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 |
|---|---|---|---|---|---|
Let $S$ be the set of 10-tuples $(a_0, a_1, \dots, a_9),$ where each entry is 0 or 1, so $S$ contains $2^{10}$ 10-tuples. For each 10-tuple $s = (a_0, a_1, \dots, a_9)$ in $S,$ let $p_s(x)$ be the polynomial of degree at most 9 such that
\[p_s(n) = a_n\]for $0 \le n \le 9.$ For example, $p(x) = p_{(0,1,0,0,1,0,1,0,0,... | tiger_math | [
0.027244947850704193,
0.027232438325881958,
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0.029056528583168983,
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0.06112797185778618,
0.017630163580179214,
-0.00816892459988594,
0.015246502123773098,
0.03993520885705948,
-0.0429118387401104,
-0.01... | 0.842461 | 0.817775 | 0.024686 |
A cube has edges of length 2 cm and has a dot marked in the centre of the top face. The cube is sitting on a flat table. The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices are always touching the table. The cube is rolled until the dot is again on the top face. Th... | tiger_math | [
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0.03490061312913895,
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0.03197159618139267,
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0.04462166503071785,
0.03940761089324951,
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0.013095015659928322,
0.0236884206533432,
0.004041285254061222,
-0.0065... | 0.841064 | 0.825912 | 0.015152 |
Given vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ with magnitudes of 5, 6, and 7 respectively, and the condition $\mathbf{a} - \mathbf{b} + \mathbf{c} = \mathbf{0}$, find the value of $\mathbf{a} \cdot \mathbf{b} - \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}$. | tiger_math | [
0.05064448341727257,
0.049898888915777206,
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0.05078602582216263,
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0.0674259141087532,
0.041630007326602936,
-0.03051653690636158,
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0.020802361890673637,
-0.028822172433137894,
-0.02... | 0.828096 | 0.825029 | 0.003066 |
A jar contains quarters (worth $\$0.25$ each), nickels (worth $\$0.05$ each) and pennies (worth $\$0.01$ each). The total value of all the coins in the jar is $\$30.00$. The number of nickels is twice the number of quarters. The number of pennies is 10 times the number of quarters. If Judith randomly chooses one coin f... | tiger_math | [
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-0.0... | 0.827757 | 0.82229 | 0.005467 |
All the numbers 2, 3, 4, X, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. The greatest possible value of the sum of t... | tiger_math | [
0.035005584359169006,
0.042675457894802094,
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0.05118812620639801,
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-0.01... | 0.828713 | 0.823981 | 0.004732 |
Points $A$, $B$, $C$, and $D$ are located on $\overline{AB}$ such that $AB = 3AD = 9BC$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $B$ and $D$? Express your answer as a common fraction.
[asy]
draw((0,0)--(12,.0000000001));
dot((0,0)); dot((4,0)); dot((11,0)); dot(... | tiger_math | [
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0.02422676794230938,
0.02951919287443161,
-0.004745247308164835,
-0.020... | 0.840572 | 0.829479 | 0.011093 |
A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\{x:a \le x \le b \}$.) Let
\[g(x) = 1 - f(x + 1).\]Then the domain of $g(x)$ is $[a,b],$ and the range of $g(x)$ is $[c,d].$ Enter the ordered quadruple $(a,b,c,d).$ | tiger_math | [
0.04081741347908974,
0.030780455097556114,
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0.04964740574359894,
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0.013087170198559761,
0.02851923182606697,
-0.026780961081385612,
-0.0275... | 0.837389 | 0.818847 | 0.018543 |
In Mrs. Reed's English class, the students are all reading a 760-page novel. Alice, Bob, and Chandra are friends in the class, with Alice taking 20 seconds to read a page, Bob taking 45 seconds, and Chandra taking 30 seconds. Chandra and Bob decide to "team read" the book to save time. In this plan, Chandra will read f... | tiger_math | [
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0.002490737708285451,
0.007030953653156757,
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-0.00... | 0.806055 | 0.802028 | 0.004026 |
Let $\mathbf{M}$ be a matrix, and let $\mathbf{v}$ and $\mathbf{w}$ be vectors, such that
\[\mathbf{M} \mathbf{v} = \begin{pmatrix} 5 \\ Y \end{pmatrix} \quad \text{and} \quad \mathbf{M} \mathbf{w} = \begin{pmatrix} 3 \\ 9 \end{pmatrix}.\]If $\mathbf{M} (\mathbf{v} - 2 \mathbf{w}) = \begin{pmatrix} -1 \\ 10 \end{pmatri... | tiger_math | [
0.03486453369259834,
0.02152523770928383,
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0.05106370896100998,
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0.019430603832006454,
0.029117412865161896,
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-0.0099... | 0.832143 | 0.825639 | 0.006504 |
In spherical coordinates, the point $\left( X, \frac{2 \pi}{7}, \frac{8 \pi}{5} \right)$ is equivalent to what other point, in the standard spherical coordinate representation? Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$
If we know the answer ... | tiger_math | [
-0.011729992926120758,
0.021212194114923477,
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0.0064256926998496056,
0.028011750429868698,
-0.029314721003174782,
... | 0.823291 | 0.815599 | 0.007692 |
The graph of $y = f(x)$ is shown below. Which is the graph of $y = f(|x|)$?
[asy]
unitsize(0.5 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 = -5; i <= 5; ++i) {... | tiger_math | [
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0.020554194226861,
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-0.007236026227474213,
-0.009221... | 0.819745 | 0.816622 | 0.003123 |
On a distant planet, there are two cities, Vertexville and Edgetown, which are planning to build train stations. The distance function between any two points \( (x, y) \) and \( (x', y') \) on the planet is given by \( d = |x - x'| + |y - y'| \). The cities decide to locate their train stations in such a manner that th... | tiger_math | [
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0.012792142108082771,
0.01978202722966671,
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-0... | 0.824411 | 0.814328 | 0.010083 |
Ten identical crates each of dimensions $3\mathrm{ft}\times 4\mathrm{ft}\times 6\mathrm{ft}$. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $\frac {m}{n}$ be the probability... | tiger_math | [
0.019452642649412155,
0.03767433017492294,
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0.03144306316971779,
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0.051695141941308975,
0.016679106280207634,
0.0007290736539289355,
0.012165364809334278,
0.03147788718342781,
-0.05912911146879196,
-0.01... | 0.839618 | 0.818775 | 0.020843 |
You have $5$ red shirts, $5$ green shirts, $6$ pairs of pants, $8$ green hats and $8$ red hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats, but allowing the hat and pants to be the same color? | tiger_math | [
0.012112833559513092,
0.02041979320347309,
-0.04205344617366791,
-0.018929993733763695,
0.009784381836652756,
-0.02211686782538891,
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0.05297410488128662,
0.02902473323047161,
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0.008393383584916592,
0.021563338115811348,
-0.02599966712296009,
-0.02... | 0.846778 | 0.838682 | 0.008097 |
Forty cards are placed into a box, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four cards. Three cards are drawn from the box at random and without replacement. Let $p$ be the probability that all three cards bear the same number. Let $q$ be the probability that two of the cards ... | tiger_math | [
0.011798168532550335,
0.018088461831212044,
-0.06164355203509331,
-0.015563261695206165,
0.0390160009264946,
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0.050488103181123734,
0.03174961358308792,
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0.014559713192284107,
0.022960888221859932,
-0.03944870084524155,
-0.... | 0.84352 | 0.828299 | 0.015221 |
A math club has students attempting different activity booths where they can earn points ranging from 1 to 10 points at each booth. Clara visits enough booths to score exactly 30 points. If there needs to be exactly 5 combinations of scores that could give Clara exactly 30 points, identify the maximum number of booths ... | tiger_math | [
0.0419541634619236,
0.010027431882917881,
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0.03392763435840607,
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0.03756754845380783,
0.027767717838287354,
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0.004331531468778849,
0.04066967964172363,
-0.037154536694288254,
-0.0132... | 0.836973 | 0.822608 | 0.014365 |
In Farmer Lu’s avocado orchard, there is a peculiar arrangement of trees. Farmer Lu records the number of avocados each tree produces, and has noticed a pattern over the years in sections of his orchard. In Section 1, the first tree produced 3 avocados, in Section 2, each of the two trees produced 5 avocados, in Sectio... | tiger_math | [
0.0026900821831077337,
0.016944438219070435,
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0.021563030779361725,
-0.035445794463157654,
... | 0.822489 | 0.815295 | 0.007194 |
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 $x^2-px+q=0.$ What is $q?$ | tiger_math | [
0.027528708800673485,
0.03729971870779991,
-0.05721684545278549,
-0.038814932107925415,
0.0385558195412159,
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0.018042486160993576,
0.06979027390480042,
0.034185685217380524,
-0.030535316094756126,
0.0028742023278027773,
0.013533446937799454,
-0.034756701439619064,
-0.0... | 0.846104 | 0.835095 | 0.011009 |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-2)(b-2)(c-2) \neq 0$$ | tiger_math | [
0.02480447292327881,
0.008681404404342175,
-0.04855732247233391,
-0.027880126610398293,
0.0322834737598896,
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0.05338379368185997,
0.025858134031295776,
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0.014320079237222672,
0.02883870154619217,
-0.02742805704474449,
-0.01... | 0.850833 | 0.845959 | 0.004875 |
Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 5$. Find the set of all possible values of $xy + xz + yz$ if it is known that $x^2 + y^2 + z^2 = 9$. | tiger_math | [
0.012770500965416431,
0.022942114621400833,
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0.040736112743616104,
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0.00008685754437465221,
0.04545391723513603,
0.05335612967610359,
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0.0380857028067112,
0.040572814643383026,
-0.045802321285009384,
-0.0... | 0.841119 | 0.826445 | 0.014674 |
A container holds blue, green, and white marbles. The probability of selecting a blue marble at random is $\frac{1}{4}$, and the probability of selecting a green marble at random is $\frac{1}{3}$. If there are 36 marbles in the container, how many white marbles are there? | tiger_math | [
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0.03251771256327629,
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0.03788904845714569,
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0.05019950494170189,
0.03248225525021553,
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0.011168763972818851,
0.021043945103883743,
-0.008065452799201012,
-0.... | 0.822618 | 0.819792 | 0.002825 |
Given that $\angle1+\angle2=180^\circ$ and $\angle3=\angle4$, find $\angle3$. Express your answer in degrees.
[asy]
/* AMC8 1997 #12 Problem */
pair A=(0,0), B=(24,0), C=(48,0), D=(18,24), E=(12,48);
pen p=1mm+black;
draw(A--C);
draw(A--E);
draw(B--E);
draw(D--C);
label("70", A, NE);
label("40", shift(0,-7)*E, S);
labe... | tiger_math | [
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0.041953396052122116,
0.02672319859266281,
0.004240664187818766,
0.022078847512602806,
0.024209750816226006,
-0.04561583697795868,
-0.017... | 0.830523 | 0.820832 | 0.009691 |
Lauren solved the equation $|2x-10| = 4$. Meanwhile Jane solved an equation of the form $x^2+ bx + c = 0$ that had the same two solutions for $x$ as Lauren's equation. What is the ordered pair $(b, c)$? | tiger_math | [
0.034024953842163086,
0.035449642688035965,
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0.040791936218738556,
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0.01590864546597004,
0.02102355659008026,
-0.028240196406841278,
-0.... | 0.821612 | 0.815027 | 0.006585 |
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 | [
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0.06870350241661072,
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0.033128827810287476,
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-0.00... | 0.823103 | 0.818169 | 0.004934 |
A pirate is searching for buried treasure on 6 islands. On each island, there is a $\frac{1}{4}$ chance that the island has buried treasure and no traps, a $\frac{1}{12}$ chance that the island has traps but no treasure, and a $\frac{2}{3}$ chance that the island has neither traps nor treasure. What is the probability ... | tiger_math | [
0.018087977543473244,
0.03076636977493763,
-0.039124589413404465,
-0.03592879697680473,
0.038306232541799545,
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0.030485009774565697,
0.013201838359236717,
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0.022198494523763657,
0.011408261023461819,
-0.017564019188284874,
-0... | 0.840257 | 0.828809 | 0.011448 |
Two similar right hexagonal pyramids have cross-sectional areas of $108\sqrt{3}$ square units and $243\sqrt{3}$ square units, respectively. If the distance between the planes containing the cross sections in the smaller pyramid is $6$ units, what is the distance between the corresponding planes in the larger pyramid? | tiger_math | [
0.0195233803242445,
0.01430157944560051,
-0.04969458281993866,
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0.036722298711538315,
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0.05190572142601013,
0.037255216389894485,
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0.025290053337812424,
0.03026394173502922,
-0.039223987609148026,
-0.0... | 0.829632 | 0.81906 | 0.010573 |
Jenny and Kenny are walking in the same direction, Kenny at 5 feet per second and Jenny at 3 foot per second, on parallel paths that are 300 feet apart. A tall circular building 150 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and K... | tiger_math | [
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0.006640221457928419,
0.015670442953705788,
-0.03018699772655964,
-0.01... | 0.845813 | 0.842336 | 0.003478 |
In the magical land of Mathemagica, there are three kinds of magical fruits: Glowberries, Shineapples, and Twinklepears. During the annual Mathemagica Fruit Festival, 60 participants reported their favorite fruits. 25 loved Glowberries, 30 loved Shineapples, and 20 didn’t favor any fruit. If everyone had only one favor... | tiger_math | [
0.014332862570881844,
0.015804030001163483,
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0.005389034282416105,
0.03385760635137558,
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-0.0... | 0.824131 | 0.820436 | 0.003695 |
A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.
What is the value of $8a-4b+2c-d$ if the graph passes through the point $(1, 2)$?
[asy]
import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=4.25,ymin=-9.25,ymax=4.25;
pen cqcqcq... | tiger_math | [
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A road has four intersections. At the first intersection, there are 40 road signs showing different types of information mounted on the street poles. At the second intersection, 1/4 times more road signs than the first intersection are mounted on the street poles. The number of road signs at the third intersection is t... | tiger_math | [
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Two concentric circles are centered at point P. The sides of a 30-degree angle at P form an arc on the larger circle that is twice the length of an arc on the smaller circle formed by the sides of a 45-degree angle at P. Find the ratio of the radius of the larger circle to the radius of the smaller circle. | tiger_math | [
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For the two quadratic equations $x^2 + mx + n = 0$ and $x^2 + px + m = 0$, it is known that the sum of the roots of the first equation is twice the sum of the roots of the second equation. If $m$, $n$, and $p$ are all non-zero, find the value of $m/n$. | tiger_math | [
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Samantha is planning an arts and crafts fair. She decides to arrange booths in a circular pattern, equidistant from each other. If she arranges 9 unique booths, each item is to be featured at the booth closest to the main entrance. The angle between each booth from the center of the circle is uniform. How many degrees ... | tiger_math | [
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In a fantasy kingdom, there are three types of coins: gold, silver, and copper. 1 gold is worth 5 silver coins, and 1 silver is worth 10 copper coins. One day, a merchant came into the kingdom with a bag containing 120 copper coins. He found that if he exchanges all his copper coins for as many gold coins as possible a... | tiger_math | [
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Let $a,$ $b,$ $c$ be the roots of $x^3 + px + q = 0.$ Express
\[\begin{vmatrix} 1 + a & X & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{vmatrix}\]in terms of $p$ and $q.$
If we know the answer to the above question is p-q, what is the value of unknown variable X? | tiger_math | [
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Rationalize the denominator of the fraction $\frac{5}{2+\sqrt{6}}$. The resulting fraction can be written in the form $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers. $D$ is positive and $B$ is not divisible by the square of any prime number. The greatest common divisor of $A$, $C$, and $D$ is 1. Fin... | tiger_math | [
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Farmer Brown has a field where he plants three types of crops: corn, wheat, and barley. This year, he decided to use different sections of his field for each crop, dedicating \(a\), \(b\), and \(c\) acres respectively. The total area of his field is 120 acres. Farmer Brown notes that the yield from his crops is inverse... | tiger_math | [
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0.0011... | 0.821809 | 0.811678 | 0.010131 |
In the diagram, rectangle $PQRS$ is divided into three identical squares. If $PQRS$ has perimeter X cm, what is its area, in square centimeters? [asy]
size(4cm);
pair p = (0, 1); pair q = (3, 1); pair r = (3, 0); pair s = (0, 0);
draw(p--q--r--s--cycle);
draw(shift(1) * (p--s)); draw(shift(2) * (p--s));
label("$P$... | tiger_math | [
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Convert the point $(2,\sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < X \pi.$ If the answer is $(4,\frac{\pi}{3})$, what is the value of X? | tiger_math | [
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Emma decides to bake a rectangular cake for her friend's birthday. The cake has a topping that forms a pathway diagonally from one corner to the opposite as a decoration. Suppose the size of the cake is such that it has a length of 15 inches and a width of 8 inches. Emma plans to place chocolate garnish along the diago... | tiger_math | [
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Farmer Bob has discovered that his new hens are laying eggs in very playful patterns. On Monday, they laid a total of $A + B$ eggs in one coop, and in another coop, they laid $C + D$ eggs, where $A \leq B$ and $C \leq D$. On Tuesday, in the first coop, they laid $A \times B$ eggs, and in the second coop, they laid $C \... | tiger_math | [
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On a particular day, Mandy decides to treat her friends by delivering homemade cookies using her bicycle. She prepares two types of cookies: chocolate chip and oatmeal raisin. If Mandy has a total of 15 friends, and she prepares $x$ chocolate chip cookies and $y$ oatmeal raisin cookies with the equation in mind that $x... | tiger_math | [
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Math City has eight streets, all of which are straight. No street is parallel to another street, but some of the streets may intersect at the same point. One police officer is stationed at each intersection. What is the greatest number of police officers needed? | tiger_math | [
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In a city's annual running competition, there are four segments varying in length. The distances (in km) of these segments are consecutive integers. Surprisingly, the sum of the squares of these distances is equal to ten times the sum of the distances themselves. What are the lengths of the segments?
### | tiger_math | [
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0.040228042751550674,
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-0.01... | 0.853646 | 0.845449 | 0.008196 |
A car driving cross-country starts its journey with an initial speed of v1 miles per hour. After driving for t1 hours, the car accelerates at a constant rate of a1 for t2 hours, reaching a final speed of v2. The car then maintains this speed (v2) for another t3 hours. Let D1 be the distance covered during the initial c... | tiger_math | [
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0.012891477905213833,
0.015531837940216064,
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-0.00... | 0.814779 | 0.808218 | 0.006561 |
In triangle $ABC$, $AB = 5$, $BC = 8$, and the length of median $AM$ is 4. Find $AC$. | tiger_math | [
0.03594101965427399,
0.005770022515207529,
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0.03533210605382919,
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0.016621477901935577,
0.030153164640069008,
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-0.0... | 0.836651 | 0.833272 | 0.003379 |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x - \cos x = \sqrt{2}.$ | tiger_math | [
-0.0019719323609024286,
0.03220384195446968,
-0.034325215965509415,
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0.027166062965989113,
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0.041965071111917496,
0.05028339847922325,
-0.023615321144461632,
0.03928937017917633,
0.0227674413472414,
-0.021572599187493324,
-0.... | 0.834557 | 0.826211 | 0.008346 |
a metallic sheet is of rectangular shape with dimensions 40 m x 30 m . from each of its corners , a square is cut off so as to make an open box . if the length of the square is 8 m , the volume of the box ( in m cube ) is : | tiger_math | [
0.038282833993434906,
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0.04234032332897186,
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0.07323417067527771,
0.04376494511961937,
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0.017264535650610924,
0.026133839040994644,
-0.02843530662357807,
-0.0084... | 0.816647 | 0.810241 | 0.006405 |
Let $w$ be a complex number satisfying $w^2 + 2w + 2 = 0.$ Compute
\[w^{97} + w^{98} + w^{99} + w^{100} + w^{101}.\] | tiger_math | [
0.02733854576945305,
0.015874015167355537,
-0.03191813826560974,
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0.03061390481889248,
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0.055173441767692566,
0.05253826454281807,
-0.012308749370276928,
0.01708318665623665,
0.03341848403215408,
-0.04069061949849129,
-0.0208... | 0.84582 | 0.839579 | 0.006241 |
Let $f(x)=x^4+14x^3+52x^2+56x+16$. Let $z_1,z_2,z_3,z_4$ be the four roots of $f$. Find the smallest possible value of $|z_{a}z_{b}+z_{c}z_{d}+z_{e}z_{f}|$ where $\{a,b,c,d,e,f\}=\{1,2,3,4\}$. | tiger_math | [
0.028060855343937874,
0.04049704223871231,
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0.05137143284082413,
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0.0036738647613674402,
0.0672544464468956,
0.03988749161362648,
0.0027566615026444197,
0.014906182885169983,
0.005047112703323364,
-0.02709437720477581,
-0.03... | 0.847875 | 0.839302 | 0.008573 |
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}.$ Given $\mathbf{m} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 6,$ find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$ | tiger_math | [
0.05241038650274277,
0.03225589171051979,
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0.04356398433446884,
0.0014968774048611522,
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0.058566126972436905,
0.033792007714509964,
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0.009136059321463108,
0.03512587770819664,
-0.028970956802368164,
-0.... | 0.838644 | 0.820115 | 0.018529 |
How many possible functions $g(x)$ are there if $T$ is the set of positive real numbers and $g: T \to \mathbb{R}$ satisfies the following conditions: (i) $g(1) = 2$, (ii) $g \left( \frac{x}{x + y} \right) = g \left( \frac{x}{x} \right) - g \left( \frac{x}{y} \right)$ for all $x$, $y \in T$ such that $x + y \in T$, and ... | tiger_math | [
0.018549595028162003,
0.03162309154868126,
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0.034461766481399536,
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0.0560762956738472,
0.027756959199905396,
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0.005516621749848127,
0.02097850665450096,
-0.025015946477651596,
-0.0... | 0.838462 | 0.827187 | 0.011275 |
Farmer Eliza has a field that is perfectly rectangular and measures 30 meters by 40 meters. She wants to install a sprinkler system that requires pipes laid out in a diagonal across the field for maximum reach. If each section of pipe is 10 meters long, how many sections does she need to cover the diagonal without any ... | tiger_math | [
-0.012367822229862213,
0.025390995666384697,
-0.04673022776842117,
-0.04122417792677879,
0.011340545490384102,
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0.056346673518419266,
0.0277764443308115,
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0.005079569295048714,
0.04212425649166107,
-0.020632535219192505,
-0.02... | 0.82829 | 0.82087 | 0.00742 |
In the Grand Chess Tournament, a beginner chess player named Alice has decided to challenge herself by facing off against multiple opponents in a row. Alice plans to face one opponent per day, and she wants to win at least 70% of her matches to consider her streak successful. After calculating, Alice figures out that t... | tiger_math | [
0.022361883893609047,
0.0012524734484031796,
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0.02330566942691803,
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0.025194954127073288,
0.03512643650174141,
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0.004558576736599207,
0.024924149736762047,
-0.04031681269407272,
-0.02... | 0.802768 | 0.789809 | 0.012959 |
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 10$ and $\angle O_{1}PO_{2} = 150^{\circ}$, then $AP = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are positive integers. Find $a + b$. | tiger_math | [
0.013519479893147945,
0.027982329949736595,
-0.04417265206575394,
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0.027540262788534164,
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0.0011006937129423022,
0.07525034248828888,
0.047749247401952744,
-0.0025356928817927837,
0.013605568557977676,
0.029314205050468445,
-0.04632792994379997,
-... | 0.842809 | 0.81711 | 0.025699 |
At a magical forest, there are mystical trees that produce algebraic fruits. Each tree has two types of fruits: apples and oranges. Today, John picked a mix of fruits. The first tree produced x apples and y oranges, while the second tree produced twice as many apples as the first but only half as many oranges. If John ... | tiger_math | [
0.03269708901643753,
0.023581668734550476,
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0.006257819943130016,
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0.013109738007187843,
0.026506386697292328,
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-0.0... | 0.834629 | 0.820985 | 0.013644 |
Farmer Jill is creating a new flower garden. The garden's shape will be rectangular, surrounded by a wooden fence. To determine how much fencing material she needs, she measures the garden to be twice as long as it is wide. Jill has already 30 meters of fence material in her storage. Today, she bought twice as much as ... | tiger_math | [
0.01193032506853342,
0.02943205088376999,
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0.026486119255423546,
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0.037585169076919556,
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-0.... | 0.802607 | 0.791023 | 0.011584 |
In a magical land, there are two lakes. These lakes are fed by rainfall which increases the water level by the same amount each day. On day one, Lake Dazzle has 1000 liters and Lake Sparkle has 700 liters. Every day, each lake receives enough rainfall to add 150% of its previous day's rainfall to its current volume. Af... | tiger_math | [
0.015936829149723053,
0.03081580623984337,
-0.03320407494902611,
-0.05841706693172455,
0.02071201615035534,
-0.03268054127693176,
-0.03486982733011246,
0.06040263921022415,
0.02224779687821865,
-0.03340641036629677,
0.012149542570114136,
0.028619447723031044,
-0.04404227435588837,
-0.01312... | 0.826789 | 0.823228 | 0.003561 |
Below is the graph of $y = a \sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$ Find $b.$
[asy]import TrigMacros;
size(400);
real f(real x)
{
return 2*sin(3*x + pi) + 1;
}
draw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);
trig_axes(-3*pi,3*pi,-4,4,pi/2,1);
layer();
rm_trig_labels(-5,5, 2);... | tiger_math | [
0.022572437301278114,
0.03701404854655266,
-0.04494177922606468,
-0.03845909237861633,
0.03386503830552101,
-0.025311587378382683,
-0.022537795826792717,
0.06940405070781708,
0.03839631751179695,
0.009017405100166798,
0.03130641207098961,
0.041712161153554916,
-0.01586572639644146,
-0.0176... | 0.838614 | 0.827777 | 0.010837 |
There exist vectors $\mathbf{a}$ and $\mathbf{b}$ such that
\[\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 \\ 5 \\ -1 \end{pmatrix},\]where $\mathbf{a}$ is parallel to $\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix},$ and $\mathbf{b}$ is orthogonal to $\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}.$ Find $\mathbf{b}.$ | tiger_math | [
0.03444958105683327,
0.04734661802649498,
-0.05272028595209122,
-0.036428891122341156,
0.046799659729003906,
0.024613849818706512,
-0.015734892338514328,
0.08802281320095062,
0.02946852333843708,
-0.0011741181369870901,
0.006590606179088354,
0.03182506933808327,
-0.027316318824887276,
-0.0... | 0.834834 | 0.820887 | 0.013947 |
The quadratic equation $px^2+qx+r=0$ has exactly one solution. If $p-r=5$, and $p>r$, find the ordered pair $(p,r)$. | tiger_math | [
0.01996443420648575,
0.02034747414290905,
-0.04672662168741226,
-0.048659954220056534,
0.039575476199388504,
-0.010317421518266201,
0.005223683547228575,
0.06301511824131012,
0.048387546092271805,
-0.03209667280316353,
0.01249920204281807,
0.0059412699192762375,
-0.02446649596095085,
-0.03... | 0.845529 | 0.827999 | 0.01753 |
[asy] draw((0,0)--(2,2)--(5/2,1/2)--(2,0)--cycle,dot); MP("A",(0,0),W);MP("B",(2,2),N);MP("C",(5/2,1/2),SE);MP("D",(2,0),S); MP("a",(1,0),N);MP("b",(17/8,1/8),N); MP("E",(1,1),NE); [/asy]
In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice that of angle $B$, and the measur... | tiger_math | [
0.027564022690057755,
0.007270096335560083,
-0.04541277885437012,
-0.03625606745481491,
0.03403668478131294,
-0.015760235488414764,
-0.026919038966298103,
0.05047882720828056,
0.02671912871301174,
0.01662943698465824,
0.013432949781417847,
0.020269902423024178,
-0.014273568987846375,
-0.00... | 0.841473 | 0.81406 | 0.027413 |
Someone observed that $6! = 8 \cdot X \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.
If we know the answer to the above question is 23, what is the value of unknown variable X? | tiger_math | [
0.018467718735337257,
0.025476882234215736,
-0.05082035809755325,
-0.029856491833925247,
0.0463494174182415,
-0.005811717361211777,
-0.021391812711954117,
0.06229526549577713,
0.040891747921705246,
-0.027862705290317535,
0.019873837009072304,
0.03883923217654228,
-0.05641797184944153,
-0.0... | 0.850217 | 0.846771 | 0.003446 |
Convert the point $(\rho,\theta,\phi) = \left( 12, \pi, \frac{\pi}{4} \right)$ in spherical coordinates to rectangular coordinates. | tiger_math | [
0.014744496904313564,
0.043247271329164505,
-0.02331935055553913,
-0.04752644523978233,
0.018688710406422615,
-0.025147469714283943,
-0.027237597852945328,
0.05276304483413696,
0.05808454006910324,
0.00040747213643044233,
0.00875770952552557,
0.036821044981479645,
-0.022364411503076553,
-0... | 0.828403 | 0.822715 | 0.005688 |
In an exciting development at the local zoo, a family of Scarlet Macaws was observed to fly around their large aviary every day. The zookeeper noticed that if the youngest Macaw starts flying at one corner of the square aviary and continues straight, it completes exactly 5 laps around the aviary in the same time that i... | tiger_math | [
0.010284767486155033,
0.028547033667564392,
-0.05014979839324951,
-0.029974332079291344,
0.033547572791576385,
0.00233330181799829,
-0.046115849167108536,
0.07905372232198715,
0.06734263896942139,
-0.018118051812052727,
-0.002759974915534258,
0.006373562850058079,
-0.04727725684642792,
-0.... | 0.821346 | 0.807549 | 0.013797 |
A club has $5$ members from each of $3$ different schools, for a total of $15$ members. How many possible ways are there to arrange a presidency meeting under the following conditions:
i. The club must choose one of the $3$ schools at which to host the meeting,
ii. The host school sends $3$ representatives to the mee... | tiger_math | [
0.015172027982771397,
0.009225490503013134,
-0.03769400343298912,
-0.039041150361299515,
0.03821194916963577,
-0.026909183710813522,
-0.039015721529722214,
0.03785111755132675,
0.028402121737599373,
-0.004753045737743378,
0.01461927779018879,
0.04216279461979866,
-0.03587211295962334,
-0.0... | 0.831435 | 0.822038 | 0.009396 |
Compute
\[\begin{pmatrix} 1 & c & -b \\ -c & 1 & a \\ b & -a & 1 \end{pmatrix} \begin{pmatrix} a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2 \end{pmatrix}.\] | tiger_math | [
0.04493344575166702,
0.05629565194249153,
-0.04628138989210129,
-0.035054806619882584,
0.056957028806209564,
-0.03490472957491875,
-0.010041851550340652,
0.058128196746110916,
0.028177864849567413,
-0.010482607409358025,
0.008179842494428158,
0.023557322099804878,
-0.04415899142622948,
-0.... | 0.817248 | 0.810613 | 0.006635 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{x}{y} + \frac{y}{z} + \frac{z}{x}.\] | tiger_math | [
0.02126898430287838,
0.035394635051488876,
-0.04024311900138855,
-0.04423223435878754,
0.04882078617811203,
-0.00023557695385534316,
-0.0009120384347625077,
0.03139781206846237,
0.04388688877224922,
-0.02274199202656746,
0.009350324980914593,
0.011720270849764347,
-0.02492949366569519,
-0.... | 0.84259 | 0.832676 | 0.009914 |
A farmer is creating a rectangular garden and has a total of 120 meters of fencing. She uses part of an existing wall for one side of the rectangle. She wants to find what lengths to make the other sides to maximize the area of the garden. Determine the dimensions of the rectangle that yield the maximum area and confir... | tiger_math | [
0.022234346717596054,
0.026990676298737526,
-0.03408099338412285,
-0.03949793055653572,
0.008007405325770378,
-0.03937936946749687,
-0.03222090005874634,
0.06482616066932678,
0.035849783569574356,
-0.001231663511134684,
-0.011762975715100765,
0.06259717792272568,
-0.03535989671945572,
-0.0... | 0.801875 | 0.792225 | 0.00965 |
Let $P$ be the product of the first $200$ positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k .$ | tiger_math | [
0.022468741983175278,
0.02957603521645069,
-0.04897226393222809,
-0.044198472052812576,
0.023595191538333893,
-0.019538631662726402,
-0.02289627306163311,
0.04964436590671539,
0.04137546941637993,
-0.013241404667496681,
0.01800946705043316,
0.03474779054522514,
-0.03037356585264206,
-0.021... | 0.859004 | 0.846536 | 0.012468 |
Tom and Elizabeth are mountain climbers who decided to climb three mountains of different heights and terrains. The first mountain is a rocky terrain, and it takes Elizabeth 30 minutes to climb it. The second mountain is forest terrain with a steeper slope, so it takes Elizabeth 45 minutes. The third mountain is a snow... | tiger_math | [
0.032894913107156754,
0.027210505679249763,
-0.0566575713455677,
-0.04818638786673546,
0.03590618073940277,
-0.023831743746995926,
-0.03658309578895569,
0.06254201382398605,
0.024660876020789146,
-0.02397976443171501,
0.008123247884213924,
0.01757010631263256,
-0.03302641212940216,
-0.0166... | 0.794033 | 0.783589 | 0.010444 |
In a magical land, Farmer McDonald has a plot of land dedicated to growing magical berries, which he fences using magic threads. The plots are rectangular, and the enchantment adds to its perimeter based on the mystic forces. For one plot that measures 5 meters by 7 meters, the magical enhancement adds 50% to its origi... | tiger_math | [
0.025272229686379433,
0.039209336042404175,
-0.028889749199151993,
-0.06252750009298325,
0.006285512819886208,
-0.034556932747364044,
-0.03437533974647522,
0.059143297374248505,
0.04011493921279907,
-0.019340652972459793,
-0.011638124473392963,
0.044187117367982864,
-0.04098963364958763,
-... | 0.827626 | 0.8125 | 0.015126 |
Let $\mathbf{A}$ be a matrix such that
\[\mathbf{A} \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} -9 \\ 3 \end{pmatrix}.\]Find $\mathbf{A}^3 \begin{pmatrix} 3 \\ 1 \end{pmatrix}.$ | tiger_math | [
0.04612530395388603,
0.020287353545427322,
-0.05579688400030136,
-0.043763596564531326,
0.04117874056100845,
-0.025146590545773506,
-0.005441099870949984,
0.07224123179912567,
0.031453944742679596,
-0.009575020521879196,
0.015382571145892143,
0.025582879781723022,
-0.06708741188049316,
-0.... | 0.82922 | 0.820942 | 0.008278 |
There exists a positive real number $x$ such that $ \sin (\arccos (x)) = x $. Find the value of $x^2$. | tiger_math | [
0.024667654186487198,
0.02371145971119404,
-0.047434188425540924,
-0.03197125345468521,
0.027973804622888565,
-0.016828149557113647,
-0.006374277174472809,
0.05903058871626854,
0.05326496809720993,
-0.018287768587470055,
0.03380229324102402,
0.02466331608593464,
-0.028128821402788162,
-0.0... | 0.838762 | 0.828171 | 0.010591 |
In trapezoid $ABCD$, sides $\overline{AB}$ and $\overline{CD}$ are parallel, $\angle A = 3\angle D$, and $\angle C = 2\angle B$. Find $\angle B$. | tiger_math | [
0.021019265055656433,
0.009760252200067043,
-0.0328218936920166,
-0.05183163657784462,
0.011985855177044868,
-0.0024513110984116793,
-0.01634368859231472,
0.042947765439748764,
0.024592727422714233,
-0.00936926994472742,
0.036752425134181976,
0.032627444714307785,
-0.040319014340639114,
-0... | 0.829017 | 0.813225 | 0.015792 |
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 32?
$\textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ X \qquad \textbf{(D)}\ 1498 \qquad \t... | tiger_math | [
-0.00046249301522038877,
0.021477418020367622,
-0.06276553869247437,
-0.016383830457925797,
0.027848144993185997,
-0.0014594246167689562,
-0.012753062881529331,
0.05534264072775841,
0.026934050023555756,
-0.0065015037544071674,
0.025982482358813286,
0.03913476690649986,
-0.030469296500086784... | 0.839361 | 0.827554 | 0.011807 |
Marcelle and Jaclyn each have a monic polynomial of degree 4. Both polynomials have the same positive constant term and the same coefficient of z. When their polynomials are multiplied together, the resulting polynomial is $z^8 +3z^7 +z^6 +3z^5 +4z^4 +6z^3 +2z^2 +4$. What is the constant term of Jaclyn's polynomial? | tiger_math | [
0.006355767138302326,
0.021731223911046982,
-0.02734682336449623,
-0.05078037455677986,
0.04423241689801216,
-0.01950693130493164,
-0.011636856012046337,
0.029954956844449043,
0.05310863256454468,
-0.0022163589019328356,
0.0030712932348251343,
0.016939861699938774,
-0.008229580707848072,
-... | 0.826002 | 0.817357 | 0.008645 |
Given a rectangle where the length is thrice its breadth and its perimeter is 104 m, find the area of the rectangle. Additionally, find the length of the diagonal of the rectangle and determine the angle between the diagonal and the length of the rectangle. | tiger_math | [
0.02697652205824852,
0.024292759597301483,
-0.02850549854338169,
-0.04338599368929863,
0.004333162680268288,
-0.020728198811411858,
-0.024849168956279755,
0.04831668362021446,
0.03393734246492386,
0.005278870463371277,
0.02278367057442665,
0.027056636288762093,
-0.040783822536468506,
-0.02... | 0.804577 | 0.801306 | 0.003271 |
In triangle $ABC,$ $\cot A \cot C = \frac{1}{3}$ and $\cot B \cot C = \frac{1}{9}.$ Find $\tan C.$ | tiger_math | [
0.026409469544887543,
0.03826485574245453,
-0.039519522339105606,
-0.035206619650125504,
0.03171784430742264,
-0.03222263604402542,
-0.01474367082118988,
0.03763524442911148,
0.03952658176422119,
-0.01764628104865551,
0.027779221534729004,
0.024551643058657646,
-0.028405901044607162,
0.001... | 0.835118 | 0.822373 | 0.012744 |
For each integer $n\geq 5$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_5a_6 \dotsm a_{100}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$? | tiger_math | [
0.03344506770372391,
0.02688608504831791,
-0.04076462611556053,
-0.025712205097079277,
0.04519955441355705,
-0.0198447797447443,
-0.00906391628086567,
0.06672638654708862,
0.048813506960868835,
-0.018966853618621826,
-0.00535318348556757,
0.044232968240976334,
-0.038460418581962585,
-0.034... | 0.849898 | 0.843866 | 0.006032 |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(-1,0),(2009,0),(2009,2010),$ and $(-1,2010)$. What is the probability that $x > 2y$? Express your answer as a common fraction. | tiger_math | [
0.001154478988610208,
0.008887809701263905,
-0.05320025235414505,
-0.028783418238162994,
0.013778907246887684,
-0.019426146522164345,
-0.021446486935019493,
0.06342139095067978,
0.03781780228018761,
0.013856923207640648,
0.01596294529736042,
0.03747890517115593,
-0.03771768510341644,
-0.00... | 0.834948 | 0.826895 | 0.008053 |
Wanda is trying to locate the Fermat point $P$ of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(6,-2)$, and $C$ is at $(3,6)$. She guesses that the point is at $P = (3,3)$, and computes the sum of the distances from $P$ to the vertices of $\triangle ABC$. If she obtains $m + n\sqrt{3}$, where $m$ and $n$ are... | tiger_math | [
0.029324466362595558,
0.031928665935993195,
-0.040787938982248306,
-0.047848179936409,
0.035021763294935226,
-0.024985631927847862,
-0.03230144828557968,
0.0607529915869236,
0.040561575442552567,
-0.005812619812786579,
0.018551301211118698,
0.04887250438332558,
-0.01424960233271122,
-0.037... | 0.84398 | 0.829121 | 0.014859 |
In a colorful garden, Sally is comparing the number of purple tulips to red tulips. There are X purple tulips arranged in perfect sequential squares along a path (1 tulip for the 1st position, 4 tulips for the 2nd, 9 for the 3rd, and so on). There are 16 more tulips in each of the red tulip square patches than in the c... | tiger_math | [
-0.01076626218855381,
0.03673277050256729,
-0.03602210432291031,
-0.05264808610081673,
0.030766967684030533,
-0.03149678185582161,
-0.02949889376759529,
0.07103759795427322,
0.039199911057949066,
-0.007150039542466402,
0.0037816513795405626,
0.035832203924655914,
-0.02683861553668976,
-0.0... | 0.835479 | 0.829322 | 0.006157 |
Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains 8 unit squares. The second ring contains 16 unit squares. If we continue this process, then what is the number of unit squares in the $100^{th}$ ring?
[asy]
fill((2,2)--(7,2)--(7,7... | tiger_math | [
0.017129547894001007,
0.035714585334062576,
-0.01911727339029312,
-0.03467254340648651,
0.029150603339076042,
-0.026952309533953667,
-0.025469623506069183,
0.07074248045682907,
0.025661960244178772,
-0.012995890341699123,
0.03311982378363609,
0.014076766557991505,
-0.024608928710222244,
-0... | 0.840682 | 0.831951 | 0.008731 |
Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(3,2)$ in seven or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$... | tiger_math | [
0.015257510356605053,
0.020077267661690712,
-0.03917459771037102,
-0.02856183983385563,
0.03430265560746193,
0.007950847037136555,
-0.04209062457084656,
0.05600212514400482,
0.032476767897605896,
-0.00022298074327409267,
0.002277905587106943,
0.03719398379325867,
-0.026408320292830467,
-0.... | 0.851288 | 0.830206 | 0.021082 |
A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\ldots,101$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in... | tiger_math | [
0.02354256995022297,
0.02032550796866417,
-0.04065326973795891,
-0.03488999977707863,
0.020400650799274445,
-0.015616132877767086,
-0.030352188274264336,
0.05242906138300896,
0.031732622534036636,
-0.006333815865218639,
0.022179126739501953,
0.024032974615693092,
-0.038195718079805374,
-0.... | 0.852306 | 0.838054 | 0.014252 |
Find the maximum value of
\[\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 + \cos \theta & 1 \\ 1 + \sin \theta & 1 & 1 \end{vmatrix},\]as $\theta$ ranges over all angles. | tiger_math | [
0.03586731478571892,
0.03339681774377823,
-0.03870159760117531,
-0.038823653012514114,
0.036498308181762695,
-0.03082616813480854,
-0.028407031670212746,
0.05154866725206375,
0.02615809626877308,
-0.012566003017127514,
0.02287021093070507,
0.02513064257800579,
-0.030429506674408913,
-0.015... | 0.83381 | 0.810425 | 0.023384 |
Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u},$ $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto the vector $\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}.$ | tiger_math | [
0.015215516090393066,
0.0452149398624897,
-0.043551769107580185,
-0.04173925146460533,
0.04146185517311096,
-0.012424406595528126,
-0.006375859957188368,
0.07010576874017715,
0.037951383739709854,
-0.023600388318300247,
0.021850116550922394,
0.0247822143137455,
-0.0255600456148386,
-0.0162... | 0.832542 | 0.813985 | 0.018556 |
In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss delive... | tiger_math | [
0.017506690695881844,
0.019644087180495262,
-0.030069658532738686,
-0.02053237333893776,
0.016868187114596367,
-0.019690290093421936,
-0.02666645497083664,
0.060989752411842346,
0.04373539984226227,
-0.026699095964431763,
0.03686555102467537,
0.024769088253378868,
-0.034263718873262405,
-0... | 0.839901 | 0.83015 | 0.009751 |
Tommy's mother lets him ride his bike 2 blocks north of his block, 3 blocks east, 2 blocks west, and 2 blocks south. Each block is rectangular in shape, with a length of 200 meters and a width of 150 meters. Due to a special rule enforced for Tommy, he needs to alternate between going north and east or west for every b... | tiger_math | [
0.041583627462387085,
0.025756021961569786,
-0.03707164153456688,
-0.036201898008584976,
0.010332813486456871,
-0.016710074618458748,
-0.046601735055446625,
0.06705290079116821,
0.046760279685258865,
-0.008700844831764698,
-0.009605102241039276,
0.03491954877972603,
-0.04347846657037735,
-... | 0.815775 | 0.809384 | 0.006391 |
A school is arranging a math competition involving a relay race in teams. Each team is made up of four students, and each member of the team solves one math problem before passing the baton to the next member. The team’s total time is calculated by adding the time taken by each member to solve their problem. Team A has... | tiger_math | [
0.027228940278291702,
0.010715564712882042,
-0.046397868543863297,
-0.04501483961939812,
0.04838457331061363,
-0.013789881952106953,
-0.04617522284388542,
0.05172460526227951,
0.050880394876003265,
-0.012568383477628231,
-0.00679213460534811,
0.01479942724108696,
-0.009513105265796185,
-0.... | 0.823662 | 0.813896 | 0.009766 |
In a local gardening competition, different participants use various geometric figures to create unique gardens, with the total edge length for each shape adding up in meters. Abby creates a triangular garden, Ben designs a square garden, and Cara decides on a pentagonal garden. Each individual uses the same length for... | tiger_math | [
0.012556778267025948,
0.015412066131830215,
-0.04884321615099907,
-0.061564527451992035,
0.022854309529066086,
-0.039751745760440826,
-0.03842131048440933,
0.031379543244838715,
0.028014810755848885,
-0.004622922278940678,
-0.002218292560428381,
0.03352911397814751,
-0.03951050713658333,
-... | 0.819539 | 0.809597 | 0.009942 |
Below is the graph of $y = a \sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$ Find $a.$
[asy]import TrigMacros;
size(400);
real f(real x)
{
return 2*sin(3*x + pi) + 1;
}
draw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);
trig_axes(-3*pi,3*pi,-4,4,pi/2,1);
layer();
rm_trig_labels(-5,5, 2);... | tiger_math | [
0.023043053224682808,
0.038284871727228165,
-0.04420113191008568,
-0.03746766969561577,
0.03341542184352875,
-0.02401227504014969,
-0.02185635268688202,
0.07148755341768265,
0.035310935229063034,
0.008062385953962803,
0.029527509585022926,
0.041328560560941696,
-0.016023971140384674,
-0.01... | 0.838717 | 0.826673 | 0.012044 |
Alice wants to organize her collection of marbles into a giant glass container shaped like a cube. The side length of the cubic container is 50 cm. Alice individually places spherical marbles with a diameter of 8 cm into the container in a single layer along the bottom, without stacking any marbles on top of each other... | tiger_math | [
0.009926850907504559,
0.046195801347494125,
-0.024520903825759888,
-0.018067892640829086,
0.02003398910164833,
-0.0036230054683983326,
-0.04532584920525551,
0.062353070825338364,
0.02457728050649166,
-0.0072504268027842045,
0.0036755227483808994,
0.03529201075434685,
-0.011312063783407211,
... | 0.800762 | 0.787986 | 0.012776 |
Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\tfrac{m}{n},$ where $m$ and $n$ are ... | tiger_math | [
0.015018069185316563,
0.01797914132475853,
-0.06099812686443329,
-0.025517459958791733,
0.02974451147019863,
-0.006869926117360592,
-0.030354056507349014,
0.0552932508289814,
0.024669839069247246,
-0.015399216674268246,
0.027977218851447105,
0.026664523407816887,
-0.04268984496593475,
0.00... | 0.846669 | 0.824818 | 0.021852 |
The function $f(x)$ satisfies
\[f(2^x) - xf(2^{-x}) = 1\]for all real numbers $x.$ Find $f(2).$ | tiger_math | [
0.04871325567364693,
0.049916818737983704,
-0.04335363954305649,
-0.040273040533065796,
0.027388256043195724,
-0.03325166925787926,
-0.014707614667713642,
0.07075829803943634,
0.048106178641319275,
-0.014477362856268883,
0.019361183047294617,
0.026270288974046707,
-0.013490306213498116,
-0... | 0.839519 | 0.830349 | 0.00917 |
Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $\$1$ each, begonias $\$1.50$ each, cannas $\$2$ each... | tiger_math | [
0.018643265590071678,
0.030621707439422607,
-0.06843513250350952,
-0.03759201243519783,
0.019780857488512993,
-0.045519206672906876,
-0.03543398901820183,
0.06588176637887955,
0.045005980879068375,
0.013382726348936558,
0.019793307408690453,
0.042991753667593,
-0.013983523473143578,
-0.006... | 0.827778 | 0.817499 | 0.010279 |
Given that the solution to the inequality $\left| \frac{2x - 1}{x - X} \right| > 2$ is $\left(\frac{3}{4},1\right)\cup(1,\infty)$, find the range of possible values for the expression $\frac{x - X}{2x - 1}$. | tiger_math | [
0.022359736263751984,
0.021075133234262466,
-0.02567528747022152,
-0.031106751412153244,
0.027571571990847588,
-0.011810487136244774,
-0.030523303896188736,
0.08223549276590347,
0.06487080454826355,
-0.027939679101109505,
0.023578304797410965,
0.04318447783589363,
-0.0189034603536129,
-0.0... | 0.81783 | 0.811122 | 0.006708 |
In a magical land, there are three types of flowers – sunroses, moonlilies, and starpetals. Elena, a botanist in this land, finds that if she plants one sunrose, four moonlilies will grow around it. If she plants one moonlily, two starpetals will grow around it. If she initially plants $p$ sunroses and $m$ moonlilies, ... | tiger_math | [
0.01463930681347847,
0.04693415388464928,
-0.04709197208285332,
-0.05016350746154785,
0.03428250923752785,
-0.04349512606859207,
-0.047329530119895935,
0.06287748366594315,
0.0405130609869957,
-0.013378150761127472,
-0.0057074278593063354,
0.04553268849849701,
-0.04488609731197357,
-0.0083... | 0.827572 | 0.818757 | 0.008815 |
If $x$ and $y$ are positive real numbers such that $3x^2 + 6xy = x^3 + 2x^2 y,$ find the value of $\frac{x}{y}$. | tiger_math | [
0.028567833825945854,
0.02121448889374733,
-0.04659198597073555,
-0.0298826415091753,
0.0448547825217247,
-0.03291616216301918,
-0.0002270786790177226,
0.04564006254076958,
0.04525576904416084,
-0.04593990370631218,
0.019374124705791473,
0.028548678383231163,
-0.04328368604183197,
-0.02719... | 0.840141 | 0.833469 | 0.006672 |
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