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-267ac7f9e6da5835e703892f3dcbfd741c37ef2e28280d1a9100bc90547cf5dc | Verify that for $\alpha = \frac{5}{6}$, $\beta = \frac{6}{7}$, $\gamma = \frac{71}{72}$, $\Delta_{1} = 78$, $\Delta_{2} = -7$, and $\Delta_{3} = 72$, the value of $b_{1}$ is \[ b_{1} = \left(\frac{1}{42}\right)^{2} + \left(\frac{11}{432}\right)^{2} + \left(\frac{1}{504}\right)^{2}. \] | [
"\\(\\left(\\frac{1}{42}\\right)^2 + \\left(\\frac{11}{432}\\right)^2 + \\left(\\frac{1}{504}\\right)^2\\)"
] | [
"Verify that for $\\alpha = \\frac{5}{6}$, $\\beta = \\frac{6}{7}$, $\\gamma = \\frac{71}{72}$, $\\Delta_{1} = 78$, $\\Delta_{2} = -7$, and $\\Delta_{3} = 72$, the value of $b_{1}$ is \n\\[\nb_{1} = \\left(\\frac{1}{42}\\right)^{2} + \\left(\\frac{11}{432}\\right)^{2} + \\left(\\frac{1}{504}\\right)^{2}.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 21
} | 35 | 0 | 0 |
omr-ws-028f25b323181b15a5bc096f17a13ac4a9146684259acbfe21f5bf5dd281b2c6 | Verify that the diagonals of the quadrilateral with vertices K(-1,4), L(2,2), M(0,-1), and N(-3,1) are equal in length. | [
"\\sqrt{26}"
] | [
"Verify that the diagonals of the quadrilateral with vertices K(-1,4), L(2,2), M(0,-1), and N(-3,1) are equal in length."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-be01a1a004b8c16651175c126a724afcef79c9b4c79f7e2823eb5a22146a04c1 | Verify that the equation \(X^2 - 34y^2 = \pm 33\) has integer solutions using a naive method. | [
"\\((1, 1)\\) and \\((13, 2)\\)"
] | [
"Verify that the equation \\(X^2 - 34y^2 = \\pm 33\\) has integer solutions using a naive method."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 4,
"tir": 25
} | 58 | 0 | 0 |
omr-ws-170cdcf79f29caeb61be45ac02c99dcb1e00b0c0133ac0e592f4a35eec8c580b | Verify that the quadrilateral with vertices K(-1,4), L(2,2), M(0,-1), and N(-3,1) is a square. | [
"\\text{The quadrilateral is a square.}"
] | [
"Verify that the quadrilateral with vertices K(-1,4), L(2,2), M(0,-1), and N(-3,1) is a square."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7315bc46470038bd90cab1098457bb87cc00701a496ccbe1f6948b1fbc88189e | Verify that the sequence $p_i = 199 + 210i$ for $i \in \{0, 1, \dots, 9\}$ consists of 10 prime numbers. | [
"The sequence \\( p_i = 199 + 210i \\) for \\( i \\in \\{0, 1, \\dots, 9\\} \\) consists of 10 prime numbers."
] | [
"Verify that the sequence $p_i = 199 + 210i$ for $i \\in \\{0, 1, \\dots, 9\\}$ consists of 10 prime numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-0a8aefb7fd563019d86d00650882bc0e7834fec2e98dcd3d5df08d3032300132 | Verify the calculation: \[ I = \int_0^\infty 2\pi \int_z^\infty re^{-r^2} \, dr \, dz + \int_0^\infty e^{-2z^2} 2\pi \int_0^z re^{r^2} \, dr \, dz \] | [
"\\frac{\\pi^{3/2}}{2} \\left(2 - \\frac{1}{\\sqrt{2}}\\right)"
] | [
"Verify the calculation:\n\\[ I = \\int_0^\\infty 2\\pi \\int_z^\\infty re^{-r^2} \\, dr \\, dz + \\int_0^\\infty e^{-2z^2} 2\\pi \\int_0^z re^{r^2} \\, dr \\, dz \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e0c158a35df30d6226c1917c5aec512724f571289274d49c370258ac75e3f9b5 | Verify the claims for the minimal and characteristic polynomials of the matrix $A_n$ representing the $n$-dimensional hypercube for $n=1, 2, 3, 4$. | [
"Verified."
] | [
"Verify the claims for the minimal and characteristic polynomials of the matrix $A_n$ representing the $n$-dimensional hypercube for $n=1, 2, 3, 4$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-a6625b0e1ae9e15f04f90bbc0aa41d97c95a60bb71e4f00cb58279e827888ffe | Verify the closure property for the pair of elements $B$ and $E$ in the group $M$ by finding $BE$ and $EB$. | [
"\\( BE = D \\) and \\( EB = F \\)"
] | [
"Verify the closure property for the pair of elements $B$ and $E$ in the group $M$ by finding $BE$ and $EB$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e30a706ecc703c817f345475e8cadd82b21491d6d65766276627323a779a578 | Verify the computation \(5^{100} \equiv x_1 + 1 \equiv -285 \pmod{1001}\). | [
"-285"
] | [
"Verify the computation \\(5^{100} \\equiv x_1 + 1 \\equiv -285 \\pmod{1001}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-56b691bc2aa3792811b8f21668796c5907776bcd3a2497bb6ee9d40d48ba11cf | Verify the correctness of the information that the beam of light from the Gibb's Hill Lighthouse, which is 362 feet above sea level, can be seen on the horizon about 26 miles distant. | [
"23.29 miles"
] | [
"Verify the correctness of the information that the beam of light from the Gibb's Hill Lighthouse, which is 362 feet above sea level, can be seen on the horizon about 26 miles distant."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-5b0beef4843860b14160d577e03d0b85ec331f329abf8668fc0a6709da2fabe4 | Verify the derived formula for the second derivative by calculating $f''(x)$ for $f(x) = x^3$ at $x = 1$ using $h = 0.01$ and $h = -0.01$. | [
"6.06"
] | [
"Verify the derived formula for the second derivative by calculating $f''(x)$ for $f(x) = x^3$ at $x = 1$ using $h = 0.01$ and $h = -0.01$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fac1a9f6e447d8bee2f1e4d4bda89443dfea9d0ef94c9ae87fc290f0bffcdb0e | Verify the equation for Fermat's Last Theorem for \( n = 3 \): \[ A^2 = \sum_{x=\sqrt{\frac{C}{A}}}^{C*\sqrt{\frac{C}{A}}} \left(2x\sqrt{\frac{C}{A}}-\frac{C}{A}\right) - \left(\frac{B}{C}\right) \sum_{x=\sqrt{\frac{C}{A}}}^{B*\sqrt{\frac{C}{A}}} \left(2x\sqrt{\frac{C}{A}}-\frac{C}{A}\right) \] Test with \( A = 5 \), \... | [
"25"
] | [
"Verify the equation for Fermat's Last Theorem for \\( n = 3 \\):\n\n\\[ A^2 = \\sum_{x=\\sqrt{\\frac{C}{A}}}^{C*\\sqrt{\\frac{C}{A}}} \\left(2x\\sqrt{\\frac{C}{A}}-\\frac{C}{A}\\right) - \\left(\\frac{B}{C}\\right) \\sum_{x=\\sqrt{\\frac{C}{A}}}^{B*\\sqrt{\\frac{C}{A}}} \\left(2x\\sqrt{\\frac{C}{A}}-\\frac{C}{A}\\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-cb9c623b4db3dc621dc500f6220e467d7691c0e89313e447c34566e8d45a16fa | Verify the equation for Fermat's Last Theorem for \( n = 3 \): \[ A^2 = \sum_{x=\sqrt{\frac{C}{A}}}^{C*\sqrt{\frac{C}{A}}} \left(2x\sqrt{\frac{C}{A}}-\frac{C}{A}\right) - \left(\frac{B}{C}\right) \sum_{x=\sqrt{\frac{C}{A}}}^{B*\sqrt{\frac{C}{A}}} \left(2x\sqrt{\frac{C}{A}}-\frac{C}{A}\right) \] Test with \( A = 7 \), \... | [
"49"
] | [
"Verify the equation for Fermat's Last Theorem for \\( n = 3 \\):\n\n\\[ A^2 = \\sum_{x=\\sqrt{\\frac{C}{A}}}^{C*\\sqrt{\\frac{C}{A}}} \\left(2x\\sqrt{\\frac{C}{A}}-\\frac{C}{A}\\right) - \\left(\\frac{B}{C}\\right) \\sum_{x=\\sqrt{\\frac{C}{A}}}^{B*\\sqrt{\\frac{C}{A}}} \\left(2x\\sqrt{\\frac{C}{A}}-\\frac{C}{A}\\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-428f6a71077dd94221d81e1ab5343fc03344d1fd792b3ac40029996890f9f2ec | Verify the identity \((2+i)(3+i) = 5(1+i)\). | [
"\\(5(1+i)\\)"
] | [
"Verify the identity \\((2+i)(3+i) = 5(1+i)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 1,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-ad1ad2291545d0dbf196462eaee588c1349f007a16db73e3ed898e29323d66e7 | Verify the identity \((2+i)(5+i)(8+i) = 65(1+i)\). | [
"\\(65(1+i)\\)"
] | [
"Verify the identity \\((2+i)(5+i)(8+i) = 65(1+i)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-b95c372633c30c6b1657b0e6c2a6893c1b09050108df05221dc860080b7edf6b | Verify the inequality $(a+b+c)^3 \ge 9(a^2b + b^2c + c^2a)$ by testing the values $a = b = 1$ and $c \rightarrow 0^+$. | [
"8 < 9"
] | [
"Verify the inequality $(a+b+c)^3 \\ge 9(a^2b + b^2c + c^2a)$ by testing the values $a = b = 1$ and $c \\rightarrow 0^+$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-503236b54d2d1f63dd21f2f3ce23ef89650f81dc63191c43c161a4cfc7bec30d | Verify the linear approximation $(1+4x)^3 \approx 1-12x$ at $a=0$. Then determine the values of $x$ for which the approximation is accurate to within $0.1$. Provide the answer in interval notation, with decimals rounded to 3 places. | [
"(-0.004, 0.004)"
] | [
"Verify the linear approximation $(1+4x)^3 \\approx 1-12x$ at $a=0$. Then determine the values of $x$ for which the approximation is accurate to within $0.1$. Provide the answer in interval notation, with decimals rounded to 3 places."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-5565f6a73be8c6ae47a398a4a099c929e1b3a254cef3b768a58d4102f970ab13 | Verify the linear approximation $\sqrt[4]{1+2x} \approx 1 + \frac{x}{2}$ at $a=0$. Then determine the values of $x$ for which the approximation is accurate to within $0.1$. Provide the answer in interval notation, with decimals rounded to 3 places. | [
"\\((-0.045, 0.055)\\)"
] | [
"Verify the linear approximation $\\sqrt[4]{1+2x} \\approx 1 + \\frac{x}{2}$ at $a=0$. Then determine the values of $x$ for which the approximation is accurate to within $0.1$. Provide the answer in interval notation, with decimals rounded to 3 places."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bdc3463e8974c2d5578f7b0cc64a363289aa450d532a6e6f3b7f5e731824daf2 | Verify the matrix multiplication: \[ \begin{pmatrix} 1 & 0.16 & 0.81 & 0.06 \\ 0.47 & 1 & 0.16 & 0.47 \\ 0.47 & 0.81 & 1 & 0.47 \\ 0.06 & 0.03 & 0.03 & 0.01 \end{pmatrix} \times \begin{pmatrix} 0.12\\ 0.42\\ 0.42\\ 0.04 \end{pmatrix} = \begin{pmatrix} 0.26\\ 0.28\\ 0.42\\ 0.04 \end{pmatrix} \] | [
"\\(\\begin{pmatrix} 0.53\\\\0.56\\\\0.84\\\\0.03 \\end{pmatrix}\\)"
] | [
"Verify the matrix multiplication:\n\\[\n\\begin{pmatrix}\n1 & 0.16 & 0.81 & 0.06 \\\\\n0.47 & 1 & 0.16 & 0.47 \\\\\n0.47 & 0.81 & 1 & 0.47 \\\\\n0.06 & 0.03 & 0.03 & 0.01\n\\end{pmatrix} \\times \\begin{pmatrix}\n0.12\\\\\n0.42\\\\\n0.42\\\\\n0.04\n\\end{pmatrix} = \\begin{pmatrix}\n0.26\\\\\n0.28\\\\\n0.42\\\\\n0... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 13
} | 22 | 0 | 0 |
omr-ws-198fa7ddc3909aa7a1e506f1f18d8e67643dc398fc17d9bd5c2524a1df748622 | Verify the matrix multiplication: \[ \begin{pmatrix} 1 & 0.72 & 0.77 & 0 \\ 0.62 & 1 & 0 & 0.56 \\ 0.38 & 0 & 1 & 0.44 \\ 0 & 0.28 & 0.23 & 1 \end{pmatrix} \times \begin{pmatrix} 0.49\\ 0.21\\ 0.13\\ 0.15 \end{pmatrix} = \begin{pmatrix} 0.38\\ 0.3\\ 0.19\\ 0.13 \end{pmatrix} \] | [
"\\begin{pmatrix} 0.74 \\\\ 0.60 \\\\ 0.38 \\\\ 0.24 \\end{pmatrix}"
] | [
"Verify the matrix multiplication:\n\\[\n\\begin{pmatrix}\n1 & 0.72 & 0.77 & 0 \\\\\n0.62 & 1 & 0 & 0.56 \\\\\n0.38 & 0 & 1 & 0.44 \\\\\n0 & 0.28 & 0.23 & 1\n\\end{pmatrix} \\times \\begin{pmatrix}\n0.49\\\\\n0.21\\\\\n0.13\\\\\n0.15\n\\end{pmatrix} = \\begin{pmatrix}\n0.38\\\\\n0.3\\\\\n0.19\\\\\n0.13\n\\end{pmatr... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-74ff99d0583c7583086a82e4849142a833568fe3d4a0a0916e48c085d11f470c | Verify the result from Problem 1 in the case \( f(x) = \tan x \). Hence, find \[ \int \frac{\sin^4 x}{\cos^8 x} \, dx. \] | [
"\\(\\frac{1}{5}(\\tan x)^5 \\sec^2 x - \\frac{2}{35}(\\tan x)^7\\)"
] | [
"Verify the result from Problem 1 in the case \\( f(x) = \\tan x \\). Hence, find\n\\[ \\int \\frac{\\sin^4 x}{\\cos^8 x} \\, dx. \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-928f92441eca1c7a20801916ca816e93c713c6c6816dfcb0b5585fe3d482acf5 | Verify the row echelon form of the matrix $$\begin{pmatrix} 1 & -1 &0 &1& 1&-1&0&1\\ 0&0&1&-1&0&0&1&-1\\5&0&-3&3&0&0&0&0\\0&5&-3&-2&0&0&0&0\end{pmatrix}$$ and determine if the form $$\begin{pmatrix}1&0&0&0&*&*&*&*\\0&1&0&-1&*&*&*&*\\0&0&1&-1&*&*&*&*\\0&0&0&0&1&-1&0&1\end{pmatrix}$$ is correct. | [
"\\(\\begin{pmatrix}1&0&0&0&*&*&*&*\\\\0&1&0&-1&*&*&*&*\\\\0&0&1&-1&*&*&*&*\\\\0&0&0&0&1&-1&0&1\\end{pmatrix}\\)"
] | [
"Verify the row echelon form of the matrix $$\\begin{pmatrix} 1 & -1 &0 &1& 1&-1&0&1\\\\ 0&0&1&-1&0&0&1&-1\\\\5&0&-3&3&0&0&0&0\\\\0&5&-3&-2&0&0&0&0\\end{pmatrix}$$ and determine if the form $$\\begin{pmatrix}1&0&0&0&*&*&*&*\\\\0&1&0&-1&*&*&*&*\\\\0&0&1&-1&*&*&*&*\\\\0&0&0&0&1&-1&0&1\\end{pmatrix}$$ is correct."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 29
} | 50 | 0 | 0 |
omr-ws-3cfc6cf44ca2adfb1190ad980d9a561e16d02311b380e64fb134438ac372316b | Verify the sum of the series when \( b = 1 \), which should be \( n \). | [
"\\( n \\)"
] | [
"Verify the sum of the series when \\( b = 1 \\), which should be \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ecde924ab589067079faaa6a715635eee4cfad83a4514117acfc30886c72f0a1 | Verify the sum of the series when \( b = 3 \), which should be \( \left\lfloor \frac{3^{n+1}}{2} \right\rfloor \), where \( \left\lfloor x \right\rfloor \) denotes the floor function. | [
"\\(\\left\\lfloor \\frac{3^{n+1}}{2} \\right\\rfloor\\)"
] | [
"Verify the sum of the series when \\( b = 3 \\), which should be \\( \\left\\lfloor \\frac{3^{n+1}}{2} \\right\\rfloor \\), where \\( \\left\\lfloor x \\right\\rfloor \\) denotes the floor function."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-5114a96fb2dcd25c240399ae6b668509714d14501bbeb92a4a41a02fe3fcea2f | Verify the validity of the inequality for the specific values \(b = c = -k\) and \(a = k\). | [
"\\text{Valid}"
] | [
"Verify the validity of the inequality for the specific values \\(b = c = -k\\) and \\(a = k\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-27ca04cb168a518b692858de490a252419057b79d7ba83a8bc239e662365ba27 | Verify whether the following sequences are bounded. $x_n=\frac{3n+5}{2n-3}$. | [
"Yes, it is, $x_n\\in [-8,11]$"
] | [
"Verify whether the following sequences are bounded. $x_n=\\frac{3n+5}{2n-3}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0670c4bab28424a02cf01c57f54e0902bac5fde90bc8d9edfb884705a1c1641e | Verify whether the following sequences are bounded. Assume that $x_1$ and $x_2$ are roots of the equation $x^2-3x+A=0$, and $x_3$ and $x_4$ are roots of the equation $x^2-12x+B=0$. The sequence $x_1,x_2,x_3,x_4$ is known to be an increasing geometric progression. Find $A$ and $B$. | [
"$A = 2$, $B = 32$"
] | [
"Verify whether the following sequences are bounded. Assume that $x_1$ and $x_2$ are roots of the equation $x^2-3x+A=0$, and $x_3$ and $x_4$ are roots of the equation $x^2-12x+B=0$. The sequence $x_1,x_2,x_3,x_4$ is known to be an increasing geometric progression. Find $A$ and $B$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b5ceae94b64d0bb8d9e0877cce79bba8f60fa8bb5bb960068441b7952b7472a | Verify whether the following sequences are bounded. Solve the following system of equations: $$\left\{\begin{array}{lll} \frac{x}{y}=\frac{y}{z}=\frac{z}{u}=\frac{u}{s}=\frac{s}{t},\medskip \\ x=8u,\medskip \\ x+y+z+u+s+t=15\frac{3}{4}. \end{array}\right.$$ | [
"$(8, 4, 2, 1, 1/2, 1/4)$"
] | [
"Verify whether the following sequences are bounded. Solve the following system of equations: $$\\left\\{\\begin{array}{lll} \\frac{x}{y}=\\frac{y}{z}=\\frac{z}{u}=\\frac{u}{s}=\\frac{s}{t},\\medskip \\\\ x=8u,\\medskip \\\\ x+y+z+u+s+t=15\\frac{3}{4}. \\end{array}\\right.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6df436796d3af3c59f278ecd7e12fbfe97608ead85d3c69c70b7becb106d2a48 | Verify whether the following sequences are bounded. The general term of a sequence is represented as $$x_n=\frac{1}{2}+\frac{1}{4}+\ldots +\frac{1}{2n}.$$ How many terms of the sequence are smaller than $\frac{1023}{1024}$? | [
"Nine terms"
] | [
"Verify whether the following sequences are bounded. The general term of a sequence is represented as $$x_n=\\frac{1}{2}+\\frac{1}{4}+\\ldots +\\frac{1}{2n}.$$ How many terms of the sequence are smaller than $\\frac{1023}{1024}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9920f6b7a9974fe18e2f2836ba7de7cf00b9998219f4f95f8b78035484486baf | Verify whether the following sequences are bounded. The leg of an isosceles right-angled triangle is divided into $n$ equal parts and inscribed rectangles are constructed on the resulting line segments. Find the limit of the sequence $(S_n)$, $n\in \mathbb{N}$, of the areas formed by the stepped figures. | [
"$\\lim\\limits_{n\\to \\infty }S_n=a^2/2$"
] | [
"Verify whether the following sequences are bounded. The leg of an isosceles right-angled triangle is divided into $n$ equal parts and inscribed rectangles are constructed on the resulting line segments. Find the limit of the sequence $(S_n)$, $n\\in \\mathbb{N}$, of the areas formed by the stepped figures."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8edeb22fce9f8f5c381db45a3ecf3d9029d64587518d9f707c40d40c2e623601 | Veronica writes \( N \) integers between 2 and 2020 (inclusive) on a blackboard, and she notices that no number on the board is an integer power of another number on the board. What is the largest possible value of \( N \)? | [
"1964"
] | [
"Veronica writes \\( N \\) integers between 2 and 2020 (inclusive) on a blackboard, and she notices that no number on the board is an integer power of another number on the board. What is the largest possible value of \\( N \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-9076c7c83adfc24568f14a015302ef79960f7c81f55b1bd15822ae80859b0a37 | Vertex $A$ of the acute-angled triangle $ABC$ is equidistant from the circumcentre $O$ and the orthocentre $H$. Determine all possible measures of the angle $A$. | [
"$\\frac{\\pi}{3}$ or $60^\\circ$"
] | [
"Vertex $A$ of the acute-angled triangle $ABC$ is equidistant from the circumcentre $O$ and the orthocentre $H$. Determine all possible measures of the angle $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-407233fa27ba6b3fd134b4a0ed3856914e16429c2c2f2f7ac43cf0c3bfbd1626 | Vertex A of square ABCD is placed at the center of square MNPQ, and AB trisects MN. If AB = MN = 1, find the area of the overlap between squares ABCD and MNPQ. | [
"\\dfrac{1}{4}"
] | [
"Vertex A of square ABCD is placed at the center of square MNPQ, and AB trisects MN. If AB = MN = 1, find the area of the overlap between squares ABCD and MNPQ."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-a73ad20c3a285046a1024f9d04f8f277cc9770cc1d510fc82b1c39e15f8fbd25 | Vertex E of equilateral triangle ABE is in the interior of unit square ABCD. Let R be the region consisting of all points inside square ABCD and outside triangle ABE whose distance from AD is between 1/3 and 2/3. What is the area of R? | [
"\\dfrac{12 - 5\\sqrt{3}}{36}"
] | [
"Vertex E of equilateral triangle ABE is in the interior of unit square ABCD. Let R be the region consisting of all points inside square ABCD and outside triangle ABE whose distance from AD is between 1/3 and 2/3. What is the area of R?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-db233d879cf18a89a06a8174080a3f05b5c43a7cd496d51299b9ae9d8596280d | Vertices $A, B, C$ of an equilateral triangle with side length $1$ are on the surface of a sphere with radius $1$ and center $O$. Let $D$ be the orthogonal projection of $A$ onto the plane $\alpha$ determined by points $B, C, O$. Let $N$ be one of the intersections of the line perpendicular to $\alpha$ passing through ... | [
"\\(30^{\\circ}\\)"
] | [
"Vertices $A, B, C$ of an equilateral triangle with side length $1$ are on the surface of a sphere with radius $1$ and center $O$. Let $D$ be the orthogonal projection of $A$ onto the plane $\\alpha$ determined by points $B, C, O$. Let $N$ be one of the intersections of the line perpendicular to $\\alpha$ passing t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-50098d1fde4efdce24b3a6ee26ed0a06a0389a0f300302782f2d4bad32daa537 | Vertices \(A\) and \(B\) of triangle \(ABC\) lie on a circle with center \(O\). Points \(C\) and \(O\) are on one side of the line \(AB\). By rotating the triangle \(ABC\) around the point \(O\), we obtained the triangle \(A_1B_1C_1\), and the ray \(B_1C_1\) passes through the vertex \(C\) and intersects the circle at ... | [
"10"
] | [
"Vertices \\(A\\) and \\(B\\) of triangle \\(ABC\\) lie on a circle with center \\(O\\). Points \\(C\\) and \\(O\\) are on one side of the line \\(AB\\). By rotating the triangle \\(ABC\\) around the point \\(O\\), we obtained the triangle \\(A_1B_1C_1\\), and the ray \\(B_1C_1\\) passes through the vertex \\(C\\) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a431c5d996489166f918e757f2a33414dd1143db59ebccc7b9dac1ac78fc0dd | Vertices of a regular 45-gon are colored in three colors, with each color equally represented. For each color, select three vertices such that the three triangles formed by the selected one-color vertices are congruent. If the vertices of the 45-gon are labeled in order as $1, 2, \ldots, 45$, calculate the sum of all p... | [
"330"
] | [
"Vertices of a regular 45-gon are colored in three colors, with each color equally represented. For each color, select three vertices such that the three triangles formed by the selected one-color vertices are congruent. If the vertices of the 45-gon are labeled in order as $1, 2, \\ldots, 45$, calculate the sum of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-e6f836c15d32df0f817c7f51ee67aa311b5aeef95a06e5b7c241b55d7577ced9 | Vertices of a square $ABCD$ with side length $\frac{25}{4}$ lie on a sphere. Parallel lines passing through points $A, B, C,$ and $D$ intersect the sphere at points $A_1, B_1, C_1,$ and $D_1$, respectively. Given that $AA_1=2$, $BB_1=10$, and $CC_1=6$, determine the length of the segment $DD_1$. | [
"14"
] | [
"Vertices of a square $ABCD$ with side length $\\frac{25}{4}$ lie on a sphere. Parallel lines passing through points $A, B, C,$ and $D$ intersect the sphere at points $A_1, B_1, C_1,$ and $D_1$, respectively. Given that $AA_1=2$, $BB_1=10$, and $CC_1=6$, determine the length of the segment $DD_1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-90e6a4cf7d748c4587c0b04f4b653998e287561fe7d850c130b230538b1404f6 | Vertices of a square \(ABCD\) with side length \(\frac{25}{4}\) lie on a sphere. Parallel lines passing through points \(A, B, C,\) and \(D\) intersect the sphere at points \(A_1, B_1, C_1,\) and \(D_1\), respectively. Given that \(AA_1 = 2\), \(BB_1 = 10\), and \(CC_1 = 6\), determine the length of the segment \(DD_1\... | [
"2"
] | [
"Vertices of a square \\(ABCD\\) with side length \\(\\frac{25}{4}\\) lie on a sphere. Parallel lines passing through points \\(A, B, C,\\) and \\(D\\) intersect the sphere at points \\(A_1, B_1, C_1,\\) and \\(D_1\\), respectively. Given that \\(AA_1 = 2\\), \\(BB_1 = 10\\), and \\(CC_1 = 6\\), determine the lengt... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-a592cdcf46144d0ebfc48683ea5768ec01208b1ec8777cb79a796c970a29ba3d | Vfire and algebrastar1234 live on opposite vertices of a cube-shaped planet. They start walking along the edges of the cube at a rate of one edge per minute. At each vertex, they randomly choose one of the three neighboring vertices with equal probability. What is the expected number of minutes it will take for them to... | [
"13"
] | [
"Vfire and algebrastar1234 live on opposite vertices of a cube-shaped planet. They start walking along the edges of the cube at a rate of one edge per minute. At each vertex, they randomly choose one of the three neighboring vertices with equal probability. What is the expected number of minutes it will take for th... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 5,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-d0096146bad822fe903f3b8cc0ecb053cccaaad1a08618788edf8166d0627d10 | Vfire is standing at the edge of a cliff. If he moves one unit towards the edge, he will fall off the cliff. Every twenty seconds, Vfire flips two fair coins. If both coins land heads, he moves one unit towards the edge of the cliff; otherwise, he moves one unit away from the edge of the cliff. What is the probability ... | [
"\\(\\frac{1}{3}\\)"
] | [
"Vfire is standing at the edge of a cliff. If he moves one unit towards the edge, he will fall off the cliff. Every twenty seconds, Vfire flips two fair coins. If both coins land heads, he moves one unit towards the edge of the cliff; otherwise, he moves one unit away from the edge of the cliff. What is the probabi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-c9d5188150d51f8aa01f96c71e8dc58d63c8befd106e52f2d288ce6dffd327ab | Vicente wrote 2005 digits on a single line such that each pair of adjacent digits forms a number divisible by either 17 or 23. If the last digit is 8, what is the first digit? | [
"2"
] | [
"Vicente wrote 2005 digits on a single line such that each pair of adjacent digits forms a number divisible by either 17 or 23. If the last digit is 8, what is the first digit?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-66f51934049033d77a6352cc72bb8860a1f4fc10821ccfc00246dfe02cc9d963 | Vicki has 6 cards. Four are Red on each side, one is Red on one side and Green on the other, and the sixth card is Red on one side and Blue on the other. Vicki selects one card at random and looks at one side of that card. If the side she is looking at is Red, find the probability that the other side of this card is Gr... | [
"\\(\\frac{1}{10}\\)"
] | [
"Vicki has 6 cards. Four are Red on each side, one is Red on one side and Green on the other, and the sixth card is Red on one side and Blue on the other. Vicki selects one card at random and looks at one side of that card. If the side she is looking at is Red, find the probability that the other side of this card ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-4379b421837fd6efb74ef2d61e37925823294f83691e7f6a84ca94701631d0fa | Victor excavates a disk of radius 2 meters to a depth of 30 cm and uses the excess earth to raise the section at a horizontal distance between 2 and 3 meters from the center O according to a triangular profile. Calculate the height difference D between the bottom and the top edge of the pit. Round to within 1 cm. | [
"81 cm"
] | [
"Victor excavates a disk of radius 2 meters to a depth of 30 cm and uses the excess earth to raise the section at a horizontal distance between 2 and 3 meters from the center O according to a triangular profile. Calculate the height difference D between the bottom and the top edge of the pit. Round to within 1 cm."... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 4,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-c2971d6ec30991d79d252a793ee5195274a9d32c532c42f058c2bea1a192a245 | Victor is ordering a burger and fries at Five Guys. There are 3 types of burgers and 2 types of fries. He can choose any combination of the following toppings: mayo, Bar-B-Q sauce, lettuce, pickles, tomatoes, jalapeno peppers, grilled onions, A1 steak sauce, grilled mushrooms, ketchup, green peppers, mustard, relish, o... | [
"384"
] | [
"Victor is ordering a burger and fries at Five Guys. There are 3 types of burgers and 2 types of fries. He can choose any combination of the following toppings: mayo, Bar-B-Q sauce, lettuce, pickles, tomatoes, jalapeno peppers, grilled onions, A1 steak sauce, grilled mushrooms, ketchup, green peppers, mustard, reli... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-7aa3d46cf934b2b8736e77c8f0bb66720675331f0ec5ab861ad340b534d9d228 | Victor's hexagonal cookie has an area of \( 6\sqrt{3} \) square inches and is perfectly inscribed in a circular cookie. What is the diameter of the circular cookie? | [
"4"
] | [
"Victor's hexagonal cookie has an area of \\( 6\\sqrt{3} \\) square inches and is perfectly inscribed in a circular cookie. What is the diameter of the circular cookie?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87bb478548ebb81c7f6873a6aca4d9ddba7a6e19a78e0658d8fe62a30de8a96a | Viktor and Natalia bought $2020$ buckets of ice-cream and want to organize a degustation schedule with $2020$ rounds such that: - In every round, both of them try $1$ ice-cream, and those $2$ ice-creams tried in a single round are different from each other. - At the end of the $2020$ rounds, both of them have tried eac... | [
"875"
] | [
"Viktor and Natalia bought $2020$ buckets of ice-cream and want to organize a degustation schedule with $2020$ rounds such that:\n- In every round, both of them try $1$ ice-cream, and those $2$ ice-creams tried in a single round are different from each other.\n- At the end of the $2020$ rounds, both of them have tr... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-249242411c3605c941e60c089133f9a8d104916beaff1236c2eb7cc7be336778 | Viktor and Natalia play a colouring game with a 3-dimensional cube taking turns alternatingly. Viktor goes first, and on each of his turns, he selects an unpainted edge and paints it violet. On each of Natalia's turns, she selects an unpainted edge or at most once during the game a face diagonal and paints it neon gree... | [
"384"
] | [
"Viktor and Natalia play a colouring game with a 3-dimensional cube taking turns alternatingly. Viktor goes first, and on each of his turns, he selects an unpainted edge and paints it violet. On each of Natalia's turns, she selects an unpainted edge or at most once during the game a face diagonal and paints it neon... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-e5bc8a787ebdab20a89ec8a5c4c0dd3fc7cc82eeced7506ec5566dc6893f8da1 | Vincent chooses two (not necessarily distinct) positive divisors of $360$ independently and at random. What is the probability that they are relatively prime? | [
"\\dfrac{35}{192}"
] | [
"Vincent chooses two (not necessarily distinct) positive divisors of $360$ independently and at random. What is the probability that they are relatively prime?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-55862f01975614dad125e8c0cdaac5035cfcf2d1fbcfc16a8c8972d1063c5640 | Vincent the Bug draws all the diagonals of a regular hexagon with area 720, splitting it into many pieces. Compute the area of the smallest piece. | [
"30"
] | [
"Vincent the Bug draws all the diagonals of a regular hexagon with area 720, splitting it into many pieces. Compute the area of the smallest piece."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-9eb6fa06888102c837b93b64ee0de949e9bdc8bf0aefd44f59df4b9b7b40092c | Vincent wishes to place three knights on a $4 \times 4$ board such that: 1. For every knight $N$, there exists another knight $N'$ that can move to the square occupied by $N$ in one move. 2. No two knights lie on the same row or column. Find the number of ways Vincent can accomplish this. | [
"36"
] | [
"Vincent wishes to place three knights on a $4 \\times 4$ board such that:\n1. For every knight $N$, there exists another knight $N'$ that can move to the square occupied by $N$ in one move.\n2. No two knights lie on the same row or column.\nFind the number of ways Vincent can accomplish this."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 17
} | 26 | 0 | 0 |
omr-ws-48f012cc448995a7e7b4417a8ca42a85e634a4deed333a218d8aea5181453ff5 | Vines must prepare envelopes containing a positive integer amount of memes such that he can distribute exactly \( p \) memes to Bruh and exactly \( q \) memes to Really N, where \( p \) and \( q \) are positive integers with \( p + q \leq 2016 \). What is the minimum number of envelopes Vines must take from Mount Meme ... | [
"21"
] | [
"Vines must prepare envelopes containing a positive integer amount of memes such that he can distribute exactly \\( p \\) memes to Bruh and exactly \\( q \\) memes to Really N, where \\( p \\) and \\( q \\) are positive integers with \\( p + q \\leq 2016 \\). What is the minimum number of envelopes Vines must take ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-dc59500e10165a6ae3bc42a466810daa4bd76caaa39492c7fe0b1120b6f01727 | Vinuta and Katy are standing on the points $(3,6)$ and $(3,-10)$ respectively. They decide to meet at some point on the ellipse $\frac{(y+2)^2}{289} + \frac{(x-3)^2}{225} = 1$. What is the sum of the distances that Vinuta and Katy must walk before they are reunited? | [
"34"
] | [
"Vinuta and Katy are standing on the points $(3,6)$ and $(3,-10)$ respectively. They decide to meet at some point on the ellipse $\\frac{(y+2)^2}{289} + \\frac{(x-3)^2}{225} = 1$. What is the sum of the distances that Vinuta and Katy must walk before they are reunited?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-bfdc33264b7c41ebb255db28af5b04f26a821b6d701ce96783b1d648ba363c38 | Vishal starts with $n$ copies of the number $1$ written on the board. Every minute, he takes two numbers $a, b$ and replaces them with either $a+b$ or $\min(a^2, b^2)$. After $n-1$ operations, there is $1$ number on the board. Let the maximal possible value of this number be $f(n)$. Calculate the value of $f(9)$. | [
"17"
] | [
"Vishal starts with $n$ copies of the number $1$ written on the board. Every minute, he takes two numbers $a, b$ and replaces them with either $a+b$ or $\\min(a^2, b^2)$. After $n-1$ operations, there is $1$ number on the board. Let the maximal possible value of this number be $f(n)$. Calculate the value of $f(9)$.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c6f557994ea684126128735779ca093aa4b3dc3bb943ed4cb1b348010f15a814 | Viswam walks half a mile to get to school each day, consisting of 10 city blocks of equal length, taking one minute per block. Today, after walking 5 blocks, he takes a detour and walks 3 blocks instead of one block to reach the next corner. From the time he starts his detour, what speed, in mph, must he walk to get to... | [
"4.2"
] | [
"Viswam walks half a mile to get to school each day, consisting of 10 city blocks of equal length, taking one minute per block. Today, after walking 5 blocks, he takes a detour and walks 3 blocks instead of one block to reach the next corner. From the time he starts his detour, what speed, in mph, must he walk to g... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 3,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-6cf6bf1791790523cff805ab4b12090786402a8d4341df58a2f6447a21064712 | Vitya cut a chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get? | [
"21"
] | [
"Vitya cut a chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-d196a87534b170233cdfea132163332b94292efbdc62aec1b0454b75c8744c2c | Vitya thought of $5$ integers and told Vanya all their pairwise sums: $$0, 1, 5, 7, 11, 12, 18, 24, 25, 29.$$ Help Vanya guess the numbers Vitya has in mind. | [
"\\{–3, 3, 4, 8, 21\\}"
] | [
"Vitya thought of $5$ integers and told Vanya all their pairwise sums:\n$$0, 1, 5, 7, 11, 12, 18, 24, 25, 29.$$\nHelp Vanya guess the numbers Vitya has in mind."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 27
} | 33 | 0 | 0 |
omr-ws-03328e7e0106d5b250373c5eae61a551471954913515ef3a5e726b2402f84176 | Vitya wrote down \( n \) different natural numbers in his notebook. For each pair of numbers from the notebook, he wrote out their smallest common multiple on the board. If \( n = 101 \), what is the maximum number of consecutive terms of a non-constant arithmetic progression that can appear among the \( \frac{n(n-1)}{... | [
"100"
] | [
"Vitya wrote down \\( n \\) different natural numbers in his notebook. For each pair of numbers from the notebook, he wrote out their smallest common multiple on the board. If \\( n = 101 \\), what is the maximum number of consecutive terms of a non-constant arithmetic progression that can appear among the \\( \\fr... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f10508a05117584c8d76c002d38034451bcc0291284fbeb6815fbb5be443ea11 | Vivek and Daniel are playing a game that ends when one person wins $5$ rounds. The probability that either wins the first round is $1/2$. In each subsequent round, the players have a probability of winning equal to the fraction of games that the player has lost. What is the probability that Vivek wins in six rounds? | [
"\\dfrac{1}{120}"
] | [
"Vivek and Daniel are playing a game that ends when one person wins $5$ rounds. The probability that either wins the first round is $1/2$. In each subsequent round, the players have a probability of winning equal to the fraction of games that the player has lost. What is the probability that Vivek wins in six round... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4fd88f57dcaf1c8e44f15aa3c612f6413054ea949a5253e99a47ae3ca4112230 | Vivek picks a number and then plays a game. At each step of the game, he takes the current number and replaces it with a new number according to the following rule: if the current number \( n \) is divisible by \( 3 \), he replaces \( n \) with \( \frac{n}{3} + 2 \), and otherwise he replaces \( n \) with \( \lfloor 3 ... | [
"1"
] | [
"Vivek picks a number and then plays a game. At each step of the game, he takes the current number and replaces it with a new number according to the following rule: if the current number \\( n \\) is divisible by \\( 3 \\), he replaces \\( n \\) with \\( \\frac{n}{3} + 2 \\), and otherwise he replaces \\( n \\) wi... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c2882e85e3d1a817ca86c2d5b3f275bc80e1620c4ff7dabbb5bd6e9c4db607cb | Vivienne is deciding on what courses to take for Spring $2021$, and she must choose from four math courses, three computer science courses, and five English courses. Vivienne decides that she will take one English course and two additional courses that are either computer science or math. How many choices does Vivienne... | [
"105"
] | [
"Vivienne is deciding on what courses to take for Spring $2021$, and she must choose from four math courses, three computer science courses, and five English courses. Vivienne decides that she will take one English course and two additional courses that are either computer science or math. How many choices does Viv... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-b1d30956267a1e536993259ef28770aa77f00a2043de3387f668eca3c44c0166 | Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. What is the largest amount of money that Vlad can get from David? | [
"49"
] | [
"Vlad draws 100 rays in the Euclidean plane. David then draws a line $\\ell$ and pays Vlad one pound for each ray that $\\ell$ intersects. What is the largest amount of money that Vlad can get from David?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aae787df481f39f15004a113059503e3557a86d8647cb510bffd744fde9c1dc1 | Vojta has 5 fair six-sided dice. What is the probability that Vojta rolls a sum of 6 or 7? | [
"\\dfrac{5}{1944}"
] | [
"Vojta has 5 fair six-sided dice. What is the probability that Vojta rolls a sum of 6 or 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f40b4b7e12a9254ab12a39c77a9e16fa9b8a2df4276c4c8f816228a7e27e832 | Voldemort bought a book for $\$5$. It was one-tenth of its original price. What was the original price in dollars? | [
"\\$50"
] | [
"Voldemort bought a book for $\\$5$. It was one-tenth of its original price. What was the original price in dollars?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-808c01413b47a5ef1c921be0e4c020dd3dda71f9aaed57d8d09964e01d862767 | Vulcan and Neptune play a turn-based game on an infinite grid of unit squares. Before the game starts, Neptune chooses a finite number of cells to be flooded. Vulcan builds a levee, which is a subset of unit edges forming a connected, non-self-intersecting path or loop. On each of Vulcan’s turns, he may add up to three... | [
"4"
] | [
"Vulcan and Neptune play a turn-based game on an infinite grid of unit squares. Before the game starts, Neptune chooses a finite number of cells to be flooded. Vulcan builds a levee, which is a subset of unit edges forming a connected, non-self-intersecting path or loop. On each of Vulcan’s turns, he may add up to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-a71bb641265b736b1746729d02a947a037d87105ee7fbb7bdac5b6ca5e97db68 | W, X, Y, and Z are the midpoints of the four sides of parallelogram ABCD. P is a point on the line segment YZ. What percent of the area of parallelogram ABCD is triangle PXW? | [
"25%"
] | [
"W, X, Y, and Z are the midpoints of the four sides of parallelogram ABCD. P is a point on the line segment YZ. What percent of the area of parallelogram ABCD is triangle PXW?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-c4492653c2a80bbb02f2c3e954e8998afdad231ca66f77c8c0f3dd1399ab636e | Wabou has two $100$ dollar bills: one is marked with blue sharpie on both sides and the other bill is marked with blue sharpie on one side and red sharpie on the other side. Wabou randomly picks a bill and places it in his palm. If the side Wabou sees is blue, then what is the probability that the other side of the bil... | [
"\\dfrac{2}{3}"
] | [
"Wabou has two $100$ dollar bills: one is marked with blue sharpie on both sides and the other bill is marked with blue sharpie on one side and red sharpie on the other side. Wabou randomly picks a bill and places it in his palm. If the side Wabou sees is blue, then what is the probability that the other side of th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-28d805226e4763c47f2131b6236c61f4b27ebdd1f64b2c0ba5cd3c3be4fba079 | Wafa is writing a sequence of four positive integers whose units digits are non-zero. She calculates all pairwise products of these integers. Let \( S \) be the set of all distinct units digits that appear in the obtained products. Find the maximum possible value of \( |S| \). | [
"6"
] | [
"Wafa is writing a sequence of four positive integers whose units digits are non-zero. She calculates all pairwise products of these integers. Let \\( S \\) be the set of all distinct units digits that appear in the obtained products. Find the maximum possible value of \\( |S| \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66d4d145f8b31051df62fdf491305f1f8db0d3f7aa56b6c21f75e178013a3a49 | Wafa is writing a sequence of positive integers whose units digits are non-zero. She calculates all pairwise products of these integers. Her goal is to make all the digits from 0 to 9 appear in the units digit of the obtained products. What is the minimum number of integers which allows Wafa to achieve her goal? | [
"6"
] | [
"Wafa is writing a sequence of positive integers whose units digits are non-zero. She calculates all pairwise products of these integers. Her goal is to make all the digits from 0 to 9 appear in the units digit of the obtained products. What is the minimum number of integers which allows Wafa to achieve her goal?"
... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-562a1e5f4c4d8900dcfb2cdaa21f38389f257d70842b1c7c972a61f37e0f652e | Walking without stopping a postman went from a point $A$ through a point $B$ to a point $C$. The distance from $A$ to $B$ was covered with a speed of 3.5 km/h and from $B$ to $C$ of 4 km/h. To get back from $C$ to $A$ in the same time following the same route with a constant speed he was to walk 3.75 km/h. However, aft... | [
"x=14 \\text{ km}, y=16 \\text{ km}"
] | [
"Walking without stopping a postman went from a point $A$ through a point $B$ to a point $C$. The distance from $A$ to $B$ was covered with a speed of 3.5 km/h and from $B$ to $C$ of 4 km/h. To get back from $C$ to $A$ in the same time following the same route with a constant speed he was to walk 3.75 km/h. However... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2cc92492f3eeccec02e03d82d82d7b3ebcd35b7f69b575c76b151cede81d425b | Wallace and Gromit are waiting in a queue. There are \( x \) people behind Wallace, who is \( y \) places in front of Gromit. If there are \( n \) people in front of Gromit, how long, in terms of \( n \), \( x \), and \( y \), is the queue? | [
"\\( n + x - y + 1 \\)"
] | [
"Wallace and Gromit are waiting in a queue. There are \\( x \\) people behind Wallace, who is \\( y \\) places in front of Gromit. If there are \\( n \\) people in front of Gromit, how long, in terms of \\( n \\), \\( x \\), and \\( y \\), is the queue?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 2
} | 25 | 0 | 0 |
omr-ws-f3456d36abeb17e888a9ad8c6eef17a9dd2c88e965787410ef089bf4e6b43837 | Wally's Key Company makes and sells two types of keys. Mr. Porter buys a total of 12 keys from Wally's. Determine the number of possible arrangements of Mr. Porter's 12 new keys on his keychain (where rotations are considered the same and any two keys of the same type are identical). | [
"352"
] | [
"Wally's Key Company makes and sells two types of keys. Mr. Porter buys a total of 12 keys from Wally's. Determine the number of possible arrangements of Mr. Porter's 12 new keys on his keychain (where rotations are considered the same and any two keys of the same type are identical)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-ad3908cf41a71045093ddc5a4a1b0cfde997d9f087c11e5cc57d548424fa8e5b | Wanda and Lucas are playing a game with squares for each integer from $0$ to $35$, inclusive. They both start at square $0$. Every minute, a fair standard coin is flipped. If it turns up heads, Wanda moves from square $a$ to square $a + 1$, and if it turns up tails, Lucas moves from square $b$ to square $b + 1$. If Wan... | [
"\\dfrac{127}{128}"
] | [
"Wanda and Lucas are playing a game with squares for each integer from $0$ to $35$, inclusive. They both start at square $0$. Every minute, a fair standard coin is flipped. If it turns up heads, Wanda moves from square $a$ to square $a + 1$, and if it turns up tails, Lucas moves from square $b$ to square $b + 1$. I... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-877b4089c55f15c713ef4000cd5a1cde0e15a8bd1f5e2e5014f89c02913a93f8 | Wanda is trying to locate the Fermat point $P$ of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,0)$, and $C$ is at $(3,5)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point is at $P = (4,2)$, and computes the sum of ... | [
"5"
] | [
"Wanda is trying to locate the Fermat point $P$ of $\\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,0)$, and $C$ is at $(3,5)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point is at $P = (4,2)$, and computes the s... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-81af8ccb3a9c66552992da7c72b1b05d26d182be43323affd131d6b28cb3f4f9 | Wanda is trying to locate the Fermat point $P$ of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(8,-1)$, and $C$ is at $(5,4)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point is at $P = (4,2)$, and computes the sum of ... | [
"8"
] | [
"Wanda is trying to locate the Fermat point $P$ of $\\triangle ABC$, where $A$ is at the origin, $B$ is at $(8,-1)$, and $C$ is at $(5,4)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point is at $P = (4,2)$, and computes the s... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-601ff60468c987a161eb50e5650967f883be2b20ed3bf33cf59b2320aca85bdb | Wanda the Worm starts at the top of Pascal's triangle and eats $\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a, b, c$ such that $a + b = c$, what is the maximum number of entries she... | [
"2011"
] | [
"Wanda the Worm starts at the top of Pascal's triangle and eats $\\binom{0}{0}=1$. Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a, b, c$ such that $a + b = c$, what is the maximum number of entri... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-9ef1f68676652395543d37ed8e638feb2b42aa83fc353947da46a669c6ec851f | Wanda the witch agrees to trade one of her magic broomsticks to Gaspar the ghost in exchange for one of his gold chains. Gaspar insists on paying by the installment plan, one gold link per day for 63 days, with the balance to be forfeit if the broomstick malfunctions during the guarantee period. Wanda insists that the ... | [
"The chain should be cut at the 1st, 3rd, and 7th links."
] | [
"Wanda the witch agrees to trade one of her magic broomsticks to Gaspar the ghost in exchange for one of his gold chains. Gaspar insists on paying by the installment plan, one gold link per day for 63 days, with the balance to be forfeit if the broomstick malfunctions during the guarantee period. Wanda insists that... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-6b0c4d0b56f50ff3701878cacfc1784614f37f447ba86bb75eeb89101dc3ecd9 | Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days f... | [
"84"
] | [
"Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school d... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9fb5878e109e1c4caebe4390e86bc1cbdcad1a5800a656b3c5c181f421a07cc0 | Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternately, with Ward having the first turn. During their turn, a player must pick two numbers $a$ and $b$ on the sheet such that $\gcd(a, b) = 1$, e... | [
"Gabrielle"
] | [
"Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternately, with Ward having the first turn. During their turn, a player must pick two numbers $a$ and $b$ on the sheet such that $\\gcd(a, b) =... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f483c49be2330a098ebda4591bbe8d8320ef47ff5acbddb6e969986ff9f9ee82 | Warren has a strip of paper 10 meters long. He wishes to cut from it as many pieces as possible, not necessarily using all the paper, with each piece of paper a whole number of centimetres long. The second piece must be 10 cm longer than the first, the third 10 cm longer than the second, and so on. What is the length i... | [
"136"
] | [
"Warren has a strip of paper 10 meters long. He wishes to cut from it as many pieces as possible, not necessarily using all the paper, with each piece of paper a whole number of centimetres long. The second piece must be 10 cm longer than the first, the third 10 cm longer than the second, and so on. What is the len... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-ebd0d8fc46261db07c172d583b0642cd01c0ce9b1a0ad3406d5fd5981e71c376 | Water falls from the tip of a cone with a height of 50 inches and a radius of 30 inches into a cube with a base area of 400 square inches. The rate of change of the water that falls down is proportional to the height of the water (x inches). The rate of change of the height of the water lost from the cone is 50-x inche... | [
"\\dfrac{18\\pi}{5}"
] | [
"Water falls from the tip of a cone with a height of 50 inches and a radius of 30 inches into a cube with a base area of 400 square inches. The rate of change of the water that falls down is proportional to the height of the water (x inches). The rate of change of the height of the water lost from the cone is 50-x ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-9d9ed1d09adebdacfd28f98a24e26bd151f016a1a8d4bcb04af5f238b8e81872 | Water flows into Bernard’s cylindrical tank at 15 cubic meters per minute. The tank initially contains $1000\pi$ cubic meters of water and has a height of 15 m. The tank holds 50 pounds of salt. The tank is well-mixed and drains at a rate of 15 cubic meters per second. Bernard drops a pebble that will entirely cover th... | [
"50"
] | [
"Water flows into Bernard’s cylindrical tank at 15 cubic meters per minute. The tank initially contains $1000\\pi$ cubic meters of water and has a height of 15 m. The tank holds 50 pounds of salt. The tank is well-mixed and drains at a rate of 15 cubic meters per second. Bernard drops a pebble that will entirely co... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f47388b5b713365deaa2bda9fce1e93250494d99d0901d15e482d7456881283 | Water flows into a tank at a rate of 1 gallon per second. Water leaves the tank at a rate of 1 gallon per second for each 100 gallons in the tank. The tank is initially empty. How long will it take for the tank to fill with 50 gallons of water? | [
"69.3 seconds"
] | [
"Water flows into a tank at a rate of 1 gallon per second. Water leaves the tank at a rate of 1 gallon per second for each 100 gallons in the tank. The tank is initially empty. How long will it take for the tank to fill with 50 gallons of water?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 24
} | 24 | 0 | 0 |
omr-ws-89ea05f57cc61232539a61542c2c75623f6defaf757a1df601aa4df8d468aede | Water flows through a cylindrical pipe with an internal diameter of 7 cm at a rate of 36 km/hr. Calculate the time in minutes it would take to fill a cylindrical tank with a radius of 35 cm and height of 1 meter. | [
"$\\frac{1}{6}$ minute or 10 seconds."
] | [
"Water flows through a cylindrical pipe with an internal diameter of 7 cm at a rate of 36 km/hr. Calculate the time in minutes it would take to fill a cylindrical tank with a radius of 35 cm and height of 1 meter."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-8c4940743977a91b961afcc5b6f346aa7995d71b2a3be4cd9f54ad8086002365 | Water gradually enters a pit. Ten pumps of equal capacity operating together can pump the water out of the full pit in 12 hours, while 15 such pumps would need six hours. How much time would it take 25 such pumps? | [
"3 hours"
] | [
"Water gradually enters a pit. Ten pumps of equal capacity operating together can pump the water out of the full pit in 12 hours, while 15 such pumps would need six hours. How much time would it take 25 such pumps?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b3ea6e575d3058c719dfad78620590db1609f74d13f501c3aad210a4f03971c | Water is being drained from a conical container with a height of $10$ feet and a base radius of $4$ feet into a cylindrical container with a radius of $6$ feet and a height of $8$ feet at a rate of $15$ ft^3/min. How fast is the water level falling in the conical tank when the water level in the conical tank is $5$ fee... | [
"$\\frac{15}{16\\pi}$ feet per minute."
] | [
"Water is being drained from a conical container with a height of $10$ feet and a base radius of $4$ feet into a cylindrical container with a radius of $6$ feet and a height of $8$ feet at a rate of $15$ ft^3/min. How fast is the water level falling in the conical tank when the water level in the conical tank is $5... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-2b7d93af2fe7c60b2e8d9e79f5e7a0f043e1f7d4e1b2fb87922e983a4a008d03 | Water is being drained from a conical container with a height of $10$ feet and a base radius of $4$ feet into a cylindrical container with a radius of $6$ feet and a height of $8$ feet at a rate of $15$ ft^3/min. How fast is the water level rising in the cylindrical container when the water level in the conical tank is... | [
"$\\frac{5}{36\\pi}$"
] | [
"Water is being drained from a conical container with a height of $10$ feet and a base radius of $4$ feet into a cylindrical container with a radius of $6$ feet and a height of $8$ feet at a rate of $15$ ft^3/min. How fast is the water level rising in the cylindrical container when the water level in the conical ta... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca25f650a91a5669d65b9622789d5f0cbc2fc704ca35c629b47853d0cc5d4421 | Water is being poured into a cone with a semi-vertical angle of 45 degrees at a rate of 2 cubic meters per second. What is the rate at which the perimeter of the water surface changes when the height of the water in the cone is 2 meters? | [
"1 meter per second"
] | [
"Water is being poured into a cone with a semi-vertical angle of 45 degrees at a rate of 2 cubic meters per second. What is the rate at which the perimeter of the water surface changes when the height of the water in the cone is 2 meters?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 11
} | 25 | 0 | 0 |
omr-ws-a5010af0be00cb1974886bfe8df96e89f615723f07fc5fc304ffdb7cb092d6f4 | Water is being poured into a conical reservoir at the rate of 20 cubic feet per second. The reservoir has a radius of 10 feet across the top and a height of 24 feet. At what rate is the depth of the water increasing when the depth is 16 feet? | [
"\\(\\frac{9}{20 \\pi}\\)"
] | [
"Water is being poured into a conical reservoir at the rate of 20 cubic feet per second. The reservoir has a radius of 10 feet across the top and a height of 24 feet. At what rate is the depth of the water increasing when the depth is 16 feet?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-dcdcdcdfbe47d6d28b235c912d2bade3208e3fa6ede2a7a7dc10a124c7f7c40e | Water is being pumped into a spherical tank of radius 60 feet at a constant rate of 10 ft^3/s. Find the height at which the height of the water in the tank changes at the same rate as the radius. | [
"\\(60 - 30\\sqrt{2}\\)"
] | [
"Water is being pumped into a spherical tank of radius 60 feet at a constant rate of 10 ft^3/s. Find the height at which the height of the water in the tank changes at the same rate as the radius."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-abe22f7f7dd522b4c92f42b4e71bd8a08c8011f4da409f68907785331aaab8cc | Water is being pumped into a spherical tank of radius 60 feet at a constant rate of 10 ft^3/s. Find the rate at which the radius of the top level of water in the tank changes when the tank is half full. | [
"0"
] | [
"Water is being pumped into a spherical tank of radius 60 feet at a constant rate of 10 ft^3/s. Find the rate at which the radius of the top level of water in the tank changes when the tank is half full."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-373d265db4a6fcfa1ecca2eefc082b6125ef9c91b6d79e29e5a0cc090f87ddc2 | Water is flowing into a hemispherical pool with a diameter of 24 feet at a rate of 2 cubic feet per minute. The volume of water in the pool when the water is \( h \) feet deep is given by \( V = \frac{\pi}{3} h^2 (36 - h) \). How fast is the radius of the water's surface changing when the water is 6 feet deep? | [
"\\dfrac{\\sqrt{3}}{162\\pi}"
] | [
"Water is flowing into a hemispherical pool with a diameter of 24 feet at a rate of 2 cubic feet per minute. The volume of water in the pool when the water is \\( h \\) feet deep is given by \\( V = \\frac{\\pi}{3} h^2 (36 - h) \\). How fast is the radius of the water's surface changing when the water is 6 feet dee... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5fed6ceb366928912e781c31094c8e0ec99de28c6a88d8c7ee3666a3f353aca | Water is leaking out of an inverted conical tank at a rate of 10000 cubic centimeters per minute while water is being pumped into the tank at a constant rate. The tank has a height of 6 meters and a diameter at the top of 4 meters. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 me... | [
"\\( 10000 + \\frac{800000 \\pi}{9} \\)"
] | [
"Water is leaking out of an inverted conical tank at a rate of 10000 cubic centimeters per minute while water is being pumped into the tank at a constant rate. The tank has a height of 6 meters and a diameter at the top of 4 meters. If the water level is rising at a rate of 20 cm/min when the height of the water is... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 27
} | 44 | 0 | 0 |
omr-ws-ce7c2b6359c0d1d496c9452766a16c02dea53c8952b01343daf7e1c13eb0d49a | Water is leaking out of an inverted conical tank at a rate of 10600.0 cubic centimeters per minute at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 6.0 meters and the diameter at the top is 7.0 meters. If the water level is rising at a rate of 27.0 centimeters per m... | [
"\\( 10715.45 \\)"
] | [
"Water is leaking out of an inverted conical tank at a rate of 10600.0 cubic centimeters per minute at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 6.0 meters and the diameter at the top is 7.0 meters. If the water level is rising at a rate of 27.0 centimeters ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba124a7af8eecdd849fff40503923aff8a7b3fd843a10dceae3cf75278d07fea | Water is leaking out of an inverted conical tank at a rate of \(10,000 \, \text{cm}^3/\text{min}\) while water is being pumped into the tank at a constant rate. The tank has a height of \(6 \, \text{m}\) and a diameter at the top of \(4 \, \text{m}\). If the water level is rising at a rate of \(200 \, \text{cm}/\text{m... | [
"\\(\\frac{800,000\\pi}{9} + 10,000\\)"
] | [
"Water is leaking out of an inverted conical tank at a rate of \\(10,000 \\, \\text{cm}^3/\\text{min}\\) while water is being pumped into the tank at a constant rate. The tank has a height of \\(6 \\, \\text{m}\\) and a diameter at the top of \\(4 \\, \\text{m}\\). If the water level is rising at a rate of \\(200 \... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 18
} | 41 | 0 | 0 |
omr-ws-a37908f9a4f39dad9302c9578f36683f333fd20af21b9027a02054f88f411c03 | Water is poured into a cone at a rate of 2/3 cubic inch per second. The cone is 6 inches tall with a radius of 2 inches. Find the rate of change of the height of the water when the height is 4 inches. | [
"\\(\\frac{3}{8\\pi}\\)"
] | [
"Water is poured into a cone at a rate of 2/3 cubic inch per second. The cone is 6 inches tall with a radius of 2 inches. Find the rate of change of the height of the water when the height is 4 inches."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-ab00449f39ca4eb254862847a9975edd9ba920124f1c91781b38b90ed0a4eb9f | Water is pumped into a cylindrical tank, standing vertically, at a decreasing rate given at time $t$ minutes by the function $r(t) = 100 - 5t$ cubic feet per minute for $0 \le t \le 8$. The tank has a radius of $4$ ft and is empty when $t = 0$ minutes. Find the depth of the water in the tank at $t = 6$ minutes. | [
"10.15"
] | [
"Water is pumped into a cylindrical tank, standing vertically, at a decreasing rate given at time $t$ minutes by the function $r(t) = 100 - 5t$ cubic feet per minute for $0 \\le t \\le 8$. The tank has a radius of $4$ ft and is empty when $t = 0$ minutes. Find the depth of the water in the tank at $t = 6$ minutes."... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-4bd4c2d19fe0f1f1b61a0a60d7e0ee65d45eb0e34958f44f87f09cb33fc40aed | Water leaks out of a tank at a rate proportional to the square root of the depth of water at that time. The water level starts at 100 inches and drops to 64 inches in 1 hour. What is the equation that models the depth of water remaining in the tank as a function of time? | [
"\\( h(t) = (10 - 2t)^2 \\)"
] | [
"Water leaks out of a tank at a rate proportional to the square root of the depth of water at that time. The water level starts at 100 inches and drops to 64 inches in 1 hour. What is the equation that models the depth of water remaining in the tank as a function of time?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-ee3a131d1ad162aeb6555062dfe78749831b29b566f2313b02e9e25d085ace11 | Watson received a telegram from Sherlock Holmes to catch a specified train from London to Ciifton. The train consists of \( p \) carriages, each holding \( q \) passengers, and in total, \( pq - m \) passengers have occupied their seats. If Watson is equally likely to get any vacant seat, find the probability that he w... | [
"\\dfrac{q - 1}{pq - 1}"
] | [
"Watson received a telegram from Sherlock Holmes to catch a specified train from London to Ciifton. The train consists of \\( p \\) carriages, each holding \\( q \\) passengers, and in total, \\( pq - m \\) passengers have occupied their seats. If Watson is equally likely to get any vacant seat, find the probabilit... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3fb4b48d7ee89599564e94a3f2f3bdf5d1399fa41121025ce58bed04abf26dfe | Wayne drove 3 hours on a freeway at 55 mi/h and then drove 10 miles in the city at 35 mi/h. What was his average speed? | [
"\\(53 \\frac{6}{7}\\) miles per hour."
] | [
"Wayne drove 3 hours on a freeway at 55 mi/h and then drove 10 miles in the city at 35 mi/h. What was his average speed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
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