conversations listlengths 2 2 | source stringclasses 1
value | score int64 5 5 |
|---|---|---|
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In quadrilateral $ABCD$, sides $\\overline{AB}$ and $\\overline{BC}$ both have length 10, sides $\\overline{CD}$ and $\\overline{DA}$ both have length 17, and the measure of angle $ADC$ is $60^\\circ$. What is the length of di... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The side lengths of a triangle are 14 cm, 48 cm and 50 cm. How many square centimeters are in the area of the triangle? [/INST] Shrinking the triangle by dividing every side length by 2, we recognize the resulting set $$\\{7,2... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In rectangle $ABCD$, $AB=5$ and $BC =3$. Points $F$ and $G$ are on $\\overline{CD}$ so that $DF = 1$ and $GC=2$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\\triangle AEB$. Express your answer as a common fraction... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has diameter 10 cm. A side view of the first four rows of each of the two different methods of packing is shown below.\n\n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. What percent larger is the area of the large semicircles ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $A=(0,9)$ and $B=(0,12)$. Points $A'$ and $B'$ are on the line $y=x$, and $\\overline{AA'}$ and $\\overline{BB'}$ intersect at $C=(2,8)$. What is the length of $\\overline{A'B'}$? [/INST] Line $AC$ has slope $-\\frac{1}{2}... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $ABCDEFGH$ shown below is a cube. Find $\\sin \\angle GAC$.\n\n[asy]\n\nimport three;\n\ntriple A,B,C,D,EE,F,G,H;\n\nA = (0,0,0);\n\nB = (1,0,0);\n\nC = (1,1,0);\n\nD= (0,1,0);\n\nEE = (0,0,1);\n\nF = B+EE;\n\nG = C + EE;\n\nH... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Stuart has drawn a pair of concentric circles, as shown. He draws chords $\\overline{AB}$, $\\overline{BC}, \\ldots$ of the large circle, each tangent to the small one. If $m\\angle ABC=75^\\circ$, then how many segments wil... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ is a right isosceles triangle. Points $D$, $E$ and $F$ are the midpoints of the sides of the triangle. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. What is the ratio o... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Points $A$, $B$, $Q$, $D$, and $C$ lie on the circle shown and the measures of arcs $BQ$ and $QD$ are $42^\\circ$ and $38^\\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$, in degrees.\n\n[asy]\nimport... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A cylinder has a radius of 3 cm and a height of 8 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? [/INST] The longest segment stretches from the bottom to the top of the cylinder and across... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A parallelogram has adjacent sides of lengths $s$ units and $2s$ units forming a 45-degree angle. The area of the parallelogram is $8\\sqrt 2$ square units. What is the value of $s$? Express your answer in simplest radical for... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In right triangle $ABC$ with $\\angle B = 90^\\circ$, we have $$2\\sin A = 3\\cos A.$$What is $\\sin A$? [/INST] The triangle is shown below:\n\n[asy]\npair A,B,C;\nA = (0,0);\nB = (10,0);\nC = (10,15);\ndraw(A--B--C--A);\ndra... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Point $P$ is inside equilateral $\\triangle ABC$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CA}$, respectively. Given that $PQ=1$, $PR=2$, and ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The image of the point with coordinates $(1,1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(9,5)$. Find $m+b$. [/INST] The line of reflection is the perpendicular bisector of the segment conn... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The congruent sides of an isosceles triangle are each 5 cm long, and the perimeter is 17 cm. In centimeters, what is the length of the base? [/INST] If the length of the base is $b$ centimeters, then the perimeter of the trian... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The length of the median to the hypotenuse of an isosceles, right triangle is $10$ units. What is the length of a leg of the triangle, in units? Express your answer in simplest radical form. [/INST] The length of the median t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In an isosceles triangle, one of the angles opposite an equal side is $40^{\\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label(\"$\\backslash$\",(1.5,1))... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\\sqrt{2}$, what is the volume of the solid?\n[asy] size(180); i... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal lengt... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is $5$, that $BC=6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only ch... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is $2$ cm. The angle $ABC$ is a right angle. Find the square o... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A point $P$ is chosen in the interior of $\\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\\triangle ABC$, the resulting smaller triangles $t_{1}$, $t_{2}$, and $t_{3}$ in the figure, have ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is $15\\mbox{cm}^2$ and the area of face $ABD$ is $12 \\mbox { cm}^2$. These two faces meet each other at a $30^\\circ$ angle. Find the volume of the tet... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: When a right triangle is rotated about one leg, the volume of the cone produced is $800\\pi \\;\\textrm{ cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\\pi \\;\\textrm{ cm}^3... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\\alpha$, $\\beta$, and $\\alpha + \\beta$ radians, respectively, where $\\alpha + \\beta < \\pi$. If $\\cos \\alpha$, which is a positive ration... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In rectangle $ABCD$, side $AB$ measures $6$ units and side $BC$ measures $3$ units, as shown. Points $F$ and $G$ are on side $CD$ with segment $DF$ measuring $1$ unit and segment $GC$ measuring $2$ units, and lines $AF$ and $B... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is $60$, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respecti... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight li... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The rectangle $ABCD$ below has dimensions $AB = 12 \\sqrt{3}$ and $BC = 13 \\sqrt{3}$. Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\\overline{AP}$ and $\\o... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $P_1$ be a regular $r~\\mbox{gon}$ and $P_2$ be a regular $s~\\mbox{gon}$ $(r\\geq s\\geq 3)$ such that each interior angle of $P_1$ is $\\frac{59}{58}$ as large as each interior angle of $P_2$. What's the largest possible... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$.\n[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arra... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Rhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P$, $Q$, $R$, and $S$ are interior points on sides $\\overline{AB}$, $\\overline{BC}$, $\\overline{CD}$, and $\\overline{DA}$, respectively. It is given that $P... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$.\n [/INST] [asy]... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Rectangle $ABCD$ has sides $\\overline {AB}$ of length 4 and $\\overline {CB}$ of length 3. Divide $\\overline {AB}$ into 168 congruent segments with points $A=P_0, P_1, \\ldots, P_{168}=B$, and divide $\\overline {CB}$ into 1... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that $\\frac{AO}{OA'}+\\frac{BO}{OB'}+\\frac{CO}{OC'}=92$, fi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.\n [/INST] Since the area $BCD=80=\\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\\frac mn$, where $m$ and $n$ are relatively pri... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V-E+F=2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vert... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\overline{CH}$ be an altitude of $\\triangle ABC$. Let $R\\,$ and $S\\,$ be the points where the circles inscribed in the triangles $ACH\\,$ and $BCH$ are tangent to $\\overline{CH}$. If $AB = 1995\\,$, $AC = 1994\\,$, a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC,\\,$ angle $C$ is a right angle and the altitude from $C\\,$ meets $\\overline{AB}\\,$ at $D.\\,$ The lengths of the sides of $\\triangle ABC\\,$ are integers, $BD=29^3,\\,$ and $\\cos B=m/n\\,$, where $m\\,$ ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire fi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Given a point $P$ on a triangular piece of paper $ABC,\\,$ consider the creases that are formed in the paper when $A, B,\\,$ and $C\\,$ are folded onto $P.\\,$ Let us call $P$ a fold point of $\\triangle ABC\\,$ if these creas... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The graphs of the equations\n$y=k, \\qquad y=\\sqrt{3}x+2k, \\qquad y=-\\sqrt{3}x+2k,$\nare drawn in the coordinate plane for $k=-10,-9,-8,\\ldots,9,10.\\,$ These 63 lines cut part of the plane into equilateral triangles of si... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The points $(0,0)\\,$, $(a,11)\\,$, and $(b,37)\\,$ are the vertices of an equilateral triangle. Find the value of $ab\\,$.\n [/INST] Consider the points on the complex plane. The point $b+37i$ is then a rotation of $60$ degre... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\\overline{OA}, \\overline{OB}, \\overline{OC},$ and $\\overline{OD},$ and $\\angle AOB=45^\\circ.$ Let $\\theta$ be the measure of the dihedral angle formed by faces $O... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In a circle of radius $42$, two chords of length $78$ intersect at a point whose distance from the center is $18$. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segmen... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circles of radius $3$ and $6$ are externally tangent to each other and are internally tangent to a circle of radius $9$. The circle of radius $9$ has a chord that is a common external tangent of the other two circles. Find the... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surfac... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form. [/INST] We... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\\sqrt{... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A car travels due east at $\\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\\frac 12\\sqrt{2}$ mile per minute. At time $t=0$, the center... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Three of the edges of a cube are $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ and $\\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ respe... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The inscribed circle of triangle $ABC$ is tangent to $\\overline{AB}$ at $P,$ and its radius is $21$. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle.\n [/INST] [asy] pathpen = black + linewidth(0.65); point... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relativel... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The two squares shown share the same center $O$ and have sides of length 1. The length of $\\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. C... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A regular polygon has exterior angles each measuring 15 degrees. How many sides does the polygon have? [/INST] The measure of each exterior angle in a regular $n$-gon is $360/n$ degrees. Setting this expression equal to 15, w... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $u$ and $v$ be integers satisfying $0 < v < u$. Let $A = (u,v)$, let $B$ be the reflection of $A$ across the line $y = x$, let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.\n[asy]d... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A circle is inscribed in quadrilateral $ABCD$, tangent to $\\overline{AB}$ at $P$ and to $\\overline{CD}$ at $Q$. Given that $AP=19$, $PB=26$, $CQ=37$, and $QD=23$, find the square of the radius of the circle.\n [/INST] Call t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\\overline{AB}$ and $\\overline{CD}$ are the only para... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The points $A$, $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$. It is given that $AB=13$, $BC=14$, $CA=15$, and that the distance from $O$ to $\\triangle ABC$ is $\\frac{m\\sqrt{n}}k$, where $m$, $n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$. Let $x$ be the length of the segme... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In trapezoid $ABCD$, leg $\\overline{BC}$ is perpendicular to bases $\\overline{AB}$ and $\\overline{CD}$, and diagonals $\\overline{AC}$ and $\\overline{BD}$ are perpendicular. Given that $AB=\\sqrt{11}$ and $AD=\\sqrt{1001}$... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\\frac{m}{n}$, w... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The line joining $(3,2)$ and $(6,0)$ divides the square shown into two parts. What fraction of the area of the square is above this line? Express your answer as a common fraction.\n\n[asy]\ndraw((-2,0)--(7,0),linewidth(1),Arro... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $\\overline{DE}$ is parallel to $\\overline{BC}$ and contains the center of the ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $AB=13$, $BC=15$ and $CA=17$. Point $D$ is on $\\overline{AB}$, $E$ is on $\\overline{BC}$, and $F$ is on $\\overline{CA}$. Let $AD=p\\cdot AB$, $BE=q\\cdot BC$, and $CF=r\\cdot CA$, where $p$, $q$, and $r$ ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In quadrilateral $ABCD$, $\\angle{BAD}\\cong\\angle{ADC}$ and $\\angle{ABD}\\cong\\angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\\frac {m}{n}$, where $m$ and $n$ are relatively pr... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\\overline{EF}$, $\\overline{EH}$, and $\\over... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $R = (8,6)$. The lines whose equations are $8y = 15x$ and $10y = 3x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\\overline{PQ}$. The length of $PQ$ equals $\\frac {m}{n}$, where $m$ and $n$... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\\overline{CD}$ and vertices $G$ and $H$ on the circle. If the area of square $ABCD$ is $1$, then the area of square $EFGH$ can be expressed as... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\triangle{PQR}$ be a right triangle with $PQ = 90$, $PR = 120$, and $QR = 150$. Let $C_{1}$ be the inscribed circle. Construct $\\overline{ST}$ with $S$ on $\\overline{PR}$ and $T$ on $\\overline{QR}$, such that $\\overl... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12$. A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 7 units from $\\overline{BG}$ and 5 units from $\\overline{BC}$. The beam ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$ the medians $\\overline{AD}$ and $\\overline{CE}$ have lengths $18$ and $27$, respectively, and $AB=24$. Extend $\\overline{CE}$ to intersect the circumcircle of $ABC$ at $F$. The area of triangle $AFB$ is $m... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $AB = 13$, $AC = 15$, and $BC = 14$. Let $I$ be the incenter. The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively. Find the length of $BI$. [/INST] Since $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\\overline{AP}$ is drawn so that it is tangent to $\\overline{AM}$ and $\\overline{PM}$. Given that $O... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circles $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6)$, and the product of the radii is $68$. The x-axis and the line $y = mx$, where $m > 0$, are tangent to both circles. It is give... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Three vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the surface area of the cube?\n [/INST] $PQ=\\sqrt{(8-7)^2+(8-12)^2+(1-10)^2}=\\sqrt{98}$\n$PR=\\sqrt{(11-7)^2+(3-12)^2+(9-10)^2}=\\sqrt{98}$\n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ is isosceles with $AC = BC$ and $\\angle ACB = 106^\\circ.$ Point $M$ is in the interior of the triangle so that $\\angle MAC = 7^\\circ$ and $\\angle MCA = 23^\\circ.$ Find the number of degrees in $\\angle CMB... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In convex quadrilateral $ABCD, \\angle A \\cong \\angle C, AB = CD = 180,$ and $AD \\neq BC.$ The perimeter of $ABCD$ is $640$. Find $\\lfloor 1000 \\cos A \\rfloor.$ (The notation $\\lfloor x \\rfloor$ means the greatest inte... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\\frac{m + n\\pi}{p},$ where $m, n,$ and $p$ are ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Point $B$ is on $\\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\\overline{AC}$ so that $AD = CD,$ and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\\triangle ACD$. Find $s.$... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $C... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\\angle FAB = 120^\\circ,$ $\\overline{AB}\\parallel \\overline{DE},$ $\\overline{BC}\\parallel \\overl... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circle $\\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\\omega$ such that $OA=13$. The two tangents to $\\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tang... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\\circ$ rotation about $G.$ Wha... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.\n [/INST] The diagonals of the rhombus perpendicularly bisect each other. Ca... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into tw... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A unicorn is tethered by a $20$-foot silver rope to the base of a magician's cylindrical tower whose radius is $8$ feet. The rope is attached to the tower at ground level and to the unicorn at a height of $4$ feet. The unicorn... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A convex polyhedron $P$ has $26$ vertices, $60$ edges, and $36$ faces, $24$ of which are triangular and $12$ of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belo... | custom_source | 5 |
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