id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-cfa0256f89abe9ef7290f0d7d0c694c5efcdb4243a8b55631c9e52c7a67a6c84 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{\sqrt {x+\sqrt {2-x^2}}\, \sqrt [6]{1-x\sqrt {2-x^2}}} {\sqrt [3]{1-x^2}}dx$. | [
"$\\sqrt [6]{2x}+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{\\sqrt {x+\\sqrt {2-x^2}}\\, \\sqrt [6]{1-x\\sqrt {2-x^2}}} {\\sqrt [3]{1-x^2}}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aee21af0e005778c95fddf2d3589fa5c76f0bd5bff6eebc65a5fff8e134f0aa3 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \left(\frac{1-x^{-2}}{x^{1/2}-x^{-1/2}}-\frac{2}{x^{3/2}} +\frac{x^{-2}-x}{x^{1/2}-x^{-1/2}}\right)dx$. | [
"$-\\frac{2}{3}x^{3/2}+4x^{-1/2}+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\left(\\frac{1-x^{-2}}{x^{1/2}-x^{-1/2}}-\\frac{2}{x^{3/2}} +\\frac{x^{-2}-x}{x^{1/2}-x^{-1/2}}\\right)dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c56d2bc0b202828a356c873a3ba66ee382983af6231033c51cabf4dd4c595e38 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \left(\frac{\sqrt x}{2}-\frac{1}{2\sqrt x}\right)^2 \left(\frac{\sqrt x-1}{\sqrt x+1}-\frac{\sqrt x+1}{\sqrt x-1}\right)dx$. | [
"$2x^{1/2}-\\frac{2}{3}x^{3/2}+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\left(\\frac{\\sqrt x}{2}-\\frac{1}{2\\sqrt x}\\right)^2 \\left(\\frac{\\sqrt x-1}{\\sqrt x+1}-\\frac{\\sqrt x+1}{\\sqrt x-1}\\right)dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-215bc835fb8fab48b92622210d834c9a05400786f2c741f9b604db06957572e1 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \left(\frac{x^{-6}-64}{4+2x^{-1}+x^{-2}}-\frac{x^2}{4-4/x+1/x^2} -\frac{4x^2(2x+1)}{1-2x}\right)dx$. | [
"$x+x^2+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\left(\\frac{x^{-6}-64}{4+2x^{-1}+x^{-2}}-\\frac{x^2}{4-4/x+1/x^2} -\\frac{4x^2(2x+1)}{1-2x}\\right)dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a3a73078e6adc3d3aa68efa8cc2fd1655921268faa7a9ecadba2da5d2d875d9 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \left[\frac{1+\sin 2\alpha }{\cos (2\alpha -2\pi ) \cot\left(\alpha -\frac{5}{4}\pi \right)}+\cos ^2 \alpha \right]d\alpha $. | [
"$\\frac{1}{4}\\sin 2\\alpha -\\frac{1}{2}\\alpha +c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\left[\\frac{1+\\sin 2\\alpha }{\\cos (2\\alpha -2\\pi ) \\cot\\left(\\alpha -\\frac{5}{4}\\pi \\right)}+\\cos ^2 \\alpha \\right]d\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33f89e0da9a8ce6c30952b416b9689ccd5bb890a987d973251b256038ac0f009 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \left[\frac{\cot^2 2x-1}{2\cot 2x}-\cos 8x\cot 4x\right]dx$. | [
"$-\\frac{1}{7}\\cos 8x+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\left[\\frac{\\cot^2 2x-1}{2\\cot 2x}-\\cos 8x\\cot 4x\\right]dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-313979dec9ea5f632ce5f8aaf36f7f75bcf85d143352c1fdf41e9efebcfdfe3c | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \left[\sin \alpha \sin (x-\alpha )+\sin ^2 \left(\frac{x}{2} -\alpha \right)\right]dx$. | [
"$\\frac{1}{2}x-\\frac{\\sin x}{2}+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\left[\\sin \\alpha \\sin (x-\\alpha )+\\sin ^2 \\left(\\frac{x}{2} -\\alpha \\right)\\right]dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a1a501e66d9743cb10ee481917abc03edb46789f806629d1b03a3a34e15846f6 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \sqrt {\frac{2x}{(1+x)\sqrt [3]{1+x}}}\ \sqrt [3]{\frac{4+8/x+4/x^2}{\sqrt 2}}dx$. | [
"$\\frac{12x^{5/6}}{5}+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\sqrt {\\frac{2x}{(1+x)\\sqrt [3]{1+x}}}\\ \\sqrt [3]{\\frac{4+8/x+4/x^2}{\\sqrt 2}}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-146f775bf345f06324232f85527ccc08f9ef1f89dae5bfa2f358268d1bbffe24 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $f(x)=\frac{\sqrt {2+x^2}-\sqrt {2-x^2}}{\sqrt {4-x^4}}$. | [
"$\\arcsin \\frac{x}{\\sqrt 2}-\\ln\\left(x+\\sqrt {x^2+2}\\right)+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $f(x)=\\frac{\\sqrt {2+x^2}-\\sqrt {2-x^2}}{\\sqrt {4-x^4}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7657afbf25ad8e0f5f31cf243801899bc921d88cec074f023b2abcf1bb002239 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $f(x)=\frac{x-1}{(2x-1)^3}$. | [
"$-\\frac{1}{4}(2x-1)^{-1}+\\frac{1}{8}(x-1)^{-2}+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $f(x)=\\frac{x-1}{(2x-1)^3}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cdf8ef34412ce00e394557e76480b12cf60440143116dedc0bd1f30527b41cb | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $f(x)=\frac{x}{\sqrt {1-x/2}}$. | [
"$-8\\left(1-\\frac{x}{2}\\right)^{1/2}+\\frac{8}{3}\\left(1-\\frac{x}{2}\\right)^{3/2}$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $f(x)=\\frac{x}{\\sqrt {1-x/2}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b84032045f5263faed061d50f5c469a1abe83822d247539044b960b2cb27dcd4 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $f(x)=x\sqrt {1-x}$. | [
"$-\\frac{2}{3}(1-x)^{3/2}+\\frac{2}{5}(1-x)^{5/2}+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $f(x)=x\\sqrt {1-x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e428df2052fd1fe545ccac5a8c8b9aa03613cd428f2339daa4c65cd024511d94 | Using the rules: (1) $b^2 \rightarrow b$ (2) $a \rightarrow a^2$ (3) $ab \rightarrow ba^2$ (4) $ba^2 \rightarrow ab$ Determine the number of distinct sequences of transformations that can convert $ab$ into $ba$ in exactly 4 steps. | [
"0"
] | [
"Using the rules:\n(1) $b^2 \\rightarrow b$\n(2) $a \\rightarrow a^2$\n(3) $ab \\rightarrow ba^2$\n(4) $ba^2 \\rightarrow ab$\n\nDetermine the number of distinct sequences of transformations that can convert $ab$ into $ba$ in exactly 4 steps."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-37021c5699408288253d2045ec4fa8d507648e6e4fcd619c22fa6dc0b2c5d432 | Using the same 160m fence and the width of the shed as part of the width of the sheep pen, find the dimensions of the sheep pen that maximize the enclosed area. | [
"\\( L = 120 \\text{ meters}, W = 20 \\text{ meters} \\)"
] | [
"Using the same 160m fence and the width of the shed as part of the width of the sheep pen, find the dimensions of the sheep pen that maximize the enclosed area."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dca190fefe9419e84bf08bf1c8d1997ab38a7a051ee74d5f08705e50d0043d2d | Using the same 160m fence, find the maximum enclosed area if three rectangular sheep pens are constructed. | [
"800"
] | [
"Using the same 160m fence, find the maximum enclosed area if three rectangular sheep pens are constructed."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-70d12ba07ed1820d61e89637bf9785019fe03b4425f579eb5246aa14b5b17502 | Using the same numbers \( x, y, \) and \( z \) from Problem 1, find numbers \( a, b, \) and \( c \), independent of \( n \), such that: \[ S_{n+1} = aS_n + bS_{n-1} + cS_{n-2} \] where \( S_n = x^n + y^n + z^n \). | [
"\\(a = 1\\), \\(b = \\frac{1}{2}\\), \\(c = \\frac{1}{6}\\)"
] | [
"Using the same numbers \\( x, y, \\) and \\( z \\) from Problem 1, find numbers \\( a, b, \\) and \\( c \\), independent of \\( n \\), such that:\n\\[ S_{n+1} = aS_n + bS_{n-1} + cS_{n-2} \\]\nwhere \\( S_n = x^n + y^n + z^n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c25f4732a2394cdc6072697515819fb8c2b7a0c686b7e2718859a61d9068ca7 | Using the segment $AB$ of length $10$ as the diameter, a circle $\omega$ is constructed. A tangent to $\omega$ is drawn through point $A$, at which point $K$ is selected. Through point $K$, a straight line is drawn, different from $AK$, tangent to circle $\omega$ at point $C$. Altitude $CH$ of triangle $ABC$ intersects... | [
"8"
] | [
"Using the segment $AB$ of length $10$ as the diameter, a circle $\\omega$ is constructed. A tangent to $\\omega$ is drawn through point $A$, at which point $K$ is selected. Through point $K$, a straight line is drawn, different from $AK$, tangent to circle $\\omega$ at point $C$. Altitude $CH$ of triangle $ABC$ in... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eab822256b15316a8b1f4516593d582ab3ad37e5aa42d38ec8d466b59b13a860 | Using the sequence $v_n$ defined as $v_n = u_n - \frac{1}{2}$, determine an expression for $u_n$ in terms of $n$. | [
"\\( u_n = \\frac{3}{2} \\left( \\frac{1}{3} \\right)^{n-1} + \\frac{1}{2} \\)"
] | [
"Using the sequence $v_n$ defined as $v_n = u_n - \\frac{1}{2}$, determine an expression for $u_n$ in terms of $n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d94d78f3719e05614b2565db7717cfa9c787a0e48caa4feeaac265f1b327b02 | Using the sequence of prime numbers 2, 3, 5, 7, 11, 13, 17, find the number 19 by performing operations (multiplication and subtraction) involving all these primes. | [
"19"
] | [
"Using the sequence of prime numbers 2, 3, 5, 7, 11, 13, 17, find the number 19 by performing operations (multiplication and subtraction) involving all these primes."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-f1527487449d4cbd93f389c2c16083ed6d9b9c370ac8cc0bd2359b5cebcf5509 | Using the series $e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \ldots$, find the smallest integer $n$ such that the sum of the first $n$ terms of the series for $e^1$ is greater than $\frac{8}{3}$. | [
"5"
] | [
"Using the series $e^{x} = 1 + x + \\frac{x^{2}}{2!} + \\frac{x^{3}}{3!} + \\frac{x^{4}}{4!} + \\ldots$, find the smallest integer $n$ such that the sum of the first $n$ terms of the series for $e^1$ is greater than $\\frac{8}{3}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d629ced80fca861a1dc851eaade201ccec8298891916545a5c073238daee0697 | Using the series definitions $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ and $e^y = \sum_{n=0}^{\infty} \frac{y^n}{n!}$, find the coefficient of $x^2y^3$ in the expansion of $e^{x+y}$. | [
"\\dfrac{1}{12}"
] | [
"Using the series definitions $e^x = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!}$ and $e^y = \\sum_{n=0}^{\\infty} \\frac{y^n}{n!}$, find the coefficient of $x^2y^3$ in the expansion of $e^{x+y}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90ebaa00c50782a55e2032096f57cf557a306db6830b6980c54af17beb61d623 | Using the series expansion and the bounds for the remainder term, find the bounds for $\sin\left(\frac{1}{5}\pi\right)$ correct to seven decimal places. | [
"0.5878481 < \\sin\\left(\\frac{\\pi}{5}\\right) < 0.5878511"
] | [
"Using the series expansion and the bounds for the remainder term, find the bounds for $\\sin\\left(\\frac{1}{5}\\pi\\right)$ correct to seven decimal places."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43c381e6b8f3f7636d3d28be0c7b0aead30b203bebdab74e32d135e627eaff29 | Using the series expansion for \(\arctan(x)\), determine how many terms are needed to approximate \(\pi\) to 1000 decimal places using the formula. | [
"5 \\times 10^{999}"
] | [
"Using the series expansion for \\(\\arctan(x)\\), determine how many terms are needed to approximate \\(\\pi\\) to 1000 decimal places using the formula."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b507d21dcd2f2fa4d1ad39baaaf2cbac619733a201678121a1e20fe3df584719 | Using the series expansion of $\log(1+x)$, calculate the coefficient of $x^3$ in the expansion of $\frac{1}{2}[\log(1+x)]^2$. | [
"-\\dfrac{1}{2}"
] | [
"Using the series expansion of $\\log(1+x)$, calculate the coefficient of $x^3$ in the expansion of $\\frac{1}{2}[\\log(1+x)]^2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0466fd6448fbb5f39208c8a701771fa5082b07f86ec5c586a8eb7bef590476ec | Using the series representations \[ e = \sum_{k=0}^{\infty} \frac{1}{k!} \] and \[ \pi = 4 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1}, \] find the value of \( \frac{e}{\pi} \). | [
"\\dfrac{e}{\\pi}"
] | [
"Using the series representations \n\\[ e = \\sum_{k=0}^{\\infty} \\frac{1}{k!} \\]\nand \n\\[ \\pi = 4 \\sum_{k=0}^{\\infty} \\frac{(-1)^k}{2k+1}, \\]\nfind the value of \\( \\frac{e}{\\pi} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-eb1ade3a47747d30e4fc9d48f4878f561f40571eb572d4ec250e9310ecc03e35 | Using the simplex algorithm with \( n \) variables (both control and slack), \( m \) non-trivial constraints, and \( v \) non-zero variables (both control and slack) in the optimum solution, calculate the value of \( v \) when the system is in a degenerate state and \( n = 10 \), \( m = 5 \). | [
"4"
] | [
"Using the simplex algorithm with \\( n \\) variables (both control and slack), \\( m \\) non-trivial constraints, and \\( v \\) non-zero variables (both control and slack) in the optimum solution, calculate the value of \\( v \\) when the system is in a degenerate state and \\( n = 10 \\), \\( m = 5 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5424d4efc2b5b5553bfb8d34cf07989105c067edea1636cf808eb2d1867f631 | Using the six digits 1, 3, 5, 6, 7, 9 exactly once, how many 6-digit numbers can be formed that are divisible by 11? | [
"72"
] | [
"Using the six digits 1, 3, 5, 6, 7, 9 exactly once, how many 6-digit numbers can be formed that are divisible by 11?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-015b6341e3dc29fca8e04fc9a31d1a5d106c9c3d7f49468e33026e9ba1f5bf42 | Using the solution from part (a), find the limiting amount of salt in the tank as \( t \) approaches infinity. | [
"40 kg"
] | [
"Using the solution from part (a), find the limiting amount of salt in the tank as \\( t \\) approaches infinity."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-31d9e00cfe801858c35a43219579a6e2e2f3d3a1accb9dd19f11123bfdd8cae8 | Using the squeeze theorem, find the limit of the sum \(\frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n}\) as \(n\) approaches infinity. | [
"\\(\\ln{2}\\)"
] | [
"Using the squeeze theorem, find the limit of the sum \\(\\frac{1}{n+1} + \\frac{1}{n+2} + \\cdots + \\frac{1}{2n}\\) as \\(n\\) approaches infinity."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-96cd1512fda489fa8ecc323b364c175cac0233f5346c84eeec2e5e82d3b37875 | Using the substitution $t = \tan(x)$, derive the expression for $\sin(x)$ in terms of $t$. If $\sin(x) = \frac{t}{\sqrt{1+t^{2}}}$, find the value of $\sin^2(x) + \cos^2(x)$ in terms of $t$. | [
"1"
] | [
"Using the substitution $t = \\tan(x)$, derive the expression for $\\sin(x)$ in terms of $t$. If $\\sin(x) = \\frac{t}{\\sqrt{1+t^{2}}}$, find the value of $\\sin^2(x) + \\cos^2(x)$ in terms of $t$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f1c127d7ee47d1dfa15f9c755374fba35f3015f1eaf319f9975759238c55abf | Using the substitution \( u = \sin{x} \), solve the differential equation \( y = y' \tan{x} - (y')^2 \sec^2{x} \). | [
"\\( y = C \\sin x - C^2 \\) or \\( y = \\frac{1}{4} \\sin^2 x \\)"
] | [
"Using the substitution \\( u = \\sin{x} \\), solve the differential equation \\( y = y' \\tan{x} - (y')^2 \\sec^2{x} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 17
} | 39 | 0 | 0 |
omr-ws-a258c0dfe18b4bed5596d0df42f3fedd6b0fa196319ac800377b53af97df9d2f | Using the substitution \( x = X + a \) and \( y = Y + b \) for appropriate values of \( a \) and \( b \), or otherwise, solve the differential equation \[ \frac{dy}{dx} = \frac{x - 2y - 4}{2x + y - 3} \] given that the solution curve passes through the point \((1, 1)\). | [
"\\( 5(y+1)^2 - (x-2-2(y+1))^2 = \\frac{5}{4}(y+1) \\)"
] | [
"Using the substitution \\( x = X + a \\) and \\( y = Y + b \\) for appropriate values of \\( a \\) and \\( b \\), or otherwise, solve the differential equation\n\n\\[\n\\frac{dy}{dx} = \\frac{x - 2y - 4}{2x + y - 3}\n\\]\n\ngiven that the solution curve passes through the point \\((1, 1)\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43f5fc04ebef06ba1cef56555a9e5ca6ef269b7f43e85cd30e3e5c73596467b8 | Using the substitution \((1 - a \cos \theta)(1- a \cos \phi) = 1- a^{2}\) where \(a < 1\), transform the integral \[ I_{n} = \int (1 + a \cos \theta)^{-n} \, d\theta \] into an integral involving \(\phi\). Evaluate the integrals \(I_{0}\), \(I_{1}\), \(I_{2}\), and \(I_{3}\) in terms of \(\theta\). | [
"\\frac{\\sin \\theta (3 + a \\cos \\theta)}{2(1 - a^2)^2 (1 + a \\cos \\theta)^2} + \\frac{3a \\sin \\theta}{2(1 - a^2)^2 (1 + a \\cos \\theta)} + \\frac{3(2 + a^2)}{2(1 - a^2)^{5/2}} \\arctan\\left( \\sqrt{\\frac{1 - a}{1 + a}} \\tan\\frac{\\theta}{2} \\right) + C"
] | [
"Using the substitution \\((1 - a \\cos \\theta)(1- a \\cos \\phi) = 1- a^{2}\\) where \\(a < 1\\), transform the integral\n\n\\[ I_{n} = \\int (1 + a \\cos \\theta)^{-n} \\, d\\theta \\]\n\ninto an integral involving \\(\\phi\\). Evaluate the integrals \\(I_{0}\\), \\(I_{1}\\), \\(I_{2}\\), and \\(I_{3}\\) in term... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ce87129bf507100e50b7a37db91cb0ee1af18a439bc505a2720050ab2b9542e | Using the substitution \(\sqrt{x} = \sin y\), transform the integral \(\int_0^{1/2} \frac{\sqrt{x}}{\sqrt{1-x}} \, dx\) into another integral in terms of \(y\). Calculate the value of the transformed integral and express your answer as \(A \int_0^{\pi/4} \sin^2 y \, dy\). Find \(A\). | [
"2"
] | [
"Using the substitution \\(\\sqrt{x} = \\sin y\\), transform the integral \\(\\int_0^{1/2} \\frac{\\sqrt{x}}{\\sqrt{1-x}} \\, dx\\) into another integral in terms of \\(y\\). Calculate the value of the transformed integral and express your answer as \\(A \\int_0^{\\pi/4} \\sin^2 y \\, dy\\). Find \\(A\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-046691861d5feff8e81a77971ab08cb276e45b09b0b6f0e74c6738c763c0527c | Using the system of equations, find the distance Thanos travels if his velocity is represented by $k(t) = 4\sin(t)\sinh(t) + 10\cos(t)\cosh(t)$. | [
"7\\sin(t)\\cosh(t) + 3\\cos(t)\\sinh(t)"
] | [
"Using the system of equations, find the distance Thanos travels if his velocity is represented by $k(t) = 4\\sin(t)\\sinh(t) + 10\\cos(t)\\cosh(t)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-942d2209de5621700d945bd88c25e09aa338e0e18e2a0cfc83a23defb6182528 | Using the tangent line from part (b), approximate the value of \(k\) such that the point \((2.1, k)\) is on the curve \(-x^2 + 3xy + y^3 = 3\). | [
"\\dfrac{91}{90}"
] | [
"Using the tangent line from part (b), approximate the value of \\(k\\) such that the point \\((2.1, k)\\) is on the curve \\(-x^2 + 3xy + y^3 = 3\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9e57d74b30b72b558452e4077d4deeacb10cca7f7d250e42bfc301a039755e44 | Using the theorem that the positive primitive solutions of \( x^2 + y^2 = z^2 \) with \( y \) even are \( x = r^2 - s^2 \), \( y = 2rs \), \( z = r^2 + s^2 \), where \( r \) and \( s \) are arbitrary integers of opposite parity with \( r > s > 0 \) and \(\gcd(r, s) = 1\), find all solutions of the equation \( x^2 + y^2... | [
"\\((x, y, z) = \\left( \\frac{d(m^2 - n^2 + 2mn)}{2}, \\frac{d(m^2 - n^2 - 2mn)}{2}, \\frac{d(m^2 + n^2)}{2} \\right)\\)"
] | [
"Using the theorem that the positive primitive solutions of \\( x^2 + y^2 = z^2 \\) with \\( y \\) even are \\( x = r^2 - s^2 \\), \\( y = 2rs \\), \\( z = r^2 + s^2 \\), where \\( r \\) and \\( s \\) are arbitrary integers of opposite parity with \\( r > s > 0 \\) and \\(\\gcd(r, s) = 1\\), find all solutions of t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 3
} | 9 | 0 | 0 |
omr-ws-894a37a6b89a6b20bc463065cdae5fbd3846ab98829a1ddd4a8f173af2eb8486 | Using the transformation of graphs, solve the equation $\cos(|x|) = \min(x, x^2)$ and find the number of real roots. | [
"1"
] | [
"Using the transformation of graphs, solve the equation $\\cos(|x|) = \\min(x, x^2)$ and find the number of real roots."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-3baf7b43f4ebbcdbd64ca3631da2e1cd6259dbd20a7f473ba7784de2815bc231 | Using the trapezoidal formula for \( \int_0^1 \sqrt{x} \, dx \) with \( 2^n \) intervals, the approximation is given by \( \frac{2}{3} + \delta_n \), where \( \delta_n \to 0 \). Find \( \lim_{n \to \infty} \frac{\delta_n}{\delta_{n+1}} \). | [
"\\( 2\\sqrt{2} \\)"
] | [
"Using the trapezoidal formula for \\( \\int_0^1 \\sqrt{x} \\, dx \\) with \\( 2^n \\) intervals, the approximation is given by \\( \\frac{2}{3} + \\delta_n \\), where \\( \\delta_n \\to 0 \\). Find \\( \\lim_{n \\to \\infty} \\frac{\\delta_n}{\\delta_{n+1}} \\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 25
} | 42 | 0 | 0 |
omr-ws-2be2949a258a6ff93681aa5cd06c14b688f9bfc32cbc271b45b5c2c64ed47da3 | Using the trapezoidal method (numeric analysis), approximate the value of the double integral \( \int_{0}^{1}\int_{0}^{1} f(x,y) \, dx \, dy \). | [
"\\dfrac{f(0,0) + f(0,1) + f(1,0) + f(1,1)}{4}"
] | [
"Using the trapezoidal method (numeric analysis), approximate the value of the double integral \\( \\int_{0}^{1}\\int_{0}^{1} f(x,y) \\, dx \\, dy \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b978fdfe1aeac04c15d05fc7e2aa0814d04c2a2ba57d6c7f9da9185cf0f15fa | Using the truth table for the NAND-connector (|), calculate the number of distinct truth assignments for \( A \) and \( B \) that satisfy the equivalence \( A \rightarrow B \equiv A | (B | B) \). | [
"4"
] | [
"Using the truth table for the NAND-connector (|), calculate the number of distinct truth assignments for \\( A \\) and \\( B \\) that satisfy the equivalence \\( A \\rightarrow B \\equiv A | (B | B) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e939ef71eccddca168d59506dccb9e69dddb666ba22a258f263b7328c5382f7 | Using the two operations (i) double and (ii) add one: (a) Find the least number of operations to convert $1$ to $63$ and show the corresponding sequence. (b) For a given positive integer $K$, give an algorithm (step by step procedure) for converting $1$ to $K$ using the least number of operations and prove it gives the... | [
"2(N-1)"
] | [
"Using the two operations (i) double and (ii) add one:\n(a) Find the least number of operations to convert $1$ to $63$ and show the corresponding sequence.\n(b) For a given positive integer $K$, give an algorithm (step by step procedure) for converting $1$ to $K$ using the least number of operations and prove it gi... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-dcbd64c21643277b59db74d646149a958b894bb11686b0abd65ee80b8b4363da | Using the values \( S = 292.618 \), \( n = 15 \), and \( t = 50 \), find the common ratio \( r \) of the geometric progression. | [
"\\( r \\approx 0.953472 \\)"
] | [
"Using the values \\( S = 292.618 \\), \\( n = 15 \\), and \\( t = 50 \\), find the common ratio \\( r \\) of the geometric progression."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2fca87fd2aca4d63cdaba40a747dcbbd0ec96684c9675e3f8c8c4049e30969c | Using the values of \(\sin(9^\circ)\) and \(\cos(9^\circ)\), find the exact value of \(\tan(9^\circ)\). | [
"\\sqrt{5} + 1 - \\sqrt{5 + 2\\sqrt{5}}"
] | [
"Using the values of \\(\\sin(9^\\circ)\\) and \\(\\cos(9^\\circ)\\), find the exact value of \\(\\tan(9^\\circ)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-0d1ea3cebe829c0777511bf1f1236b9ff22d9c298a404c92fc11e5c9a3a0f624 | Using the values of \(q\) and \(s\) from the quartic equation \(x^{4} + 3x^{2} - 6x + 10 = 0\), find the value of \((\alpha + \beta)(\gamma + \delta)\) and the value of \(\alpha \beta\) given that \(\alpha \beta > \gamma \delta\). | [
"\\((\\alpha + \\beta)(\\gamma + \\delta) = -4\\) and \\(\\alpha \\beta = 5\\)"
] | [
"Using the values of \\(q\\) and \\(s\\) from the quartic equation \\(x^{4} + 3x^{2} - 6x + 10 = 0\\), find the value of \\((\\alpha + \\beta)(\\gamma + \\delta)\\) and the value of \\(\\alpha \\beta\\) given that \\(\\alpha \\beta > \\gamma \\delta\\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-537e428d71c7ef9ddfc5a08533b1b5608cf4369c91626d5b84171e5bcbf5433e | Using the vertices of an n-sided polygon, how many triangles with no common sides can be drawn? | [
"\\dfrac{n(n-4)(n-5)}{6}"
] | [
"Using the vertices of an n-sided polygon, how many triangles with no common sides can be drawn?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01630024eea2946c70c19be4fa2c89550ba230d5c212e0c7f416e5b673c22011 | Using the whole numbers from $1$ to $22$ inclusive, form eleven fractions by choosing one number as the numerator and one number as the denominator, with each number used exactly once. What is the maximum number of these fractions that could have an integer value? | [
"11"
] | [
"Using the whole numbers from $1$ to $22$ inclusive, form eleven fractions by choosing one number as the numerator and one number as the denominator, with each number used exactly once. What is the maximum number of these fractions that could have an integer value?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-77d7ab415a4dd3b95fd4003e4012fc78de87add052fe3ac5b2e364ce402bded7 | Using the whole numbers from 1 to 22 inclusive, Jack wants to form eleven fractions by choosing one number as the numerator and one number as the denominator, with each number used exactly once. What is the maximum number of Jack's fractions that could have an integer value? | [
"10"
] | [
"Using the whole numbers from 1 to 22 inclusive, Jack wants to form eleven fractions by choosing one number as the numerator and one number as the denominator, with each number used exactly once. What is the maximum number of Jack's fractions that could have an integer value?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 4
} | 24 | 0 | 0 |
omr-ws-6844e6090e94b6bd4f53dd4531ab3a34b8d4078d1cd460e67f5aaab78b449c18 | Using three different weights (1lb, 3lb, and 9lb), how many different weights can be measured if the weights and the objects to be weighed can be placed in either pan of the scale? | [
"13"
] | [
"Using three different weights (1lb, 3lb, and 9lb), how many different weights can be measured if the weights and the objects to be weighed can be placed in either pan of the scale?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-463f086af7759d4aeb74e4825be19f9e41ea8f8e82c4ae0435dc7c972597407b | Using three different weights of 1 lb., 3 lb., and 9 lb., how many different object weights can be measured if the weights can be placed in either pan of the scale? | [
"13"
] | [
"Using three different weights of 1 lb., 3 lb., and 9 lb., how many different object weights can be measured if the weights can be placed in either pan of the scale?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-c4a8f5a845dfabaeb68c977cb9b253f41c31fa8e23069d9a0708c5cba1fc0f64 | Using tiles made by combining $3$ unit squares and allowing rotations of the tiles by multiples of $90$ degrees, determine the total number of distinct ways to construct a $5 \times 6$ rectangle. Two constructions are considered distinct if one cannot be obtained from the other by rotating or reflecting the entire rect... | [
"2"
] | [
"Using tiles made by combining $3$ unit squares and allowing rotations of the tiles by multiples of $90$ degrees, determine the total number of distinct ways to construct a $5 \\times 6$ rectangle. Two constructions are considered distinct if one cannot be obtained from the other by rotating or reflecting the entir... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-5b4240e84b8b21223cc1e4583c8497ab4b8dc82091540b5b5458ad36c8bd7da5 | Using tiles made by combining $3$ unit squares, allowing rotations of the tiles by multiples of $90$ degrees, calculate the total number of distinct ways to tile a $5 \times 9$ rectangle, considering two tilings distinct if one cannot be obtained from the other by rotation or reflection. | [
"12"
] | [
"Using tiles made by combining $3$ unit squares, allowing rotations of the tiles by multiples of $90$ degrees, calculate the total number of distinct ways to tile a $5 \\times 9$ rectangle, considering two tilings distinct if one cannot be obtained from the other by rotation or reflection."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-d56f67b9a90f1086f305fd2af536e00bf58f048319e01904ac347e8f4ee1fb77 | Using two buckets with capacities of 3 litres and 5 litres, how can you measure exactly 1 litre of a mixture that contains 40% pure alcohol and 60% water? | [
"1"
] | [
"Using two buckets with capacities of 3 litres and 5 litres, how can you measure exactly 1 litre of a mixture that contains 40% pure alcohol and 60% water?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2ba720fa2cd7d9c8e08fb3a35b795ced5b2e482f920b28982b96e6a060b77564 | Using two containers that hold up to 11 liters and 7 liters respectively, and a fountain to draw and dump water, can you measure exactly 2 liters? Describe the steps. | [
"2"
] | [
"Using two containers that hold up to 11 liters and 7 liters respectively, and a fountain to draw and dump water, can you measure exactly 2 liters? Describe the steps."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7db03da89ab4a763adb6362b34c5b2f8fa55de4991da92d2fff011875a088206 | Using unit cubes and six colors, color each cube such that each face of the cube is a different color. Glue together faces of the same color to form an \( r \times s \times t \) box. For which values of \( r \), \( s \), and \( t \) is it possible to form such a box where each face of the box has a different color? | [
"1 \\times 1 \\times 1"
] | [
"Using unit cubes and six colors, color each cube such that each face of the cube is a different color. Glue together faces of the same color to form an \\( r \\times s \\times t \\) box. For which values of \\( r \\), \\( s \\), and \\( t \\) is it possible to form such a box where each face of the box has a diffe... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0f1722524f2e78e6b9c0392667294fcde047e98ae3862876856330d4303a2111 | Using unlimited pennies, nickels, dimes, quarters, and half-dollars, how many ways are there to create 50 cents? | [
"50"
] | [
"Using unlimited pennies, nickels, dimes, quarters, and half-dollars, how many ways are there to create 50 cents?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-804120c96cb851cf3fb2a3910b0e48bb519d4326aa5cb0d643f9b23f824bb102 | Using weights of $1g, 3g, 9g, 27g, 81g$, determine the largest value that can be weighed exactly using any combination of these weights on a balance scale. | [
"121"
] | [
"Using weights of $1g, 3g, 9g, 27g, 81g$, determine the largest value that can be weighed exactly using any combination of these weights on a balance scale."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-54e0a129296b9062a9b553550cf6d24da3ecc29b2a1765af94e72701538a4e84 | Using weights of masses 1, 3, 3^2, 3^3, ..., what is the minimum number of weights needed to determine the weight of an object with a mass of 2023? | [
"5"
] | [
"Using weights of masses 1, 3, 3^2, 3^3, ..., what is the minimum number of weights needed to determine the weight of an object with a mass of 2023?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c75f6b375f15373170e42f96c28cea1c6cb1afee2a761368a3594ae3f8190e64 | VABC is a pyramid with vertex V and base ABC, which is an equilateral triangle with sides equal to 3. The lengths VA = 4, VB = 5, and VC = 6. What is the height of the pyramid? | [
"\\(\\frac{\\sqrt{933}}{9}\\)"
] | [
"VABC is a pyramid with vertex V and base ABC, which is an equilateral triangle with sides equal to 3. The lengths VA = 4, VB = 5, and VC = 6. What is the height of the pyramid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-e0d6aa7230dd17d281d7502870e9f25a59415c4d1aa602fbaa31c96c4003545a | Vadim wished to rotate a point in the Cartesian plane \( 120^\circ \) counter-clockwise around the point \((3,4)\). First, he reflected the point across the line \( y = 4 \), then he reflected it across the line \( y = mx + b \). If the point ends up where he wanted it, compute \((m, b)\). | [
"(m, b) = (\\(\\sqrt{3}\\), 4 - 3\\(\\sqrt{3}\\))"
] | [
"Vadim wished to rotate a point in the Cartesian plane \\( 120^\\circ \\) counter-clockwise around the point \\((3,4)\\). First, he reflected the point across the line \\( y = 4 \\), then he reflected it across the line \\( y = mx + b \\). If the point ends up where he wanted it, compute \\((m, b)\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 26
} | 73 | 0 | 0 |
omr-ws-765522344ae5164d43cc66584167775177ecc840a09406514b72efc24ad63f1c | Vaggelis has a box containing 2015 white and 2015 black balls. In each step, he randomly selects two balls and follows these rules: - If both balls are black, he paints one white and keeps it in the box, throwing the other out. - If both balls are white, he keeps one in the box and throws the other out. - If one ball i... | [
"2 white balls and 1 black ball."
] | [
"Vaggelis has a box containing 2015 white and 2015 black balls. In each step, he randomly selects two balls and follows these rules:\n- If both balls are black, he paints one white and keeps it in the box, throwing the other out.\n- If both balls are white, he keeps one in the box and throws the other out.\n- If on... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-ac1822c2b9634c0c73e9dfe57cb16d6c74dd4c08b15abaa9028acd227fa0b173 | Valentine wants to build an imaginary country consisting of $n$ cities with the following conditions: - Exactly two roads depart from each city. - It should be possible to travel from any city to any other city with at most three roads. Find the maximum value of $n$. | [
"7"
] | [
"Valentine wants to build an imaginary country consisting of $n$ cities with the following conditions:\n- Exactly two roads depart from each city.\n- It should be possible to travel from any city to any other city with at most three roads.\nFind the maximum value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-fb3bd8a9194b6369c85e3c5d30a28dd78d4ed4b750edaad6a2e8c93e13bd5a75 | Valery has rods of lengths $1$ cm, $2$ cm, $3$ cm, ..., $n$ cm. She chooses $k$ rods such that any three of the chosen rods can form a triangle. Determine the greatest value $k$ can take. | [
"\\( k = n - \\left\\lceil \\frac{n}{2} \\right\\rceil \\)"
] | [
"Valery has rods of lengths $1$ cm, $2$ cm, $3$ cm, ..., $n$ cm. She chooses $k$ rods such that any three of the chosen rods can form a triangle. Determine the greatest value $k$ can take."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 8,
"tir": 27
} | 77 | 0 | 0 |
omr-ws-a30b7a43b7701f0ddaea903aad2cf4a5665fa8beeca0280295d1a8b91b0bc3fb | Values for $A, B, C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6 \}$ without replacement. How many ways are there to make such choices that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? | [
"180"
] | [
"Values for $A, B, C,$ and $D$ are to be selected from $\\{1, 2, 3, 4, 5, 6 \\}$ without replacement. How many ways are there to make such choices that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-45d3fab6b428871448985e4dfbac5803bc0b8db9a42622a17dc00f700586340e | Values for $A, B, C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e., no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, th... | [
"90"
] | [
"Values for $A, B, C,$ and $D$ are to be selected from $\\{1, 2, 3, 4, 5, 6\\}$ without replacement (i.e., no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for exam... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-61c48bf023406c664cac87a559b845caa3ad939b1f1449302a38a9b0782cf996 | Values for $A,B,C,$ and $D$ are to be selected from ${1, 2, 3, 4, 5, 6}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter.) | [
"90"
] | [
"Values for $A,B,C,$ and $D$ are to be selected from ${1, 2, 3, 4, 5, 6}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter.)"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 17
} | 35 | 0 | 0 |
omr-ws-31122932c817cb430a9ab42854bd6dadae50f66d5683e1c04187533b72f0b3ee | Van was asked to add 46 to 37 and round the result to the nearest ten. He figured that since he was going to have to round anyway, he might as well do it before adding the numbers. He rounded 46 to 50 and 37 to 40, thus getting an answer of $50+40=90$. This answer is not correct. What is the correct answer? | [
"80"
] | [
"Van was asked to add 46 to 37 and round the result to the nearest ten. He figured that since he was going to have to round anyway, he might as well do it before adding the numbers. He rounded 46 to 50 and 37 to 40, thus getting an answer of $50+40=90$. This answer is not correct. What is the correct answer?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a60deea1f32856a78d7b8609d1b5dd69e311e1f643011f3738896fe29e11261 | Vanessa set a school record for most points in a single basketball game when her team scored $48$ points. The six other players on her team averaged $3.5$ points each. How many points did Vanessa score to set her school record? | [
"27"
] | [
"Vanessa set a school record for most points in a single basketball game when her team scored $48$ points. The six other players on her team averaged $3.5$ points each. How many points did Vanessa score to set her school record?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8fc8bee473f1afb42de06fd0d18eb797516256a7dddd5d88898ab7f820867ab4 | Vanya has chosen two positive numbers, \( x \) and \( y \). He wrote the numbers \( x+y \), \( x-y \), \( x/y \), and \( xy \), and showed these numbers to Petya. If Petya knows that \( x+y = 10 \) and \( xy = 21 \), find the value of \( x^2 + y^2 \). | [
"58"
] | [
"Vanya has chosen two positive numbers, \\( x \\) and \\( y \\). He wrote the numbers \\( x+y \\), \\( x-y \\), \\( x/y \\), and \\( xy \\), and showed these numbers to Petya. If Petya knows that \\( x+y = 10 \\) and \\( xy = 21 \\), find the value of \\( x^2 + y^2 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61d8dd082911aec4149304b148424fce2960ba0b800c91d315d6e06b0f837620 | Vanya needs to write an expression equal to $10$ using only the numbers $1$, the signs $+$ and $-$, and brackets. The expression must remain equal to $10$ even if all $+$ signs are changed to $-$ and vice versa. Help Vanya compose such an expression. | [
"\\((1+1)-(1-1)+(1+1)-(1-1)+(1+1)-(1-1)+(1+1)-(1-1)+(1+1)-(1-1)\\)"
] | [
"Vanya needs to write an expression equal to $10$ using only the numbers $1$, the signs $+$ and $-$, and brackets. The expression must remain equal to $10$ even if all $+$ signs are changed to $-$ and vice versa. Help Vanya compose such an expression."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 2
} | 20 | 0 | 0 |
omr-ws-3f4767393e43fb8013f1922106fbc9a89b57c50ea5898ef72ad2ebe4a20e732b | Variable points $M$ and $N$ are considered on the arms $\left[ OX \right.$ and $\left[ OY \right.$, respectively, of an angle $XOY$ so that $\left| OM \right|+\left| ON \right|$ is constant. Determine the locus of the midpoint of $\left[ MN \right]$. | [
"The locus of the midpoint of \\([MN]\\) is the segment \\([M_{\\omega}N_{\\omega}]\\)."
] | [
"Variable points $M$ and $N$ are considered on the arms $\\left[ OX \\right.$ and $\\left[ OY \\right.$, respectively, of an angle $XOY$ so that $\\left| OM \\right|+\\left| ON \\right|$ is constant. Determine the locus of the midpoint of $\\left[ MN \\right]$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 10
} | 18 | 0 | 0 |
omr-ws-84321128deb7e8080c180a43cdc6707a2dd156441f8f471f159aa119c24312e7 | Variables $x$ and $y$ are related by the equation $2xy = 195$. If $x$ is increasing at the rate of $\frac{3}{2}$ units per second, find the rate of change of $y$ when $x = 15$ units. | [
"\\(-0.65\\)"
] | [
"Variables $x$ and $y$ are related by the equation $2xy = 195$. If $x$ is increasing at the rate of $\\frac{3}{2}$ units per second, find the rate of change of $y$ when $x = 15$ units."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-69f2e1bc7f2e00338ac7826936aefacd52f6e1687fa00f2745b08c30e1a1ac3b | Vasya and Petya take turns laying out on a single segment smaller cuts. On the first move, Vasya places a segment of length $1/2$, and each next segment is twice as long as the previous one. Laid-out segments cannot have common interior points and go beyond the boundaries of the original segment. Loses with no move. Wh... | [
"Petya"
] | [
"Vasya and Petya take turns laying out on a single segment smaller cuts. On the first move, Vasya places a segment of length $1/2$, and each next segment is twice as long as the previous one. Laid-out segments cannot have common interior points and go beyond the boundaries of the original segment. Loses with no mov... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a6d04e8af0afb12ee877a5c8ef6f95f8edcbb6b4765deb95404c670d68df939f | Vasya chose $2000$ numbers from $1$ to $2000^2$ such that their sum equals two thousandth of the sum of all numbers. Let the sum of the remaining numbers be $S$. Petya colors each of the remaining numbers with one of $1999$ other colors such that the sum of numbers with each of the total $2000$ colors are the same. Fin... | [
"3992000998"
] | [
"Vasya chose $2000$ numbers from $1$ to $2000^2$ such that their sum equals two thousandth of the sum of all numbers. Let the sum of the remaining numbers be $S$. Petya colors each of the remaining numbers with one of $1999$ other colors such that the sum of numbers with each of the total $2000$ colors are the same... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5056f4534199f82162a21981ec64dbad48119b8c2a4b61ef5ac1cd95f32b8ce7 | Vasya chose a certain number $x$ and calculated the sequence $a_n = 1 + x^{n+1} + x^{n+2}$ for $n \geq 1$. Given that $a_2^2 = a_1a_3$, find the value of $a_5^2 - a_4a_6$. | [
"0"
] | [
"Vasya chose a certain number $x$ and calculated the sequence $a_n = 1 + x^{n+1} + x^{n+2}$ for $n \\geq 1$. Given that $a_2^2 = a_1a_3$, find the value of $a_5^2 - a_4a_6$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-975a46c6f9aed327b6183ce41644201c28b1d31b0f17431a7b64dad12f4e77d5 | Vasya conceives a two-digit number $a$, and Petya tries to guess it. Petya tells Vasya a natural number $k$, and Vasya responds with the sum of the digits of the number $ka$. What is the smallest number of questions Petya must ask to be certain of determining Vasya’s number? | [
"3"
] | [
"Vasya conceives a two-digit number $a$, and Petya tries to guess it. Petya tells Vasya a natural number $k$, and Vasya responds with the sum of the digits of the number $ka$. What is the smallest number of questions Petya must ask to be certain of determining Vasya’s number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-f01af0badb04cfd6a4d6b774835f760c67b240686309b55f2bbea03944e7e119 | Vasya has $n$ candies of several types, where $n > 145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n$. | [
"160"
] | [
"Vasya has $n$ candies of several types, where $n > 145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-e549e2ef68ef75919b5966b5b082d88604a39b30460a442a06d7d822544d6147 | Vasya has 100 cards of 3 colors, with no more than 50 cards of the same color. He arranges these cards into a 10x10 square such that no two cards of the same color share a common side. Let $N$ be the number of distinct color patterns possible under these conditions, assuming rotations and reflections of the square are ... | [
"342"
] | [
"Vasya has 100 cards of 3 colors, with no more than 50 cards of the same color. He arranges these cards into a 10x10 square such that no two cards of the same color share a common side. Let $N$ be the number of distinct color patterns possible under these conditions, assuming rotations and reflections of the square... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-fc9cc48345fad38cb48804a1b31f020c0a08f777a5ee66dd846d95407de277b3 | Vasya has a calculator that can transform a pair of numbers $(x, y)$ into either $(x+y, x)$ or $(2x+y+1, x+y+1)$. Starting with the pair $(1, 1)$, for a given natural number $n$, let $k_n$ be the unique integer such that the pair $(n, k_n)$ can be obtained using the calculator. Find the value of $k_{2023}$. | [
"no_valid_answer_found"
] | [
"Vasya has a calculator that can transform a pair of numbers $(x, y)$ into either $(x+y, x)$ or $(2x+y+1, x+y+1)$. Starting with the pair $(1, 1)$, for a given natural number $n$, let $k_n$ be the unique integer such that the pair $(n, k_n)$ can be obtained using the calculator. Find the value of $k_{2023}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fad2ef309806e1825b2abde2880850b136300005a870699f2755f5cbfbeedc0e | Vasya has a numeric expression \( \square \cdot \square + \square \cdot \square \) and 4 cards with numbers \(a\), \(b\), \(c\), and \(d\) that can be placed in the 4 free places in the expression. Vasya tried all possible ways to place the cards and always obtained the same value as a result. If \(a\), \(b\), \(c\), a... | [
"8"
] | [
"Vasya has a numeric expression \\( \\square \\cdot \\square + \\square \\cdot \\square \\) and 4 cards with numbers \\(a\\), \\(b\\), \\(c\\), and \\(d\\) that can be placed in the 4 free places in the expression. Vasya tried all possible ways to place the cards and always obtained the same value as a result. If \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-2736222b58c61d6ac366f1236f223e86a18143098c684406962d68f9adc0bc58 | Vasya has a numeric expression \(\Box \cdot \Box + \Box \cdot \Box\) and 4 cards with numbers \(a, b, c,\) and \(d\) that can be placed in the 4 free places in the expression. Vasya tried all possible ways to place the cards and always obtained the same value as a result. If \(a, b, c,\) and \(d\) are distinct integers... | [
"0"
] | [
"Vasya has a numeric expression \\(\\Box \\cdot \\Box + \\Box \\cdot \\Box\\) and 4 cards with numbers \\(a, b, c,\\) and \\(d\\) that can be placed in the 4 free places in the expression. Vasya tried all possible ways to place the cards and always obtained the same value as a result. If \\(a, b, c,\\) and \\(d\\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e73fe4d648ab472758a08b6b5e5704c31d5c896f8cdc3a0d8106120a2c0960dc | Vasya marked a dot on the plane with sympathetic ink and drew a square with ordinary ink. Petya sees the square but not the point. He can draw a straight line on the plane and ask Vasya on which side of the straight line the point is. How many questions does Petya need to determine if the point is inside or outside the... | [
"4"
] | [
"Vasya marked a dot on the plane with sympathetic ink and drew a square with ordinary ink. Petya sees the square but not the point. He can draw a straight line on the plane and ask Vasya on which side of the straight line the point is. How many questions does Petya need to determine if the point is inside or outsid... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-9718b08b2cdf2775ae8b600a75019f3a1b00dc068b279e655ac7a688d10380fb | Vasya moves on a straight line at a speed of $u$ and encounters three cockroaches initially positioned at the vertices of a regular $n$-gon, moving clockwise at a speed of $v$ along the sides of the polygon. If Vasya crushes all three cockroaches, find the ratio $\frac{v}{u}$. | [
"\\sqrt{3}"
] | [
"Vasya moves on a straight line at a speed of $u$ and encounters three cockroaches initially positioned at the vertices of a regular $n$-gon, moving clockwise at a speed of $v$ along the sides of the polygon. If Vasya crushes all three cockroaches, find the ratio $\\frac{v}{u}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-e933a1c79474410460068e88d1a3ec725550ad41549baeb3121bb9c15a6afce6 | Vasya started swimming in a basin with a boundary defined by a closed non-self-intersecting broken line $A_1A_2\dots A_nA_1$ of length $\ell$ meters. Vasya visited all points $A_2, \dots, A_n$ and finished at $A_1$. If the total distance Vasya swam is denoted by $D$, find the minimum possible value of $D - \ell$. | [
"-\\dfrac{\\ell}{2}"
] | [
"Vasya started swimming in a basin with a boundary defined by a closed non-self-intersecting broken line $A_1A_2\\dots A_nA_1$ of length $\\ell$ meters. Vasya visited all points $A_2, \\dots, A_n$ and finished at $A_1$. If the total distance Vasya swam is denoted by $D$, find the minimum possible value of $D - \\el... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-3030d729f71334f86a90d3a72f6785d5600f206803482098e3cde84b1c513a87 | Vasya thinks of two different positive integers, both not greater than 13. Peter writes 5 sets of numbers, and Vasya writes under each set the number of his chosen integers that are in that set (0, 1, or 2). Let $N$ be the number of different pairs of integers Vasya could have chosen such that Peter cannot uniquely det... | [
"3"
] | [
"Vasya thinks of two different positive integers, both not greater than 13. Peter writes 5 sets of numbers, and Vasya writes under each set the number of his chosen integers that are in that set (0, 1, or 2). Let $N$ be the number of different pairs of integers Vasya could have chosen such that Peter cannot uniquel... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-03dad5584435cfacfc5bd542906274a57937aa95219814375383c86d737fa5ef | Vecindad Island has 2011 inhabitants, divided into three types: the innocent, the wicked, and the capricious. The innocent always tell the truth, the wicked always lie, and the capricious alternately tell lies one day and truths the next. A reporter visits the island for two days. On the first day, the reporter intervi... | [
"2"
] | [
"Vecindad Island has 2011 inhabitants, divided into three types: the innocent, the wicked, and the capricious. The innocent always tell the truth, the wicked always lie, and the capricious alternately tell lies one day and truths the next. A reporter visits the island for two days. On the first day, the reporter in... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-6d5a462f8bfe6bb6c8931ac9c7db4372e9d8b80ab690c5abca0f90fe42767ab4 | Vehicles arriving at an intersection are either cars or trucks. Cars arrive at a Poisson rate of \( x \) per minute and trucks at a Poisson rate of \( y \) per minute. Given that the next vehicle arriving is a car, find the expected value of the time until the next arrival. | [
"\\dfrac{1}{x + y}"
] | [
"Vehicles arriving at an intersection are either cars or trucks. Cars arrive at a Poisson rate of \\( x \\) per minute and trucks at a Poisson rate of \\( y \\) per minute. Given that the next vehicle arriving is a car, find the expected value of the time until the next arrival."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1e0b66b237f5d7a8f7e4ce53d18329e9e0fe51a7e68cad474491f743d48e8a89 | Velociraptor $A$ is located at $x = 10$ on the number line and runs at $4$ units per second. Velociraptor $B$ is located at $x = -10$ on the number line and runs at $3$ units per second. If the velociraptors run towards each other, at what point do they meet? | [
"\\(-\\frac{10}{7}\\)"
] | [
"Velociraptor $A$ is located at $x = 10$ on the number line and runs at $4$ units per second. Velociraptor $B$ is located at $x = -10$ on the number line and runs at $3$ units per second. If the velociraptors run towards each other, at what point do they meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-51ffa4146ea1da6b5267690500899aac03168f0a460dc8b30aaf97fd9ba6069c | Veni writes down finitely many real numbers (possibly one), squares them, and then subtracts 1 from each of them, resulting in the same set of numbers as in the beginning. What were the starting numbers? | [
"\\( \\frac{1 + \\sqrt{5}}{2} \\) and \\( \\frac{1 - \\sqrt{5}}{2} \\)"
] | [
"Veni writes down finitely many real numbers (possibly one), squares them, and then subtracts 1 from each of them, resulting in the same set of numbers as in the beginning. What were the starting numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-c1be0f96e5d9804526c0228424f731780a5cf72a3be1ee68874363c6073cfa49 | Vera has a row of 2023 coins alternately tails-up and heads-up, with the leftmost coin tails-up. In a move, Vera may flip over one of the coins in the row, subject to the following rules: - On the first move, Vera may flip over any of the 2023 coins. - On all subsequent moves, Vera may only flip over a coin adjacent to... | [
"4044"
] | [
"Vera has a row of 2023 coins alternately tails-up and heads-up, with the leftmost coin tails-up. In a move, Vera may flip over one of the coins in the row, subject to the following rules:\n- On the first move, Vera may flip over any of the 2023 coins.\n- On all subsequent moves, Vera may only flip over a coin adja... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ef3d628b8f9bf62ef3fd779ed0ef7dc06c896ba415d33bcc5fc5953f2a3f1fc | Vera has several identical matches and wants to form a triangle such that any two sides differ in length by at least 10 matches. It is impossible to form such a triangle without having extra matches. What is the maximum number of matches Vera can have? | [
"62"
] | [
"Vera has several identical matches and wants to form a triangle such that any two sides differ in length by at least 10 matches. It is impossible to form such a triangle without having extra matches. What is the maximum number of matches Vera can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-aa868e74ba18e7f7a2ea4e256661d53f1f44289310d841bad0438e1a2ee450f9 | Verify if the expression for \(y'(x)\) given by \(y'(x) = \frac{2e^{y}+2x}{e^{y}(1+y-2x)}\) is correct, and then find \(y'(1)\) using this expression. | [
"\\dfrac{2e^{y(1)} + 2}{e^{y(1)}(y(1) - 1)}"
] | [
"Verify if the expression for \\(y'(x)\\) given by \\(y'(x) = \\frac{2e^{y}+2x}{e^{y}(1+y-2x)}\\) is correct, and then find \\(y'(1)\\) using this expression."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-27353b74a507240303b445e1c9cb18737e5dbc9ec438cac8a11b55b8fc120662 | Verify if the function \( f(x) = x \sin \frac{1}{\sqrt{x}} \) for \( x \neq 0 \) and \( f(0) = 0 \) is continuously differentiable, differentiable but not continuously differentiable, or not differentiable at the origin. | [
"not differentiable."
] | [
"Verify if the function \\( f(x) = x \\sin \\frac{1}{\\sqrt{x}} \\) for \\( x \\neq 0 \\) and \\( f(0) = 0 \\) is continuously differentiable, differentiable but not continuously differentiable, or not differentiable at the origin."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d23bbcb2da2554c2b4f0ce6400721db06c77dad1240f1a395bf754c14ff31431 | Verify if the function \( f(x) = x \sin |x| \) is continuously differentiable, differentiable but not continuously differentiable, or not differentiable at the origin. | [
"continuously differentiable"
] | [
"Verify if the function \\( f(x) = x \\sin |x| \\) is continuously differentiable, differentiable but not continuously differentiable, or not differentiable at the origin."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-e2f9fd0981db7ca4ccb12857805f3d364012fbe1dacda9f71e62c06326d61d4c | Verify if the function \( f(x) = |x| \) is continuously differentiable, differentiable but not continuously differentiable, or not differentiable at the origin. | [
"not differentiable"
] | [
"Verify if the function \\( f(x) = |x| \\) is continuously differentiable, differentiable but not continuously differentiable, or not differentiable at the origin."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-cf17914058236b40f5701ecabc771e8a2075bc08b4aee788825d188411f08853 | Verify that $(1376)_8 = (53t)_{12}$. | [
"766"
] | [
"Verify that $(1376)_8 = (53t)_{12}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 7
} | 10 | 0 | 0 |
omr-ws-dbdb9f292796d1ca7e8828f73a05c50cee1683f0b3b6f2e1171810156b950afe | Verify that a 7x7 square with one 1x1 square removed can be covered with 16 L-shaped pieces, each composed of 3 squares. | [
"The 7x7 square with one 1x1 square removed can be covered with 16 L-shaped pieces, each composed of 3 squares."
] | [
"Verify that a 7x7 square with one 1x1 square removed can be covered with 16 L-shaped pieces, each composed of 3 squares."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-5d6b5ec6d05dd3b37a88f18b216da830e3c9aae391d7e7ed336f03020bf359ea | Verify that among $16^{17}$ crocodiles, there are no two crocodiles with the same order of teeth, given that each crocodile has 68 teeth. | [
"\\(68! > 16^{17}\\)"
] | [
"Verify that among $16^{17}$ crocodiles, there are no two crocodiles with the same order of teeth, given that each crocodile has 68 teeth."
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-60fd7b3efd3a91c6bb7acc5509cba55d434c51401f7d88b382454adac8668040 | Verify that for $\alpha = \frac{2}{3}$, $\beta = \frac{3}{4}$, $\gamma = \frac{17}{18}$, $\Delta_{1} = 21$, $\Delta_{2} = -4$, and $\Delta_{3} = 18$, the value of $b_{1}$ is \[ b_{1} = \left(\frac{1}{12}\right)^{2} + \left(\frac{5}{54}\right)^{2} + \left(\frac{1}{72}\right)^{2}. \] | [
"\\(\\left(\\frac{1}{12}\\right)^{2} + \\left(\\frac{5}{54}\\right)^{2} + \\left(\\frac{1}{72}\\right)^{2}\\)"
] | [
"Verify that for $\\alpha = \\frac{2}{3}$, $\\beta = \\frac{3}{4}$, $\\gamma = \\frac{17}{18}$, $\\Delta_{1} = 21$, $\\Delta_{2} = -4$, and $\\Delta_{3} = 18$, the value of $b_{1}$ is \n\\[\nb_{1} = \\left(\\frac{1}{12}\\right)^{2} + \\left(\\frac{5}{54}\\right)^{2} + \\left(\\frac{1}{72}\\right)^{2}.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 5
} | 9 | 0 | 0 |
omr-ws-722f46aba0d51c00c668edd32e3b8366078f65e6c0055d1794059179d53ddc3f | Verify that for $\alpha = \frac{3}{5}$, $\beta = -\frac{5}{8}$, $\gamma = -\frac{1}{25}$, $\Delta_{1} = 15$, $\Delta_{2} = -16$, and $\Delta_{3} = 25$, the value of $b_{1}$ is \[ b_{1} = \left(\frac{1}{40}\right)^{2} + \left(\frac{8}{125}\right)^{2} + \left(\frac{13}{200}\right)^{2}. \] | [
"\\(\\left(\\frac{1}{40}\\right)^{2} + \\left(\\frac{8}{125}\\right)^{2} + \\left(\\frac{13}{200}\\right)^{2}\\)"
] | [
"Verify that for $\\alpha = \\frac{3}{5}$, $\\beta = -\\frac{5}{8}$, $\\gamma = -\\frac{1}{25}$, $\\Delta_{1} = 15$, $\\Delta_{2} = -16$, and $\\Delta_{3} = 25$, the value of $b_{1}$ is \n\\[\nb_{1} = \\left(\\frac{1}{40}\\right)^{2} + \\left(\\frac{8}{125}\\right)^{2} + \\left(\\frac{13}{200}\\right)^{2}.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 1
} | 7 | 0 | 0 |
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