conversations listlengths 2 2 | source stringclasses 1
value | score int64 5 5 |
|---|---|---|
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $ABCD$ be an isosceles trapezoid with bases $AB=92$ and $CD=19$. Suppose $AD=BC=x$ and a circle with center on $\\overline{AB}$ is tangent to segments $\\overline{AD}$ and $\\overline{BC}$. If $m$ is the smallest possible ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A circle of radius $r$ has chords $\\overline{AB}$ of length $10$ and $\\overline{CD}$ of length 7. When $\\overline{AB}$ and $\\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at $P$, which is out... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle{ABC}$, $\\angle ABC=120^\\circ,AB=3$ and $BC=4$. If perpendiculars constructed to $\\overline{AB}$ at $A$ and to $\\overline{BC}$ at $C$ meet at $D$, then $CD=$\n$\\text{(A) } 3\\quad \\text{(B) } \\frac{8}{\\sq... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: [asy] draw((0,0)--(0,2)--(2,2)--(2,0)--cycle,dot); draw((2,2)--(0,0)--(0,1)--cycle,dot); draw((0,2)--(1,0),dot); MP(\"B\",(0,0),SW);MP(\"A\",(0,2),NW);MP(\"D\",(2,2),NE);MP(\"C\",(2,0),SE); MP(\"E\",(0,1),W);MP(\"F\",(1,0),S);... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: [asy] draw(circle((0,6sqrt(2)),2sqrt(2)),black+linewidth(.75)); draw(circle((0,3sqrt(2)),sqrt(2)),black+linewidth(.75)); draw((-8/3,16sqrt(2)/3)--(-4/3,8sqrt(2)/3)--(0,0)--(4/3,8sqrt(2)/3)--(8/3,16sqrt(2)/3),dot); MP(\"B\",(-8... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: [asy] draw((0,0)--(0,3)--(4,0)--cycle,dot); draw((4,0)--(7,0)--(7,10)--cycle,dot); draw((0,3)--(7,10),dot); MP(\"C\",(0,0),SW);MP(\"A\",(0,3),NW);MP(\"B\",(4,0),S);MP(\"E\",(7,0),SE);MP(\"D\",(7,10),NE); [/asy]\nTriangle $ABC$... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that s... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: [asy] pair A = (0,0), B = (7,4.2), C = (10, 0), D = (3, -5), E = (3, 0), F = (7,0); draw(A--B--C--D--cycle,dot); draw(A--E--F--C,dot); draw(D--E--F--B,dot); markscalefactor = 0.1; draw(rightanglemark(B, A, D)); draw(rightangle... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is\n$\\mathrm{(A) \\frac{\\sqrt{3}}{12} } \\qquad \\mathrm{(B) \\frac{\\sq... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many cubic centimeters are in the volume of a cone having a diameter of 10cm and a height of 6cm? Express your answer in terms of $\\pi$. [/INST] The radius of the base of the cone is $10/2 = 5$, so the volume of the cone ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $P$ be a point outside of circle $O.$ A segment is drawn from $P$ such that it is tangent to circle $O$ at point $T.$ Meanwhile, a secant from $P$ intersects $O$ at $A$ and $B,$ such that $PA < PB.$ If $PA = 3$ and $PT = A... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red ar... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 3, 4, and 5. What is the area of the triangle?\n$\\mathrm{(A) \\ 6 } \\qquad \\mathrm{(B) \\frac{18}{\\pi^2} } \\qq... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the figure, $AB \\perp BC, BC \\perp CD$, and $BC$ is tangent to the circle with center $O$ and diameter $AD$. In which one of the following cases is the area of $ABCD$ an integer?\n[asy] pair O=origin, A=(-1/sqrt(2),1/sqrt... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$, and one of the base angles is $\\arcsin(.8)$. Find the area of the trapezoid.\n$\\textbf{(A)}\\ 72\\qquad \\textbf{(B)}\\ 75\\q... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The six edges of a tetrahedron $ABCD$ measure $7, 13, 18, 27, 36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is\n$\\textbf{(A)}\\ 7\\qquad \\textbf{(B)}\\ 13\\qquad \\textbf{(C)}\\ 18\\qqu... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $ABC$ is a triangle: $A=(0,0), B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\\triangle ABC$ can have?\n$\\textbf{(A)}\\ \\frac{1}{2} \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ \\fra... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)?\n$\\textbf{(A)}\\ ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is $441 \\text{cm}^2$. What is the area (in $\\text{cm}^2$) of... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be?\n$\\textbf{(A)}\\ 4\\qquad \\textbf{(B)}\\ 5\\qquad \\textbf{(... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the adjoining figure, $AB$ is a diameter of the circle, $CD$ is a chord parallel to $AB$, and $AC$ intersects $BD$ at $E$, with $\\angle AED = \\alpha$. The ratio of the area of $\\triangle CDE$ to that of $\\triangle ABE$ ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the configuration below, $\\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are line segments and $AB$ is tangent to the circle at $A$.\n[asy] defaultpen(fontsize(10pt)+linewidth(.8pt)); pair... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?\n$\\textbf{(A)}\\ \\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse of $50\\%$ longer than the minor axis, the length of the major axis is\n$\\textbf{(A)}\\ 1\\qquad \\textbf{(B)}\\ ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle ABC, AB = 8, BC = 7, CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\\triangle PAB$ is similar to $\\triangle PCA$. The length of $PC$ is\n[asy] defaultpen(linewidth(0.7)+font... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The length of the hypotenuse of a right triangle is $h$ , and the radius of the inscribed circle is $r$. The ratio of the area of the circle to the area of the triangle is\n$\\textbf{(A) }\\frac{\\pi r}{h+2r}\\qquad \\textbf{(... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A sector with acute central angle $\\theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is\n$\\textbf{(A)}\\ 3\\cos\\theta \\qquad \\textbf{(B)}\\ 3\\sec\\theta \\qquad \\textbf{(C... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: [asy] draw((0,0)--(2,2)--(5/2,1/2)--(2,0)--cycle,dot); MP(\"A\",(0,0),W);MP(\"B\",(2,2),N);MP(\"C\",(5/2,1/2),SE);MP(\"D\",(2,0),S); MP(\"a\",(1,0),N);MP(\"b\",(17/8,1/8),N); [/asy]\nIn the accompanying figure, segments $AB$ a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle?\n$\\text{(A) } 2\\quad \\text{(B) } 3\\quad \\text{(C) } 4\\quad \\text{(D) } 5\\quad \\text{(E) } 6$\n [/INST] One of ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, angle $ACB$ is 50 degrees, and angle $CBA$ is 70 degrees. Let $D$ be the foot of the perpendicular from $A$ to $BC$, $O$ the center of the circle circumscribed about triangle $ABC$, and $E$ the other end of... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: [asy] fill(circle((4,0),4),grey); fill((0,0)--(8,0)--(8,-4)--(0,-4)--cycle,white); fill(circle((7,0),1),white); fill(circle((3,0),3),white); draw((0,0)--(8,0),black+linewidth(1)); draw((6,0)--(6,sqrt(12)),black+linewidth(1)); ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Increasing the radius of a cylinder by $6$ units increased the volume by $y$ cubic units. Increasing the height of the cylinder by $6$ units also increases the volume by $y$ cubic units. If the original height is $2$, then the... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle ABC$ lines $CE$ and $AD$ are drawn so that $\\dfrac{CD}{DB}=\\dfrac{3}{1}$ and $\\dfrac{AE}{EB}=\\dfrac{3}{2}$. Let $r=\\dfrac{CP}{PE}$ where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals:\n[as... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 32$ and $GF = 24$, then $BE$ equals:\n[asy] size(7cm); pair A = (0, 0), B = (7, 0), ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Acute-angled $\\triangle ABC$ is inscribed in a circle with center at $O$; $\\stackrel \\frown {AB} = 120^\\circ$ and $\\stackrel \\frown {BC} = 72^\\circ$.\nA point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicul... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $\\triangle ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:\n$\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Any five points are taken inside or on a square with side length $1$. Let a be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance betwe... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.\n$\\textbf{(A)}\\ 4\\qquad \\textbf{(B)}\\ \\sqrt{17} \\qquad \\textbf{(C)}\\ 4.25\\qquad \\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The base of a triangle is of length $b$, and the altitude is of length $h$. A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:\n$\\t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Points $A$ and $B$ are selected on the graph of $y = -\\frac{1}{2}x^2$ so that triangle $ABO$ is equilateral. Find the length of one side of triangle $ABO$. [asy]\nsize(150);\ndraw( (-4, -8) -- (-3.4641, -6)-- (-3, -9/2)-- (-5... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: [asy] unitsize(27); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E,F,X,Y,Z; A=(3,3); B=(0,0); C=(6,0); D=(4,0); E=(4,2); F=(1,1); draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(C--F); X=intersectionpoint(A--D,C... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $P_{1}: y=x^{2}+\\frac{101}{100}$ and $P_{2}: x=y^{2}+\\frac{45}{4}$ be two parabolas in the Cartesian plane. Let $\\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\\mathcal{L}... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $\\triangle DEF$ is inscribed inside $\\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\\triangle DEC, \\triangle BFD, \\triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $ABCD$ is a rectangular sheet of paper. $E$ and $F$ are points on $AB$ and $CD$ respectively such that $BE < CF$. If $BCFE$ is folded over $EF$, $C$ maps to $C'$ on $AD$ and $B$ maps to $B'$ such that $\\angle{AB'C'} \\cong \\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\\cos{B} = \\cos{A} + \\cos{C}$, then the area of triangle $ABC$ can be expressed as $\\frac{a\\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers s... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $A, B, C, D,$ and $E$ are collinear in that order such that $AB = BC = 1, CD = 2,$ and $DE = 9$. If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$?\n [/INST] Let the al... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $ABCD$, a rectangle with $AB = 12$ and $BC = 16$, is the base of pyramid $P$, which has a height of $24$. A plane parallel to $ABCD$ is passed through $P$, dividing $P$ into a frustum $F$ and a smaller pyramid $P'$. Let $X$ de... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has sides $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CA}$ of length 43, 13, and 48, respectively. Let $\\omega$ be the circle circumscribed around $\\triangle ABC$ and let $D$ be the intersection of $\\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \\ln x$, with $0 < x_1 < x_2$. The points $C$ and $D$ trisect $\\overline{AB}$, with $AC < CB$. Through $C$ a horizontal line is drawn to cut the cu... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Five identical rectangles are arranged to form a larger rectangle $PQRS$, as shown. The area of $PQRS$ is $4000$. What is the length, $x$, rounded off to the nearest integer? [asy]\nreal x = 1; real w = 2/3;\n\n// Draw outer... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $\\cos 150^\\circ$. [/INST] Let $P$ be the point on the unit circle that is $150^\\circ$ counterclockwise from $(1,0)$, and let $D$ be the foot of the altitude from $P$ to the $x$-axis, as shown below.\n\n[asy]\npair A... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the area, in square units, of the interior region formed by the lines $y = 2x - 4, y = -3x +16$ and the $y$-axis? [/INST] To find the vertices of the triangle, we find where the two lines $y=2x-4$ and $y=-3x+16$ inters... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A circular sheet of paper with radius of $6$ cm is cut into three congruent sectors. What is the height of the cone in centimeters that can be created by rolling one of the sections until the edges meet? Express your answer in... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the ratio of the volume of cone $A$ to the volume of cone $B$? Express your answer as a common fraction. [asy]\nsize(260);\ndraw(ellipse((0,0),14.8,6),S);\nlabel(\"Cone $A$\", (0,-6), S);\ndraw((0,28.3)--(0,0),dashed);... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two sides of an isosceles triangle are 10 inches and 20 inches. If the shortest side of a similar triangle is 50 inches, what is the perimeter of the larger triangle? [/INST] First of all, the third side of the smaller triangl... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the slope of the line that is tangent to a circle at point (5,5) if the center of the circle is (3,2)? Express your answer as a common fraction. [/INST] If a line can be drawn tangent to a circle a the point $(5,5)$, t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle ABC,$ $AB=AC=25$ and $BC=23.$ Points $D,E,$ and $F$ are on sides $\\overline{AB},$ $\\overline{BC},$ and $\\overline{AC},$ respectively, such that $\\overline{DE}$ and $\\overline{EF}$ are parallel to $\\overli... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $BCD$, $\\angle C = 90^\\circ$, $CD = 3$, and $BD = \\sqrt{13}$. What is $\\tan B$? [/INST] [asy]\npair B,C,D;\nB = (0,0);\nC = (2,0);\nD = (2,-3);\ndraw(B--C--D--B);\ndraw(rightanglemark(B,C,D,7));\nlabel(\"$D$\",... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the circle with center $O$, the measure of $\\angle RIP$ is $36^\\circ$ and $OR=10$ cm. Find the number of centimeters in the length of arc $RP$. Express your answer in terms of $\\pi$. [asy]\ndraw((1,0)..(0,1)..(-1,0)..(0,... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\Delta ABC$, $AC = BC$, $m\\angle DCB = 40^{\\circ}$, and $CD \\parallel AB$. What is the number of degrees in $m\\angle ECD$?\n\n[asy] pair A,B,C,D,E; B = dir(-40); A = dir(-140); D = (.5,0); E = .4 * dir(40);\ndraw(C--B... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, four circles of radius 4 units intersect at the origin. What is the number of square units in the area of the shaded region? Express your answer in terms of $\\pi$. [asy]\nimport olympiad; import geometry; size... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the height of Jack's house, in feet, if the house casts a shadow 56 feet long at the same time a 21-foot tree casts a shadow that is 24 feet long? Express your answer to the nearest whole number. [/INST] The ratio of s... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: One angle of a parallelogram is 120 degrees, and two consecutive sides have lengths of 8 inches and 15 inches. What is the area of the parallelogram? Express your answer in simplest radical form. [/INST] [asy]\npair A,B,C,D,X;... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A pipe with inside diameter 10'' is to carry water from a reservoir to a small town in an arid land. Neglecting the friction and turbulence of the water against the inside of the pipes, what is the minimum number of 2''-inside... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A circle with a radius of 2 units has its center at $(0, 0)$. A circle with a radius of 7 units has its center at $(15, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What i... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the radius of the circle inscribed in triangle $ABC$ if $AB = AC=7$ and $BC=6$? Express your answer in simplest radical form. [/INST] Let $r$ be the radius of the inscribed circle. Let $s$ be the semiperimeter of the t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC =1:2$. Point $F$ is on $AD$ such that $AF:FD=3:1$. If the area of $\\triangle DEF$ is 17, determine the area of $\\triangle A... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A circle is circumscribed about an equilateral triangle with side lengths of $9$ units each. What is the area of the circle, in square units? Express your answer in terms of $\\pi$. [/INST] [asy]\nsize(100);\ndefaultpen(linew... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $ABCDEFGH$ shown below is a cube with volume 1. Find the volume of pyramid $ABCH$.\n\n[asy]\nimport three;\ntriple A,B,C,D,EE,F,G,H;\nA = (0,0,0);\nB = (1,0,0);\nC = (1,1,0);\nD= (0,1,0);\nEE = (0,0,1);\nF = B+EE;\nG = C + EE... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Parallelogram $ABCD$ with $A(2,5)$, $B(4,9)$, $C(6,5)$, and $D(4,1)$ is reflected across the $x$-axis to $A'B'C'D'$ and then $A'B'C'D'$ is reflected across the line $y=x+1$ to $A''B''C''D''$. This is done such that $D'$ is the... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The diagonals of a regular hexagon have two possible lengths. What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in simplest radical form. [/INST] The two possible diagonal l... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\\sqrt{89}$ units and diagonals that differ by 6 units? [/INST] Because the diagonals of a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The sides of a triangle have lengths $11, 15,$ and $k,$ where $k$ is a positive integer. For how many values of $k$ is the triangle obtuse? [/INST] The longest side of the triangle either has length $15$ or has length $k.$ Tak... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the figure, circle $O$ has radius 6 units. Chord $CD$ has length 8 units and is parallel to segment $KB$. If $KA$ = 12 units and points $K$, $A$, $O$ and $B$ are collinear, what is the area of triangle $KDC$? Express your a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Inside a square with side length 10, two congruent equilateral triangles are drawn such that they share one side and each has one vertex on a vertex of the square. What is the side length of the largest square that can be ins... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Charlyn walks completely around the boundary of a square whose sides are each 5 km long. From any point on her path she can see exactly 1 km horizontally in all directions. What is the area of the region consisting of all poin... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the figure, $\\angle EAB$ and $\\angle ABC$ are right angles, $AB=4,$ $BC=6,$ $AE=8,$ and $\\overline{AC}$ and $\\overline{BE}$ intersect at $D.$ What is the difference between the areas of $\\triangle ADE$ and $\\triangle ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The convex pentagon $ABCDE$ has $\\angle A = \\angle B = 120^\\circ$, $EA = AB = BC = 2$ and $CD = DE = 4$. What is the area of $ABCDE$?\n\n[asy]\n\nunitsize(1 cm);\n\npair A, B, C, D, E;\n\nA = (0,0);\n\nB = (1,0);\n\nC = B ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An equilateral triangle has an area of $64\\sqrt{3}$ $\\text{cm}^2$. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased? [/INST] We first consider an equilateral triangle wi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A cube has edges of length 1 cm and has a dot marked in the centre of the top face. The cube is sitting on a flat table. The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum? [/INST] We begin by finding the equation of... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A line with slope of $-2$ intersects the positive $x$-axis at $A$ and the positive $y$-axis at $B$. A second line intersects the $x$-axis at $C(8,0)$ and the $y$-axis at $D$. The lines intersect at $E(4,4)$. What is the area o... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $AB = 3$, $BC = 4$, $AC = 5$, and $BD$ is the angle bisector from vertex $B$. If $BD = k \\sqrt{2}$, then find $k$. [/INST] By Pythagoras, $\\angle ABC = 90^\\circ$. Let $P$ and $Q$ be the projections of $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, $RSP$ is a straight line and $\\angle QSP = 80^\\circ$. What is the measure of $\\angle PQR$, in degrees?\n\n[asy]\ndraw((.48,-.05)--(.48,.05)); draw((.52,-.05)--(.52,.05)); draw((1.48,-.05)--(1.48,.05)); draw... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A wooden model of a square pyramid has a base edge of 12 cm and an altitude of 8 cm. A cut is made parallel to the base of the pyramid that separates it into two pieces: a smaller pyramid and a frustum. Each base edge of the s... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Points $(1,-5)$ and $(11,7)$ are the opposite vertices of a parallelogram. What are the coordinates of the point where the diagonals of the parallelogram intersect? [/INST] The diagonals of a parallelogram intersect at the mi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A regular hexagon is inscribed in a circle of radius 2 units. In square units, what is the area of the hexagon? Express your answer in simplest radical form. [/INST] Connect opposite pairs of vertices of the regular hexagon wi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The nine points of this grid are equally spaced horizontally and vertically. The distance between two neighboring points is 1 unit. What is the area, in square units, of the region where the two triangles overlap?\n\n[asy]\n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A jar of peanut butter which is 3 inches in diameter and 4 inches high sells for $\\$$0.60. At the same rate, what would be the price for a jar that is 6 inches in diameter and 6 inches high? [/INST] The first jar has a volume... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many integer side lengths are possible to complete a triangle in which the other sides measure 6 units and 3 units? [/INST] By the Triangle Inequality, the unknown side must measure less than $6+3=9$ units. Also, the lengt... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the figure, $m\\angle A = 28^{\\circ}$, $m\\angle B = 74^\\circ$ and $m\\angle C = 26^{\\circ}$. If $x$ and $y$ are the measures of the angles in which they are shown, what is the value of $x + y$? [asy]\nsize(150);\ndraw((... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the minimum number of equilateral triangles, of side length 1 unit, needed to cover an equilateral triangle of side length 10 units? [/INST] The ratio of the sides of the small to big equilateral triangle (note that th... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circle $B$ has its center at $(-6, 2)$ and a radius of $10$ units. What is the sum of the $y$-coordinates of the two points on circle $B$ that are also on the $y$-axis? [/INST] Note that if the $y$-coordinate of one of the two... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find $\\cos C$ in the right triangle shown below.\n\n\n[asy]\npair A,B,C;\nA = (0,0);\nB = (6,0);\nC = (0,8);\ndraw(A--B--C--A);\ndraw(rightanglemark(B,A,C,10));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,SE);\nlabel(\"$C$\",C,N);... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A large sphere has a volume of $288\\pi$ cubic units. A smaller sphere has a volume which is $12.5\\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sph... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The side lengths of both triangles to the right are given in centimeters. What is the length of segment $AB$?\n\n[asy]\npair A,B,C,D,E,F,G;\n\nA=(0,0);\nB=12*dir(0);\nC=20*dir(120);\nD=8+B;\nE=D+6*dir(0);\nF=D+10*dir(120);\n\n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ with vertices $A(-2, 0)$, $B(1, 4)$ and $C(-3, 2)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $C$ to $C'$? [/INST] Reflecting a point over the $y$-axis n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $\\sin(-60^\\circ)$. [/INST] Rotating $60^\\circ$ clockwise is the same as rotating $360^\\circ - 60^\\circ = 300^\\circ$ counterclockwise, so $\\sin(-60^\\circ) = \\sin (360^\\circ - 60^\\circ) = \\sin 300^\\circ$.\n\... | custom_source | 5 |
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