Dataset Viewer
messages
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| dataset
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value | source
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|---|---|---|---|
[
{
"content": "How many solutions does the equation 2(3|((3x + 2)/(1x + 3))|)^2 - 2 = 4(1|x + 1|) - 2 have for x in the interval [-10, 10]?",
"role": "user"
}
] |
2
|
math
| null |
[
{
"content": "Compute the rank of the given 5x5 matrix:\n[[12, -20, 11, 2, 11],\n [30, -17, 22, 5, 22],\n [-24, -8, -14, -4, -14],\n [-18, 27, -16, -3, -16],\n [24, -16, 18, 4, 18]]",
"role": "user"
}
] |
2
|
math
| null |
[
{
"content": "Let v(q) = 2*q - 8. Let d be v(5). Express the second largest root of 1920/11*r**3 - 2016/11*r**d + 702/11*r - 81/11 = 0 as n/m where n and m are coprime and calculate n + m.",
"role": "user"
}
] |
13
|
math
| null |
[
{
"content": "Calculate the rank of matrix A where A =\n[[1, -4, -9, 2, -23],\n [14, -10, -4, 19, 1],\n [18, -3, 8, 16, 25],\n [27, -16, -1, 35, 18],\n [-13, -17, -27, 7, -49]]",
"role": "user"
}
] |
3
|
math
| null |
[
{
"content": "What is the trace (sum of diagonal elements) when matrix A is multiplied by matrix B?\nA =\n[[-8, 7, 4, 0],\n [-7, 5, -3, 8],\n [4, 0, -2, 4],\n [2, 1, -4, -8],\n [7, -8, 1, -6]]\nB =\n[[-8, -2, -2, -2, 7],\n [-8, 6, -4, -6, 0],\n [5, 6, -7, -6, 7],\n [8, 0, 6, -1, -6]]",
"role": "user"
}
] |
198
|
math
| null |
[
{
"content": "let N be the greatest 3-digit positive integer with the property that both the number and the number formed by reversing its digits are divisible by 11. Let Q and R be the quotient and remainder, respectively, when N is divided by 100. Find Q + R.",
"role": "user"
}
] |
99
|
math
| null |
[
{
"content": "If 17472*z**4 - 4756 + 6154*z - 7352*z**3 - 9428*z**2 + 2124*z + 3796 - 2070*z = 0, what is n + m where n/m is the second largest root in simplified form?",
"role": "user"
}
] |
11
|
math
| null |
[
{
"content": "What is the second-largest prime factor of 3958696?",
"role": "user"
}
] |
223
|
math
| null |
[
{
"content": "Find the second-largest prime factor of 5431438.",
"role": "user"
}
] |
257
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {k: 1, u: 1, j: 1, v: 1}, shuffle the letters in the word, what is the probability of exact 1 letter 'v' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
5
|
math
| null |
[
{
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is uncolored, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is uncolored. How many valid colorings of the uncolored vertices exist such that the 6-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?",
"role": "user"
}
] |
2
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {l: 6, v: 2}, shuffle the letters in the word, what is the probability of at least 1 letter 'v' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
37
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {s: 2, i: 2, o: 2}, shuffle the letters in the word, what is the probability of no letter 'i' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
23
|
math
| null |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(1|(3(2x + 4)^2 - 5(2x + 4) + 3) - 1|) - 2)/(-2(1|(3(2x + 4)^2 - 5(2x + 4) + 3) - 1|) + 3) and g(x) = (3(-2|(-3x) - 3| - 3))/(-1(-2|(-3x) - 3| - 3) + 2) where -10 ≤ x ≤ 10.",
"role": "user"
}
] |
2
|
math
| null |
[
{
"content": "Let p be the smallest prime where n^7+-1=0 (mod p^3) has a solution, then find the minimal m for this p.",
"role": "user"
}
] |
1
|
math
| null |
[
{
"content": "Find the rank of the matrix:\n[[-31, 19, -12, 49, 29],\n [-22, 8, 28, 24, -18],\n [-32, -10, 64, 8, -80],\n [49, -16, 15, 8, 20],\n [-45, 3, 28, 6, -53]]",
"role": "user"
}
] |
4
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {a: 1, s: 1, k: 2, w: 2}, shuffle the letters in the word, what is the probability of no letter 'a' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
11
|
math
| null |
[
{
"content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 0a^2 + 0b^2 + 0c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000.",
"role": "user"
}
] |
512
|
math
| null |
[
{
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 2 and from vertex 1 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?",
"role": "user"
}
] |
6
|
math
| null |
[
{
"content": "How many solutions does the equation (-2(3|(4(-1x + 3)^2 + 2(-1x + 3) - 4) - 2| - 3) + 1)/(-1(3|(4(-1x + 3)^2 + 2(-1x + 3) - 4) - 2| - 3)) = 1sin(1π(-2|(-4x^2 + 2x + 4)| - 1) - 2) - 3 have for x in the interval [-10, 10]?",
"role": "user"
}
] |
4
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {e: 1, d: 3, h: 3}, shuffle the letters in the word, what is the probability of no letter 'e' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
13
|
math
| null |
[
{
"content": "Let r(d) be the first derivative of 1815/19*d**5 + 4 - 1296/19*d**2 + 3795/38*d**4 - 432/19*d - 767/19*d**3. For r(s) = 0, find the second largest root in simplified form n/m and return n + m.",
"role": "user"
}
] |
8
|
math
| null |
[
{
"content": "Circle D has center O and radius 2. OE = 6. Let W be the rotation of point E by angle pi/2 around point O. Point V is the midpoint of segment EO. Let A be the midpoint of segment OV. Let P be the inversion of point A in circle D. Let Z be the polar line of point W relative to circle D. Let L be the rotation of point O by angle 11pi/12 around point E. Let R be the line through points W and P. Circle I has center L and radius 3. Circle K is the reflection of circle I across line Z. Circle Q is the reflection of circle K across point E. Let C be the polar line of point V relative to circle Q. Point S is the intersection of lines R and C. Find the distance between points P and S. Express your answer as a decimal number rounded to 2 decimal places.",
"role": "user"
}
] |
4.14
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {z: 5, c: 3}, shuffle the letters in the word, what is the probability of exact 3 letter 'c' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
57
|
math
| null |
[
{
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 4 and from vertex 5 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?",
"role": "user"
}
] |
7
|
math
| null |
[
{
"content": "Supposing you start in the bottom left (0, 0) cell of a 9x8 grid, what is minimum number of steps you can take to get to (6, 4), if your only valid moves are to move over 4 squares in one direction and over 3 squares in a perpendicular direction (at once)? (Assume you cannot move off the grid.)",
"role": "user"
}
] |
6
|
math
| null |
[
{
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 5 and from vertex 7 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?",
"role": "user"
}
] |
5
|
math
| null |
[
{
"content": "How many zeros does the function f(x) = -3sin(2π(2|(-5x + 4) - 3| + 2) - 1) + 1 have in the interval [-10, 10]?",
"role": "user"
}
] |
400
|
math
| null |
[
{
"content": "In a 6x4 grid, how many different paths are there from the bottom left (0, 0) to the top right (5, 3), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (0, 3), (5, 2), (4, 0), (4, 1), (1, 2)?",
"role": "user"
}
] |
17
|
math
| null |
[
{
"content": "YS = 8. Circle J has center S and radius 12. Let D be the reflection of point S across point Y. Chord IK of circle J has length 1. Circle M passes through points K, I, and Y. Point N is the center of circle M. Let C be the midpoint of segment IK. Point R is the reflection of point C across segment SN. Let E be the midpoint of segment RD. Find the length of segment YE. Express your answer as a decimal number rounded to 2 decimal places.",
"role": "user"
}
] |
5.99
|
math
| null |
[
{
"content": "Form a word by randomly choosing 2 letters from the multiset {m: 2, z: 2, x: 1}. What is the expected number of occurrences of the pattern 'x.*z' in each word? Round your answer to 2 decimal places.",
"role": "user"
}
] |
0.1
|
math
| null |
[
{
"content": "Determine the rank of the following matrix:\n[[0, -5, -14, -23, 33],\n [-45, -4, -30, -4, 0],\n [-48, -3, -21, 0, -12],\n [11, 7, -6, 35, -25],\n [-48, -5, -18, -11, -3]]",
"role": "user"
}
] |
3
|
math
| null |
[
{
"content": "When randomly selecting 3 letters from the multiset {l: 1, b: 3} to form a word, what is the expected number of matches of the pattern 'l+l'? Round to 2 decimal places.",
"role": "user"
}
] |
0.0
|
math
| null |
[
{
"content": "Supposing you start in the bottom left (0, 0) cell of a 7x9 grid, what is minimum number of steps you can take to get to (7, 4), if your only valid moves are to move over 1 squares in one direction and over 6 squares in a perpendicular direction (at once)? (Assume you cannot move off the grid.)",
"role": "user"
}
] |
9
|
math
| null |
[
{
"content": "Let a(v) be the first derivative of 2 + 63*v**3 + 1605/4*v**4 + 280*v**5 - 297/2*v**2 - 81*v. If a(b) = 0, what is n + m where n/m is the second largest root in simplified form?",
"role": "user"
}
] |
5
|
math
| null |
[
{
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 2, from vertex 2 to vertex 5, from vertex 4 to vertex 1, from vertex 7 to vertex 10, from vertex 10 to vertex 8, from vertex 8 to vertex 4, from vertex 6 to vertex 3, from vertex 10 to vertex 6, from vertex 5 to vertex 1, from vertex 6 to vertex 1, and from vertex 2 to vertex 10. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?",
"role": "user"
}
] |
6
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {h: 2, d: 1, m: 1, t: 3}, shuffle the letters in the word, what is the probability of no letter 'm' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
13
|
math
| null |
[
{
"content": "Determine the rank of the following matrix:\n[[-14, -10, -15, 6, 61],\n [-6, -13, -16, 3, 46],\n [15, 9, 6, -3, -37],\n [-31, -8, -3, 21, 35],\n [41, 0, 5, -30, -64]]",
"role": "user"
}
] |
4
|
math
| null |
[
{
"content": "Supposing you start in the bottom left (0, 0) cell of a 8x9 grid, what is minimum number of steps you can take to get to (8, 5), if your only valid moves are to move over 1 squares in one direction and over 6 squares in a perpendicular direction (at once)? (Assume you cannot move off the grid.)",
"role": "user"
}
] |
7
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {h: 5, s: 1, l: 2}, shuffle the letters in the word, what is the probability of no letter 'l' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
47
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {p: 1, r: 2, z: 2, a: 1}, shuffle the letters in the word, what is the probability of no letter 'z' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
23
|
math
| null |
[
{
"content": "Calculate the greatest common factor of 5377482 and 1542024.",
"role": "user"
}
] |
6534
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {j: 5, a: 2}, shuffle the letters in the word, what is the probability of no letter 'j' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
22
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {r: 4, b: 1}, shuffle the letters in the word, what is the probability of no letter 'b' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
9
|
math
| null |
[
{
"content": "Let y(n) be the first derivative of -13 - 792/13*n + 1089/13*n**2 + 4774/39*n**3 - 2695/13*n**4. For the polynomial y(s), find the second largest root in the form n/m where n and m are coprime integers. What is n + m?",
"role": "user"
}
] |
13
|
math
| null |
[
{
"content": "For the function f(x) = (3(-4(-1(1|x + 1| - 3) - 3)^2 - 4(-1(1|x + 1| - 3) - 3) + 3) - 1)/(1(-4(-1(1|x + 1| - 3) - 3)^2 - 4(-1(1|x + 1| - 3) - 3) + 3) - 2), find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.",
"role": "user"
}
] |
2
|
math
| null |
[
{
"content": "Compute the sum of all entries of the product matrix AB given:\nA =\n[[0, 7, -10, 3, -3],\n [-3, 1, 9, 7, 9],\n [-5, -7, 10, 10, 5],\n [-1, 8, 9, -9, -4],\n [-3, 5, 5, -3, -8]]\nB =\n[[2, -7, -9, -9, 8],\n [-2, -1, 4, 10, -1],\n [-10, 1, -4, -7, 5],\n [3, 4, -7, 6, 8],\n [6, -6, 7, 0, 1]]",
"role": "user"
}
] |
79
|
math
| null |
[
{
"content": "Suppose you have a 14-gon, with vertices numbered 1 through 14 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 12, from vertex 5 to vertex 3, from vertex 9 to vertex 5, from vertex 4 to vertex 6, from vertex 2 to vertex 7, from vertex 6 to vertex 12, and from vertex 3 to vertex 12. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?",
"role": "user"
}
] |
2
|
math
| null |
[
{
"content": "If you add up all the x-coordinates where the functions f(x) = 1(3|((2x - 1)/(-1x + 3))| - 3) and g(x) = 2|((2x + 1)/(2x - 1)) - 2| - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?",
"role": "user"
}
] |
-5
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {o: 3, z: 4}, shuffle the letters in the word, what is the probability of no letter 'z' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
44
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {i: 2, a: 4}, shuffle the letters in the word, what is the probability of exact 3 letter 'a' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
6
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {x: 3, f: 2, j: 1}, shuffle the letters in the word, what is the probability of exact 2 letter 'x' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
23
|
math
| null |
[
{
"content": "Calculate the rank of matrix A where A =\n[[18, 4, -15, -17, -7],\n [-18, 8, 2, 46, -15],\n [-16, 4, 41, 7, 16],\n [20, -2, -6, -29, 5],\n [-8, -60, -45, 25, 21]]",
"role": "user"
}
] |
4
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {i: 3, m: 2}, shuffle the letters in the word, what is the probability of no letter 'm' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
8
|
math
| null |
[
{
"content": "What is the rank of the matrix:\n[[-15, -12, -17, -8, -2],\n [12, -24, -5, 23, -31],\n [-18, 43, 52, -14, 21],\n [45, 8, -10, 11, 15],\n [21, 10, 14, 11, 1]]?",
"role": "user"
}
] |
3
|
math
| null |
[
{
"content": "What is the second-largest prime factor of 574603?",
"role": "user"
}
] |
71
|
math
| null |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 4(-3|x - 2| + 2) - 3 and g(x) = (2x - 2)/(2x + 1) from x = 1.6 to x = 2.4. Round to 1 decimal place.",
"role": "user"
}
] |
1.8
|
math
| null |
[
{
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 1, from vertex 2 to vertex 10, from vertex 8 to vertex 3, from vertex 5 to vertex 2, from vertex 1 to vertex 6, from vertex 8 to vertex 2, from vertex 5 to vertex 10, from vertex 3 to vertex 1, from vertex 4 to vertex 1, and from vertex 9 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?",
"role": "user"
}
] |
4
|
math
| null |
[
{
"content": "Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 2 and from vertex 3 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?",
"role": "user"
}
] |
8
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {g: 2, o: 3}, shuffle the letters in the word, what is the probability of no letter 'o' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
7
|
math
| null |
[
{
"content": "If -928*m**2 - 360 - 178*m**4 + 573*m - 524*m**3 - 1067*m**4 + 3459*m**2 - 1455*m**4 - 136*m**3 = 0, what is n + m where n/m is the second largest root in simplified form?",
"role": "user"
}
] |
13
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {b: 3, a: 3, w: 2}, shuffle the letters in the word, what is the probability of at least 1 letter 'a' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
43
|
math
| null |
[
{
"content": "What is the rank of the matrix:\n[[14, -22, 22, 26, -12],\n [-20, 30, -40, -35, 0],\n [-4, 10, 16, -13, 48],\n [16, -28, 8, 34, -48],\n [6, -8, 18, 9, 12]]?",
"role": "user"
}
] |
2
|
math
| null |
[
{
"content": "Let d(a) = -2008 + 1457*a - 54898*a**4 - 6433*a**3 + 12881*a**4 + 720*a + 20981*a**2. Let i(r) = 14006*r**4 + 2144*r**3 - 6993*r**2 - 726*r + 669. Let n(w) = -6*d(w) - 17*i(w). Suppose n(j) = 0. Express the second largest root as n/m where n and m are coprime and return n + m.",
"role": "user"
}
] |
13
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {y: 2, i: 2}, shuffle the letters in the word, what is the probability of no letter 'i' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
3
|
math
| null |
[
{
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 3 and from vertex 5 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?",
"role": "user"
}
] |
6
|
math
| null |
[
{
"content": "Compute the rank of the given 5x5 matrix:\n[[12, -20, 11, 2, 11],\n [30, -17, 22, 5, 22],\n [-24, -8, -14, -4, -14],\n [-18, 27, -16, -3, -16],\n [24, -16, 18, 4, 18]]",
"role": "user"
}
] |
2
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {k: 5, b: 2}, shuffle the letters in the word, what is the probability of at least 2 letter 'k' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
31
|
math
| null |
[
{
"content": "Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 1, from vertex 7 to vertex 10, from vertex 4 to vertex 1, from vertex 10 to vertex 6, from vertex 8 to vertex 3, from vertex 3 to vertex 1, from vertex 8 to vertex 2, and from vertex 10 to vertex 4. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?",
"role": "user"
}
] |
0
|
math
| null |
[
{
"content": "Let n(c) be the second derivative of -243/2*c**2 + 11264/5*c**6 + 2*c + 8064/5*c**5 - 783/2*c**3 - 11 - 162*c**4. For the polynomial n(b), find the second largest root in the form n/m where n and m are coprime integers. What is n + m?",
"role": "user"
}
] |
13
|
math
| null |
[
{
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 5, from vertex 2 to vertex 6, from vertex 4 to vertex 2, from vertex 2 to vertex 8, from vertex 8 to vertex 5, from vertex 2 to vertex 7, and from vertex 3 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?",
"role": "user"
}
] |
3
|
math
| null |
[
{
"content": "Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 8, from vertex 3 to vertex 1, and from vertex 9 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 7 vertices counterclockwise (so that vertex 1 ends up where vertex 8 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?",
"role": "user"
}
] |
9
|
math
| null |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(3|((-1(-5x^2 + 4x - 4) + 1)/(1(-5x^2 + 4x - 4) - 2))| - 1) and g(x) = 1|(-1(-1x - 4)^2 + 2(-1x - 4) + 4) - 3| - 1 where -10 ≤ x ≤ 10.",
"role": "user"
}
] |
2
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {z: 1, k: 5}, shuffle the letters in the word, what is the probability of no letter 'z' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
11
|
math
| null |
[
{
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 5, from vertex 10 to vertex 8, from vertex 6 to vertex 3, from vertex 7 to vertex 9, from vertex 9 to vertex 4, from vertex 9 to vertex 11, from vertex 4 to vertex 12, from vertex 10 to vertex 1, from vertex 10 to vertex 12, from vertex 4 to vertex 8, and from vertex 3 to vertex 8. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?",
"role": "user"
}
] |
3
|
math
| null |
[
{
"content": "Compute the rank of the given 4x4 matrix:\n[[10, -7, 7, 26],\n [18, -17, 11, 10],\n [-8, 9, -18, -6],\n [-16, 13, -4, -20]]",
"role": "user"
}
] |
3
|
math
| null |
[
{
"content": "Let R be the circle with center B and radius 7. F has center K and radius 9 and is externally tangent to circle R. Let C be the inversion of point B in circle F. Let H be the rotation of point C by angle pi/12 around point K. Let W be the midpoint of segment CH. Point P is the midpoint of segment WC. Line J is the angle bisector of angle WCK. Point Q is the reflection of point H across line J. Find the distance between points P and Q. Express your answer as a decimal number rounded to 2 decimal places.",
"role": "user"
}
] |
1.32
|
math
| null |
[
{
"content": "Calculate the greatest common divisor of 16533205 and 4485.",
"role": "user"
}
] |
1495
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {p: 2, h: 2, i: 1, v: 1}, shuffle the letters in the word, what is the probability of at least 1 letter 'h' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
4
|
math
| null |
[
{
"content": "Let p be the smallest prime number for which there exists a positive integer n where n^2+(2) is a multiple of p^3, and n^2+(2) is also a multiple of p^4. Find the largest positive integer m such that m^2+(2) is a multiple of p^3.",
"role": "user"
}
] |
59
|
math
| null |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -3|(2e^(1x)) - 2| - 1 and g(x) = 1x^2 + 4x - 3 between the vertical lines x = -2.6 and x = 0.2. Round to 1 decimal place.",
"role": "user"
}
] |
3.0
|
math
| null |
[
{
"content": "Let z be 227700/368*80/78. For the polynomial -270/13 - 2610/13*i - z*i**2 - 8470/13*i**3, find the second largest root in the form n/m where n and m are coprime integers. What is n + m?",
"role": "user"
}
] |
4
|
math
| null |
[
{
"content": "Let I be the circle with center O and radius 5. R has center M and radius 1 and is externally tangent to circle I. Let N be the inversion of point M in circle I. Let C be the polar line of point N relative to circle I. Let D be the reflection of line C through point O. Point E lies on line D at distance 1 from point O. Point J lies on line C at distance 11 from point O. Let B be the circle through points E, J, and N. Let H be the polar line of point O relative to circle B. Line W passes through point E parallel to line H. Lines W and C intersect at point A. Point T is the reflection of point A across line D. What is the length of segment TA? Express your answer as a decimal number rounded to 2 decimal places.",
"role": "user"
}
] |
24.0
|
math
| null |
[
{
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is red, vertex 2 is uncolored, vertex 3 is red, vertex 4 is uncolored, vertex 5 is red. How many valid colorings of the uncolored vertices exist such that the 6-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?",
"role": "user"
}
] |
8
|
math
| null |
[
{
"content": "Let XK = 1. Let P be the midpoint of segment XK. Let C be the rotation of point X by angle pi/3 around point P. Line T is the angle bisector of angle CKX. Let D be the circle with center P passing through point K. Point R is the reflection of point C across line T. Circle E is the reflection of circle D across point C. Line M is the tangent from point C to circle E. Let L be the line through point R parallel to line M. Point Q is the center of circle E. Circle F is the reflection of circle D across line L. Point V is the inversion of point Q with respect to circle F. What is the length of segment RV? Express your answer as a decimal number rounded to 2 decimal places.",
"role": "user"
}
] |
0.66
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {q: 2, r: 1, l: 1, m: 2}, shuffle the letters in the word, what is the probability of no letter 'r' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
11
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {v: 1, c: 4, n: 1}, shuffle the letters in the word, what is the probability of no letter 'n' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
11
|
math
| null |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(2sin(1πx + 2) - 2) + 3 and g(x) = (-1x - 3)/(3x) where -10 ≤ x ≤ 10.",
"role": "user"
}
] |
19
|
math
| null |
[
{
"content": "Suppose 48 = 8*n + 16*n. Express the second largest root of 28662*q**4 + 810*q + 8827*q**4 + 324 - 309*q**4 - 6768*q**n - 9594*q**3 = 0 as n/m where n and m are coprime and calculate n + m.",
"role": "user"
}
] |
13
|
math
| null |
[
{
"content": "Let IO = 5. Let Y be the rotation of point I by angle pi/12 around point O. S is placed 6 units away from O. Let Q be the angle bisector of angle SOI. Point Z is the reflection of point S across line Q. Let A be the rotation of point I by angle pi/3 around point Z. Point U is the reflection of point Y across segment ZI. Let M be the rotation of point O by angle 11pi/12 around point U. Let N be the midpoint of segment YM. Point K is the intersection of segments NO and UA. Find the length of segment IK. Express your answer as a decimal number rounded to 2 decimal places.",
"role": "user"
}
] |
1.02
|
math
| null |
[
{
"content": "There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Peter`, `Eric`, `Arnold`\n - Each person has a unique hobby: `gardening`, `photography`, `cooking`\n - Each person has a unique type of pet: `dog`, `fish`, `cat`\n\n## Clues:\n1. The person who owns a dog is in the first house.\n2. The person who loves cooking is in the second house.\n3. Eric is the person who has a cat.\n4. Arnold is the person who enjoys gardening.\n5. The person who enjoys gardening is in the third house.\n\nChoices:\n0. dog\n1. cat\n2. fish\n\nWhen answering, please specify the number corresponding to the correct choice, instead of the choice itself.",
"role": "user"
}
] |
2
|
math
| null |
[
{
"content": "Supposing you start in the bottom left (0, 0) cell of a 6x5 grid, what is minimum number of steps you can take to get to (2, 5), if your only valid moves are to move over 4 squares in one direction and over 1 squares in a perpendicular direction (at once)? (Assume you cannot move off the grid.)",
"role": "user"
}
] |
7
|
math
| null |
[
{
"content": "Find the total number of local minima and maxima for f(x) = -2e^(1(1|((-3(1x^2 + 2x - 4) - 2)/(1(1x^2 + 2x - 4) + 2)) - 2|)) - 2 where -10 ≤ x ≤ 10.",
"role": "user"
}
] |
5
|
math
| null |
[
{
"content": "There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Eric`, `Arnold`\n - Each person has a unique level of education: `associate`, `high school`\n\n## Clues:\n1. The person with a high school diploma is in the second house.\n2. Eric is not in the second house.\n\nChoices:\n0. Arnold\n1. Eric\n\nWhen answering, please specify the number corresponding to the correct choice, instead of the choice itself.",
"role": "user"
}
] |
0
|
math
| null |
[
{
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {n: 3, q: 3}, shuffle the letters in the word, what is the probability of exact 2 letter 'n' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.",
"role": "user"
}
] |
23
|
math
| null |
[
{
"content": "Calculate the rank of matrix A where A =\n[[-30, -20, -9, -17, 0],\n [-20, 17, 22, -9, -21],\n [-24, -4, 4, -14, -4],\n [-36, 6, 18, -30, 18],\n [6, 2, 1, -8, 38]]",
"role": "user"
}
] |
3
|
math
| null |
[
{
"content": "Suppose 2083*w**2 - 2950*w**2 + 166*w - 174*w**3 - 970*w**3 + 120 = 0. Express the second largest root as n/m where n and m are coprime and return n + m.",
"role": "user"
}
] |
7
|
math
| null |
[
{
"content": "Find the determinant of the following matrix. Round to the nearest integer, then give your answer modulo 1000:\n[[-5, -9, 7, 2],\n [8, 12, 11, -9],\n [-10, 7, -6, 12],\n [-12, 4, -3, 11]]",
"role": "user"
}
] |
378
|
math
| null |
[
{
"content": "Supposing you start in the bottom left (0, 0) cell of a 6x9 grid, what is minimum number of steps you can take to get to (6, 5), if your only valid moves are to move over 2 squares in one direction and over 3 squares in a perpendicular direction (at once)? (Assume you cannot move off the grid.)",
"role": "user"
}
] |
5
|
math
| null |
[
{
"content": "There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Arnold`, `Eric`, `Peter`, `Alice`\n - Each person has a unique favorite drink: `coffee`, `water`, `tea`, `milk`\n - Each mother is accompanied by their child: `Samantha`, `Meredith`, `Bella`, `Fred`\n\n## Clues:\n1. The person's child is named Meredith is not in the fourth house.\n2. The person who likes milk is the person's child is named Bella.\n3. The person's child is named Samantha is somewhere to the right of Alice.\n4. The tea drinker is the person's child is named Fred.\n5. There is one house between the coffee drinker and Peter.\n6. The person's child is named Samantha is somewhere to the left of the tea drinker.\n7. Eric is somewhere to the right of the person's child is named Fred.\n\nChoices:\n0. Alice\n1. Eric\n2. Peter\n3. Arnold\n\nWhen answering, please specify the number corresponding to the correct choice, instead of the choice itself.",
"role": "user"
}
] |
3
|
math
| null |
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