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-b8b6d71d3d5cc0668b9d294e345bd781ece9dee1eccb82e15fb5ee5ea27a17f7 | A line contains the points $(6,8)$, $(-2, k)$ and $(-10, 4)$. What is the value of $k$? | [
"6"
] | [
"A line contains the points $(6,8)$, $(-2, k)$ and $(-10, 4)$. What is the value of $k$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee7aec6bcac21fd59c72577362599b246f5fedc74dc66c803608b1d90d4c2969 | A line creates a 60-degree angle with the x-axis and is 4 units away from the origin. Find the equation of the line. | [
"\\( y = \\sqrt{3}x \\pm 8 \\)"
] | [
"A line creates a 60-degree angle with the x-axis and is 4 units away from the origin. Find the equation of the line."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 18
} | 19 | 0 | 0 |
omr-ws-a4cc28a15a483ecce249c418f023f76a4e3fa1b8736f21886fa9d40ed689701b | A line cuts a square into two regions such that the ratio of the areas of the two regions is 1 : 4. The part of the line inside the square is a segment of length \(L\). What is the smallest possible value of \(L^2\)? | [
"\\dfrac{4}{5}"
] | [
"A line cuts a square into two regions such that the ratio of the areas of the two regions is 1 : 4. The part of the line inside the square is a segment of length \\(L\\). What is the smallest possible value of \\(L^2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5494586f7c9a9959d1b80b9fea91b9c2577d8d3943d2fda0c7d702fc1ea294f | A line dance requires two steps forward and then one step back. If a wall is ten steps away, how many steps must the dancers take to reach the wall? | [
"30"
] | [
"A line dance requires two steps forward and then one step back. If a wall is ten steps away, how many steps must the dancers take to reach the wall?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f7209d096449d0388199a2278a49fb5247fd75570fdba060d07b51e2ed74fa1 | A line divides an equilateral triangle of side length unity into two parts with the same perimeter. Let the areas of these two parts be $S_1$ and $S_2$. Find the minimum value of $\frac{S_1}{S_2}$. | [
"$\\frac{7}{9}$"
] | [
"A line divides an equilateral triangle of side length unity into two parts with the same perimeter. Let the areas of these two parts be $S_1$ and $S_2$. Find the minimum value of $\\frac{S_1}{S_2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 8,
"tir": 5
} | 33 | 0 | 0 |
omr-ws-cca0483ae5581116d44a0df8fabd0d69b64f254a52282db03ca39a3b60d9f5c6 | A line drawn through a point $K$ in a square $ABCD$ intersects two opposite sides $AB$ and $CD$ at points $P$ and $Q$. Two circles are drawn: one through points $K, B, P$ and the other through points $K, D, Q$. Let $X$ be the second point of intersection of these two circles (the point other than $K$). If the side leng... | [
"\\dfrac{s\\sqrt{2}}{2}"
] | [
"A line drawn through a point $K$ in a square $ABCD$ intersects two opposite sides $AB$ and $CD$ at points $P$ and $Q$. Two circles are drawn: one through points $K, B, P$ and the other through points $K, D, Q$. Let $X$ be the second point of intersection of these two circles (the point other than $K$). If the side... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-a0cbf8c7b95408ae2cfd00175b375c5f52d0f894841cc1d1c55aad896a2abf06 | A line from the origin meets the lines $\frac{x-2}{1} = \frac{y-1}{-2} = \frac{z+1}{1}$ and $\frac{x-\frac{8}{3}}{2} = \frac{y+3}{-1} = \frac{z-1}{1}$ at points P and Q, respectively. Find the length of PQ. | [
"\\sqrt{6}"
] | [
"A line from the origin meets the lines $\\frac{x-2}{1} = \\frac{y-1}{-2} = \\frac{z+1}{1}$ and $\\frac{x-\\frac{8}{3}}{2} = \\frac{y+3}{-1} = \\frac{z-1}{1}$ at points P and Q, respectively. Find the length of PQ."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d55603da91244b460f6adc89e2b27d8057eecc6de4c8433b161e76b13b15b153 | A line from the right-angled vertex is drawn to the hypotenuse such that it is equidistant from the legs and it divides the hypotenuse into lengths 15 cm and 20 cm. Find the lengths of the legs of the triangle. | [
"28"
] | [
"A line from the right-angled vertex is drawn to the hypotenuse such that it is equidistant from the legs and it divides the hypotenuse into lengths 15 cm and 20 cm. Find the lengths of the legs of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-5ee17c4f43d6e0c8ab6905841f2aea2634a54ffe0c63078a38bd5917e625725c | A line has a slope of $-7$ and contains the point $(3,0)$. The equation of this line can be written in the form $y = mx+b$. What is the value of $m+b$? | [
"14"
] | [
"A line has a slope of $-7$ and contains the point $(3,0)$. The equation of this line can be written in the form $y = mx+b$. What is the value of $m+b$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87649fe76d199ab8e51dbdd52409fa50b3c3b39ea61bee11c65b080f1fa3671c | A line in the $xy$ plane is called wholesome if its equation is $y = mx + b$ where $m$ is rational and $b$ is an integer. Given a point with integer coordinates $(x, y)$ on a wholesome line $\ell$, let $r$ be the remainder when $x$ is divided by $7$, and let $s$ be the remainder when $y$ is divided by $7$. The pair $(r... | [
"50"
] | [
"A line in the $xy$ plane is called wholesome if its equation is $y = mx + b$ where $m$ is rational and $b$ is an integer. Given a point with integer coordinates $(x, y)$ on a wholesome line $\\ell$, let $r$ be the remainder when $x$ is divided by $7$, and let $s$ be the remainder when $y$ is divided by $7$. The pa... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-6e3b5c68ffa071f366339845c063c2c496919bfe7dd33b334973bc15fb258531 | A line in the first quadrant is at a distance of 6 units from the origin and makes an angle of $120^\circ$ with the positive X-axis. Find the equation of the line. | [
"\\sqrt{3}x + y = 12"
] | [
"A line in the first quadrant is at a distance of 6 units from the origin and makes an angle of $120^\\circ$ with the positive X-axis. Find the equation of the line."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8329e766db56c1d89f6c047f99a578f08a40ef066180495a98b37c3b5ef01671 | A line in the plane is called *strange* if it passes through $(a,0)$ and $(0,10-a)$ for some $a$ in the interval $[0,10]$. A point in the plane is called *charming* if it lies in the first quadrant and also lies below some *strange* line. What is the area of the set of all *charming* points? | [
"\\(\\frac{50}{3}\\)"
] | [
"A line in the plane is called *strange* if it passes through $(a,0)$ and $(0,10-a)$ for some $a$ in the interval $[0,10]$. A point in the plane is called *charming* if it lies in the first quadrant and also lies below some *strange* line. What is the area of the set of all *charming* points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-a66eb42104c545dd2e1b9ddefafefd9398f2dda8da478ae5e6810ff7b8c0f38e | A line in the plane is called strange if it passes through the points $(a, 0)$ and $(0, 10-a)$ for some $a$ in the interval $[0, 10]$. A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming points? | [
"\\(\\frac{50}{3}\\)"
] | [
"A line in the plane is called strange if it passes through the points $(a, 0)$ and $(0, 10-a)$ for some $a$ in the interval $[0, 10]$. A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming points?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-e0dad3e9b9ed34328734eb5ed5d3ff043707b549c98b1c6fa83eef4997a5efd7 | A line in the plane of a triangle \(ABC\) intersects the sides \(AB\) and \(AC\) at points \(X\) and \(Y\) respectively such that \(BX = CY\). Find the locus of the center of the circumcircle of triangle \(XAY\). | [
"The locus of the center of the circumcircle of triangle \\(XAY\\) is a line passing through the circumcenter of triangle \\(ABC\\) and parallel to the angle bisector of \\(\\angle BAC\\)."
] | [
"A line in the plane of a triangle \\(ABC\\) intersects the sides \\(AB\\) and \\(AC\\) at points \\(X\\) and \\(Y\\) respectively such that \\(BX = CY\\). Find the locus of the center of the circumcircle of triangle \\(XAY\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-74460cd9c59bb1f2cb7e30f0f739189f67b98c0578df292f9a4ecc6e7b338688 | A line in the plane of a triangle \(ABC\) intersects the sides \(AB\) and \(AC\) respectively at points \(X\) and \(Y\) such that \(BX = CY\). Find the locus of the center of the circumcircle of triangle \(XAY\). | [
"\\text{The median from } A \\text{ to the midpoint of } BC"
] | [
"A line in the plane of a triangle \\(ABC\\) intersects the sides \\(AB\\) and \\(AC\\) respectively at points \\(X\\) and \\(Y\\) such that \\(BX = CY\\). Find the locus of the center of the circumcircle of triangle \\(XAY\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-b206840f77e66c2fb5e2797d7602741ce889b739aa8d55f4c9e09a5c990be1fb | A line intersects a semicircle with diameter $AB$ and center $O$ at points $C$ and $D$, and the line $AB$ at point $M$, where $MB < MA$ and $MD < MC$. If the circumcircles of triangles $AOC$ and $DOB$ meet again at point $K$, calculate the measure of $\angle MKO$ in degrees. | [
"90"
] | [
"A line intersects a semicircle with diameter $AB$ and center $O$ at points $C$ and $D$, and the line $AB$ at point $M$, where $MB < MA$ and $MD < MC$. If the circumcircles of triangles $AOC$ and $DOB$ meet again at point $K$, calculate the measure of $\\angle MKO$ in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-de1fecd9e4f262364073ac2f4ffe04705475948bc74ee25ad5c2338ad088a4c0 | A line intersects one circle at points A and B and another circle at points C and D. Let P and Q be the points of intersection of the tangents to the first circle at points A and B, and R and S be the points of intersection of the tangents to the second circle at points C and D. If the centers of the two original circl... | [
"1"
] | [
"A line intersects one circle at points A and B and another circle at points C and D. Let P and Q be the points of intersection of the tangents to the first circle at points A and B, and R and S be the points of intersection of the tangents to the second circle at points C and D. If the centers of the two original ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a8c1f4439169ba678f8ccc920b3b06076b576ebfd7c0b43a2da69cc61ef2829 | A line intersects the curve \( y = x^3 \) at the point \( (a, a^3) \) for some \( a > 0 \), and at exactly one other point where the line and the curve are tangent. Compute the area of the region enclosed by the line and the curve. Express the answer in terms of \( a \). | [
"\\(\\frac{9a^4}{16}\\)"
] | [
"A line intersects the curve \\( y = x^3 \\) at the point \\( (a, a^3) \\) for some \\( a > 0 \\), and at exactly one other point where the line and the curve are tangent. Compute the area of the region enclosed by the line and the curve. Express the answer in terms of \\( a \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-a471df2c0e2e1c774a02915769962d2003edecec3716713a642eea43ed0c78ad | A line intersects the sides $BC$, $CA$, and $AB$ of triangle $ABC$ at points $A_1$, $B_1$, and $C_1$, respectively. The points $A_2$, $B_2$, and $C_2$ are the reflections of $A_1$, $B_1$, and $C_1$ over the midpoints of $BC$, $CA$, and $AB$, respectively. If the equation of the line passing through $A_2$, $B_2$, and $C... | [
"-1"
] | [
"A line intersects the sides $BC$, $CA$, and $AB$ of triangle $ABC$ at points $A_1$, $B_1$, and $C_1$, respectively. The points $A_2$, $B_2$, and $C_2$ are the reflections of $A_1$, $B_1$, and $C_1$ over the midpoints of $BC$, $CA$, and $AB$, respectively. If the equation of the line passing through $A_2$, $B_2$, a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e15aea819fc17ce0cc50e2dda0a57cf91f24f5336d2707b2d5876126b7314c8 | A line intersects the sides \(AB\) and \(BC\) of \(\triangle ABC\) at points \(M\) and \(K\). If the area of \(\triangle MBK\) is equal to the area of the quadrilateral \(AMKC\), find the minimum possible value of \[ \frac{|MB| + |BK|}{|AM| + |CA| + |KC|}. \] | [
"\\dfrac{\\sqrt{2}}{2}",
"\\dfrac{\\sqrt{5} - 1}{2}"
] | [
"A line intersects the sides \\(AB\\) and \\(BC\\) of \\(\\triangle ABC\\) at points \\(M\\) and \\(K\\). If the area of \\(\\triangle MBK\\) is equal to the area of the quadrilateral \\(AMKC\\), find the minimum possible value of \n\\[ \\frac{|MB| + |BK|}{|AM| + |CA| + |KC|}. \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-18e0270ca1d30740efc86a4f6dfb452b4cbe59ac5cd25d5a1d6a24345266fa1c | A line intersects the x-axis at \( A(7,0) \) and the y-axis at \( B(0,-4) \). A variable line \( Q \), which is perpendicular to \( AB \), intersects the x-axis at \( P \) and the y-axis at \( Q \). If \( AQ \) and \( BP \) intersect at \( R \), find the locus of \( R \). | [
"\\( x^2 + y^2 - 7x + 4y = 0 \\)"
] | [
"A line intersects the x-axis at \\( A(7,0) \\) and the y-axis at \\( B(0,-4) \\). A variable line \\( Q \\), which is perpendicular to \\( AB \\), intersects the x-axis at \\( P \\) and the y-axis at \\( Q \\). If \\( AQ \\) and \\( BP \\) intersect at \\( R \\), find the locus of \\( R \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-66357193cc90f76816ce7e801cb8cc6498aa81caf0bf40533aa34018c39a157f | A line intersects two sides of an equilateral triangle and is parallel to the third side. If the smaller triangle and the trapezoid formed have equal perimeters, determine the ratio of the area of the smaller triangle to that of the trapezoid. Express your answer as a fraction \( \frac{a}{b} \). | [
"\\(\\frac{9}{7}\\)"
] | [
"A line intersects two sides of an equilateral triangle and is parallel to the third side. If the smaller triangle and the trapezoid formed have equal perimeters, determine the ratio of the area of the smaller triangle to that of the trapezoid. Express your answer as a fraction \\( \\frac{a}{b} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-18015ed3d627c70aab8fc964231d05727b975656ed2a2b61ac916888d22ab36d | A line is a tangent to a circle at the point (2, 5). A parallel line is tangent to the circle at (8, 13). What is the common slope of the two lines? Express your answer as a common fraction. | [
"\\(\\frac{-3}{4}\\)"
] | [
"A line is a tangent to a circle at the point (2, 5). A parallel line is tangent to the circle at (8, 13). What is the common slope of the two lines? Express your answer as a common fraction."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-81912844b1e010e2acc14b65752e8fe91e1ccc7761efcc45700bc1ba3ef83456 | A line is described by the equation $y-4=4(x-8)$. What is the sum of its $x$-intercept and $y$-intercept? | [
"-21"
] | [
"A line is described by the equation $y-4=4(x-8)$. What is the sum of its $x$-intercept and $y$-intercept?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53fdffa44d7ee4cfb7e3ee8147aa356d1ccf1a7021373c4d970faed60829ab06 | A line is drawn from each vertex of an equilateral triangle to n points on the opposite side of the vertex such that it divides the opposite side into n+1 segments of equal length. How many non-overlapping bounded regions are formed when n = 14? | [
"1410"
] | [
"A line is drawn from each vertex of an equilateral triangle to n points on the opposite side of the vertex such that it divides the opposite side into n+1 segments of equal length. How many non-overlapping bounded regions are formed when n = 14?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-73a3f98a2576c44f3c3853bd7c575aec7e4470133364cbc3ccfc789adda77843 | A line is drawn from point \( A(-2,0) \) to intersect the curve \( y^2 = 4x \) at points \( P \) and \( Q \) in the first quadrant. If \( \frac{1}{AP} + \frac{1}{AQ} < \frac{1}{4} \), what is the slope of the line? | [
"2"
] | [
"A line is drawn from point \\( A(-2,0) \\) to intersect the curve \\( y^2 = 4x \\) at points \\( P \\) and \\( Q \\) in the first quadrant. If \\( \\frac{1}{AP} + \\frac{1}{AQ} < \\frac{1}{4} \\), what is the slope of the line?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-95792976f5dae78c912b10ffa74b5438a293b6769cf5bb2e2b84747f8ab5210b | A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e., lines connecting the midpoints of two sides) in the same ratio. Determine this ratio. | [
"\\(\\frac{1+\\sqrt{5}}{2}\\)"
] | [
"A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e., lines connecting the midpoints of two sides) in the same ratio. Determine this ratio."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 8,
"tir": 21
} | 75 | 0 | 0 |
omr-ws-7ef1b0581dea5084bbf360c8ff1a9d4588afdaea31e81f7d4db9778a87f57f7c | A line is drawn through the point \( A(1,2) \) to cut the line \( 2y = 3x - 5 \) at point \( P \) and the line \( x + y = 12 \) at point \( Q \). If \( AQ = 2AP \), find the coordinates of \( P \) and \( Q \). | [
"\\( P(4, 7/2) \\) and \\( Q(7, 5) \\)"
] | [
"A line is drawn through the point \\( A(1,2) \\) to cut the line \\( 2y = 3x - 5 \\) at point \\( P \\) and the line \\( x + y = 12 \\) at point \\( Q \\). If \\( AQ = 2AP \\), find the coordinates of \\( P \\) and \\( Q \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 12
} | 13 | 0 | 0 |
omr-ws-8cf0f0bc81aaf8d089882119772daec1abff4e627a8c85739401ffa2e2c05ae4 | A line is expressed in the form \[\begin{pmatrix} -2 \\ -5 \end{pmatrix} \cdot \left( \begin{pmatrix} x \\ y \end{pmatrix} - \begin{pmatrix} 1 \\ 11 \end{pmatrix} \right) = 0.\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the ordered pair $(m,b).$ | [
"\\left( -\\frac{2}{5}, \\frac{57}{5} \\right)"
] | [
"A line is expressed in the form\n\\[\\begin{pmatrix} -2 \\\\ -5 \\end{pmatrix} \\cdot \\left( \\begin{pmatrix} x \\\\ y \\end{pmatrix} - \\begin{pmatrix} 1 \\\\ 11 \\end{pmatrix} \\right) = 0.\\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the ordered pair $(m,b).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-114d58e30e76f269e46439e2e9ccdd4cd9ee28e4a5e66b828edf53fb5c1cb3a2 | A line is expressed in the form \[\begin{pmatrix} 1 \\ 3 \end{pmatrix} \cdot \left( \begin{pmatrix} x \\ y \end{pmatrix} - \begin{pmatrix} -2 \\ 8 \end{pmatrix} \right) = 0.\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the ordered pair $(m,b).$ | [
"\\left( -\\frac{1}{3}, \\frac{22}{3} \\right)"
] | [
"A line is expressed in the form\n\\[\\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix} \\cdot \\left( \\begin{pmatrix} x \\\\ y \\end{pmatrix} - \\begin{pmatrix} -2 \\\\ 8 \\end{pmatrix} \\right) = 0.\\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the ordered pair $(m,b).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6170d1cec106609acf3c377edf6e6352291eb84ce3ef9e62aba0c7ecd42a42ec | A line is moving so that its distance from the origin is always equal to \( a \). Find the equation of the locus of the image of the origin with respect to this line. | [
"\\( x^2 + y^2 = 4a^2 \\)"
] | [
"A line is moving so that its distance from the origin is always equal to \\( a \\). Find the equation of the locus of the image of the origin with respect to this line."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-981f6aa987971e3c67ea20d8b8473a76929370581d69ea4a28a636c0769519ba | A line is moving so that its distance from the origin is always equal to \( a \). Find the equation of the locus of the midpoint of the intercept made by this line between the coordinate axes. | [
"\\(\\frac{1}{x^2} + \\frac{1}{y^2} = \\frac{4}{a^2}\\)"
] | [
"A line is moving so that its distance from the origin is always equal to \\( a \\). Find the equation of the locus of the midpoint of the intercept made by this line between the coordinate axes."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 9
} | 31 | 0 | 0 |
omr-ws-71e197a4e7f3ef6852604d77c0a81a014a2b5f86c2014987fd70503041473183 | A line is normal (i.e., perpendicular) to the parabola \( y = x^2 \) at the point \( (a, a^2) \neq (0,0) \). The line meets the parabola at a second point. Find the coordinates of that point. | [
"\\left( -\\frac{1 + 2a^2}{2a}, \\frac{(1 + 2a^2)^2}{4a^2} \\right)"
] | [
"A line is normal (i.e., perpendicular) to the parabola \\( y = x^2 \\) at the point \\( (a, a^2) \\neq (0,0) \\). The line meets the parabola at a second point. Find the coordinates of that point."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb620cf43e938eed1048df10dc67aac11a036dce0c87a971af7cb1702885f1b6 | A line is parameterized by \[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ -2 \end{pmatrix} + t \begin{pmatrix} 3 \\ 4 \end{pmatrix}.\]A second line is parameterized by \[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -8 \\ 12 \end{pmatrix} + u \begin{pmatrix} 1 \\ 3 \end{pmatrix}.\]If $\theta$ is... | [
"\\frac{3}{\\sqrt{10}}"
] | [
"A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ -2 \\end{pmatrix} + t \\begin{pmatrix} 3 \\\\ 4 \\end{pmatrix}.\\]A second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} -8 \\\\ 12 \\end{pmatrix} + u \\begin{pmatrix} 1 \\\\... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7bc46051227aaaaf0ad92283a5b41df1469c1088942c375150365d65d5fd0df5 | A line is parameterized by \[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ -3 \end{pmatrix}.\]A second line is parameterized by \[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ -9 \end{pmatrix} + u \begin{pmatrix} 4 \\ 2 \end{pmatrix}.\]Find the point ... | [
"\\begin{pmatrix} 7 \\\\ -8 \\end{pmatrix}"
] | [
"A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} + t \\begin{pmatrix} 2 \\\\ -3 \\end{pmatrix}.\\]A second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 5 \\\\ -9 \\end{pmatrix} + u \\begin{pmatrix} 4 \\\\ ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e25f1e17a72334df659043132db02819acc1b0bddd2d18d4adf5a10764af08a6 | A line is parameterized by \[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} -1 \\ 5 \end{pmatrix}.\]A second line is parameterized by \[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 7 \end{pmatrix} + u \begin{pmatrix} -1 \\ 4 \end{pmatrix}.\]Find the point ... | [
"\\begin{pmatrix} 6 \\\\ -17 \\end{pmatrix}"
] | [
"A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix} + t \\begin{pmatrix} -1 \\\\ 5 \\end{pmatrix}.\\]A second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 7 \\end{pmatrix} + u \\begin{pmatrix} -1 \\\\ ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-589471b73cc80d2fa9fe03efeb9cc251421d2a359839dcf688b4cc6dca6853bb | A line is parameterized by a parameter $t,$ so that the vector on the line at $t = -1$ is $\begin{pmatrix} 1 \\ 3 \\ 8 \end{pmatrix},$ and the vector on the line at $t = 2$ is $\begin{pmatrix} 0 \\ -2 \\ -4 \end{pmatrix}.$ Find the vector on the line at $t = 3.$ | [
"\\begin{pmatrix} -1/3 \\\\ -11/3 \\\\ -8 \\end{pmatrix}"
] | [
"A line is parameterized by a parameter $t,$ so that the vector on the line at $t = -1$ is $\\begin{pmatrix} 1 \\\\ 3 \\\\ 8 \\end{pmatrix},$ and the vector on the line at $t = 2$ is $\\begin{pmatrix} 0 \\\\ -2 \\\\ -4 \\end{pmatrix}.$ Find the vector on the line at $t = 3.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed7fc3daa8e9b8218b74f76e414870c5b83e18ef6edd7ef09e19eb8b532de8c4 | A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 2$ is $\begin{pmatrix} 1 \\ 4 \end{pmatrix},$ and the vector on the line at $t = 3$ is $\begin{pmatrix} 3 \\ -4 \end{pmatrix}.$ Find the vector on the line at $t = -7.$ | [
"\\begin{pmatrix} -17 \\\\ 76 \\end{pmatrix}"
] | [
"A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 2$ is $\\begin{pmatrix} 1 \\\\ 4 \\end{pmatrix},$ and the vector on the line at $t = 3$ is $\\begin{pmatrix} 3 \\\\ -4 \\end{pmatrix}.$ Find the vector on the line at $t = -7.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed57ab2aad239e66863c444aa717b6b10c41fb0962f3ac8d7862f6e8b2ffa7d8 | A line is tangent at point \( A \) to a circle with center \( O \). Point \( N \) is on the line and point \( P \) is on the circle such that \( PN \) is perpendicular to \( AN \). If \( AN = 20 \) and \( PN = 8 \), determine the radius of the circle. | [
"29"
] | [
"A line is tangent at point \\( A \\) to a circle with center \\( O \\). Point \\( N \\) is on the line and point \\( P \\) is on the circle such that \\( PN \\) is perpendicular to \\( AN \\). If \\( AN = 20 \\) and \\( PN = 8 \\), determine the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 8
} | 55 | 0 | 0 |
omr-ws-3ed44d7c28c292f525c9187353c565e296562397c5e9a13ff8875e8480b3bfe3 | A line meets a segment $AB$ at point $C$. What is the maximal number of points $X$ on this line such that one of the angles $AXC$ or $BXC$ is equal to half of the other angle? | [
"2"
] | [
"A line meets a segment $AB$ at point $C$. What is the maximal number of points $X$ on this line such that one of the angles $AXC$ or $BXC$ is equal to half of the other angle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8702a35ccf5d9a1934a5c685bc32aca5bb2d4b182279c42a0600ab7d9ed8111 | A line moves in the plane so that it passes through the point (1, 1) and intersects the two coordinate axes. Find the locus of the center of the circle which passes through these two points of intersection of the line with the coordinate axes and through the origin. | [
"\\(2xy = x + y\\)"
] | [
"A line moves in the plane so that it passes through the point (1, 1) and intersects the two coordinate axes. Find the locus of the center of the circle which passes through these two points of intersection of the line with the coordinate axes and through the origin."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-72b951da59fc9c21891e83b96e8b460e8b5671376a96c5dfcf902e80ed79f875 | A line of slope $\dfrac{1}{2}$ intersects the $x$-axis at an angle $\theta$ from the horizontal. Compute the following sum: $$\sum_{n = 0}^{\infty} \dfrac{5^n \sin (n \theta) \cos (n \theta)}{7^n}$$ | [
"\\(\\frac{7}{16}\\)"
] | [
"A line of slope $\\dfrac{1}{2}$ intersects the $x$-axis at an angle $\\theta$ from the horizontal. Compute the following sum:\n\n$$\\sum_{n = 0}^{\\infty} \\dfrac{5^n \\sin (n \\theta) \\cos (n \\theta)}{7^n}$$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c94f6808b2993444de6b234068334dfb93b805e21095ca8a361d068f3d1df4f6 | A line of slope -4 forms a right triangle with the positive x and y axes. If the area of the triangle is 2013, find the square of the length of the hypotenuse of the triangle. | [
"\\dfrac{34221}{2}"
] | [
"A line of slope -4 forms a right triangle with the positive x and y axes. If the area of the triangle is 2013, find the square of the length of the hypotenuse of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b6a8f707e1d6a9483430b47aa7959745a3bbd1bc63a5ba2ab27ff28ba0c5e56 | A line parallel to the hypotenuse $BC$ passes through the incentre $I$ of a right triangle $ABC$. The line meets the sides $AB$ and $AC$ at points $D$ and $E$, respectively, such that $DI = 3$ and $IE = 4$. Find the lengths of the sides of triangle $ABC$. | [
"\\(\\frac{36}{5}, \\frac{48}{5}, 12\\)"
] | [
"A line parallel to the hypotenuse $BC$ passes through the incentre $I$ of a right triangle $ABC$. The line meets the sides $AB$ and $AC$ at points $D$ and $E$, respectively, such that $DI = 3$ and $IE = 4$. Find the lengths of the sides of triangle $ABC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 16
} | 26 | 0 | 0 |
omr-ws-9b6c92bf26a5d50f3e9bb923bde8756a769ccdc3083e5dd3d62d373589c72e0c | A line parallel to the parallel sides of a trapezium is drawn through the midpoints of the unequal sides, dividing the trapezium into two regions in the ratio of $5:2$. Find the ratio of the lengths of the parallel sides. | [
"13:1"
] | [
"A line parallel to the parallel sides of a trapezium is drawn through the midpoints of the unequal sides, dividing the trapezium into two regions in the ratio of $5:2$. Find the ratio of the lengths of the parallel sides."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-68de4f42bae709508dc365685d10812c34fe530415f8c3189a25aeefdf3902ea | A line passes through $(2,2,1)$ and $(5,1,-2).$ A point on this line has an $x$-coordinate of 4. Find the $z$-coordinate of the point. | [
"-1"
] | [
"A line passes through $(2,2,1)$ and $(5,1,-2).$ A point on this line has an $x$-coordinate of 4. Find the $z$-coordinate of the point."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee375e05e1439fdc410993f7c6b87259687ae46866fa8128dcea1bf67b125a70 | A line passes through \( A(1,1) \) and \( B(100,1000) \). How many other points with integer coordinates are on the line and strictly between \( A \) and \( B \)? | [
"8"
] | [
"A line passes through \\( A(1,1) \\) and \\( B(100,1000) \\). How many other points with integer coordinates are on the line and strictly between \\( A \\) and \\( B \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6186cabc65e22facc146688c9154905f7e869ec3d05813281fc3c09254436299 | A line passes through point \(A(1,0)\) and has exactly one common point with the parabola \(y = \frac{x^2}{2}\). How many such lines are there? | [
"3"
] | [
"A line passes through point \\(A(1,0)\\) and has exactly one common point with the parabola \\(y = \\frac{x^2}{2}\\). How many such lines are there?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-5d82a6e80a49f99d0236bec4b14b990a57e88c0de18ea13919fd31169a0503af | A line passes through points \( A(-3,0) \) and \( B(0,-4) \). A variable line perpendicular to \( AB \) intersects the \( x \)-axis at \( M \) and the \( y \)-axis at \( N \). Find the locus of the point of intersection of the lines \( AN \) and \( BM \). | [
"\\(\\left(x+\\dfrac{41}{22}\\right)^2+\\left(y+\\dfrac{30}{11}\\right)^2=\\dfrac{2465}{484}\\)"
] | [
"A line passes through points \\( A(-3,0) \\) and \\( B(0,-4) \\). A variable line perpendicular to \\( AB \\) intersects the \\( x \\)-axis at \\( M \\) and the \\( y \\)-axis at \\( N \\). Find the locus of the point of intersection of the lines \\( AN \\) and \\( BM \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f6847131b3083234a1489105673d86dadeb27756e2c536588abf50a56335789f | A line passes through points \( A(1,1) \) and \( B(100,1000) \). How many other points with integer coordinates are on the line and strictly between \( A \) and \( B \)? | [
"8"
] | [
"A line passes through points \\( A(1,1) \\) and \\( B(100,1000) \\). How many other points with integer coordinates are on the line and strictly between \\( A \\) and \\( B \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-c167dc36979c4adf6c2ac038473f980e16722774b80f7d0c6bdbcac2905d4205 | A line passes through the center $O$ of an equilateral triangle $ABC$ and intersects the side $BC$. At what angle with respect to $BC$ should this line be drawn so that its segment inside the triangle has the smallest possible length? | [
"\\( 90^\\circ \\)"
] | [
"A line passes through the center $O$ of an equilateral triangle $ABC$ and intersects the side $BC$. At what angle with respect to $BC$ should this line be drawn so that its segment inside the triangle has the smallest possible length?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d20bc4f43ef412297c86dbf9111663ae91f373c9b84323ddb5439c5546c4498c | A line passes through the distinct vectors $\mathbf{a}$ and $\mathbf{b}.$ Then for a certain value of $k,$ the vector \[k \mathbf{a} + \frac{3}{4} \mathbf{b}\]must also lie on the line. Find $k.$ | [
"\\frac{1}{4}"
] | [
"A line passes through the distinct vectors $\\mathbf{a}$ and $\\mathbf{b}.$ Then for a certain value of $k,$ the vector\n\\[k \\mathbf{a} + \\frac{3}{4} \\mathbf{b}\\]must also lie on the line. Find $k.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a198ccd15bab94d1b25e627805ccfb0387123856fef694aa020f06dfed92e95e | A line passes through the point $P(2, 1)$ and intersects the ellipse $3x^2 + 2y^2 = 6$ at two points $Q$ and $R$. Find the range of $\overline{PQ} \cdot \overline{PR}$. | [
"\\(\\frac{8}{3} \\leq \\overline{PQ} \\cdot \\overline{PR} < \\frac{160 + 16\\sqrt{2}}{49}\\)"
] | [
"A line passes through the point $P(2, 1)$ and intersects the ellipse $3x^2 + 2y^2 = 6$ at two points $Q$ and $R$. Find the range of $\\overline{PQ} \\cdot \\overline{PR}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-04061a123941daa1aa506103a885536d286cd96ce7b9e9f62cc5245770a9d338 | A line passing through the barycenter \( G \) of the triangle \( \triangle OAB \) intersects \( OA \) at \( P \) and \( OB \) at \( Q \). Let \( \overrightarrow{OP} = h\overrightarrow{OA} \) and \( \overrightarrow{OQ} = k\overrightarrow{OB} \). Denote the areas of \( \triangle OAB \) and \( \triangle OPQ \) by \( S \) ... | [
"\\dfrac{2}{9}"
] | [
"A line passing through the barycenter \\( G \\) of the triangle \\( \\triangle OAB \\) intersects \\( OA \\) at \\( P \\) and \\( OB \\) at \\( Q \\). Let \\( \\overrightarrow{OP} = h\\overrightarrow{OA} \\) and \\( \\overrightarrow{OQ} = k\\overrightarrow{OB} \\). Denote the areas of \\( \\triangle OAB \\) and \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d6eb4d96bd1af4f1618ffd02e8a531d8f174fecd2bc6f037d862e8d4de738fb | A line passing through the barycenter \( G \) of the triangle \( \triangle OAB \) intersects \( OA \) at \( P \) and \( OB \) at \( Q \). Let \( \overrightarrow{OP} = h\overrightarrow{OA} \) and \( \overrightarrow{OQ} = k\overrightarrow{OB} \). If \( h = \frac{1}{4} \), find the value of \( k \). | [
"-1"
] | [
"A line passing through the barycenter \\( G \\) of the triangle \\( \\triangle OAB \\) intersects \\( OA \\) at \\( P \\) and \\( OB \\) at \\( Q \\). Let \\( \\overrightarrow{OP} = h\\overrightarrow{OA} \\) and \\( \\overrightarrow{OQ} = k\\overrightarrow{OB} \\). If \\( h = \\frac{1}{4} \\), find the value of \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42cc4fa3a3a355e503b50b4a20af384a74687e0e1d4e04e0ba7489f2db4f578d | A line passing through the center \( M \) of the equilateral triangle \( ABC \) intersects sides \( BC \) and \( CA \) at points \( D \) and \( E \), respectively. The circumcircles of triangles \( AEM \) and \( BDM \) intersect at points \( M \) and \( P \). If \( O \) is the center of the circumcircle of triangle \( ... | [
"\\dfrac{\\sqrt{3}}{9}"
] | [
"A line passing through the center \\( M \\) of the equilateral triangle \\( ABC \\) intersects sides \\( BC \\) and \\( CA \\) at points \\( D \\) and \\( E \\), respectively. The circumcircles of triangles \\( AEM \\) and \\( BDM \\) intersect at points \\( M \\) and \\( P \\). If \\( O \\) is the center of the c... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-da72f6d60d3b006922bbe130b12eafc15e33bc4521e07d78058a53cbc55e2e9c | A line passing through the centroid \( G \) of a triangle \( ABC \) meets sides \( AB \) and \( AC \) and line \( BC \) at points \( D \), \( E \), and \( F \) respectively, where \( C \) lies on the segment \( BF \). If \( GD = 3 \), \( GE = 4 \), and \( GF = x \), find the value of \( x \). | [
"12"
] | [
"A line passing through the centroid \\( G \\) of a triangle \\( ABC \\) meets sides \\( AB \\) and \\( AC \\) and line \\( BC \\) at points \\( D \\), \\( E \\), and \\( F \\) respectively, where \\( C \\) lies on the segment \\( BF \\). If \\( GD = 3 \\), \\( GE = 4 \\), and \\( GF = x \\), find the value of \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-24b136c298cf1424bb97bc32358d7c861aa37a5bcfdfa164c5ce920b2243fb4e | A line passing through the incenter \(I\) of a triangle \(ABC\) meets its sides \(AB\) and \(BC\) in points \(M\) and \(N\) respectively. Triangle \(BMN\) is acute. Points \(K\) and \(L\) are chosen on the side \(AC\) so that \(\angle ILA = \angle IMB\) and \(\angle IKC = \angle INB\). If \(AM = 3\), \(KL = 5\), and \(... | [
"15"
] | [
"A line passing through the incenter \\(I\\) of a triangle \\(ABC\\) meets its sides \\(AB\\) and \\(BC\\) in points \\(M\\) and \\(N\\) respectively. Triangle \\(BMN\\) is acute. Points \\(K\\) and \\(L\\) are chosen on the side \\(AC\\) so that \\(\\angle ILA = \\angle IMB\\) and \\(\\angle IKC = \\angle INB\\). ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6d05eff372888adf753a0ab79ea877c08859f1605b867bed024ea5df7924387 | A line passing through the point $M(-2,10)$ intersects the circle $x^2 + y^2 + 10x - 18y + 80 = 0$ at two points $A$ and $B$ such that $\dfrac{AM}{MB} = 2$. Find the equation of the line. | [
"\\( x + y = 8 \\) or \\( 7x - y + 24 = 0 \\)"
] | [
"A line passing through the point $M(-2,10)$ intersects the circle $x^2 + y^2 + 10x - 18y + 80 = 0$ at two points $A$ and $B$ such that $\\dfrac{AM}{MB} = 2$. Find the equation of the line."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 12
} | 19 | 0 | 0 |
omr-ws-b5c19fbcf7b41076db9b9b675e806ebc4c5ecb05f04c5f13d738c313f049baa6 | A line passing through the point (4,0) is tangent to a circle with radius 1 centered at (0,1). Find the point of intersection between the line and the circle. | [
"\\(\\left(\\frac{8}{17}, \\frac{32}{17}\\right)\\)",
"\\left( \\dfrac{8}{17}, \\dfrac{32}{17} \\right)"
] | [
"A line passing through the point (4,0) is tangent to a circle with radius 1 centered at (0,1). Find the point of intersection between the line and the circle."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 12
} | 43 | 0 | 0 |
omr-ws-596cb2752fa242ee5c130817404333514c0e56c43d263aca862b9d4f5ee015fd | A line segment $AB$ lies on the edge of a dihedral angle. $M$ is the point of one of the faces of the angle and is at a distance $l$ from the edge. A perpendicular dropped from $M$ on the other face of the dihedral angle is seen from $A$ at an angle of $\alpha $ and from point $B$ at an angle of $\beta $. The distance ... | [
"$\\sqrt {9m^2{\\rm cosec}^2 \\alpha -l^2} +\\sqrt {9m^2{\\rm cosec}^2 \\beta -l^2}$ if the angles of the triangle $ABM$ at the side $AB$ are both acute; $\\left|\\sqrt {9m^2{\\rm cosec}^2 \\alpha -l^2}-\\sqrt {9m^2{\\rm cosec}^2 \\beta -l^2} \\right|$ if one of these angles is obtuse"
] | [
"A line segment $AB$ lies on the edge of a dihedral angle. $M$ is the point of one of the faces of the angle and is at a distance $l$ from the edge. A perpendicular dropped from $M$ on the other face of the dihedral angle is seen from $A$ at an angle of $\\alpha $ and from point $B$ at an angle of $\\beta $. The di... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f3d8f4719ad9df9f03530ef9e58c81e8c12543b09d265daa965a18535417ab4 | A line segment (pencil) is inserted between two tennis balls of radius 1 in a can such that it is tangent to both balls and its ends touch the can. The top ball remains at its usual height. Determine the length of the pencil. | [
"4"
] | [
"A line segment (pencil) is inserted between two tennis balls of radius 1 in a can such that it is tangent to both balls and its ends touch the can. The top ball remains at its usual height. Determine the length of the pencil."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5e66908d9cfc3b7e7e73cbc89e65397045e5887e397795e30882e8964a44d888 | A line segment 6 units long is placed at random in the first quadrant of a coordinate system such that its endpoints are on the two coordinate axes. What is the probability that the area of the right triangle formed by this segment and the two axes is greater than or equal to 8.5 square units? Give an exact answer. | [
"\\(\\frac{\\sqrt{2}}{6}\\)"
] | [
"A line segment 6 units long is placed at random in the first quadrant of a coordinate system such that its endpoints are on the two coordinate axes. What is the probability that the area of the right triangle formed by this segment and the two axes is greater than or equal to 8.5 square units? Give an exact answer... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e6c2019dd070ae85ef63eb8aa074d4e7f2695acbec8d2c9e8a40adde321f35d0 | A line segment BC has length 6. Point A is chosen such that angle BAC is a right angle. For any position of A, a point D is chosen on BC so that AD is perpendicular to BC. A circle with AD as diameter has tangents drawn from C and B to touch the circle at M and N, respectively, with these tangents intersecting at Z. Ca... | [
"10"
] | [
"A line segment BC has length 6. Point A is chosen such that angle BAC is a right angle. For any position of A, a point D is chosen on BC so that AD is perpendicular to BC. A circle with AD as diameter has tangents drawn from C and B to touch the circle at M and N, respectively, with these tangents intersecting at ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-a436ce6a24766e4810b2c57ad0ec04ccca2a19616f7bb9a057b39066059834c1 | A line segment BC has length 6. Point A is chosen such that angle BAC is a right angle. For any position of A, a point D is chosen on BC such that AD is perpendicular to BC. A circle with AD as diameter has tangents drawn from C and B to touch the circle at M and N, respectively, with these tangents intersecting at Z. ... | [
"10"
] | [
"A line segment BC has length 6. Point A is chosen such that angle BAC is a right angle. For any position of A, a point D is chosen on BC such that AD is perpendicular to BC. A circle with AD as diameter has tangents drawn from C and B to touch the circle at M and N, respectively, with these tangents intersecting a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-a9d14e07940c4e3c6cc83770cc61182f6e96f8c28ede1a7c1579f5316a021be6 | A line segment BC has length 6. Point A is chosen such that ∠BAC is a right angle. For any position of A, a point D is chosen on BC so that AD is perpendicular to BC. A circle with AD as diameter has tangents drawn from C and B to touch the circle at M and N, respectively, with these tangents intersecting at Z. Find th... | [
"10"
] | [
"A line segment BC has length 6. Point A is chosen such that ∠BAC is a right angle. For any position of A, a point D is chosen on BC so that AD is perpendicular to BC. A circle with AD as diameter has tangents drawn from C and B to touch the circle at M and N, respectively, with these tangents intersecting at Z. Fi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-dd665944816d3d9096aca1b0c898c0231f2bfb13fbe43f6e733b13aca369a56f | A line segment begins at $(1, 3)$. It is 5 units long and ends at the point $(4, x)$ and $x > 0$. What is the value of $x$? | [
"7"
] | [
"A line segment begins at $(1, 3)$. It is 5 units long and ends at the point $(4, x)$ and $x > 0$. What is the value of $x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-adad3c6fd4e3eda2590ec36a54fd36dbcff288ad3540baf0f7f4d6b4ee2dd7c1 | A line segment begins at $(2, 5)$. It is 10 units long and ends at the point $(-6, y)$ where $y > 0$. What is the value of $y$? | [
"11"
] | [
"A line segment begins at $(2, 5)$. It is 10 units long and ends at the point $(-6, y)$ where $y > 0$. What is the value of $y$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5347717bdb7e68dbbf352fb5a7621bd40a34852b0a04c6716d54c57fa7050fe0 | A line segment is broken at two random points along its length. What is the probability that the three new segments can form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"A line segment is broken at two random points along its length. What is the probability that the three new segments can form a triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 59
} | 89 | 0 | 0 |
omr-ws-7fdc51feee4104141a50146b89009d8ddbea2cc65bc12e8f8f371fbea90e8e98 | A line segment is broken at two random points along its length. What is the probability that the three new segments form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"A line segment is broken at two random points along its length. What is the probability that the three new segments form a triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 88
} | 136 | 0 | 0 |
omr-ws-778a96ac0453436e0ef3cbfa7e2b696f99bf33779a933d65aec946513baaa5e7 | A line segment is broken at two random points along its length. What is the probability that the three resulting segments can form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"A line segment is broken at two random points along its length. What is the probability that the three resulting segments can form a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2ad516e7d73c0dd24b8d3ef12a63b6a58c292c98d8234ae6ab4ece9423c0fb8d | A line segment is divided into two unequal lengths so that the ratio of the length of the short part to the length of the long part is the same as the ratio of the length of the long part to the length of the whole line segment. Let $D$ be this ratio. Compute $$D^{-1} + D^{[D^{-1}+D^{(D^{-1}+D^2)}]}.$$ | [
"2"
] | [
"A line segment is divided into two unequal lengths so that the ratio of the length of the short part to the length of the long part is the same as the ratio of the length of the long part to the length of the whole line segment. Let $D$ be this ratio. Compute $$D^{-1} + D^{[D^{-1}+D^{(D^{-1}+D^2)}]}.$$"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-607b266a940af699cbbbb0239fbb7fd1bf686f6366ccb484a7c5091cf8c145f8 | A line segment moves in the plane with its endpoints on the coordinate axes such that the sum of the cubes of the lengths of its intercepts on the coordinate axes is a constant $c$. Find the locus of the midpoint of this segment. | [
"|x|^3 + |y|^3 = \\dfrac{c}{8}"
] | [
"A line segment moves in the plane with its endpoints on the coordinate axes such that the sum of the cubes of the lengths of its intercepts on the coordinate axes is a constant $c$. Find the locus of the midpoint of this segment."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-6b4ccf69e5c794c320bf511a896905ab814f9cfb0aaa4f7a2ab75ea685041f29 | A line segment of length $6$ is divided at random into $3$ parts with the restriction that the longest part is at least $0.50$ more than the next longest part and that the next longest part is twice the shortest part. Find the probability that the three parts can serve as the $3$ sides of a triangle. | [
"\\(\\frac{1}{11}\\)"
] | [
"A line segment of length $6$ is divided at random into $3$ parts with the restriction that the longest part is at least $0.50$ more than the next longest part and that the next longest part is twice the shortest part. Find the probability that the three parts can serve as the $3$ sides of a triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ac5d8b4ae666112fc8ca07b43e7e44e0392e44893ccb58486040595153f7a072 | A line segment of length $q$ is dropped on a plane consisting of parallel lines, with the distance between each line being $3q$. What is the probability that the line segment will touch one of the parallel lines? | [
"\\dfrac{2}{3\\pi}"
] | [
"A line segment of length $q$ is dropped on a plane consisting of parallel lines, with the distance between each line being $3q$. What is the probability that the line segment will touch one of the parallel lines?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a363884dbfdf8af85b2091a342e8893307bcf9edc1b86889e899826456adc7d | A line segment of length 36 cm is divided into four unequal parts. If the distance between the midpoints of the first and the fourth parts is 30 cm, what is the distance between the midpoints of the second and the third parts? | [
"12"
] | [
"A line segment of length 36 cm is divided into four unequal parts. If the distance between the midpoints of the first and the fourth parts is 30 cm, what is the distance between the midpoints of the second and the third parts?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-109b9dd6d1b0b1c2c9dfb74a4214f69b27898d7d3b1bad44d7fa4001720852ae | A line segment of length 6 is divided at random into 3 parts with the restriction that the longest part is at least 0.5 more than the next longest part, and that the next longest part is twice the shortest part. Find the probability that the three parts can serve as the 3 sides of a triangle. | [
"\\(\\frac{1}{11}\\)"
] | [
"A line segment of length 6 is divided at random into 3 parts with the restriction that the longest part is at least 0.5 more than the next longest part, and that the next longest part is twice the shortest part. Find the probability that the three parts can serve as the 3 sides of a triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-59d0bb8baf481c120163e0890c14802575a41f0abe9fd7f0e15000a691a48414 | A line segment of length 6 is divided at random into three parts such that the longest part has a length greater than 2 and less than 4, and one of the remaining parts is twice the other remaining part. Find the probability that the three lengths can serve as the lengths of the three sides of a triangle. | [
"\\(\\frac{3}{8}\\)"
] | [
"A line segment of length 6 is divided at random into three parts such that the longest part has a length greater than 2 and less than 4, and one of the remaining parts is twice the other remaining part. Find the probability that the three lengths can serve as the lengths of the three sides of a triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-92de4e4cbec8f2dec77d3bce2cf59f99908a2bab724c66b12a2cee06cd3f6b56 | A line segment of length 6 is divided at random into three parts with the restriction that the longest part is at least 0.5 more than the next longest part and the next longest part is twice the shortest part. Find the probability that the three parts can form a triangle. | [
"$\\frac{1}{11}$"
] | [
"A line segment of length 6 is divided at random into three parts with the restriction that the longest part is at least 0.5 more than the next longest part and the next longest part is twice the shortest part. Find the probability that the three parts can form a triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-289f7d7477f9928cb0ff998b5bca88cf6d3a7d2d200de03ef9f3e35845aa3f44 | A line segment of length 6 is divided at random into three parts with the restrictions that the longest part has a length greater than 2 and less than 4, and that one of the remaining parts is twice the other remaining part. Find the probability that the three lengths of the three parts can serve as the lengths of the ... | [
"\\(\\frac{3}{8}\\)"
] | [
"A line segment of length 6 is divided at random into three parts with the restrictions that the longest part has a length greater than 2 and less than 4, and that one of the remaining parts is twice the other remaining part. Find the probability that the three lengths of the three parts can serve as the lengths of... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-d985d664fade81a0bc4b2d55264097fafaa23d76a28aabf8e1c8637a6b3b6f44 | A line segment parallel to an edge of a regular hexagon with side length 1 divides the hexagon into two pieces whose ratio of areas is 1:2. What is the length of that line segment? | [
"\\(\\sqrt{3}\\)"
] | [
"A line segment parallel to an edge of a regular hexagon with side length 1 divides the hexagon into two pieces whose ratio of areas is 1:2. What is the length of that line segment?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-5e9f198063a6b3c324743ddf278e164b26ad5c36fe89349779d0ea7edeca619b | A line that passes through the origin intersects both the line \( x = 1 \) and the line \( y = 1 + \frac{\sqrt{3}}{3} x \). The three lines create an equilateral triangle. What is the perimeter of the triangle? | [
"\\( 2\\sqrt{6} \\)"
] | [
"A line that passes through the origin intersects both the line \\( x = 1 \\) and the line \\( y = 1 + \\frac{\\sqrt{3}}{3} x \\). The three lines create an equilateral triangle. What is the perimeter of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-edfb98bda6db7a94c687a12ea3b93d208a3e20e217ac6ac82aaaef0ea1f87ed4 | A line that passes through the origin intersects both the line \(x=1\) and the line \(y=1+\frac{\sqrt{3}}{3}x\). The three lines create an equilateral triangle. What is the perimeter of the triangle? | [
"\\(3 + 2\\sqrt{3}\\)"
] | [
"A line that passes through the origin intersects both the line \\(x=1\\) and the line \\(y=1+\\frac{\\sqrt{3}}{3}x\\). The three lines create an equilateral triangle. What is the perimeter of the triangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 18
} | 66 | 0 | 0 |
omr-ws-05824a5da73f0afe3d112841b72355c0c05c96cd236fbf7613a955d9672135d1 | A line through a point $A$ intersects a circle at points $B$ and $C$, with $B$ lying between $A$ and $C$. From point $A$, draw the two tangents to the circle, meeting the circle at points $S$ and $T$. Let $P$ be the intersection of the lines $ST$ and $AC$. If $AB = 4$ and $BC = 6$, calculate the value of $\frac{AP}{PC}... | [
"\\dfrac{4}{3}"
] | [
"A line through a point $A$ intersects a circle at points $B$ and $C$, with $B$ lying between $A$ and $C$. From point $A$, draw the two tangents to the circle, meeting the circle at points $S$ and $T$. Let $P$ be the intersection of the lines $ST$ and $AC$. If $AB = 4$ and $BC = 6$, calculate the value of $\\frac{A... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d04a6b44314bff00427c71404fe450480bd0fffe053838b8e281aa5ce57b112b | A line through a vertex of a non-degenerate triangle cuts it into two similar triangles with $\sqrt{3}$ as the ratio between corresponding sides. Find the angles of the given triangle. | [
"\\(30^\\circ, 60^\\circ, 90^\\circ\\)"
] | [
"A line through a vertex of a non-degenerate triangle cuts it into two similar triangles with $\\sqrt{3}$ as the ratio between corresponding sides. Find the angles of the given triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-648406b153b1f001a2e994edc00a09c6d302655b3d23b49c41bb4559500f2422 | A line through point \( A \) intersects a circle at points \( B \) and \( C \) with \( B \) between \( A \) and \( C \). The two tangents from \( A \) intersect the circle at points \( S \) and \( T \). The line \( ST \) and the line \( AC \) intersect at point \( P \). If \( AB = 3 \) and \( BC = 6 \), calculate the r... | [
"1"
] | [
"A line through point \\( A \\) intersects a circle at points \\( B \\) and \\( C \\) with \\( B \\) between \\( A \\) and \\( C \\). The two tangents from \\( A \\) intersect the circle at points \\( S \\) and \\( T \\). The line \\( ST \\) and the line \\( AC \\) intersect at point \\( P \\). If \\( AB = 3 \\) an... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-079ecb95912c323bd9c9bc0a75f22a8f61c4536951d688d97ce3f958a1529fee | A line through point \( P \) outside of circle \( O \) meets the circle at points \( B \) and \( C \) (\( PB < PC \)). Let \( PO \) meet circle \( O \) at points \( Q \) and \( D \) (\( PQ < PD \)). Let the line passing through \( Q \) and perpendicular to \( BC \) meet circle \( O \) at point \( A \). Given that \( BD... | [
"1"
] | [
"A line through point \\( P \\) outside of circle \\( O \\) meets the circle at points \\( B \\) and \\( C \\) (\\( PB < PC \\)). Let \\( PO \\) meet circle \\( O \\) at points \\( Q \\) and \\( D \\) (\\( PQ < PD \\)). Let the line passing through \\( Q \\) and perpendicular to \\( BC \\) meet circle \\( O \\) at ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-0508ac9dc04f19ea8483b8b8a7386d36965df86c4642b42376f7233e3c8aa2e0 | A line through the center \( T \) of an equilateral triangle \( ABC \) intersects sides \( AB \), \( AC \), and \( BC \) at points \( M \), \( L \), and \( K \) respectively. If \( MT = 3 \), \( LT = 4 \), and \( KT = x \), find the value of \( x \). | [
"\\dfrac{12}{7}"
] | [
"A line through the center \\( T \\) of an equilateral triangle \\( ABC \\) intersects sides \\( AB \\), \\( AC \\), and \\( BC \\) at points \\( M \\), \\( L \\), and \\( K \\) respectively. If \\( MT = 3 \\), \\( LT = 4 \\), and \\( KT = x \\), find the value of \\( x \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-2c1edaf94de2e53366732ba4491cc489d95eb098257cd96f4248442c3c284970 | A line through the centroid \( G \) of \(\Delta ABC\) intersects the sides of the triangle at points \( X, Y, Z \). Using directed line segments, calculate the value of \[\frac{1}{GX} + \frac{1}{GY} + \frac{1}{GZ}.\] | [
"0"
] | [
"A line through the centroid \\( G \\) of \\(\\Delta ABC\\) intersects the sides of the triangle at points \\( X, Y, Z \\). Using directed line segments, calculate the value of \\[\\frac{1}{GX} + \\frac{1}{GY} + \\frac{1}{GZ}.\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dad01361c163d01ffd445537081d49b891116b0a74b2a8089384369e1931a236 | A line through the origin meets the hyperbola \(x^2 - y^2 = 1\) at point \(P\) and the circle \(x^2 + y^2 = 1\) at point \(Q\). Find the locus of the intersection of the tangents at \(P\) and \(Q\) to the hyperbola and the circle, respectively. | [
"\\( x^{2}(4y^{4}+1)=1 \\)"
] | [
"A line through the origin meets the hyperbola \\(x^2 - y^2 = 1\\) at point \\(P\\) and the circle \\(x^2 + y^2 = 1\\) at point \\(Q\\). Find the locus of the intersection of the tangents at \\(P\\) and \\(Q\\) to the hyperbola and the circle, respectively."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 9
} | 22 | 0 | 0 |
omr-ws-c7235603c18647087f226458c46cb6977930c1f4d9ee1fe03025ebeaf3b88949 | A line through the point \( P(2, -3) \) meets the lines \( x - 2y + 7 = 0 \) and \( x + 3y - 3 = 0 \) at points \( A \) and \( B \). If \( P \) divides \( AB \) externally in the ratio \( 3:2 \), find the equation of the line \( AB \). | [
"\\( 2x + y - 1 = 0 \\)"
] | [
"A line through the point \\( P(2, -3) \\) meets the lines \\( x - 2y + 7 = 0 \\) and \\( x + 3y - 3 = 0 \\) at points \\( A \\) and \\( B \\). If \\( P \\) divides \\( AB \\) externally in the ratio \\( 3:2 \\), find the equation of the line \\( AB \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-1254f727c977cbca3845ff65d17489870aebde14b8d93dad8bee08983d8ddfb4 | A line through the point \( P(a, 2) \) intersects the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) at points \( A \) and \( D \), and intersects the coordinate axes at points \( B \) and \( C \). Given that \( PA, PB, PC, PD \) are in geometric progression, determine the possible values of \( a \). | [
"5, 8, 10, -7"
] | [
"A line through the point \\( P(a, 2) \\) intersects the ellipse \\(\\frac{x^2}{9} + \\frac{y^2}{4} = 1\\) at points \\( A \\) and \\( D \\), and intersects the coordinate axes at points \\( B \\) and \\( C \\). Given that \\( PA, PB, PC, PD \\) are in geometric progression, determine the possible values of \\( a \... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41e791b30ad0974085a77b397faac9be9fc89de0d2e1f0fab6dfdf7642a01ba2 | A line through the point \( P(l, 3) \) meets the ellipse \(\frac{x^2}{16} + \frac{y^2}{9} = 1\) at points \( A \) and \( D \), and meets the \( x \)-axis and \( y \)-axis at points \( B \) and \( C \) respectively. Given that \( PA \times PD = PB \times PC \), find the minimum value of \( |l| \). | [
"8"
] | [
"A line through the point \\( P(l, 3) \\) meets the ellipse \\(\\frac{x^2}{16} + \\frac{y^2}{9} = 1\\) at points \\( A \\) and \\( D \\), and meets the \\( x \\)-axis and \\( y \\)-axis at points \\( B \\) and \\( C \\) respectively. Given that \\( PA \\times PD = PB \\times PC \\), find the minimum value of \\( |l... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 46,
"genselect": 7
} | 53 | 0 | 0 |
omr-ws-1c277c93b9a0984fad6ba952317a1bbde9be6909510c121c062f7c8de2563c5e | A line through the points $(2, -9)$ and $(j, 17)$ is parallel to the line $2x + 3y = 21$. What is the value of $j$? | [
"-37"
] | [
"A line through the points $(2, -9)$ and $(j, 17)$ is parallel to the line $2x + 3y = 21$. What is the value of $j$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e5525dbc0cc2cd0cfcd7bfe9060a69a0a180044acfd0e66eb5582abdbd9a0a2c | A line whose $y$-intercept is $(0,5)$ intersects the ellipse $9x^2 + 16y^2 = 144.$ Find all possible slopes of this line. | [
"(-\\infty,-1] \\cup [1,\\infty)"
] | [
"A line whose $y$-intercept is $(0,5)$ intersects the ellipse $9x^2 + 16y^2 = 144.$ Find all possible slopes of this line."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9253ee4c15794766c76d7749e7999becd959c4d54a76b729ab540addbec9b40e | A line with an X-intercept of 5 cuts both axes and delimits a triangle. If this line is tangent to the function $f(x)=\frac{1}{x}$ at a point, find the area of the delimited triangle. | [
"2"
] | [
"A line with an X-intercept of 5 cuts both axes and delimits a triangle. If this line is tangent to the function $f(x)=\\frac{1}{x}$ at a point, find the area of the delimited triangle."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d121de237ebefbe24026f03389bc561aae0a791314c90d51ce01921a0be2398c | A line with equation \( y = 2x + 105 \) is the angle bisector of a line with equation \( y = x + 304 \) and another line \( \ell \). What is the y-intercept of \( \ell \)? | [
"890"
] | [
"A line with equation \\( y = 2x + 105 \\) is the angle bisector of a line with equation \\( y = x + 304 \\) and another line \\( \\ell \\). What is the y-intercept of \\( \\ell \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f2c7d670c21dea413c1074c8bb4930542978686397d35bdfcf00954552e068e | A line with equation \( y = mx + b \) is tangent to the parabolas \( y = x^2 + 5 \) and \( y = -x^2 + 2 \). If \( m \) is positive, find \( m^2 + 4b^2 \). | [
"55"
] | [
"A line with equation \\( y = mx + b \\) is tangent to the parabolas \\( y = x^2 + 5 \\) and \\( y = -x^2 + 2 \\). If \\( m \\) is positive, find \\( m^2 + 4b^2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7268384f8a931e1636c72668ab3aa408b951e5cf74a9925dd2f37bb765875f6e | A line with slope $1$ intersects the ellipse $x^2+\frac{y^2}{4}=1$ at two distinct points $A$ and $B$. Suppose that $P$ lies on segment $AB$ such that $\frac{AP}{PB}=2$. Determine the locus of $P$. | [
"\\( 148x^2 + 64xy + 13y^2 = 20 \\)"
] | [
"A line with slope $1$ intersects the ellipse $x^2+\\frac{y^2}{4}=1$ at two distinct points $A$ and $B$. Suppose that $P$ lies on segment $AB$ such that $\\frac{AP}{PB}=2$. Determine the locus of $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 14
} | 61 | 0 | 0 |
omr-ws-6ef2f903cc538ddeadf6a56b740e4feee3ff981756e98c43ef5eb7b6fb8ec355 | A line with slope 1 passes through the point \( P \) on the negative x-axis and intersects the parabola \( y = x^2 \) at points \( Q \) and \( R \). If \( PQ = QR \), find the ordinate of the line \( PR \). | [
"\\( 4 \\pm 3\\sqrt{2} \\)"
] | [
"A line with slope 1 passes through the point \\( P \\) on the negative x-axis and intersects the parabola \\( y = x^2 \\) at points \\( Q \\) and \\( R \\). If \\( PQ = QR \\), find the ordinate of the line \\( PR \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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