.02304750122129917,
0.07074985653162003,
0.041992951184511185,
-0.01677200384438038,
0.0068831187672913074,
0.02940216101706028,
-0.0454714260995388,
-0.014... | 0.847426 | 0.833189 | 0.014237 |
Two distinct positive integers from 1 to 60 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is two less than a multiple of 5? | tiger_math | [
0.014563917182385921,
0.005463701672852039,
-0.04721946641802788,
-0.015934281051158905,
0.04071681946516037,
-0.013465022668242455,
-0.01751822791993618,
0.053786952048540115,
0.044392917305231094,
-0.011646139435470104,
0.012181461788713932,
0.030509959906339645,
-0.015326612628996372,
-... | 0.846253 | 0.840749 | 0.005504 |
For the equation $2x^2 - 5x - 4 = 0$, the roots can be expressed as $x = \frac{m \pm \sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers with a greatest common divisor of 1. What is the value of $n$? | tiger_math | [
0.038673035800457,
0.03081267885863781,
-0.06370684504508972,
-0.05870472639799118,
0.038693301379680634,
-0.041161153465509415,
-0.0021639850456267595,
0.0728524774312973,
0.052089571952819824,
-0.01668531447649002,
0.017163021489977837,
0.025350036099553108,
-0.0259565357118845,
-0.03463... | 0.833647 | 0.830043 | 0.003603 |
You are given that $x$ is directly proportional to $y^2$, and $y$ is inversely proportional to $\sqrt[3]{z}$. If the value of $x$ is 2 when $z$ is $8$, what is the value of $x$ when $z$ is equal to $125$? Express your answer as a common fraction. | tiger_math | [
0.020787831395864487,
0.03589019551873207,
-0.04295560345053673,
-0.05048806220293045,
0.02556898444890976,
-0.009788557887077332,
0.008981888182461262,
0.05899711325764656,
0.03970402851700783,
0.006575011648237705,
0.02056356519460678,
0.04272167384624481,
-0.034357838332653046,
-0.00798... | 0.850545 | 0.842104 | 0.008441 |
A group of 26 friends were discussing a large positive integer. ``It can be divided by 1,'' said the first friend. ``It can be divided by 2,'' said the second friend. ``And by 3,'' said the third friend. ``And by 4,'' added the fourth friend. This continued until everyone had made such a comment. If exactly three frien... | tiger_math | [
0.03331036865711212,
0.025124317035079002,
-0.03650744631886482,
-0.04840220883488655,
0.036217812448740005,
-0.021283486858010292,
-0.026318982243537903,
0.042408738285303116,
0.020630544051527977,
-0.025919033214449883,
0.016280971467494965,
0.025815287604928017,
-0.048224832862615585,
-... | 0.846916 | 0.843499 | 0.003417 |
A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\spadesuit$ and $\clubsuit$, called 'spades' and 'clubs') are black. The cards in the deck are placed in random order (usu... | tiger_math | [
0.015364590100944042,
0.014739248901605606,
-0.05466405674815178,
-0.025360435247421265,
0.0166291706264019,
-0.018333759158849716,
-0.03900960832834244,
0.0303301103413105,
0.026247955858707428,
-0.013592872768640518,
0.014758104458451271,
0.02908506989479065,
-0.04132735729217529,
-0.010... | 0.837237 | 0.82593 | 0.011306 |
Find the number of solutions to the equation
\[\tan (4 \pi \cos \theta) = \cot (4 \pi \sin \theta)\]where $\theta \in (0, 2 \pi).$ | tiger_math | [
0.00952936988323927,
0.03841633349657059,
-0.0421677827835083,
-0.03426143527030945,
0.036332931369543076,
-0.011370654217898846,
-0.029421458020806313,
0.05069594457745552,
0.04626194015145302,
0.002190154045820236,
0.023579295724630356,
0.004245707765221596,
-0.011808786541223526,
-0.023... | 0.833735 | 0.828622 | 0.005113 |
Let $a,$ $b,$ $c$ be real numbers such that $a + b + c = 0.$ Find the set of all possible values of $ab + ac + bc.$ | tiger_math | [
0.03433592617511749,
0.020899461582303047,
-0.04291045665740967,
-0.02716412954032421,
0.03902621567249298,
-0.02225682884454727,
0.0018685319228097796,
0.05585412308573723,
0.053245026618242264,
-0.03690781816840172,
0.011565611697733402,
0.03739762678742409,
-0.04196283593773842,
-0.0468... | 0.844945 | 0.834055 | 0.01089 |
The matrix for projecting onto a certain line $\ell,$ which passes through the origin, is given by
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{2}{15} & -\frac{1}{15} & -\frac{1}{3} \\ -\frac{1}{15} & \frac{1}{30} & \frac{1}{6} \\ -\frac{1}{3} & \frac{1}{6} & \frac{5}{6} \end{pmatrix} \renewcommand{\arrays... | tiger_math | [
0.006385426968336105,
0.035944655537605286,
-0.05137231573462486,
-0.042446937412023544,
0.05838480964303017,
-0.010674160905182362,
-0.02116740494966507,
0.07472719997167587,
0.04124011844396591,
-0.005952510982751846,
0.01624772511422634,
0.032349325716495514,
-0.021558502689003944,
-0.0... | 0.828494 | 0.82483 | 0.003665 |
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