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Given the set of integers Z+, define a function g(x) = x^2 - |x-2| for all x in Z+. Is the function g(x) injective, surjective, or bijective? Explain your answer.
To determine whether the function $g(x) = x^2 - |x-2|$ is injective, surjective, or bijective, we need to analyze its properties. Injective (One-to-One): A function is injective if each output value corresponds to exactly one input value. Let's consider two cases for $x \in Z^+$: Case 1: $x \leq 2$ In this case, $|x...
\text{injective}
augmented_math
data/train_1M-00002-of-00003.parquet
106,035
902007bad332b1492320947b8625f61aef83efdb15329e085cc6065fbe0f3f22
72d6c75b4a3b8f2833e6ffe59e63ff5bcd184c29481ea2fbbf9d009efdcbeea7
816
0.441834
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Given the set of integers Z+, define a function g(x) = x^2 - |x-2| for all x in Z+. Is the function g(x) injective, surjective, or bijective? Explain your answer.", "role"...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Emily baked cookies for a party. She gave 3 cookies to her neighbor and ate 2 cookies herself. Her sister came over and ate 4 cookies. Then, her two friends arrived and each had 2 cookies. There were 5 cookies left. How many cookies did Emily bake?
To find out how many cookies Emily baked, we need to sum up all the cookies that were eaten and add them to the cookies that were left. Emily gave 3 cookies to her neighbor, ate 2 cookies herself, her sister ate 4 cookies, and her two friends ate 2 * 2 = 4 cookies. So the total number of cookies eaten is $3 + 2 + 4 + ...
18
augmented_gsm8k
data/train_1M-00000-of-00003.parquet
111,121
7ae2e2640ff36c1f9aaa0782dee914a73e54664f9644ce786c376e74bcea38f3
b8ea922f151a675d3d6d2115954040aa863300df3d79aba8b59c30f90cce0c59
240
0.769237
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Emily baked cookies for a party. She gave 3 cookies to her neighbor and ate 2 cookies herself. Her sister came over and ate 4 cookies. Then, her two friends arrived and each h...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha^2}{\beta}$ is a real number and $|\alpha + \beta| = 4.$ Find $|\alpha|.$
Let $\alpha = a + bi$ and $\beta = a - bi$, where $a$ and $b$ are real numbers. Then we have \[ \frac{\alpha^2}{\beta} = \frac{(a + bi)^2}{a - bi} \] To simplify this expression, multiply the numerator and denominator by the conjugate of the denominator, $a + bi$: \[ \frac{(a + bi)^2(a + bi)}{(a - bi)(a + bi)} = \fra...
4
augmented_math
data/train_1M-00002-of-00003.parquet
160,333
da079945d60647203d7e69da585d7c39816cfe7bf667f4dd336de9486a6c8126
ecc2a30364433d618155750ea46656512ade924b07c7e517a208c343259c3e13
672
0.44454
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let $\\alpha$ and $\\beta$ be conjugate complex numbers such that $\\frac{\\alpha^2}{\\beta}$ is a real number and $|\\alpha + \\beta| = 4.$ Find $|\\alpha|.$", "role": "...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
In triangle $ABC$, $A$ is a right angle, and the length of side $AB$ is $x$, where $x$ is a positive integer. The length of side $AC$ is $3$ units longer than the length of side $BC$. If the area of the triangle is $12$ square units, find the perimeter of the triangle.
Let the length of side $BC$ be $y$. Then the length of side $AC$ is $y + 3$. The area of the triangle is given by: \[ \frac{1}{2} \cdot x \cdot (y + 3) = 12 \Rightarrow x(y + 3) = 24 \] Since $x$ is a positive integer, we can list the possible values of $x$ and $y + 3$ that multiply to $24$: \begin{align*} x &=...
13
augmented_math
data/train_1M-00002-of-00003.parquet
41,694
eff25013aa551f1b17e7ce88de943c345fe3c2676e9c0f270f3323d530b91f2b
0ea3d593e0634bf78f4a6b1f59b087c2456c135e23a62f79fcfc4a9cfaefa25d
910
0.644559
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "In triangle $ABC$, $A$ is a right angle, and the length of side $AB$ is $x$, where $x$ is a positive integer. The length of side $AC$ is $3$ units longer than the length of si...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Alex bikes 8 miles per hour. He bikes for 45 minutes on Monday, 2 hours on Tuesday, 30 minutes on Wednesday, and 1 hour on Thursday. If he wants to bike 30 miles for the week, how many minutes should he bike on Friday?
Let's analyze the problem step by step. Alex bikes 8 miles per hour. To calculate the distance he bikes each day, we need to convert the time he bikes each day to hours and then multiply it by his speed. On Monday, he bikes for 45 minutes, which is equal to 0.75 hours. So, he bikes 0.75 * 8 = 6 miles. On Tuesday, he...
0
augmented_gsm8k
data/train_1M-00000-of-00003.parquet
76,027
8ccf8534bcec5bf986c4cf0adf0d015232db0c68a2a519382a8c098ebd403c76
1d2195383bba6402876988a4ff75c6f37ae719d98b3b29670e064fa48b88e8b8
394
0.665546
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Alex bikes 8 miles per hour. He bikes for 45 minutes on Monday, 2 hours on Tuesday, 30 minutes on Wednesday, and 1 hour on Thursday. If he wants to bike 30 miles for the week,...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
A water park has 120 people waiting to ride a water slide. The water slide has 8 tubes, and each tube can accommodate 3 people. How many times will the lifeguard need to send the tubes down the slide to give everyone in line a turn?
The water slide can accommodate 8 * 3 = 24 people at a time. To give everyone a turn, we need to divide the total number of people (120) by the number of people that can be accommodated at a time (24). 120 / 24 = 5. So the lifeguard needs to send the tubes down the slide \boxed{5} times.
5
augmented_gsm8k
data/train_1M-00000-of-00003.parquet
246,193
d7b6de84001174c4cf04e1a9339a035675b9859c5377334b3e68e3705019ef29
385c9fee1167bc4dffeda461d9ad99b9beb884ce8a4dd264f25e793ef50e7abd
176
0.563347
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "A water park has 120 people waiting to ride a water slide. The water slide has 8 tubes, and each tube can accommodate 3 people. How many times will the lifeguard need to send ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let $f(x) = 5x + 1$ and $g(x) = 2x^2 - 4$. If $f(g(x)) = 21$, find the value of $x$.
To find the value of $x$, we first need to find the expression for $f(g(x))$. \[ f(g(x)) = f(2x^2 - 4) \] Substitute $g(x) = 2x^2 - 4$ into the function $f$: \[ f(g(x)) = 5(2x^2 - 4) + 1 \] \[ f(g(x)) = 10x^2 - 20 + 1 \] \[ f(g(x)) = 10x^2 - 19 \] Now set $f(g(x))$ equal to $21$: \[ 10x^2 - 19 = 21 \] Solve for $...
2
augmented_math
data/train_1M-00001-of-00003.parquet
267,414
841f0588fefb2c24ffba5a95537a2a41ec1eb0ef972027ce23b09977517f6c97
eaa44cd01bae0cde6669fd44115db1bb0ae094561c6f83f47a2fc69cf1911b08
377
0.825324
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let $f(x) = 5x + 1$ and $g(x) = 2x^2 - 4$. If $f(g(x)) = 21$, find the value of $x$.", "role": "user" }, { "content": "To find the value of $x$, we first need to ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Maya has 36 pencils in her pencil case. She has 18 red pencils, 10 black pencils, and the rest are yellow pencils. How many yellow pencils does Maya have?
To find the number of yellow pencils Maya has, we need to subtract the number of red and black pencils from the total number of pencils. Total pencils = 36 Red pencils = 18 Black pencils = 10 Total pencils that are not yellow = Red pencils + Black pencils = 18 + 10 = 28 Number of yellow pencils = Total pencils - Tot...
8
augmented_gsm8k
data/train_1M-00000-of-00003.parquet
23,028
5b3c75b8a63280ca61079a9192cb41ffee78092eba55b57c56e0db21a98d8d50
abfef4886c4c1056ef8a01877d22db3e9950c41c07b2f68a538c33d27716517a
187
0.680411
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Maya has 36 pencils in her pencil case. She has 18 red pencils, 10 black pencils, and the rest are yellow pencils. How many yellow pencils does Maya have?", "role": "user"...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Given the quadratic equation $x^2 - 6x + 2 = 0,$ for what value of $k$ is the expression $\frac{k-3\sqrt{2}}{2}$ a root of this equation?
## Step 1: Identify the given quadratic equation and the expression for the root. The given quadratic equation is $x^2 - 6x + 2 = 0$, and the expression for the root is $\frac{k-3\sqrt{2}}{2}$. ## Step 2: Recall the formula for the roots of a quadratic equation. The roots of a quadratic equation $ax^2 + bx + c = 0$ ar...
8
augmented_math
data/train_1M-00002-of-00003.parquet
70,300
1016aa65df1be93609ead12aece430f5e73316830c7b2bfd847b3c05a59f2764
e9cdc1a521fcd558a986ace134d21f542ce2fa471ba99b764c3eadfda39b490a
920
0.517137
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Given the quadratic equation $x^2 - 6x + 2 = 0,$ for what value of $k$ is the expression $\\frac{k-3\\sqrt{2}}{2}$ a root of this equation?", "role": "user" }, { "...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Ethan is preparing gift boxes of cookies for a holiday sale. He packs 6 boxes with 8 cookies in each box. How many boxes would he have if he put 10 cookies in each box instead?
To find the number of boxes, first, we need to find the total number of cookies. Since Ethan packs 6 boxes with 8 cookies in each box, the total number of cookies is $8 * 6 = 48$. If Ethan puts 10 cookies in each box, the total number of boxes will be $48 / 10 = 4.8$. Since a fraction of a box does not make sense, we n...
4
augmented_gsm8k
data/train_1M-00001-of-00003.parquet
167,076
ed1fef9cdb03cc88b3ba0723336f544b8d01d2cddd7379c3917d876e0ce706b7
94b2e86e26ec0bec9620a27b80a08351336965c9150b47babb942b1a7beb74f6
194
0.720741
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Ethan is preparing gift boxes of cookies for a holiday sale. He packs 6 boxes with 8 cookies in each box. How many boxes would he have if he put 10 cookies in each box instead...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Maya discovered 3 beehives with 6 honeycombs each in a nearby forest and 2 hives with 4 honeycombs each in her backyard. There was also a hive with 5 honeycombs in her neighbor's garden. How many honeycombs did Maya find in total?
In the nearby forest, Maya found 3 beehives with 6 honeycombs each. So there were 3 * 6 = 18 honeycombs in the forest. In her backyard, she found 2 hives with 4 honeycombs each. So there were 2 * 4 = 8 honeycombs in her backyard. Additionally, she found a hive with 5 honeycombs in her neighbor's garden. So the total nu...
31
augmented_gsm8k
data/train_1M-00000-of-00003.parquet
176,091
ded47f853723f94d892c5708c81f8dcea09e5a25aed064b164c05ddb7f7f446a
5c6f4da33d219555fd207ad0cc0382f3ae0d99778c0b1068fd7c9677a03323e7
238
0.649241
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Maya discovered 3 beehives with 6 honeycombs each in a nearby forest and 2 hives with 4 honeycombs each in her backyard. There was also a hive with 5 honeycombs in her neighbo...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let $r(x)$ be a rational function such that the graph of $y = r(x)$ has vertical asymptotes at $x = -2$ and $x = 3$, and a horizontal asymptote at $y = 2$. If $r(-3) = 0$, find $r(x)$.
A rational function $r(x)$ with vertical asymptotes at $x = -2$ and $x = 3$ must have the form: \[ r(x) = \frac{a(x - p)}{(x + 2)(x - 3)} \] where $p$ is the root of the numerator and $a$ is a constant. Given that $r(-3) = 0$, we know that: \[ r(-3) = \frac{a(-3 - p)}{(-3 + 2)(-3 - 3)} = 0 \Rightarrow a(-3 - p) = 0 \...
\frac{2(x + 3)}{(x + 2)(x - 3)}
augmented_math
data/train_1M-00002-of-00003.parquet
329,330
a97ed549c8e6c57e404ea94b44d2494519ca4fbe3f29b892ee489065053ae461
156713bfaa91d786de9b45bca2a774290ff9413d6f6c947b1f9958b4c3f1c1bc
455
0.636771
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let $r(x)$ be a rational function such that the graph of $y = r(x)$ has vertical asymptotes at $x = -2$ and $x = 3$, and a horizontal asymptote at $y = 2$. If $r(-3) = 0$, fi...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
In a four-dimensional space, there are four mutually orthogonal vectors: $\mathbf{u} = \begin{pmatrix} 2 \\ 0 \\ 0 \\ 1 \end{pmatrix},$ $\mathbf{v} = \begin{pmatrix} 0 \\ 3 \\ 1 \\ 0 \end{pmatrix},$ $\mathbf{w} = \begin{pmatrix} 0 \\ 0 \\ 1 \\ -1 \end{pmatrix},$ and $\mathbf{x} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{p...
We have the following equation: \[ \begin{pmatrix} -4 \\ 7 \\ 6 \\ 8 \end{pmatrix} = a \begin{pmatrix} 2 \\ 0 \\ 0 \\ 1 \end{pmatrix} + b \begin{pmatrix} 0 \\ 3 \\ 1 \\ 0 \end{pmatrix} + c \begin{pmatrix} 0 \\ 0 \\ 1 \\ -1 \end{pmatrix} + d \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} \] We need to find $a$, $b$, $c...
\left(-\frac{47}{3}, \frac{7}{3}, \frac{11}{3}, \frac{82}{3}\right)
augmented_math
data/train_1M-00000-of-00003.parquet
201,030
36e55b76fe77fa88e2f7cfc887f29309f6ef8c2de64473ff3b67a19390758565
7b4b341eda47dd4743e38da7148df16ccf9de6111e5238a023d708f0d56ebae7
1,050
0.634456
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "In a four-dimensional space, there are four mutually orthogonal vectors: $\\mathbf{u} = \\begin{pmatrix} 2 \\\\ 0 \\\\ 0 \\\\ 1 \\end{pmatrix},$ $\\mathbf{v} = \\begin{pmatrix...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
What is the binary representation of the smallest positive integer that is divisible by 3 and has exactly 5 digits in base 8?
The smallest positive integer that has exactly 5 digits in base 8 is $10000_8$. To find its decimal representation, we use the place values for base 8: \[ 1 \cdot 8^4 + 0 \cdot 8^3 + 0 \cdot 8^2 + 0 \cdot 8^1 + 0 \cdot 8^0 = 4096 \] Now, we need to find the smallest multiple of 3 that is greater than or equal to 4096...
1000000000010_2
augmented_math
data/train_1M-00001-of-00003.parquet
47,708
65b7b40432e8a39db60737d669e83bebe40ef722a54d6ac669f47181ef40b0d2
2f7ece943a1e352753cfb62bbde3b8ec212514723c2825676e49b7ea25cc3bea
380
0.627241
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "What is the binary representation of the smallest positive integer that is divisible by 3 and has exactly 5 digits in base 8?", "role": "user" }, { "content": "The...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Lily and Finn are planning to decorate their house for a party. Lily has bought 3 packs of decorations while Finn has bought 2 packs. Lily's packs each contain 4 balloons and 6 streamers. Finn's packs each have 5 balloons and 8 streamers. In total, how many decorations do Lily and Finn have?
To find the total number of decorations, we first need to calculate the number of decorations each of them bought and then add them together. Lily has 3 packs, each containing 4 balloons and 6 streamers. So the total number of decorations Lily has is $3*(4+6) = 3*10 = 30$. Finn has 2 packs, each containing 5 balloon...
56
augmented_gsm8k
data/train_1M-00002-of-00003.parquet
287,581
1df279b81b2ba8bc20ce539c1396f86a775aa18fdef4f88a71b4fbbb0cc3f370
2a77c17b69726e9e0c2f6d98bfd4e132cf1bd058f716e22b49138f33575a1e4d
251
0.596269
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Lily and Finn are planning to decorate their house for a party. Lily has bought 3 packs of decorations while Finn has bought 2 packs. Lily's packs each contain 4 balloons and ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let $\mathbf{u} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{u},$ and $\mathbf{v}.$
The area of the triangle with vertices $\mathbf{0},$ $\mathbf{u},$ and $\mathbf{v}$ can be found using the formula: \[ \text{Area} = \frac{1}{2} \begin{vmatrix} 2 & 4 \\ -3 & 1 \end{vmatrix} = \frac{1}{2}(2(1) - 4(-3)) = \frac{1}{2}(2 + 12) = \frac{1}{2}(14) = \boxed{7} \]
7
augmented_math
data/train_1M-00002-of-00003.parquet
34,288
b74914eaea2f732b7c364a2064601af342a279d7b570081dfbd06fa53781ba7f
3e5a6d069fa8de9b5169b08faa2e2e3c576b191f7a5cd12b7a8c61f98e2e5d25
239
0.682197
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let $\\mathbf{u} = \\begin{pmatrix} 2 \\\\ -3 \\end{pmatrix}$ and $\\mathbf{v} = \\begin{pmatrix} 4 \\\\ 1 \\end{pmatrix}.$ Find the area of the triangle with vertices $\\mat...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
A brother and sister decided to open a joint savings account. The brother committed to saving $75 every week, while the sister committed to saving $150 every month. After 6 months of savings, they decided to invest one-third of their money in buying bonds. Each bond costs $25. How many bonds can they buy?
To solve this problem, we first need to calculate the total amount of money the brother and sister saved after 6 months. The brother saves $75 every week. There are approximately 4 weeks in a month, so he saves $75 * 4 = 300$ every month. In 6 months, he will save $300 * 6 = 1800$. The sister saves $150 every month. ...
36
augmented_gsm8k
data/train_1M-00001-of-00003.parquet
202,940
1be0cf6ca1d70544b8fe550c09cbae353aeef78224bf8e17ae24846a0a1894b1
7ad2f6911cd31b7e1a37a473d7e5f5160bbe3793e555aa2bd3ebf9ab6093220c
327
0.649792
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "A brother and sister decided to open a joint savings account. The brother committed to saving $75 every week, while the sister committed to saving $150 every month. After 6 mo...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Every time Mark layers his foam sheets, their thickness triples. If the initial thickness is 2 inches, how thick will the layered foam sheets be after 3 layerings?
Let's compute the thickness after each layering. The initial thickness is 2 inches. After the first layering, the thickness triples to $2 * 3 = 6$ inches. After the second layering, the thickness triples again to $6 * 3 = 18$ inches. After the third layering, the thickness triples again to $18 * 3 = 54$ inches. So, aft...
54
augmented_gsm8k
data/train_1M-00001-of-00003.parquet
128,282
8df592db8823f56172d753bb6c9208924ed86f456178e89f89b461aa1891df5b
448ce1960e708e340c3dab2bbcadb37c039033d7aae0cdc6d879f2016bc1867d
175
0.442166
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Every time Mark layers his foam sheets, their thickness triples. If the initial thickness is 2 inches, how thick will the layered foam sheets be after 3 layerings?", "role...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
What is the sum of the first 20 terms of the arithmetic sequence whose first term is -3 and common difference is 1/2? Express your answer as a decimal to the nearest tenth.
The sum of the first $n$ terms of an arithmetic sequence can be found using the formula: \[ S_n = \frac{n}{2} [2a + (n - 1)d] \] where $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms. Given: - First term $a = -3$ - Common difference $d = 1/2$ - Number of terms $n = 20$ Substitute t...
35.0
augmented_math
data/train_1M-00002-of-00003.parquet
48,710
248abd537cd71e72fb2097210cf6ce017787ad26f5dfd045e1252b118941adac
1d398ca3cb9272d807438a91ff0d46de0e1764428ba163f980bc7e85377c3ce1
310
0.612715
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "What is the sum of the first 20 terms of the arithmetic sequence whose first term is -3 and common difference is 1/2? Express your answer as a decimal to the nearest tenth.", ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
In $\triangle XYZ$, $X(1, 2)$, $Y(3, 5)$ and $Z(6, 1)$ are the vertices. If $\triangle XYZ$ is reflected about the line $y = 2x - 3$, what will be the coordinates of the image of point Z?
To reflect a point about a line, we can use the following steps: 1. Find the slope of the line. The line is $y = 2x - 3$, so the slope is $m = 2$. 2. Find the slope of the perpendicular line. The slope of the perpendicular line is the negative reciprocal of the slope of the original line, so $m_{\perp} = -1/2$. 3. F...
\left(-\frac{2}{5}, \frac{21}{5}\right)
augmented_math
data/train_1M-00000-of-00003.parquet
321,094
450e86e28590a3c0344e5d76491367840c4502e14b99cdf94332198b7b02a5a3
d43b8ddf68d2c43b1db913ee79ca9889b2e87de63bd478324efa764848cd722b
589
0.696265
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "In $\\triangle XYZ$, $X(1, 2)$, $Y(3, 5)$ and $Z(6, 1)$ are the vertices. If $\\triangle XYZ$ is reflected about the line $y = 2x - 3$, what will be the coordinates of the ima...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
A certain fruit salad contains a mixture of grapes and berries. If the grapes contain 80% water and the berries contain 60% water, what is the ratio of the weight of the grapes to the weight of the berries in the salad, if the overall water content of the salad is 72%?
Let the weight of the grapes be $g$ and the weight of the berries be $b$. The weight of water in the grapes is $0.8g$ and in the berries is $0.6b$. The total weight of water in the salad is $0.8g + 0.6b$. The total weight of the salad is $g + b$. We know that the overall water content of the salad is 72%, so we can wr...
\frac{3}{2}
augmented_math
data/train_1M-00002-of-00003.parquet
216,373
e894b972ee4e1b59171916d7325365f6e50df130f13e922ca2b4c2515382c425
783fdf44ba973beb83e8949f4d3a7fb1e8760a53e9a15cfaf9cbe1def5ee66fe
333
0.574784
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "A certain fruit salad contains a mixture of grapes and berries. If the grapes contain 80% water and the berries contain 60% water, what is the ratio of the weight of the grape...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Find the value of $x$ that satisfies the equation $\sin x + \sin 2x + \sin 3x = 0$, where $0 < x < \frac{\pi}{2}$.
We have the equation $\sin x + \sin 2x + \sin 3x = 0$. Use the sum-to-product identity to rewrite the equation: \[ \sin x + \sin 3x = 2 \sin 2x \cos x \] Now the equation becomes: \[ 2 \sin 2x \cos x + \sin 2x = 0 \] Factor out $\sin 2x$: \[ \sin 2x (2 \cos x + 1) = 0 \] We have two possible cases: Case 1: $\sin 2...
\frac{\pi}{3}
augmented_math
data/train_1M-00000-of-00003.parquet
261,301
eb1b71d2de91a103beb2906f6f64b9feb21b182c48410c8fe91c08a125002cac
e19da4a3e30a9448f3e6b69b8619c3194baf265bcec24543d10c752f2b615eea
381
0.513689
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Find the value of $x$ that satisfies the equation $\\sin x + \\sin 2x + \\sin 3x = 0$, where $0 < x < \\frac{\\pi}{2}$.", "role": "user" }, { "content": "We have t...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Jacob spent eight hours to bake 16 cakes. How many hours will he have spent in total if he decides to bake 24 more cakes?
To find out how many hours Jacob spent to bake one cake, we divide 8 hours by 16 cakes. So Jacob spent 8 / 16 = 0.5 hours to bake one cake. To bake 24 more cakes he will spend 0.5 * 24 = 12 hours. So in total Jacob will have spent 8 + 12 = 20 hours to bake all the cakes. Thus Jacob will have spent \boxed{20} hours in t...
20
augmented_gsm8k
data/train_1M-00001-of-00003.parquet
55,784
f8824704087a71551b33b68b75a574fcf42b1a3a8d7a2a93587842b2f0fa5003
379af4ff5aebd8a618861bd588ec9105dd2e462ae0a3a608eff85bfdb3400ab5
172
0.688933
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Jacob spent eight hours to bake 16 cakes. How many hours will he have spent in total if he decides to bake 24 more cakes?", "role": "user" }, { "content": "To find...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Maria has orange trees that produce 4 oranges each. If an orange can be sold for $2 and Maria needs $64, how many orange trees does she have to harvest?
Since each tree produces 4 oranges and each orange can be sold for $2, the total amount of money Maria can get from one tree is $2 * 4 = 8$. To get $64, Maria needs to harvest $64 / 8 = 8$ trees. Thus, Maria needs to harvest \boxed{8} trees.
8
augmented_gsm8k
data/train_1M-00001-of-00003.parquet
289,813
113a36403b9b8b804ababd92ff85b29c76106ab57414e896a8868b7cf82007c5
121392aa5a68cce26e6e84fbe7bc8d4649bfbf90e7e6ff959bd22abe4a00ba49
145
0.691223
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Maria has orange trees that produce 4 oranges each. If an orange can be sold for $2 and Maria needs $64, how many orange trees does she have to harvest?", "role": "user" ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
What is the greatest common multiple of 8 and 12 that is less than 90?
To find the greatest common multiple of 8 and 12 less than 90, we first find the least common multiple (LCM) of 8 and 12. 1. List the multiples of 8 and 12: - Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96,... - Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96,... 2. Identify the smallest number t...
72
math
data/train_1M-00000-of-00003.parquet
156,418
3d76d6fc9fbcdec3614184b836dbb7461c1da7e73a5e7db04a6c6a7cc1ca94cf
25ce86debd8dd56d69543372edd0f016dae7648a7d990fc0a76664bf3c8420ad
347
0.588443
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "What is the greatest common multiple of 8 and 12 that is less than 90?", "role": "user" }, { "content": "To find the greatest common multiple of 8 and 12 less than...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
The Franzia wine is three times as old as the Carlo Rosi, while the Carlo Rosi is four times older than the Twin Valley. Calculate the total age of the three brands of wine if the Carlo Rosi is 40 years old.
Let's break down the problem: Carlo Rosi is 40 years old. Franzia wine is three times as old as Carlo Rosi, so Franzia wine is 3 x 40 = 120 years old. Carlo Rosi is four times older than Twin Valley, so Twin Valley is 40 / 4 = 10 years old. The total age of the three brands of wine is 120 + 40 + 10 = 160 + 10 = 170 y...
170
gsm8k
data/train_1M-00002-of-00003.parquet
139,497
e9c041a151d182227107bc6a313b1f8774d183a661d51af7d51bd571710660fa
49044c9b876b54030a949b4d20eafc5a85293d977d2f112386e90c08231091f3
224
0.466379
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "The Franzia wine is three times as old as the Carlo Rosi, while the Carlo Rosi is four times older than the Twin Valley. Calculate the total age of the three brands of wine i...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
There were 1800 stickers that were evenly divided into 6 albums. Emily took an album and used one-third of the stickers for her art project. How many stickers did Emily use?
There are 1800 stickers evenly divided into 6 albums. This means each album has 1800 / 6 = 300 stickers. Emily took one album, so she had 300 stickers to begin with. She used one-third of these stickers, which is 300 / 3 = 100 stickers. So Emily used \boxed{100} stickers.
100
augmented_gsm8k
data/train_1M-00000-of-00003.parquet
260,303
80dd3be22d94248e9128e47c0715d6d241612ce7d207349864159fbb8d4c41dd
24479bde3c55b9698cf03dca71aa1ce2bb8cdd5aec558c4d56232e2ca4830873
162
0.706742
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "There were 1800 stickers that were evenly divided into 6 albums. Emily took an album and used one-third of the stickers for her art project. How many stickers did Emily use?",...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let g(n) be the product of all the divisors of a positive integer n. If g(g(n)) = n^2 + 1, then call n super-abundant. How many super-abundant positive integers are there?
We have that $g(n)$ is the product of all the divisors of $n$. If $g(g(n)) = n^2 + 1$, then $g(n)$ must be equal to $n + 1$, since $g(n)$ is the product of all the divisors of $n$ and $n + 1$ is the only number whose product of divisors is $n^2 + 1$. Now, if $g(n) = n + 1$, then $n$ must be a perfect square, since the...
1
augmented_math
data/train_1M-00002-of-00003.parquet
278,218
aaa1457f94eae6a70f0908bc086f9283efffa573bfdf9a98fb739bf3be2e4a30
3b746a95beea39088e18b7a0beec7df342e3854e3d06ccfa4f1e3e8b4e70ba71
369
0.486648
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let g(n) be the product of all the divisors of a positive integer n. If g(g(n)) = n^2 + 1, then call n super-abundant. How many super-abundant positive integers are there?", ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let $g(x)$ be a function defined as follows: \[g(x) = \begin{cases} \lceil{x}\rceil^2 &\text{if }x\text{ is a rational number}, \\ 2\lfloor{x}\rfloor-1 &\text{if }x\text{ is an irrational number}. \end{cases} \] Find $g(\sqrt[3]{11})$.
Since $\sqrt[3]{11}$ is an irrational number, we use the formula: \[ g(x) = 2\lfloor{x}\rfloor - 1 \] Find $\lfloor{\sqrt[3]{11}}\rfloor$. Given that $2^3 = 8$ and $3^3 = 27$, we know that: \[ 2 < \sqrt[3]{11} < 3 \] So, $\lfloor{\sqrt[3]{11}}\rfloor = 2$. Now, substitute this value into the formula: \[ g(\sqrt[3]{...
3
augmented_math
data/train_1M-00001-of-00003.parquet
42,277
332ea0841b0f81253b4a48355d97e3c3cecc71a2c482e4866e96896fb8c8b282
c152b31f24280c93bdf0c704982aa95107e7165f1f3d66b151f3694fd1a41a1c
278
0.539959
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let $g(x)$ be a function defined as follows:\n\n\\[g(x) =\n\\begin{cases}\n\\lceil{x}\\rceil^2 &\\text{if }x\\text{ is a rational number}, \\\\\n2\\lfloor{x}\\rfloor-1 &\\text...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Maya is biking 45 miles in 3 hours. How far will she go in 20 minutes?
To find out how far Maya will go in 20 minutes, we need to calculate her speed in miles per minute. She bikes 45 miles in 3 hours. We convert 3 hours to minutes: 3 hours * 60 minutes/hour = 180 minutes. Her speed is then 45 miles / 180 minutes = 0.25 miles per minute. Now, to find out how far she goes in 20 minutes,...
5
augmented_gsm8k
data/train_1M-00002-of-00003.parquet
316,635
4c64a6437ea4c1dba7c15508563236719341d5dff76873469f4a9576af06de80
a136f00fc42e646e2a376e7f08bbe06458d87ca78551202ea0693ef9f70da60f
201
0.603651
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Maya is biking 45 miles in 3 hours. How far will she go in 20 minutes?", "role": "user" }, { "content": "To find out how far Maya will go in 20 minutes, we need to...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let g be a quadratic function for which $g(5)=g(3)=0$. Find $g(7)-g(1)$.
Given that $g(5) = g(3) = 0$, we can write the quadratic function $g(x)$ in factored form as: \[ g(x) = a(x - 5)(x - 3) \] where $a$ is a constant. To find $g(7) - g(1)$, we substitute $x = 7$ and $x = 1$ into the function: \[ g(7) = a(7 - 5)(7 - 3) = a(2)(4) = 8a \] \[ g(1) = a(1 - 5)(1 - 3) = a(-4)(-2) = 8a \] Now,...
0
augmented_math
data/train_1M-00001-of-00003.parquet
204,910
87a4a9f956ab11f42c7c38f7bd7d4692c669f877ef7c27fbf3842c4fa14d0d21
47ce66d6be928f06cc9ea1d05482c9ac22cca1488696987f7be89b1ebd11d2e6
250
0.653026
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let g be a quadratic function for which $g(5)=g(3)=0$. Find $g(7)-g(1)$.", "role": "user" }, { "content": "Given that $g(5) = g(3) = 0$, we can write the quadratic...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
A bookshelf has 5 shelves, and each shelf can hold 8 rows of books, with each row containing 9 books. If the bookshelf is currently empty, what is the smallest number of books that need to be placed on it so that the total number of books is a multiple of 504?
Let's break down the information: * Each shelf can hold 8 rows of books. * Each row can hold 9 books. * There are 5 shelves. The total capacity of the bookshelf is: \[ 5 \text{ shelves} \times 8 \text{ rows/shelf} \times 9 \text{ books/row} = 360 \text{ books} \] We need to find the smallest number of books that, wh...
144
augmented_math
data/train_1M-00002-of-00003.parquet
149,244
d8a1831d26fe903a9e223f24e586169dea05a5642124158d11c1cf26b8f7b8ac
dc91901e8b4410118d326aa93e2ca1c843962bc76cf0ed395edfc2f7029c6bd8
381
0.725138
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "A bookshelf has 5 shelves, and each shelf can hold 8 rows of books, with each row containing 9 books. If the bookshelf is currently empty, what is the smallest number of books...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Find the smallest positive integer that is equal to the sum of its distinct prime divisors.
We can start by testing small positive integers. 1. $n = 1$: It has no prime divisors. 2. $n = 2$: It is prime, so its only prime divisor is 2. $2 = 2$. 3. $n = 3$: It is prime, so its only prime divisor is 3. $3 = 3$. 4. $n = 4$: Its prime divisors are 2. $4 \neq 2$. 5. $n = 5$: It is prime, so its only prime divisor...
6
augmented_math
data/train_1M-00001-of-00003.parquet
175,178
aa6125c72100f19ea81a87fe466f45166003cc2169b4dba7e03197595214e7e2
aa6125c72100f19ea81a87fe466f45166003cc2169b4dba7e03197595214e7e2
240
0.641964
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Find the smallest positive integer that is equal to the sum of its distinct prime divisors.", "role": "user" }, { "content": "We can start by testing small positiv...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let $a$, $b$, and $c$ be complex numbers such that $a + b + c = 1$ and $|a| = |b| = |c|$. Find $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$.
Let $r = |a| = |b| = |c|$. Then $a$, $b$, and $c$ lie on a circle of radius $r$ centered at the origin. Since $a + b + c = 1$, the centroid of the triangle formed by $a$, $b$, and $c$ is $1/3$. This implies that the origin (the center of the circle) is also the centroid of the triangle. Since the centroid and circumc...
1
augmented_math
data/train_1M-00001-of-00003.parquet
218,160
9f4eb405d9a32ed653cc27ca9b524fdb1a0bc05acf43f5714894a5abb91b7d9d
381fe03fb51c8cc178842b8f5eec4fe9ed8d55e75500d386a0fdd2e280c1acbc
359
0.627317
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let $a$, $b$, and $c$ be complex numbers such that $a + b + c = 1$ and $|a| = |b| = |c|$. Find $\\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a}$.", "role": "user" }, { ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
A rectangular prism has a length of 10 cm, a width of 5 cm, and a height of 4 cm. If the length and width are both decreased by the same amount, while the height is increased by that same amount, what is the amount of the change if the resulting volume is equal to the original volume? Express your answer as a decimal.
The volume of a rectangular prism is given by the formula $V = lwh$, where $l$ is the length, $w$ is the width, and $h$ is the height. The original volume is: \[ V_{\text{original}} = lwh = 10 \times 5 \times 4 = 200 \, \text{cm}^3 \] Let's denote the change in length and width by $x$. The new dimensions are: \[ l_{\...
1
augmented_math
data/train_1M-00002-of-00003.parquet
266,577
19caa1b673678b8f1d64386969257b85d885ab3d5ee5d738c830e672ba8e5be4
df5543afd01a204c4ab2d974d462a904cd3cdf46800261603f04aa66b92ece30
610
0.53606
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "A rectangular prism has a length of 10 cm, a width of 5 cm, and a height of 4 cm. If the length and width are both decreased by the same amount, while the height is increased ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
The equation \[\frac{x^2}{25} + \frac{(z-3)^2}{64} = 0\]describes a degenerate ellipse. Determine the point on the graph of this equation that has the smallest $x$-coordinate.
The equation of a degenerate ellipse is \[ \frac{x^2}{25} + \frac{(z-3)^2}{64} = 0 \] Since the equation is the sum of two squares, both squares must be zero to satisfy the equation. Thus, we have \[ \frac{x^2}{25} = 0 \quad \text{and} \quad \frac{(z-3)^2}{64} = 0 \] Solving these two equations gives us \[ x = 0 \qua...
(0, 3)
augmented_math
data/train_1M-00001-of-00003.parquet
53,739
03eab1e6772bc85ade83cd71d5306c5d6fa39a39ac6619fe347d83a077ca053b
bd0bf271f328363fe922723a470b465295c03955397da5c33a96ce1be3d3a19e
260
0.486186
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "The equation \\[\\frac{x^2}{25} + \\frac{(z-3)^2}{64} = 0\\]describes a degenerate ellipse. Determine the point on the graph of this equation that has the smallest $x$-coordin...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
A bookshelf contains $n$ books, each labeled with a unique integer from $1$ to $n$. The books are arranged on the shelf in a seemingly random order. However, it is known that for any two adjacent books, the difference between their labels is either $1$ or $2$. What is the maximum possible value of $n$ such that there e...
Let's consider the possible arrangements of the books. Since the difference between the labels of any two adjacent books is either 1 or 2, we can start by placing the book with label 1 on the shelf. There are two possible cases: Case 1: The book with label 2 is placed next to the book with label 1. In this case, the...
8
augmented_math
data/train_1M-00000-of-00003.parquet
91,181
aa38bdcfcab289649868bb7752d0a977cb9b20231785cc7fb43233a9fe695cb5
90d237f361120799ee1ce210d00af8b66fd8e1165d68c0ddfd8106347465e99c
713
0.528965
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "A bookshelf contains $n$ books, each labeled with a unique integer from $1$ to $n$. The books are arranged on the shelf in a seemingly random order. However, it is known that ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
A box contains 5 red balls, 3 blue balls, and 2 green balls. A ball is chosen at random, its color is noted, and then it is put back in the box. This process is repeated until a ball of each color has been chosen. What is the probability that this happens in exactly 5 draws?
Let's define the probability of drawing a red ball as $R$, a blue ball as $B$, and a green ball as $G$. The total number of balls is $5 + 3 + 2 = 8 + 2 = 10$. The probabilities are: \[ R = \frac{5}{10} = \frac{1}{2} \] \[ B = \frac{3}{10} \] \[ G = \frac{2}{10} = \frac{1}{5} \] We want to find the probability that w...
\frac{27}{200}
augmented_math
data/train_1M-00000-of-00003.parquet
147,515
cc779977dfcc79aee5f2375152813d3548d4be06262b931959871a8a2c3a2eaf
b458f05b0482eff5094ebe777f308fa5666cf798d5519427c34a2ca8e4bdb772
600
0.553403
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "A box contains 5 red balls, 3 blue balls, and 2 green balls. A ball is chosen at random, its color is noted, and then it is put back in the box. This process is repeated until...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let $a$, $b$ and $c$ be positive numbers satisfying \begin{align*} \log_c a & = bc - b + 1, \\ \log_b c & = ac - a - 1, \\ \log_a b & = ab - c - 1. \end{align*} Find $a + b + c$.
Since the problem involves logarithmic equations, we can start by using the properties of logarithms to simplify the given equations. Recall that $\log_a b = c$ is equivalent to $a^c = b$. 1. $\log_c a = bc - b + 1$ \[ \Rightarrow c^{bc - b + 1} = a \] 2. $\log_b c = ac - a - 1$ \[ \Rightarrow b^{ac - a - 1} =...
3
augmented_math
data/train_1M-00000-of-00003.parquet
275,415
a36be81a1ec8e8a76e183cb5bb0f38cbc2abcc199b824af2070cb9a429d9e620
af6b541282f2ce34c22f221cfaf42b057ee9bb80a8b10447354ce4f007548a40
712
0.466639
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let $a$, $b$ and $c$ be positive numbers satisfying\n\\begin{align*}\n\\log_c a & = bc - b + 1, \\\\\n\\log_b c & = ac - a - 1, \\\\\n\\log_a b & = ab - c - 1.\n\\end{align*}\...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Find the maximum value of $3\cos^2 \theta + 4\sin \theta - 2\cos \theta + 5$ for $0 \leq \theta \leq 2\pi$.
We can express $3\cos^2 \theta$ in terms of $\sin^2 \theta$ by using the identity $\cos^2 \theta + \sin^2 \theta = 1$. Substituting $3\cos^2 \theta = 3(1 - \sin^2 \theta)$, we get: \[ 3\cos^2 \theta + 4\sin \theta - 2\cos \theta + 5 = 3(1 - \sin^2 \theta) + 4\sin \theta - 2\cos \theta + 5 = 8 - 3\sin^2 \theta + 4\sin \...
\frac{34}{3}
augmented_math
data/train_1M-00001-of-00003.parquet
23,880
7895926e894e8fd01036c1651845c4914d8cf5db4c8367a471c968059e14a508
7ea35bc58d5aae659d2fa9b4779e92d7035bdc9649801cec1ec93de4b8d05837
598
0.465528
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Find the maximum value of $3\\cos^2 \\theta + 4\\sin \\theta - 2\\cos \\theta + 5$ for $0 \\leq \\theta \\leq 2\\pi$.", "role": "user" }, { "content": "We can expr...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Ethan has yard work to do. He has a lawn that will take 90 minutes to mow and 30 minutes to edge. He has a garden that will take 75 minutes to weed and 45 minutes to water. He has a patio that will take 60 minutes to sweep and 40 minutes to clean. If he does one task at a time, how long will it take him to complete all...
To find the total time taken by Ethan to complete all tasks, we need to add the time taken by each task. Time to mow the lawn: 90 minutes Time to edge the lawn: 30 minutes Total time for lawn: 90 + 30 = 120 minutes Time to weed the garden: 75 minutes Time to water the garden: 45 minutes Total time for garden: 75 + 45...
340
augmented_gsm8k
data/train_1M-00001-of-00003.parquet
64,672
df795f5f8b74f358daa8bbaa3d48ab376e706eab857eae793bfb1cc4219f488c
7fc77c671ff0537a9f05e81fe700598ba560954a128c08906cd7511c1f8ce866
333
0.684599
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Ethan has yard work to do. He has a lawn that will take 90 minutes to mow and 30 minutes to edge. He has a garden that will take 75 minutes to weed and 45 minutes to water. He...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
The ratio of $x+2$ to $2x+2$ is equal to the ratio of $4x+3$ to $7x+3$. What is the product of all real x which satisfy this statement? If the ratio of $x+2$ to $2x+2$ is equal to the ratio of $4x+3$ to $7x+3$, then we can write: $$\frac{x+2}{2x+2} = \frac{4x+3}{7x+3}$$ To solve for x, we can cross-multiply and sim...
$$\frac{x+2}{2x+2} = \frac{4x+3}{7x+3}$$ $$(x+2)(7x+3) = (4x+3)(2x+2)$$ $$7x^2 + 17x + 6 = 8x^2 + 14x + 6$$ $$-x^2 + 3x = 0$$ $$x(-x + 3) = 0$$ $$x = 0 \text{ or } x = 3$$ $$0 \times 3 = 0$$ The answer is $\boxed{0}.$
0
augmented_math
data/train_1M-00001-of-00003.parquet
228,647
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c400d78d5818f41f1c64d21b8c28a0c212324875cc30cce2583bb81cc26e8c99
465
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[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "The ratio of $x+2$ to $2x+2$ is equal to the ratio of $4x+3$ to $7x+3$. What is the product of all real x which satisfy this statement?\n\nIf the ratio of $x+2$ to $2x+2$ is ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
A popular coffee shop typically serves 120 customers per hour. However, on Fridays, this number increases by 50%. If the coffee shop is open for 10 hours on a Friday, how many customers will it serve that day?
The coffee shop typically serves 120 customers per hour. On Fridays, this number increases by 50%. 50% of 120 is 120 * 50 / 100 = 6000 / 100 = 60 So on Fridays, the coffee shop serves 120 + 60 = 180 customers per hour. Since the coffee shop is open for 10 hours on a Friday, it will serve 180 * 10 = 1800 customers t...
1800
augmented_gsm8k
data/train_1M-00000-of-00003.parquet
329,795
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230
0.586039
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "A popular coffee shop typically serves 120 customers per hour. However, on Fridays, this number increases by 50%. If the coffee shop is open for 10 hours on a Friday, how many...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let $T$ be the set of all possible remainders when a number of the form $3^n + 4^n$, $n$ a positive integer, is divided by 100. Let $U$ be the product of the elements in $T$. Find the remainder when $U$ is divided by 100.
We need to find the set $T$ of all possible remainders when $3^n + 4^n$ is divided by 100 for positive integers $n$. Notice that we can rewrite $3^n + 4^n$ as follows: \[ 3^n + 4^n = (3^n + 4^n) \mod 100 \] To find the remainders, we can start by computing the first few values of $3^n + 4^n$ modulo 100: \begin{ali...
25
augmented_math
data/train_1M-00002-of-00003.parquet
232,419
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731ade07a9de2843e8491154273f5964e7f4cea60085ee778211bfd6c462751b
1,004
0.54128
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let $T$ be the set of all possible remainders when a number of the form $3^n + 4^n$, $n$ a positive integer, is divided by 100. Let $U$ be the product of the elements in $T$. ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
The line $y = m$ intersects the graph of the parabola $y = 3x^2 + 2x - 4$ at exactly one point. What is $m$?
For the line $y = m$ to intersect the parabola at exactly one point, the equation $3x^2 + 2x - 4 = m$ must have exactly one solution. This means that the quadratic equation $3x^2 + 2x - (m + 4) = 0$ must have a discriminant of 0, so its solutions will be equal. The discriminant of a quadratic equation $ax^2 + bx + c ...
-\frac{13}{3}
augmented_math
data/train_1M-00001-of-00003.parquet
167,998
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b1d499076e0b0d9e9ae4299fefcf566f3157a8bc5866e44cc952bf65f9f66da9
295
0.561326
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "The line $y = m$ intersects the graph of the parabola $y = 3x^2 + 2x - 4$ at exactly one point. What is $m$?", "role": "user" }, { "content": "For the line $y = m...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let $A$, $B$, and $C$ be the vertices of an equilateral triangle, and let $\mathbf{P}$ be a matrix such that for any point $X$, $\mathbf{P}X$ is the image of $X$ under a $120^\circ$ counterclockwise rotation about the center of the triangle. Compute $\mathbf{P}^3$.
For an equilateral triangle, the center is also the centroid. Let the centroid be $O$. We know that $O$ is the average of the vertices $A$, $B$, and $C$, so $O = \frac{A + B + C}{3}$. Also, $A + B + C = 3O$. Now, let's consider the effect of $\mathbf{P}$ on the vertices $A$, $B$, and $C$. Since $\mathbf{P}$ is a $120^...
\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
augmented_math
data/train_1M-00002-of-00003.parquet
316,808
46e1312854029db12c287d2087f43c3085281724adb62c0ef6a2fbadaa836224
1d91e6fda01c30e301b4038b2de553d44986a6cef1593427c9e8675140ef1209
544
0.471784
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let $A$, $B$, and $C$ be the vertices of an equilateral triangle, and let $\\mathbf{P}$ be a matrix such that for any point $X$, $\\mathbf{P}X$ is the image of $X$ under a $12...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
A sequence of integers is defined recursively as follows: The first term is 5 and each subsequent term is obtained by appending 3 to the end of the previous term. For example, the second term is 53, the third term is 533, and the fourth term is 5333. How many terms of this sequence are divisible by 7?
To determine how many terms of the sequence are divisible by 7, we can start by examining the pattern and checking divisibility for each term. The terms are: \[ 5, 53, 533, 5333, 53333, \ldots \] We can check the divisibility of each term by 7: - 5 is not divisible by 7. - 53 is not divisible by 7. - 533 is not divis...
1
augmented_math
data/train_1M-00000-of-00003.parquet
283,553
b22fce3c10c267480bad59445f2d7905d8318102ddfe9e8a5c6595f89c5e78e6
55b8d124a4b20cc925bab2fedb6a7b0ada717a33097a45d079274100c07d8ee9
934
0.555007
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "A sequence of integers is defined recursively as follows: \nThe first term is 5 and each subsequent term is obtained by appending 3 to the end of the previous term. \nFor exam...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
500 students and 300 teachers attend a concert. The price of a teacher's ticket is three times that of a student's ticket. What is the price of a teacher's ticket if the total amount collected is $24,000?
Let's denote the price of a student's ticket as $x. Then, the price of a teacher's ticket is $3x. Given that 500 students and 300 teachers attended the concert and the total amount collected was $24,000, we can set up an equation: \[500x + 300(3x) = 24,000\] This simplifies to: \[500x + 900x = 24,000\] \[1400x = 24,000...
51.42
augmented_gsm8k
data/train_1M-00001-of-00003.parquet
66,406
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19d3be537cc6cb73097fd45800c078ac866b8a6f918808d7784f0f8e0a8a8fe2
316
0.702846
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "500 students and 300 teachers attend a concert. The price of a teacher's ticket is three times that of a student's ticket. What is the price of a teacher's ticket if the total...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Calculate the sum of the areas of two rectangles, one with a length of 8 feet and a width of $\frac{2}{3}$ yards, and the other with a length of 6 yards and a width of $\frac{3}{4}$ feet.
Convert both rectangles' dimensions to feet to facilitate the calculation. Rectangle 1: - Length = 8 feet - Width = $\frac{2}{3}$ yards = $\frac{2}{3} \times 3$ feet = 2 feet Rectangle 2: - Length = 6 yards = $6 \times 3$ feet = 18 feet - Width = $\frac{3}{4}$ feet Now calculate the area of each rectangle: Rectangl...
29.5
augmented_math
data/train_1M-00001-of-00003.parquet
114,733
87edcf5d187fba96684bba8aa49682423c4884665eb4fc2a28a53af217e136fc
21ba1932bfbbf4b07c325a4c46979dbc588d556a5043d111f5810731052c8f9c
264
0.521619
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Calculate the sum of the areas of two rectangles, one with a length of 8 feet and a width of $\\frac{2}{3}$ yards, and the other with a length of 6 yards and a width of $\\fra...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Real numbers $a$ and $b$ have a harmonic mean of 12 and a geometric mean of $\sqrt{50}$. Find $a^2+b^2$.
The harmonic mean of two numbers $a$ and $b$ is given by $\frac{2ab}{a+b}$ and their geometric mean is $\sqrt{ab}$. We are given that: \begin{align*} \frac{2ab}{a+b} &= 12 \\ \sqrt{ab} &= \sqrt{50} \end{align*} From the second equation, we have $ab = 50$. Substituting $ab = 50$ into the first equation, we...
\frac{275}{9}
augmented_math
data/train_1M-00001-of-00003.parquet
271,584
333b4eba01f31d71b980d1289b4bbc80bccb574da66f956944c1535572e54271
ed8d70e7ce046247f82ef488742aa860263c85530a8709d08cf58de86d5c4b7a
434
0.452031
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Real numbers $a$ and $b$ have a harmonic mean of 12 and a geometric mean of $\\sqrt{50}$. Find $a^2+b^2$.", "role": "user" }, { "content": "The harmonic mean of tw...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
In late August, the water level in the reservoir is 8 feet deep. By early September, the water level has risen by 6 feet. By early October, the water level is twice as deep as it was in early September. How many feet deep is the water level in the reservoir by early October?
In late August, the water level is 8 feet deep. By early September, the water level has risen by 6 feet, making it 8 + 6 = 14 feet deep. By early October, the water level is twice as deep as it was in early September, making it 2 * 14 = 28 feet deep. So, the water level in the reservoir by early October is \boxed{28} ...
28
augmented_gsm8k
data/train_1M-00002-of-00003.parquet
327,374
d0d39ced73db4bf0a4d5c11297a84eb37f3247eddf9aeefff992d54fdd3edb50
b356e90fcad782d9aade6aeb6108cb8f25f380a5adc47fa61993b20041bb8c7d
192
0.642936
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "In late August, the water level in the reservoir is 8 feet deep. By early September, the water level has risen by 6 feet. By early October, the water level is twice as deep as...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
The perimeter of a right triangle is 84 cm. If the length of the hypotenuse is increased by 5 cm, and the lengths of the legs remain the same, the perimeter of the new triangle is 87 cm. What is the length of the hypotenuse of the original triangle?
Let the lengths of the legs of the original triangle be $x$ and $y$, and let the length of the hypotenuse be $z$. The perimeter of the original triangle is $84$, so: \[ x + y + z = 84 \quad \text{(1)} \] The perimeter of the new triangle is $87$, and the hypotenuse is increased by $5$ cm, so: \[ x + y + (z + 5) = 87 \...
30
augmented_math
data/train_1M-00001-of-00003.parquet
23,500
6591f88da4ef0a69781cfff6aac1e1606e24c458a16c8128d32b9ce27e75e5e5
c1321db3df489466931f4aa7c3f9e1ea3609bc7bb4767772b30e4a3327eee265
668
0.649648
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "The perimeter of a right triangle is 84 cm. If the length of the hypotenuse is increased by 5 cm, and the lengths of the legs remain the same, the perimeter of the new triangl...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Ana can assemble toy cars at a rate of one every 3 minutes for the first 2 hours of her shift. After that, she can assemble them at a rate of one every 5 minutes. How many toy cars can Ana assemble in a 6-hour shift?
To find the number of toy cars Ana can assemble in 6 hours, let's calculate the number of toy cars she can assemble in the first 2 hours and in the remaining 4 hours separately and then add them up. In the first 2 hours, she can assemble 1 toy car every 3 minutes. 2 hours is 120 minutes. So in 120 minutes she can ass...
88
augmented_gsm8k
data/train_1M-00002-of-00003.parquet
121,138
9ef34ec7f16693c271652693606609c7caf88f643303e352f2b34649612b76db
627def1a7d45ce523c76ef46776879020e6e564eb7329a2cf909d35a81bacfdf
289
0.732268
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Ana can assemble toy cars at a rate of one every 3 minutes for the first 2 hours of her shift. After that, she can assemble them at a rate of one every 5 minutes. How many toy...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Alexandra purchases a set of dumbbells that weigh 40 pounds each. As she progresses in her workout routine, she decides to increase the weight by 75%. The weights can be upgraded with 5-pound metal plates. Each plate costs $8, and a 15% discount is applied if more than 5 plates are purchased. How much will it cost Alex...
To increase the weight by 75%, Alexandra needs to add 40 * 0.75 = 30 pounds to each dumbbell. Since each dumbbell needs to be upgraded by 30 pounds and each plate weighs 5 pounds, she will need 30 / 5 = 6 plates per dumbbell. Since she has 2 dumbbells (a set), she will need 6 * 2 = 12 plates in total. The total cost of...
81.6
augmented_gsm8k
data/train_1M-00001-of-00003.parquet
178,436
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4b972f3c0f5717f4b3bba620e455871d71a8b3027daaa98a40492e7770189cee
321
0.588068
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Alexandra purchases a set of dumbbells that weigh 40 pounds each. As she progresses in her workout routine, she decides to increase the weight by 75%. The weights can be upgra...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
The measures of the interior angles of a convex hexagon form an increasing geometric sequence. How many such sequences are possible if the hexagon is not equiangular and all of the angle degree measures are positive integers less than 180 degrees?
## Step 1: Recall the sum of interior angles of a hexagon The sum of the interior angles of a hexagon is given by the formula (n-2)*180 degrees, where n is the number of sides. For a hexagon, this is (6-2)*180 = 4*180 = 720 degrees. ## Step 2: Express the sum of the interior angles as a geometric series Let's denote t...
5
augmented_math
data/train_1M-00002-of-00003.parquet
53,520
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5ba7bcbabc8a024e611a17e0ee7acda98f2e3d278867fbfa4e9df23593b11739
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[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "The measures of the interior angles of a convex hexagon form an increasing geometric sequence. How many such sequences are possible if the hexagon is not equiangular and all o...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Given a sequence of real numbers $a_1, a_2,..., a_n$ satisfying $\sum_{i=1}^{n} a_i^2 = 1$, find the largest positive integer $n$ such that for all sequences $a_1, a_2,..., a_n$ and for any positive real number $x$, \[(a_1x)^n + (a_2x)^n +... + (a_nx)^n \ge x\]
Using the power mean inequality, we have: \[ \frac{(a_1x)^n + (a_2x)^n + \dots + (a_nx)^n}{n} \ge \left( \frac{a_1^2x^2 + a_2^2x^2 + \dots + a_n^2x^2}{n} \right)^{\frac{n}{2}} \] Simplifying and using the given condition $\sum_{i=1}^{n} a_i^2 = 1$, we get: \[ \frac{(a_1x)^n + (a_2x)^n + \dots + (a_nx)^n}{n} \ge \left(...
2
augmented_math
data/train_1M-00000-of-00003.parquet
105,007
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c4be68bf80ecd6b2bc7d43057b1d565a41549fc4221b936a6a14f80378c60c0f
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[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Given a sequence of real numbers $a_1, a_2,..., a_n$ satisfying $\\sum_{i=1}^{n} a_i^2 = 1$, find the largest positive integer $n$ such that for all sequences $a_1, a_2,..., a...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
There are 5 buses in a fleet and each bus has 30 seats. If each bus could accommodate 8 more passengers, how many passengers would fill up 2 fleets?
Each bus can accommodate 30 + 8 = 38 passengers. Since there are 5 buses in each fleet, each fleet can accommodate 38 * 5 = 190 passengers. Two fleets can accommodate 190 * 2 = 380 passengers. Thus, it would take \boxed{380} passengers to fill up 2 fleets.
380
augmented_gsm8k
data/train_1M-00000-of-00003.parquet
86,956
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[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "There are 5 buses in a fleet and each bus has 30 seats. If each bus could accommodate 8 more passengers, how many passengers would fill up 2 fleets?", "role": "user" }, ...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
A basketball team has five centers and eight power forwards. The team also has three times as many small forwards as power forwards, and the rest of the players are shooting guards. If the team has 50 players, how many shooting guards are on the team?
Let's break down the number of players by position: * Centers: 5 * Power forwards: 8 * Small forwards: 3 times the number of power forwards, so 3 * 8 = 24 * Total number of centers, power forwards, and small forwards: 5 + 8 + 24 = 13 + 24 = 37 Since the team has 50 players, the number of shooting guards is: 50 - 37 ...
13
augmented_gsm8k
data/train_1M-00000-of-00003.parquet
230,280
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[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "A basketball team has five centers and eight power forwards. The team also has three times as many small forwards as power forwards, and the rest of the players are shooting g...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Emily is three times younger than her grandmother, who is 6 years older than Emily's mother. If Emily's grandmother is 54 years old, what's the age difference between Emily and her mother?
Let's break it down step by step: 1. Emily's grandmother is 54 years old. 2. Emily's grandmother is 6 years older than Emily's mother, so Emily's mother is 54 - 6 = 48 years old. 3. Emily is three times younger than her grandmother, so Emily is 54 / 3 = 18 years old. 4. The age difference between Emily and her mother ...
30
augmented_gsm8k
data/train_1M-00002-of-00003.parquet
96,775
262ed5d7ace6903c1da738eec02735df9480fe892e161d0f9832a877eb2503cf
1ce27ff64076f3c421eb78cd7b8758943dd33215184d2e4ae119bed466ae49f9
200
0.743941
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Emily is three times younger than her grandmother, who is 6 years older than Emily's mother. If Emily's grandmother is 54 years old, what's the age difference between Emily an...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
In triangle $ABC$, $\angle A = \angle C = 60^\circ$, $AB = 1$, and $BC = 2$. Compute the sine of the angle formed by the intersection of the angle bisector of $\angle A$ and the perpendicular bisector of side $BC$.
Since $\angle A = \angle C = 60^\circ$, triangle $ABC$ is isosceles, so $AC = AB = 1$. Let $M$ be the midpoint of $BC$, and let $D$ be the point where the angle bisector of $\angle A$ intersects $BC$. Then, $\triangle AMD$ is a right triangle with $\angle MAD = 30^\circ$, since the angle bisector of $\angle A$ cuts it ...
\frac{1}{2}
augmented_math
data/train_1M-00001-of-00003.parquet
221,739
7607e94450b0bd17ab05984abf8d6dcede3aae9d6560f914439ce566a2c8ea25
fc350ea54950e08cf3a178c6bb48efab317a34d8955c4d0bac4afac2420556d1
233
0.563414
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "In triangle $ABC$, $\\angle A = \\angle C = 60^\\circ$, $AB = 1$, and $BC = 2$. Compute the sine of the angle formed by the intersection of the angle bisector of $\\angle A$ a...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
In the equation $x^3 + 3x^2 - 2x - 8 = 0$, one root is an integer, and the product of the other two roots is $-3$. Determine the value of the integer root.
Let the integer root be $r$. The other two roots can be denoted as $x_1$ and $x_2$, and their product is $x_1 \cdot x_2 = -3$. The sum of the roots of a cubic equation $ax^3 + bx^2 + cx + d = 0$ is given by $-\frac{b}{a}$. For our equation, this means the sum of the roots is: \[ r + x_1 + x_2 = -3 \] The product of t...
-2
augmented_math
data/train_1M-00002-of-00003.parquet
327,423
ee28d05122dd6eebf89eeffdff916d288753e245e272d90cf657d55a3cba2683
f1f97e14d827c952b4a45989472d417d668fb001596a49fa8f1a67e485b7416a
873
0.629672
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "In the equation $x^3 + 3x^2 - 2x - 8 = 0$, one root is an integer, and the product of the other two roots is $-3$. Determine the value of the integer root.", "role": "user...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
If $x, y,$ and $z$ are three non-zero positive real numbers such that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$ and $x^2 + y^2 + z^2 = 144$, compute $\frac{xy + yz + zx}{xyz}$
## Step 1: We are given the equation $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$ and $x^2 + y^2 + z^2 = 144$. We need to use these equations to find $\frac{xy + yz + zx}{xyz}$. ## Step 2: First, notice that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{xy + yz + zx}{xyz}$. This is because we can rewrite each frac...
1
augmented_math
data/train_1M-00000-of-00003.parquet
93,594
01ff8edd1274678f0ab78762bbdc639c68781c5e94103a3b1eaf81c413945cb3
a51e171ba31878d22fd511cb1468f79f4386f285d8ff29ee2c9f87984d2ccdac
773
0.699445
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "If $x, y,$ and $z$ are three non-zero positive real numbers such that $\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = 1$ and $x^2 + y^2 + z^2 = 144$, compute $\\frac{xy + yz + z...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Let $a$ and $b$ be two randomly chosen real numbers such that $a^2 + b^2 = 1.$ Find the probability that the equation $ax^2 + bx + 1 = 0$ has no real roots.
We need to find when the equation $ax^2 + bx + 1 = 0$ has no real roots. This happens when the discriminant of the quadratic equation is less than 0: \[ b^2 - 4a(1) < 0 \Rightarrow b^2 - 4a < 0 \] We also know that $a^2 + b^2 = 1$. We can use this condition to eliminate $b^2$ from the inequality: \[ b^2 = 1 - a^2 \] ...
\frac{3 - \sqrt{5}}{2}
augmented_math
data/train_1M-00001-of-00003.parquet
92,460
3baae177becd28b03af3886f8e689c43d3311923c0dbc695fc6cbb54187482f5
1bc7034683ebfb804bc3e46052f6b5ddbc118b3dcbac0624f693e0a2bc201519
634
0.505915
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Let $a$ and $b$ be two randomly chosen real numbers such that $a^2 + b^2 = 1.$ Find the probability that the equation $ax^2 + bx + 1 = 0$ has no real roots.", "role": "use...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
Two teams of five players each are participating in a friendly competition to see which team can roll the highest sum with a set of five six-sided dice. The team with the highest sum wins. If the competition is held multiple times, what is the probability that the same team wins three times in a row? Express your answe...
The probability of a team winning once is $\frac{1}{2}$, since there are only two teams. The probability of a team winning three times in a row is: \[ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \] So, the answer is: \[ \boxed{0.13} \]
0.13
augmented_math
data/train_1M-00001-of-00003.parquet
117,159
abb6233cf43ae7a45bd481865c8a06881b75cd1d85c2ce924d8a761e15611002
abb6233cf43ae7a45bd481865c8a06881b75cd1d85c2ce924d8a761e15611002
195
0.559674
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "Two teams of five players each are participating in a friendly competition to see which team can roll the highest sum with a set of five six-sided dice. The team with the high...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
John has a habit of going to the gym every third day. If he starts going to the gym on a Monday, what day of the week will he be going to the gym on his 20th visit?
John goes to the gym every third day. So, we can make a list of days he goes to the gym: - 1st visit: Monday - 2nd visit: Thursday - 3rd visit: Sunday - 4th visit: Wednesday - 5th visit: Saturday - 6th visit: Tuesday - 7th visit: Friday - 8th visit: Monday - 9th visit: Thursday - 10th visi...
\text{Tuesday}
augmented_math
data/train_1M-00000-of-00003.parquet
159,270
fe980f5e0286885faf936e711119b3141e78ae96df5940536287bce5b83ab3cf
86bd1f3c7102315e6063d08ccdf9839c2ee502f1a429712c50585070570cb81c
309
0.604985
[ { "content": "Solve the math problem step by step. Put the final answer in \\boxed{...}.", "role": "system" }, { "content": "John has a habit of going to the gym every third day. If he starts going to the gym on a Monday, what day of the week will he be going to the gym on his 20th visit?", "rol...
nvidia/OpenMathInstruct-2
469216e3f46f4dacf476b382e192485ea51a143e
train_1M
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Qwen2.5 OpenMathInstruct-2 SFT 50K

A 50,000-example, short-context mathematics SFT dataset extracted from NVIDIA OpenMathInstruct-2. It was prepared for instruction tuning mssfj/qwen25-0.5b-finemath-4plus and subsequent GSM8K and MATH-500 benchmarking, including controlled comparisons with TDT. This is an extracted and filtered dataset, not a newly generated set of solutions.

Composition

MATH-related examples make up 70% of each split; GSM8K-related examples make up 30%. These categories describe the upstream problem_source, not a newly assigned difficulty label. Each selected numeric problem template occurs only once across the entire dataset.

Split Examples MATH-related GSM8K-related Total formatted tokens Maximum tokens
train 49,000 34,300 14,700 21,212,482 1854
validation 1,000 700 300 427,265 1868

The token limit is 2,048 per complete conversation, including the system prompt, user problem, full assistant solution, and Qwen chat-template delimiters. Problems and solutions are not truncated. The published solution text is retained apart from trimming outer whitespace; no answer line or solution was regenerated or appended.

Source and provenance

  • Source: nvidia/OpenMathInstruct-2, split train_1M.
  • Source revision: 469216e3f46f4dacf476b382e192485ea51a143e.
  • Upstream fields: problem, generated_solution, expected_answer, problem_source.
  • Upstream teacher model: Llama3.1-405B-Instruct, as documented by NVIDIA.
  • Tokenizer: mssfj/qwen25-0.5b-finemath-4plus, revision 0600e6f3b418e77d7323b8934451ac66d49a1185, using its Qwen2.5 tokenizer and chat template.
  • Selection and split seed: 42.

The source paper is OpenMathInstruct-2: Accelerating AI for Math with Massive Open-Source Instruction Data, by Shubham Toshniwal, Wei Du, Ivan Moshkov, Branislav Kisacanin, Alexan Ayrapetyan, and Igor Gitman (2024).

Selection procedure

  1. Scan all 1,000,000 rows in the pinned train_1M files in sorted filename order.
  2. Reject empty required fields, unknown source categories, problems containing [asy] diagram markup, and problem/solution text containing chat control-token strings.
  3. Require a balanced final \boxed{...} or \fbox{...} and compare it with expected_answer after limited whitespace and LaTeX-format normalization. This is a string-consistency check, not verification of the mathematical reasoning.
  4. Deduplicate normalized problems. Choose the shortest answer-consistent solution by character length, with a seeded SHA-256 tie-breaker.
  5. Render complete conversations with the pinned chat template and reject lengths greater than 2,048 tokenizer tokens.
  6. Deduplicate numeric templates. Normalization uses NFKC, lowercase text, removes selected LaTeX presentation commands, and keeps letter, number, and math-operator tokens. Numeric templates replace numeric literals with NUM. Keep the problem with the lowest seeded SHA-256 rank per template.
  7. Form a deterministic candidate pool of up to 70,000 MATH-related and 30,000 GSM8K-related examples. Apply the benchmark checks below to this candidate pool.
  8. Select 35,000 MATH-related and 15,000 GSM8K-related surviving examples by seeded hash. Assign 700 MATH-related and 300 GSM8K-related examples to validation using a separate seeded hash ordering, leaving 49,000 training examples. Shuffle file order by another deterministic hash.
  9. Retokenize every exported example to verify its saved length and check that normalized problem IDs and numeric-template group IDs are disjoint between splits.

This favors short complete solutions and template diversity. It is not a representative random sample of the original corpus or its difficulty distribution. Derived problems whose templates change substantially may still occur in both train and validation.

Benchmark overlap filtering

Reference sets:

  • openai/gsm8k, main, test: 1,319 problems, revision 740312add88f781978c0658806c59bc2815b9866.
  • HuggingFaceH4/MATH-500, test: 500 problems, revision 6e4ed1a2a79af7d8630a6b768ec859cb5af4d3be.

An example is removed when its problem meets any of these conditions:

  • Exact match after problem normalization.
  • Exact numeric-template match, targeting number-substitution variants.
  • RapidFuzz normalized-problem ratio of at least 90 against any benchmark problem.
  • Maximum sentence-embedding cosine similarity of at least 0.94.
  • Maximum cosine similarity of at least 0.90, together with a normalized-problem ratio of at least 70 or a numeric-template ratio of at least 85 for that nearest benchmark.

Embedding model: sentence-transformers/all-MiniLM-L6-v2, revision 1110a243fdf4706b3f48f1d95db1a4f5529b4d41, with normalized embeddings and a maximum input length of 512 embedding-model tokens. Longer questions are truncated for this similarity check only, not in the SFT data. These thresholds are heuristics: they can remove related but distinct problems and miss paraphrases or mathematical equivalents. This dataset does not guarantee complete semantic decontamination.

The filtering checks problem text, not all mentions within generated solutions. It also does not undo possible benchmark exposure during the base model's prior FineWeb-Edu or FineMath pretraining, or during the teacher model's training. GSM8K training problems remain eligible; only the named evaluation sets are exclusion references. MATH test problems outside MATH-500 were not an additional exclusion set.

decontamination_audit.jsonl records removed candidate IDs, matched benchmark row IDs, and matching scores. It does not reproduce benchmark questions or answers. selection_stats.json records filter counts, split composition, source revisions, template/script fingerprints, token statistics, and Parquet SHA-256 checksums.

Fields

Field Description
messages System/user/assistant messages for SFT
problem Selected upstream problem
generated_solution Selected upstream solution
expected_answer Upstream reference or majority-vote answer
problem_source math, augmented_math, gsm8k, or augmented_gsm8k
problem_id SHA-256 of the normalized problem
problem_group_id SHA-256 of its numeric template
token_count Complete conversation length under the pinned tokenizer/template
source_dataset, source_revision, source_split Upstream provenance
source_file, source_row Source Parquet path and zero-based row number
nearest_benchmark_cosine Maximum embedding similarity to the exclusion references; audit metadata only

Usage

from datasets import load_dataset

dataset = load_dataset("mssfj/qwen25-openmathinstruct2-sft-50k")
train = dataset["train"]
validation = dataset["validation"]
print(train[0]["messages"])

Use messages for training and keep the reference/audit fields out of the prompt. The system prompt is:

Solve the math problem step by step. Put the final answer in \boxed{...}.

Render with the pinned Qwen chat template (chat_template.jinja is also supplied), train with loss on the assistant response and its turn-ending token, and keep the input embeddings and output weights untied to match the base checkpoint. At inference, stop on the Qwen assistant turn terminator <|im_end|> as well as the base EOS token. Use this dataset's validation split for model selection; keep GSM8K test and MATH-500 for final evaluation. For TDT comparisons, fix the dataset revision and examples, initial weights, training token budget, template, and evaluation settings.

Reproduction

The included prepare_openmath_sft.py downloads pinned input revisions and rebuilds the Parquet files and selection statistics. A CUDA GPU accelerates embedding checks. Install a suitable PyTorch build for your system, then:

pip install -r requirements.txt
python prepare_openmath_sft.py \
  --output ./rebuilt-openmath-sft-50k \
  --total 50000 --validation 1000 --math-ratio 0.70 \
  --max-tokens 2048 --seed 42 --candidate-multiplier 2

environment.json records the generation environment, including the CUDA PyTorch build. Different numerical environments can change borderline embedding decisions. Compare output checksums and statistics when reproducing the dataset. The build script creates data and statistics; this README and dependency/environment files accompany the published release separately.

Build statistics

Counters refer to the stage where each filter is applied; they are not all counts over the same population. Similarity checks apply to the hash-selected candidate pool.

Counter Count
source_rows 1,000,000
diagram_or_control_token 21,843
missing_or_mismatched_final_box 993
duplicate_normalized_problem 386,845
empty_field 733
unique_answer_checked_candidates 589,586
over_token_limit 355
duplicate_numeric_template 28,173
eligible_math 481,746
eligible_gsm8k 79,312
similarity_candidate_pool 100,000
benchmark_fuzzy_ratio_90 81
benchmark_numeric_template_exact 11
embedding_input_over_512_tokens 169
benchmark_semantic_similarity 11

License and attribution

This derived dataset follows the upstream OpenMathInstruct-2 dataset's CC BY 4.0 license. Attribute NVIDIA and the OpenMathInstruct-2 authors, and identify this release as an extracted subset with formatting, consistency, length, deduplication, benchmark overlap filtering, and split changes. Refer to the upstream dataset card for the original release and citation.

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