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khmer_math_doc_001141 | [
"images/khmer_math_doc_001141.png"
] | [
{
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"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
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"value": "$$= \\lim_{x \\to 2} (x^2 +$2x$+ 4)(\\sqrt{x + 2} + 2)$$\n\nα. $\\lim_{x \\to 0} \\frac{\\cos x - 1}{\\sin^2 x}$(ααΆαααΆααα·αααααα$\\frac{0}{0}... |
khmer_math_doc_001483 | [
"images/khmer_math_doc_001483.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "$$f'(x) = \\left[-x + 4 + \\ln\\left(\\frac{x+1}{x-1}\\right)\\right]'$$\n\nααααΆα $\\frac{x+1}{x-1} > 1$(ααΆαααΈαααα αΆαααΆαααΎ) ααααΆαα$\\ln$... |
khmer_math_doc_001130 | [
"images/khmer_math_doc_001130.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "# α
ααααΎα αα·ααααΆααΆααα·ααα·ααααΆ ααΆαααα»αααααΆα α’α α‘α¨\n\nααααΆα αααα»ααααα $2$, αααα α$4$αα·αααα$4$\n\n$$= \\frac{4!}{(4 - 2)! \\times 2!} \\... |
khmer_math_doc_000482 | [
"images/khmer_math_doc_000482.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "αααα·ααααΆαααααααΆααααΈααΆαααααΆα $C$αα·ααααΈααΆααααααΆαα$D$\nααΆαα±αα αααααΆαααααααΆααααΈααΆα $D : y = -x + 4$ ααΆ\n\nα. ααααααααΈααΆααααααΆαααααααα... |
khmer_math_doc_001284 | [
"images/khmer_math_doc_001284.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "ααΆαα±αα $S = \\sqrt{(-1)^2 + (-1)^2 + 2^2} = \\sqrt{6}$ α―αααΆαααααααα‘αΆ\nααΌα
αααα αααααααααα‘αΌααααΆα $S = \\sqrt{6}$ α―αααΆαααααααα‘αΆ\n$A(2, 2, 1)$αα·... |
khmer_math_doc_000261 | [
"images/khmer_math_doc_000261.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "ααΌα
ααα αααΈααΆααααααΆαααααααααΆα $(C)$ααααα$A$ααΊ$y =$2x$- 3$ α\n\nα’. αααα αΆαααΆαααΈααΆααααααΆαααΈααΆαα’αααΈα\nααΌα
αααα αααΈααΆααααααΆαα±αα ααΆαααΈααΆα... |
khmer_math_doc_000542 | [
"images/khmer_math_doc_000542.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "B. 9. αα·ααααΆααΈααΈαααα’αα»αααα $f$ααααα$0$αα·α$+\\infty$\nααααΆα $\\lim_{x \\to 0}$f($x$)$= \\lim_{x \\to 0} (x + 1 - \\frac{\\ln x}{x}... |
khmer_math_doc_001466 | [
"images/khmer_math_doc_001466.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "$$f'(x) = \\frac{e^{2x} +$6e$^x + 9 -$12e$^x}{(e^x + 3)^2} = \\left(\\frac{e^x - 3}{e^x + 3}\\right)^2$$\n\n$$\\begin{case... |
khmer_math_doc_001352 | [
"images/khmer_math_doc_001352.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "$$\\begin{cases}$4a$= 1 \\$4b$= 2 \\\\ 2a + 4c = -1 \\end{cases} \\Rightarrow \\begin{cases} a = 1/4 \\\\ b = 1/2 \\\\ c = -3/8 \\end{cases}... |
khmer_math_doc_001645 | [
"images/khmer_math_doc_001645.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "$$\\overrightarrow{BC} = (1 - 7, 5 + 1, -2 + 2) = \\boxed{(-6, 6, 0)}$$\n\nααΌα
ααα ααΌα’ααααααααααΌα $V(h, k)$α¬$\\boxed{V(-2, -2)}$ α\nc. αααα αΆα... |
khmer_math_doc_000721 | [
"images/khmer_math_doc_000721.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "α αΎα $(ABC)$ααΆααααΆαα
ααα»α
$A(1, -1, 4)$\nααΆαααΆα $x_0 = 1, z_0 = -1, z_0 = 4$\n\n$$\\mathbf{K} = \\int_{0}^{\\frac{\\pi}{4}} \\left[\... |
khmer_math_doc_001507 | [
"images/khmer_math_doc_001507.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "α
ααα»α
$A, B$αα·α$C$ αα·ααα
ααΎαααααΆαααααα½α αα»αααααΆαα\nααΌα’ααααααααααΌα $V(h \\pm a , k)$\nααΌα
ααα$αα
ααα»α
A ,B αα·α C αα·ααα
ααΎαααααΆαααααα½α$\nc. αααα αΆα... |
khmer_math_doc_001881 | [
"images/khmer_math_doc_001881.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "$$\\frac{x^2 + 6x + 5}{(x - 2)(x^2 + x + 1)} = \\frac{m}{x - 2} + \\frac{nx + p}{x^2 + x + 1} (x \\neq 2)$$\n\n$ab = 8\\left( \\cos \\frac{7\\pi}{12} +... |
khmer_math_doc_001353 | [
"images/khmer_math_doc_001353.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "(αα·α)\nααΌα
αααα\n$d_1$ααΆα’αΆαααΈαααΌααααααααααααΆα$(C)$ααααα$+\\infty$\nb. αααα αΆαααΆα
ααααααααα $x$ααΎ$(1, +\\infty)$κΈ$\\frac{x+1}{x-1} > ... |
khmer_math_doc_000912 | [
"images/khmer_math_doc_000912.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "$$\\overrightarrow{AC} = (1 - 1, 5 + 1, -2 - 4) = \\boxed{(0, 6, -6)}$$\n\n$$\\overrightarrow{BC} = (1 - 7, 5 + 1, -2 + 2) = \\bo... |
khmer_math_doc_000810 | [
"images/khmer_math_doc_000810.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "ααααΆα ααΌα’ααααααααα
ααα»α
$B(x_B, y_B)$\nααα αααααΆαααααααααα $B$ααααααΉααααααΆαα$y = x$\nααΌα
ααα ααΌα’ααααααααα
ααα»α
ααααΆαααΊ $B(2, 1 - \\ln 2)$ ... |
khmer_math_doc_000038 | [
"images/khmer_math_doc_000038.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "ααααααΈ α‘ α ααααΎααΆαααααααααααΆ\nααααΆα $z_1 = 3 +$3i$\\sqrt{3}$αα·α$z_2 = \\sqrt{3} + i$\nααααΆα $u = \\frac{\\pi}{3} - x$ααα$ αx = \\... |
khmer_math_doc_001598 | [
"images/khmer_math_doc_001598.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "$$= \\frac{1 + 2e^x + (e^x)^2 - 4e^x}{(1 + e^x)^2} = \\frac{1 - 2e^x + (e^x)^2}{(1 + e^x)^2}$$\n\n$$= \\frac{(1-e^x)^2}{(1+e^x)^2} = \\left(\\frac{1-e^... |
khmer_math_doc_000251 | [
"images/khmer_math_doc_000251.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "αααΈααΆααααΆαααΆαααααααααααααΆααααααΌαααααΆαααΆα\n\n$$(AB) : \\begin{cases} x = x_0 + at \\\\ y = y_0 + bt \\\\ z = z_0 + ct \\end{cases}, t \... |
khmer_math_doc_001381 | [
"images/khmer_math_doc_001381.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "ααα
αΆαααααααΌα $3$ α
ααααΈααααααα
ααααα\nααΆαα±αα α
ααα½αααααΈα’αΆα
$n(S) = C(16\\,, \\, 3)$\n\n$$n(S) = \\frac{16!}{(16-3)!\\cdot 3!} = \\frac{16... |
khmer_math_doc_000978 | [
"images/khmer_math_doc_000978.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "αααααααααΌαααα $a(x-x_0)+b(y-y_0) + c(z-z_0)=0$\nααα αααααααΆαα $A(2, 2, 1)$\nαααα $x_0 = 2 , y_0 = 2 , z_0 = 1$\nα αΎα αααααααΆααα»α
ααααααα... |
khmer_math_doc_001493 | [
"images/khmer_math_doc_001493.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "α‘. α. αα $\\lim_{x \\to +\\infty}f(x)$\n\nα. αα $\\lim_{x\\to-\\infty}$f($x$)$, ($ααααΎαααααα$\\lim_{x\\to-\\infty} xe^x = 0$ ) α\n\n##... |
khmer_math_doc_000429 | [
"images/khmer_math_doc_000429.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "α. αααααα
ααα½ααα»ααααα·α
$z_1$αα·α$z_2$ ααΆααααααααααΈαααααΆααα α\n\nα. αααα αΆαααΆ $z_1$αα·α$z_2$ααΆα«ααααααααΈααΆα$z^3 - 8 = 0$ α\n\nα. ααααΆαααααααΊ... |
khmer_math_doc_000683 | [
"images/khmer_math_doc_000683.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "$$= \\lim_{x \\to 0^+} \\left( x + 1 - \\frac{1}{x} \\ln x \\right)$$\n\n$$\\lim_{x \\to +\\infty}$f($x$)$= \\lim_{x \\to +\\inft... |
khmer_math_doc_001318 | [
"images/khmer_math_doc_001318.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "ααΆααα»α $\\begin{cases}$y(0)$= 3 \\\\ y'(0) = -3 \\end{cases} \\iff \\begin{cases} A + B = 3 \\\\ A -$5B$= -3 \\end{cases} \\Right... |
khmer_math_doc_001033 | [
"images/khmer_math_doc_001033.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "ααΌα
αααα $\\boxed{$P($C$)$= \\frac{3}{110}}$ α\n\n$$= 1 + \\frac{8 - (9 + 8 \\ln x)}{x^2} = \\frac{x^2 - 1 - 8 \\ln x}{x^2}$$\n\n## IV. ... |
khmer_math_doc_001693 | [
"images/khmer_math_doc_001693.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "$$y = \\frac{4}{3}x$$\n\nααααΆα $AC = \\sqrt{14} , AD = 2\\sqrt{6} , CD = \\sqrt{38}$\nααα $AC^2 + AD^2 = (\\sqrt{14})^2 + (2\\sqrt{6})^... |
khmer_math_doc_000816 | [
"images/khmer_math_doc_000816.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "ααα αααα»αααααΆαααα·ααααααααΆααααα $d_2$ααααΎ$-\\frac{5}{3}$\nααα ααααα $x \\in (1, +\\infty),f'(x) = \\frac{-(x^2 + 1)}{(x+1)(... |
khmer_math_doc_000706 | [
"images/khmer_math_doc_000706.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "## I. (α‘α₯αα·αααα») ααααΆααΈααΈα α. $\\lim_{x \\to 1} \\frac{1 - x^2}{x^3 - x^2 + x - 1}$α$.\\lim_{x \\to 0} \\frac{\\sin 3x}{-x}$α$.\\lim_{... |
khmer_math_doc_001632 | [
"images/khmer_math_doc_001632.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "α. $\\lim_{x \\to 0} \\frac{(e^{-x} + e^x)\\sin^2 x}{2x^2} \\left(\\text{ααΆαααΆααα·αααααα \\frac{0}{0}\\right)$}\n\n$$=\\lim... |
khmer_math_doc_000606 | [
"images/khmer_math_doc_000606.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "αααααΆααααα $d_2$αααααΉαααααΆα$C$ αααααα
ααα»α
αααααΆαα’αΆαααααΈα\nααΆαααααΆαααααααΆααααα»αααααΆαααα·αααααΎ $f'(\\ln 3) = 0$ αααααααΆ\nααΆαα±αα $ααααΆ... |
khmer_math_doc_000176 | [
"images/khmer_math_doc_000176.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "$$-\\infty$$\n\n$$f'(x)$$\n\n$$f(x)$$\n\n$$+\\infty$$\n\nαααααααΆα $C$αα·αα’αΆαααΈαααΌα$d_1$αα·α$d_2$ ααααααΆ\nααΌα
αααα\n\nα£. $a$. ααααΆααα... |
khmer_math_doc_001629 | [
"images/khmer_math_doc_001629.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "ααΎ $0 < x < 1$ααα$αx^2 + \\ln x - 1 < 0$\nααΎ $x = 1$ααα$αx^2 + \\ln x - 1 = 0$\nααΎ $x > 1$ααα$αx^2 + \\ln x - 1 > 0$\nααααΆα $f(x)... |
khmer_math_doc_001873 | [
"images/khmer_math_doc_001873.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "α $P(C) = \\frac{n(C)}{n(S)} = \\frac{24}{84} = \\frac{2}{7}$\nααΌα
αααα $\\boxed{$P($C$)$= \\frac{2}{7}}$ α\nααΆαα±αα α
ααα½αααααΈα’αΆα
$n(S) =... |
khmer_math_doc_000394 | [
"images/khmer_math_doc_000394.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "α±αα $f'(x) = 0 \\iff e^x - 3 = 0 \\Rightarrow x = \\ln 3$\n\n$$\\lim_{x \\to -\\infty}$f($x$)$= \\lim_{x \\to -\\infty} \\left( x + 2 - \\frac{4e^x}{e^... |
khmer_math_doc_001230 | [
"images/khmer_math_doc_001230.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "$$= \\lim_{x \\to 1^+} [-x + 4 + \\ln(x+1) - \\ln(x-1)]$$\n\n$$= -1 + 4 + \\ln 2 - (-\\infty) = +\\infty$$\n\nααΌα
αααα\n$d_1$ααΆα’αΆαααΈαααΌ... |
khmer_math_doc_001275 | [
"images/khmer_math_doc_001275.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "ααααΆα αααα»ααααα $2$, αααα α$4$αα·αααα$4$\n\n$$n(B) =C(4, 2)\\timesC(2, 1) +C(4, 2)\\timesC(4, 1) +C(4, 3)$$\n\nααα
αΆαααααααα»α $3$ ααααααααΆαααα
α... |
khmer_math_doc_001768 | [
"images/khmer_math_doc_001768.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "α. ααααα $\\left( \\frac{z_1}{z_2} \\right)^3$ ααΆααΈαααα·α\n\n$$\\left(\\frac{z_1}{z_2}\\right)^3 = \\left[3\\left(\\frac{\\sqrt{3}... |
khmer_math_doc_001811 | [
"images/khmer_math_doc_001811.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "ααα ααα
αΆααααΆαααΆααααΌααααα½ααααα
ααααΆα
αααΉα 4 ααα\nααΆαα±αα α
ααα½αααααΈαααα $n(C) =$C(4, 1)$\\timesC(12, 2)$\n\n$$n(C) = 4 \\times \\frac{12!}{(... |
khmer_math_doc_000806 | [
"images/khmer_math_doc_000806.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "## II. (α‘α₯αα·αααα») ααα±ααα
ααα½ααα»ααααα·α
$z_1 = -1 + i\\sqrt{3},z_2 = -1 - i\\sqrt{3}$ α\n\nα. ααααΆ $z_1 + z_2,z_1 - z_2,z_1 z_2$ α\n\nα. αααααα
ααα½ααα»ααααα·... |
khmer_math_doc_000214 | [
"images/khmer_math_doc_000214.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "αααα $a = -1 , b = -1 , c = 2$\nααααΆα $(ABD) : -(x - 2) - (y - 2) + 2(z - 1) = 0$\n\n$$x + y -$2z$- 2 = 0$$\n\nααΌα
αααα $(A... |
khmer_math_doc_001128 | [
"images/khmer_math_doc_001128.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "α’. ααααααΌααΆα $B$: \"ααα
αΆααααΆαααααΈααΆαααααΌααααααααΎ$3$\nααα
αΆααααΆαααααΈααΆαααααΌααααααααΎ $3$ααΆα$4$ ααααΈα’αΆα
\n\n$$n(B) = 4$$\n\nααααΆα $P(B) = \... |
khmer_math_doc_000705 | [
"images/khmer_math_doc_000705.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "$$-(x^3 +$0x$^2 + x)$$\n\n$$x + 1$$\n\n$$\\dfrac{\\quad}{x^2 + x + 1}$$\n\n$$-(x^2 +$0x$+ 1)$$\n\n$$= -\\infty + 1 - \\frac{4 \\cdot 0}{1 + ... |
khmer_math_doc_000245 | [
"images/khmer_math_doc_000245.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "ααααΆα $P(C) = \\frac{n(C)}{n(S)} = \\frac{90}{210} = \\frac{3}{7}$\nααΌα
ααα: $\\boxed{$P($C$)$= \\frac{3}{7}}$\n\n## II. ααααααΌααΆα... |
khmer_math_doc_001376 | [
"images/khmer_math_doc_001376.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "$$= \\lim_{x \\to 1} \\frac{-(1+x)}{x^2+1} = \\frac{-(1+1)}{1^2+1} = \\boxed{-1}$$\n\nα. $\\lim_{x \\to 0} \\frac{\\sin 3x}{-x}$(ααΆαααΆααα·ααα... |
khmer_math_doc_001959 | [
"images/khmer_math_doc_001959.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "ααΆαα±αα $g(1) = 1^2 + \\ln 1 = 1$\n\nα’. αααα αΆαααΆααααΈααααα’αα»αααα $f$ααΊ$f'(x) = \\frac{x^2 + \\ln x - 1}{x^2}$\nααααΆα $f(x) =... |
khmer_math_doc_001595 | [
"images/khmer_math_doc_001595.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "$$y =$$\n\n$$C :$f($x$)$= x + 2 -$$\n\n# α
ααααΎα αα·ααααΆααΆααα·ααα·ααααΆ ααΆαααα»αααααΆα α’α α‘α₯\n\n## I. α‘. ααααΆ $z_1 + z_2 \\ , \\ z_1 - z_2 \\ ,... |
khmer_math_doc_001246 | [
"images/khmer_math_doc_001246.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "$$AC = |\\overrightarrow{AC}| = \\sqrt{3^2 + 1^2 + 2^2} = \\sqrt{14}$$\n\n$$AD = |\\overrightarrow{AD}| = \\sqrt{2^2 + 2^2... |
khmer_math_doc_001281 | [
"images/khmer_math_doc_001281.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "$B$ Β«αααΆααα·α
ααΆααα·ααα α’ ααΆαα ααΆαα·αααααααΈΒ»\n\n$$= +\\infty(0 - \\infty) = \\boxed{-\\infty}$$\n\n$$= 0 + 0 = \\boxed{0}$$\n\... |
khmer_math_doc_001861 | [
"images/khmer_math_doc_001861.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "$y = x + 1$α’ααααΈαααΌααα
ααΉαααααΆα$C$ααααα$-\\infty$ α αα·ααααΆααΈααΆααααααααΆα$C$αα
ααΉααααααΆαα$d_1$ α\n$y = x - 3$α’ααααΈαααΌααα
ααΉαααααΆα$C$αα... |
khmer_math_doc_000123 | [
"images/khmer_math_doc_000123.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "α. αα·ααααΆα’αααααΆαααα’αα»αααα $f$αααααΉαααΆαααΈααΆα$g(x) = 0$ααΆαα
ααααΎα$x' = 1$αα·α$x'' = \\alpha$($1 < \\alpha$) α\n\nα. αααα αΆαααΆαααααΆαα $... |
khmer_math_doc_001866 | [
"images/khmer_math_doc_001866.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "α
ααα½αααααΈα’αΆα
$n(S) =$C(5, 2)$= \\frac{5!}{3! \\cdot 2!} = \\frac{5 \\cdot 4 \\cdot 3!}{3! \\cdot 2} = 10$\n\nα‘. ααααααΌααΆαααααααΉαα... |
khmer_math_doc_000646 | [
"images/khmer_math_doc_000646.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "ααΈααΈα $\\lim_{x \\to 0^+}$g($x$)$= \\lim_{x \\to 0^+} (x^2 - 1 - 8\\ln x) = +\\infty$\n\n$$\\lim_{x \\to +\\infty}$g($x$)$... |
khmer_math_doc_000610 | [
"images/khmer_math_doc_000610.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "ααα
αΆαααααααΌα $3$ α
ααααΈααααααα
ααααα\nααΆαα±αα α
ααα½αααααΈα’αΆα
$n(S) = C(16\\,, \\, 3)$\n\n$$n(S) = \\frac{16!}{(16-3)!\\cdot 3!} = \\frac{16\\cdot 15\\cdot 1... |
khmer_math_doc_000656 | [
"images/khmer_math_doc_000656.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "# α
ααααΎα αα·ααααΆααΆααα·ααα·ααααΆ ααΆαααα»αααααΆα α’α α‘α© (ααααΆαααα·ααααΆααΆααααα)\n\n$$= \\frac{2}{(+\\infty)(\\sqrt{1+0+0} + (1+0))} = \\boxed{0}$$\n\nαα
ααααααΌαααΈαα... |
khmer_math_doc_001967 | [
"images/khmer_math_doc_001967.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "β’αα·ααααΆααΈααΆααααααααΆα $C$αα
ααΉααααααΆαα$d_1$\nααΆαααααααααααΆαα $d_1$αααααααααααα$x \\in \\mathbb{R}$\nαααααΆαα $d_2 : y = x - 3$ ααΆα’αΆα... |
khmer_math_doc_001556 | [
"images/khmer_math_doc_001556.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "$A$ : Β« ααααΈααΆαα 3 ααΆαααααααα α Β»\n$C$: Β« ααααΈααΆαα$3$ ααΆαααααα»ααααααΆ Β»\nααΆααααΆαα 3 ααΆαααααα»ααααααΆ ααΆααααααΆ α
αΆααααΆαααΆαααα½ααααα»α\n\n$$= 2 \\tim... |
khmer_math_doc_001671 | [
"images/khmer_math_doc_001671.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "α
αΆααααΆααααΆααα·α
ααΆααααΌα α’ ααααααααΆααααααΆ αααα»αα
αααα\nαααΌα α£ ααααΌαα
αΆαα ααααΌαααΆααααΌαααα α’ α¬αααΌαααα α£\nααΆαα±αα $n(A) =C(3, 2)C(5, 1) +C(3, 3... |
khmer_math_doc_000621 | [
"images/khmer_math_doc_000621.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "α. 2. αααα αΆαααΆααααΈααα $ABC$ ααΆααααΈααααααααα\n\n$$= \\left( -\\frac{1}{3}\\cos\\pi + \\frac{1}{5}\\left(\\frac{\\sqrt{2}}{2}\\right)^5 \\right) - \\left... |
khmer_math_doc_000547 | [
"images/khmer_math_doc_000547.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "ααα $\\lim_{x \\to -\\infty} \\left(-\\frac{4e^x}{1 + e^x}\\right) = \\left(-\\frac{4 \\cdot 0}{1 + 0}\\right) = -\\frac{0}{1} = 0$\n\n... |
khmer_math_doc_001839 | [
"images/khmer_math_doc_001839.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "ααα $\\Delta ABC$ααΆα$AB = AC = BC = 6\\sqrt{2}$ α―αααΆααααα\nααΌα
αααα $\\boxed{\\text{ααααΈααα ABC \\text{ ααΆααααΈααααααααα }$ α}}\nαα... |
khmer_math_doc_001636 | [
"images/khmer_math_doc_001636.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "α. $\\lim_{x \\to +\\infty} \\sqrt{x^2 + x} - x$\n\nα. $\\lim_{x \\to 0} \\frac{(e^{-x} + e^x) \\sin^2 x}{2x^2}$α$.\\lim_{x \\to +\\inf... |
khmer_math_doc_001238 | [
"images/khmer_math_doc_001238.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "$$= \\lim_{x \\rightarrow +\\infty} \\frac{\\sqrt{(x^2 + 2x + 3)^2} - (x+1)^2}{\\sqrt{x^2 + 2x + 3} + (x+1)}$$\n\n$$= \\lim_{x \\rightarrow ... |
khmer_math_doc_000679 | [
"images/khmer_math_doc_000679.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "ααΆααΆαααααααααααααΌαααααΆαααααΆαααΆααΌα\n\n## VII. 1. a. αααα αΆαααΆ ααα’αΆα
ααααα $f(x) = x \\left(1 - \\frac{\\ln x}{x} - \\frac{2}{x^2}\\r... |
khmer_math_doc_001236 | [
"images/khmer_math_doc_001236.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "ααΆαα±αα ααααΌααΆαααααααΉαααα·ααΆαααα
αΆααααΆααααΆααα·α
αααα»αααααΈ $2$ ααααααααΊ\n\n$$P(B) = \\frac{n(B)}{n(S)} = \\frac{40}{120} = \\fra... |
khmer_math_doc_001497 | [
"images/khmer_math_doc_001497.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "$$n(S) =$C(10, 3)$= \\frac{10!}{(10-3)! \\times 3!}$$\n\nααΆαα±αα ααααΌααΆαααααααΉαααα·ααΆαααα
αΆααααΆααααΆααα·α
ααααΈ α’ ααααααααΊ\n\n$$P... |
khmer_math_doc_001578 | [
"images/khmer_math_doc_001578.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "$$f(x)$$\n\n$$\\nearrow +\\infty$$\n\n$$+\\infty$$\n\n$$-\\infty$$\n\n3. a. αααααΈααΆααααααΆααααααα
ααα $(C)$αααααα
ααα»α
$A$\nαα... |
khmer_math_doc_001197 | [
"images/khmer_math_doc_001197.png"
] | [
{
"from": "human",
"value": "<image>\nTranscribe the complete content of this document image into Markdown format with LaTeX math expressions."
},
{
"from": "gpt",
"value": "ααααΆα $P(C) = P(B/A) = \\frac{2/10}{2/5} = \\frac{2}{10} \\times \\frac{5}{2} = \\frac{1}{2}$\nααΌα
αααα $P(C) = \\frac{1}{2... |
khmer_math_doc_000922 | [
"images/khmer_math_doc_000922.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "$$f'(x) = 1 - \\frac{4e^x}{(1 + e^x)^2} = \\frac{(1 + e^x)^2 - 4e^x}{(1 + e^x)^2}$$\n\n$$= \\frac{1 + 2e^x + (e^x)^2 - 4e^x}{(1 +... |
khmer_math_doc_000669 | [
"images/khmer_math_doc_000669.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "$$= \\lim_{x \\to +\\infty} \\left( x + 2 - \\frac{4e^x}{e^x(1 + \\frac{3}{e^x})} \\right)$$\n\nααΌα
αααα $\\lim_{x \\to -\\infty}$... |
khmer_math_doc_001935 | [
"images/khmer_math_doc_001935.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "$$= \\lim_{x \\to +\\infty} \\left( x + 2 - \\frac{4e^x}{e^x(1 + \\frac{3}{e^x})} \\right)$$\n\nααΌα
αααα $\\lim_{x \\to -\\infty}$f($x$)$= -\\infty,\\li... |
khmer_math_doc_001314 | [
"images/khmer_math_doc_001314.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "α’. αααααααΆααααααααααΈαααααΆαααααα
ααα½ααα»ααααα·α
$z_1 - z_2,z_1 \\times z_2$\n\nα’. αααααΈααΆααααααααΆααα’αααΈα\nααα $f'(x) +$2f$(x) = \\fra... |
khmer_math_doc_001485 | [
"images/khmer_math_doc_001485.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "ααααΆα $z_1 + z_2 = (-1 + i\\sqrt{3}) + (1 - i\\sqrt{3}) = 0$\n\n$$z_1 - z_2 = (-1 + i\\sqrt{3}) - (1 - i\\sqrt{3}) = -2 + i2\\sqrt{3}$... |
khmer_math_doc_001227 | [
"images/khmer_math_doc_001227.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "c. αααα αΆαααΆ $\\vec{n} = (1, 1, 1)$ααΆααα·α
ααααααααΆαααα
ααΉα$(ABC)$\nααΌα’αααααααααα»α $F(h+p, k)$α¬$F(0, -2)$ α\nαααααΆααααααΆαααα·α $x = k - p$α¬... |
khmer_math_doc_000975 | [
"images/khmer_math_doc_000975.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "ααΆαααΌααααα $P(B/A) = \\frac{P(A \\cap B)}{P(A)}$\nααΆαα±αα $n(A \\cap B) = 2, P(A \\cap B) = \\frac{n(A \\cap B)}{n(S)} = \\frac{2}{10}$\... |
khmer_math_doc_000016 | [
"images/khmer_math_doc_000016.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "ααααΆα ααΌα’ααααααααα
ααα»α
$B(x_B, y_B)$\nααα αααααΆαααααααααα $B$ααααααΉααααααΆαα$y = x$\nααΌα
ααα ααΌα’ααααααααα
ααα»α
ααααΆαααΊ $B(2, 1 - \\ln 2)$ α... |
khmer_math_doc_000974 | [
"images/khmer_math_doc_000974.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "$b$. αα·ααααΆα’αααααΆααα$f$ααΎ$\\mathbb{R}$αα·ααααααΆααΆαα’αααααΆααα$f$\nααα $f'(x)=\\left(\\dfrac{e^x-3}{e^x+3}\\right)^2\\ge 0,\\forall x\\in\\... |
khmer_math_doc_000239 | [
"images/khmer_math_doc_000239.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "ααααΆα $\\frac{x+1}{x-1} > 1$(ααΆαααΈαααα αΆαααΆαααΎ) ααααΆαα$\\ln$\nααααΆα $\\ln\\left(\\frac{x+1}{x-1}\\right) > \\ln 1$(ααα$\\ln 1 = 0$)\n$-x + 4 ... |
khmer_math_doc_000481 | [
"images/khmer_math_doc_000481.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "α’αΆαααΈαααΌααααααααααααΆα $C$ααΆαα’αα»ααααααΆα$+\\infty$ α\n\n$$f(\\ln 2) = 4 - \\ln 2 -$2e$^{-\\ln 2} = 3 - \\ln 2 \\approx 2.3$$\n\nααΆα... |
khmer_math_doc_001984 | [
"images/khmer_math_doc_001984.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "ααΆαααΌααααα $P(B/A) = \\frac{P(A \\cap B)}{P(A)}$\nααΆαα±αα $n(A \\cap B) = 2, P(A \\cap B) = \\frac{n(A \\cap B)}{n(S)} = \\frac{2}{10}$\... |
khmer_math_doc_001876 | [
"images/khmer_math_doc_001876.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "## III. ααααΆααααΌααΆα α\n\nααΆαα±αα α
ααα½αααααΈα’αΆα
$n(S) =$C(8, 3)ααα\n\n$$C(8, 3) = \\frac{8!}{5! \\cdot 3!} = \\frac{8 \\cdot 7 \\cdot 6 \... |
khmer_math_doc_000303 | [
"images/khmer_math_doc_000303.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "ααΌα
αααα $f$ααΆα’αα»ααααααΎαααΎα
ααααα$(0, +\\infty)$ α\n$>$ αααααΆααΆαα’αααααΆαααααααΆ\n\n$$f(x_0) =$f(1)$= \\frac{1^2 - 2}{1} - \\ln 1 = -1$$\n\nααααΆα $y_B = f(x... |
khmer_math_doc_000185 | [
"images/khmer_math_doc_000185.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "ααΆααΆααααααΆαα $f'(x)$\nααααΆα $C :$f($x$)$= x + 1 - \\frac{\\ln x}{x}$αα·α$\\Delta : y = x + 1$\nαααααΆααααΆα’αα»αααα $f$ααΆαα’αααααααΆαααα... |
khmer_math_doc_001355 | [
"images/khmer_math_doc_001355.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "$$n(S) =$C(10, 3)$= \\frac{10!}{(10-3)! \\times 3!}$$\n\nααΆαα±αα ααααΌααΆαααααααΉαααα·ααΆαααα
αΆααααΆααααΆααα·α
ααααΈ α’ ααααααααΊ\n\n$$P(B) = \\frac{n(B)}... |
khmer_math_doc_000150 | [
"images/khmer_math_doc_000150.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "ααΆαα±αα α
ααα½αααααΈα’αΆα
$n(S) =$C(9, 3)ααα\n\n$$C(9, 3) = \\frac{9!}{6! \\cdot 3!} = \\frac{9 \\cdot 8 \\cdot 7 \\cdot 6!}{6! \\cdot ... |
khmer_math_doc_000367 | [
"images/khmer_math_doc_000367.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "ααΆααΆαα’αααααΆααα $f$\n\n$$\\ln 3$$\n\n$$0\\ln 3$$\n\n$$f(x) - y = \\left( x + 2 - \\frac{4e^x}{e^x + 3} \\right) - (x + 2) = -\\fra... |
khmer_math_doc_000160 | [
"images/khmer_math_doc_000160.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "ααΌα
αααα $y = Ae^x + Be^{-3x} , A , B$ ααα α\nααααΌαααΆααααΈα $V(h,k \\pm a) = V(0,0 \\pm 3)$\nααΆαα±αα $V_1(0, 3) , V_2(0, -3)$\n\nα. ... |
khmer_math_doc_001515 | [
"images/khmer_math_doc_001515.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "c. αααα αΆαααΆ $\\vec{n} = (0, 1, -1)$ααΆααα·α
ααααααααΆααααΉαααααα$(ABC)$\nααααΉαααΆ $\\vec{AB} \\times \\vec{AC}$ααΆααα·α
ααααααααΆααααααααα$... |
khmer_math_doc_000795 | [
"images/khmer_math_doc_000795.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "ααααΆα $AB = \\sqrt{6^2 + 0^2 + (-6)^2} = 6\\sqrt{2}$ α―αααΆαααααα\n$AC = \\sqrt{0^2 + 6^2 + (-6)^2} = 6\\sqrt{2}$ α―αααΆαααααα\n\n$$= \\left( \\frac{1}{3} ... |
khmer_math_doc_001087 | [
"images/khmer_math_doc_001087.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "## VI. αααααααΆααααΈααΆαααΈαααααααααααααα α\n\n$y'' + y' -$2y$= 0$ ααΆααααΈααΆαααααΆαα\nααααΈααααααααΆαααααααααΆαααααΈααΆα $(E)$ ααα α\n\n3. a. αααααΈααΆαα... |
khmer_math_doc_000838 | [
"images/khmer_math_doc_000838.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "# αα·ααααΆααΆ ααα·ααα·ααααΆ ααΆαααα»αααααΆα α’α‘α€ (ααααΆαααα·ααααΆααΆααααα) αααααα α‘α₯α ααΆααΈ αα·αααα» α‘α’α₯\n\nα. ααααΆ $z_1 + z_2,z_1 - z_2,z_1 z_2$ α\n\n... |
khmer_math_doc_000036 | [
"images/khmer_math_doc_000036.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌααααααα’ααααα αα·αααΌαααααααα·ααα·ααααΆααΆααα’ααααΈαααααα―αααΆααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "ααΌα
αααα $\\mathbf{$P($A$)$=\\frac{1}{210}}$\n$A(-2, 1, 0)$,B(0, 1, 1)$,C(1, 2, 2)$αα·α$D(0, 3, -4)$\n\n## IV. αααααααΌα α‘ ααΈαααα»ααααααΆααααΌα α‘α₯ ααααΆαα ααα... |
khmer_math_doc_000114 | [
"images/khmer_math_doc_000114.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "α. Β« ααΆααα·αααααααΆαααααα»ααα½αααααΈα Β» α\n\n3. αααα αΆαααΆααΎ $I$,f(x)$ααΆααααααΆααΌα
$h(x)$ α\n\n4. ααΆαααα’αααααΆααα $f(x)$ααΎ$I$αα·ααααααααΆα$C$αα$f(... |
khmer_math_doc_001542 | [
"images/khmer_math_doc_001542.png"
] | [
{
"from": "human",
"value": "<image>\nExtract all Khmer text and mathematical formulas from this document image as structured Markdown."
},
{
"from": "gpt",
"value": "αααααΆαα $d_1 : y = x + 1$ ααΆα’αΆαααΈαααΌαααααα\nβ’αα·ααααΆααΈααΆααααααααΆα $C$αα
ααΉααααααΆαα$d_1$\nααΆαααααααααααΆαα $d_1$αααααααααααα$x \\in \... |
khmer_math_doc_001208 | [
"images/khmer_math_doc_001208.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "ααΌα
ααα αααΈααΆααααααΆαααααααααΆα $(C)$ααααα$A$ααΊ$y =$2x$- 3$ α\n\nα’. αααα αΆαααΆαααΈααΆααααααΆαααΈααΆαα’αααΈα\nααΌα
αααα αααΈααΆααααααΆαα±αα ααΆαααΈααΆαα’αααΈα... |
khmer_math_doc_001752 | [
"images/khmer_math_doc_001752.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "$$= \\frac{x + 1 - 2x - 2\\ln x}{x^3} = \\frac{-x + 1 - 2\\ln x}{x^3}$$\n\nααα α $\\boxed{f'(x) = \\frac{-x + 1 - 2\\ln x}{x^3}}$ α\nα... |
khmer_math_doc_000210 | [
"images/khmer_math_doc_000210.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαααααΎ OCR ααΎααΌαααΆαααα ααααααααααΆααα’ααααααααα αα·αααΌαααααααα·ααα·ααααΆααΆ LaTeX αααα»ααααααα Markdownα"
},
{
"from": "gpt",
"value": "ααΌα
αααα $y =$2e$^x + e^{-5x}$ααΆα
ααααΎααα·ααααα$(E)$ α\nαα»αααααΆαα ααΆαααααααααΆαα $\\lim_{x \\to +\\infty} [$f($x$)$- (-x + 4)] = 0$\n... |
khmer_math_doc_000854 | [
"images/khmer_math_doc_000854.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαα’αΆαα’ααααα αα·αααααααααα·ααα·ααααΆααΆααα’αααααααΆααααα»αααΌαααΆαααα α αΎααααααααΆαααααα Markdownα"
},
{
"from": "gpt",
"value": "αα½αααα (α α‘ αααα αα‘ αααα‘) α α
ααα½αααααΈααααααΊα\n\n$$P(A) = \\frac{n(A)}{n(S)} = \\frac{4}{120} = \\frac{1}{30}$$\n\n$$n(C) =$C(2, 1)$\\timesC(4... |
khmer_math_doc_000043 | [
"images/khmer_math_doc_000043.png"
] | [
{
"from": "human",
"value": "<image>\nα
αΌαα
ααααααααΉαααΆαα―αααΆααααααΆ Markdown ααααααααΆαα
ααΆααααααααα
αααααΎα αα·αααΌαααααααα·ααα·ααααΆ LaTeXα"
},
{
"from": "gpt",
"value": "ααααΆα $P(C) = \\frac{n(C)}{n(S)} = \\frac{4}{15}$\nααΌα
αααα $\\boxed{$P($C$)$= \\frac{4}{15}}$ α\nααααΆα $M(2, 3, 4)$,N(3, 5, 6)$,\n$P(4,... |
khmer_math_doc_001044 | [
"images/khmer_math_doc_001044.png"
] | [
{
"from": "human",
"value": "<image>\nααΌαααααααααααααα―αααΆαααα·ααα·ααααΆααααα
ααΆα’ααααα Markdown ααααααααΆααΌααααα LaTeX α±ααααΆαααααΉαααααΌαα"
},
{
"from": "gpt",
"value": "$$= \\lim_{x \\to 0} \\frac{-2 \\sin$5x$\\cdot (\\sqrt{5} + \\sqrt{x + 5})}{-x}$$\n\n$$= \\lim_{x \\to 0} \\left[ \\frac{\\sin 5x}{5x}... |
End of preview. Expand in Data Studio
π°π khVLDB: Khmer Vision-Language Dataset for Math & Document OCR
khVLDB is a high-quality multi-line synthetic document dataset designed specifically for End-to-End Vision-Language Models (VLMs) such as Qwen2-VL and LLaMA-Factory.
It bridges the gap between single-line cropped OCR and full-page multi-modal document transcription for Khmer textbooks.
π Dataset Overview
- Total Synthetic Pages: 2,000 multi-line pages (~20,000 formula/text crops)
- Image Resolution: 1,000 px canvas width (A4 realistic aspect ratio)
- Page Layout: 8 β 14 lines per page, realistic paper textures (cream, parchment, newsprint), multiply blending, subtle noise, and anti-clipping margins.
- Semantic Coherence: Grouped by topic (Calculus, Probability, Complex Numbers, Geometry, Algebra) to prevent multi-column topic interleaving.
- Format: ShareGPT / LLaMA-Factory SFT format (
train.json,val.json) with image paths and Markdown/LaTeX annotations.
π Repository Structure
βββ README.md # Dataset card & documentation
βββ dataset_info.json # LLaMA-Factory dataset registration info
βββ dataset_stats.json # Distribution & synthesis metadata
βββ train.json # Primary training split (ShareGPT format)
βββ val.json # Validation split
βββ images/ # 2,000 PNG rendered document images
βββ khmer_math_doc_000001.png
βββ ...
π Usage with LLaMA-Factory
Register khVLDB in your dataset_info.json:
"khmer_vlm_math": {
"file_name": "train.json",
"formatting": "sharegpt",
"columns": {
"messages": "conversations",
"images": "images"
}
}
Then train with Qwen2-VL-2B-Instruct:
llamafactory-cli train qwen2_vl_lora_sft.yaml
π Citations & Acknowledgements
- Source crops derived from krotreaksmey/khmer-math-textbook
- Created with Khmer VLM Lab Studio Pro
- Downloads last month
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