conversations listlengths 2 2 | source stringclasses 1
value | score int64 5 5 |
|---|---|---|
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The hexagon with the R is colored red. Each hexagon is colored either red, yellow or green, such that no two hexagons with a common side are colored the same color. In how many different ways can the figure be colored?\n\n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A point $P$ is randomly selected from the rectangular region with vertices $(0,0), (2,0)$, $(2,1),(0,1)$. What is the probability that $P$ is closer to the origin than it is to the point $(3,1)$? [/INST] The area of the rectan... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In a single-elimination tournament, each game is between two players. Only the winner of each game advances to the next round. In a particular such tournament there are 256 players. How many individual games must be played to ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many ways are there to put 5 balls in 2 boxes if the balls are not distinguishable but the boxes are? [/INST] Since the balls are indistinguishable, we need only count the number of balls in the distinguishable boxes. We ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The number 121 is a palindrome, because it reads the same backwards as forward. How many integer palindromes are between 100 and 500? [/INST] The hundreds digit can be any one of 1, 2, 3 or 4. Whatever the hundreds digit is, t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\\frac{1}{15}$. How many balls in the bag are r... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Mr. Wong has 10 grandchildren. Assuming that the gender of each child is determined independently and with equal likelihood of male and female, what is the probability that Mr. Wong has more grandsons than granddaughters or mo... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The reality game show Survivor is played with 16 people divided into two tribes of 8. In the first episode, two people get homesick and quit. If every person has an equal chance of being one of the two quitters, and the probab... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many integers $n$ satisfy the condition $100"
},
{
"from": "gpt",
"value": "Solution placeholder"
}
] | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the $1$ at each end, row $4$ consists of only even numbers, as does row $2.$ How many of the first $20$ rows have this property? (Don... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Determine the number of ways to arrange the letters of the word PROOF. [/INST] There are two O's and five total letters, so the answer is $\\dfrac{5!}{2!} = \\boxed{60}$."
},
{
"from": "gpt",
"value": "Solution pla... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The integers $r$ and $k$ are randomly selected, where $-3 < r < 6$ and $1 < k < 8$. What is the probability that the division $r \\div k$ is an integer value? Express your answer as a common fraction. [/INST] The possible valu... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many four-digit numbers $N = \\underline{a}\\,\\underline{b}\\,\\underline{c}\\,\\underline{d}$ satisfy all of the following conditions?\n\n$4000 \\le N < 6000.$\n$N$ is a multiple of $5.$\n$3 \\le b"
},
{
"from": ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The Badgers play the Cougars in a series of seven basketball games. Each team has an equal chance of winning each game. What is the probability that the Badgers will win at least four games? Express your answer as a common fra... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace? [/INST] There are 16 cards in a st... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Square ABCD has its center at $(8,-8)$ and has an area of 4 square units. The top side of the square is horizontal. The square is then dilated with the dilation center at (0,0) and a scale factor of 2. What are the coordinates... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram below, we have $\\sin \\angle RPQ = \\frac{7}{25}$. What is $\\sin \\angle RPS$?\n\n[asy]\npair R,P,Q,SS;\nSS = (-2,0);\nP = (0,0);\nQ = (2,0);\nR = rotate(aSin(7/25))*(1.5,0);\ndot(\"$S$\",SS,S);\ndot(\"$Q$\",... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The shaded region shown consists of 11 unit squares and rests along the $x$-axis and the $y$-axis. The shaded region is rotated about the $x$-axis to form a solid. In cubic units, what is the volume of the resulting solid? Exp... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A scalene triangle has side lengths which are prime numbers and the length of its perimeter is also prime. What is its smallest possible perimeter? [/INST] The first few prime numbers are: $2, 3, 5, 7, 11, 13, 17,\\ldots$. ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The field shown has been planted uniformly with wheat. [asy]\ndraw((0,0)--(1/2,sqrt(3)/2)--(3/2,sqrt(3)/2)--(2,0)--(0,0),linewidth(0.8));\nlabel(\"$60^\\circ$\",(0.06,0.1),E);\nlabel(\"$120^\\circ$\",(1/2-0.05,sqrt(3)/2-0.1),E... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Points $A(-4,1), B(-1,4)$ and $C(-1,1)$ are the vertices of $\\triangle ABC$. What will be the coordinates of the image of point A if $\\triangle ABC$ is rotated 90 degrees clockwise about the origin? [/INST] When we rotate im... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is 56. Find the area of the polygon.\n\n[asy]\nunitsize(0.5 cm);\n\ndraw((3,0)--(4,0... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $\\sin 510^\\circ$. [/INST] Rotating $360^\\circ$ is the same as doing nothing, so rotating $510^\\circ$ is the same as rotating $510^\\circ - 360^\\circ = 150^\\circ$. Therefore, we have $\\sin 510^\\circ = \\sin (51... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A bowling ball cannot weigh more than 16 pounds and must have a diameter of $8 \\frac{1}{2}$ inches. How many square inches are in the surface area of a bowling ball before the finger holes are drilled? Express your answer as ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $AX = XY = YB = BC$ and the measure of angle $ABC$ is 120 degrees. What is the number of degrees in the measure of angle $BAC$?\n\n[asy]\npair A,X,Y,B,C;\nX = A + dir(30); Y = X + dir(0); B = Y + dir(60); C ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A right pyramid has a square base with perimeter 24 inches. Its apex is 9 inches from each of the other vertices. What is the height of the pyramid from its peak to the center of its square base, in inches? [/INST] [asy]\nimpo... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles? [/INST] Let $A$, $B$, and $C$ be the vertices of the triangle so that $AB = 5... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Regions I, II and III are bounded by squares. The perimeter of region I is 12 units and the perimeter of region II is 24 units. What is the ratio of the area of region I to the area of region III? Express your answer as a comm... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In a trapezoid $ABCD$ with $AB$ parallel to $CD$, the diagonals $AC$ and $BD$ intersect at $E$. If the area of triangle $ABE$ is 50 square units, and the area of triangle $ADE$ is 20 square units, what is the area of trapezoid... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circle $A$ is in the interior of circle $B$. The diameter of circle $B$ is 16 cm. What is the diameter of circle $A$ for which the ratio of the shaded area to the area of circle $A$ is 3:1?\n\n[asy]size(101);\nimport graph;\np... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The center of a circle has coordinates $(6,-5)$. The circle is reflected about the line $y=x$. What are the $x,y$ coordinates of the center of the image circle? State the $x$ coordinate first. [/INST] The center of the image c... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The longer leg of a right triangle is $1$ foot shorter than twice the length of the shorter leg. The area of the triangle is $60$ square feet. What is the length of the hypotenuse, in feet? [/INST] If $x$ represents the length... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The vertices of a square are the centers of four circles as shown below. Given each side of the square is 6cm and the radius of each circle is $2\\sqrt{3}$cm, find the area in square centimeters of the shaded region. [asy]\nfi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The measure of each exterior angle of a regular polygon is $30$ degrees. What is the sum of the measures of the interior angles, in degrees? [/INST] Taking one exterior angle per vertex, the sum of the exterior angles of a pol... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In trapezoid $ABCD$ the lengths of the bases $AB$ and $CD$ are 8 and 17 respectively. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Quadrilateral $CDEF$ is a parallelogram. Its area is $36$ square units. Points $G$ and $H$ are the midpoints of sides $CD$ and $EF,$ respectively. What is the area of triangle $CDJ?$ [asy]\ndraw((0,0)--(30,0)--(12,8)--(22,8)--... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 10 and altitude 12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A right circular cylinder with radius 2 is inscribed in a hemisphere with radius 5 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? [/INST] We draw and label a diagram as foll... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ c... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A circular spinner for a game has a radius of 10 cm. The probability of winning on one spin of this spinner is $\\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\\pi$.\n\n[asy]impo... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The area of an equilateral triangle is numerically equal to the length of one of its sides. What is the perimeter of the triangle, in units? Express your answer in simplest radical form. [/INST] The formula for the area of an ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, $O$ is the center of a circle with radii $OP=OQ=5$. What is the perimeter of the shaded region?\n\n[asy]\nsize(100);\nimport graph;\nlabel(\"$P$\",(-1,0),W); label(\"$O$\",(0,0),NE); label(\"$Q$\",(0,-1),S);\n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A circle of radius 1 is tangent to a circle of radius 2. The sides of $\\triangle ABC$ are tangent to the circles as shown, and the sides $\\overline{AB}$ and $\\overline{AC}$ are congruent. What is the area of $\\triangle ABC... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two right triangles share a side as follows: [asy]\npair pA, pB, pC, pD, pE;\npA = (0, 0);\npB = pA + 6 * dir(0);\npC = pA + 10 * dir(90);\npD = pB + 6 * dir(90);\npE = (6 * pA + 10 * pD) / 16;\ndraw(pA--pB--pC--pA);\ndraw(pA-... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\triangle XOY$ be a right-angled triangle with $m\\angle XOY = 90^{\\circ}$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Given that $XN=19$ and $YM=22$, find $XY$. [/INST] Let $OM = a$ and $ON =... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ with vertices $A(1, -3)$, $B(-2, 0)$ and $C(4, 3)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $A$ to $A'$? [/INST] Reflecting a point over the $y$-axis n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A right, rectangular prism has three faces with areas of $6,8$ and $12$ square inches. What is the volume of the prism, in cubic inches? [/INST] If $l$, $w$, and $h$ represent the dimensions of the rectangular prism, we look f... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A square with sides 6 inches is shown. If $P$ is a point such that the segment $\\overline{PA}$, $\\overline{PB}$, $\\overline{PC}$ are equal in length, and segment $\\overline{PC}$ is perpendicular to segment $\\overline{FD}$... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Each edge length of a rectangular solid is a prime number. If the volume of the rectangular solid is 385 cubic units, what is the total surface area, in square units, of the rectangular solid? [/INST] Prime factorize $385$ as ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A spiral staircase turns $270^\\circ$ as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth. [/INST] The ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ is a right triangle with legs $AB$ and $AC$. Points $X$ and $Y$ lie on legs $AB$ and $AC$, respectively, so that $AX:XB = AY:YC = 1:2$. If $BY = 16$ units, and $CX = 28$ units, what is the length of hypotenuse $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The sides of a triangle have lengths of $15$, $20$, and $25$. Find the length of the shortest altitude. [/INST] First notice that this is a right triangle, so two of the altitudes are the legs, whose lengths are $15$ and $20$.... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The area of one lateral face of a right pyramid with an equilateral triangular base is 75 square meters. If the slant height is 30 meters, what is the length of the side of its base, in meters? [/INST] Let $s$ represent the si... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Altitudes $\\overline{AX}$ and $\\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\\angle BAC = 61^\\circ$ and $\\angle ABC = 73^\\circ$, then what is $\\angle CHX$? [/INST] First, we build a diagram:\n\n[asy]\nsi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle? Provide your answer as a decimal rounded to the nearest hundredth. [/INST] Let the sides of the triangle ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A hexagon inscribed in a circle has three consecutive sides, each of length 3, and three consecutive sides, each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides, each of ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, $\\triangle PQR$ is right-angled at $P$ and has $PQ=2$ and $PR=2\\sqrt{3}$. Altitude $PL$ intersects median $RM$ at $F$. What is the length of $PF$? [asy]\ndraw((0,0)--(10,0)--(0,10*sqrt(3))--cycle);\ndraw((0... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the tr... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The first square below is in position ABCD. After rotating the square 90 degrees clockwise about its center point, the second square is in position DABC, as shown. Next, square DABC is reflected over its vertical line of symme... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Suppose that we have a right triangle $ABC$ with the right angle at $B$ such that $AC = \\sqrt{61}$ and $AB = 5.$ A circle is drawn with its center on $AB$ such that the circle is tangent to $AC$ and $BC.$ If $P$ is the point ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $ABCDE$ is a regular pentagon. $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively. Let $O$ be the center of the pentagon. If $OP = 1$, then find $AO + AQ + A... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A right cylinder with a base radius of 3 units is inscribed in a sphere of radius 5 units. The total volume, in cubic units, of the space inside the sphere and outside the cylinder is $W\\pi$. Find $W$, as a common fraction. ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In the diagram, $\\triangle ABE$, $\\triangle BCE$ and $\\triangle CDE$ are right-angled, with $\\angle AEB=\\angle BEC = \\angle CED = 60^\\circ$, and $AE=24$. [asy]\npair A, B, C, D, E;\nA=(0,20.785);\nB=(0,0);\nC=(9,-5.196)... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The radius of a sphere is $p$ units and the radius of a hemisphere is $2p$ units. What is the ratio of the volume of the sphere to the volume of the hemisphere? [/INST] The volume of the sphere is \\[\\frac{4}{3}\\pi p^3\\] an... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B=(0,0), C=(223,0), D=(680,380),$ and $E=(689,389).$ What is the sum of all possible $x$-coordinates of $A$? [/INST] Let $h$ be the length of the alti... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Right triangle $ABC$ has one leg of length 6 cm, one leg of length 8 cm and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the le... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $P$ be a point inside triangle $ABC$. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $PBC$, $PCA$, and $PAB$, respectively. If the area of triangle $ABC$ is 18, then find the area of triangle $G_1 G_2 G_3$.\n\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The figure shows a square in the interior of a regular hexagon. The square and regular hexagon share a common side. What is the degree measure of $\\angle ABC$? [asy]\nsize(150);\npair A, B, C, D, E, F, G, H;\nA=(0,.866);\nB=... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A cube has a surface area of 216 square centimeters. What is the volume of the cube, in cubic centimeters? [/INST] There are 6 faces to a cube, meaning that each face has area 36, and the edge has length 6, for a total volume ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the diameter of the circle inscribed in triangle $ABC$ if $AB = 11,$ $AC=6,$ and $BC=7$? Express your answer in simplest radical form. [/INST] Let $d$ be the diameter of the inscribed circle, and let $r$ be the radius ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In square $ABCD$, $AD$ is 4 centimeters, and $M$ is the midpoint of $\\overline{CD}$. Let $O$ be the intersection of $\\overline{AC}$ and $\\overline{BM}$. What is the ratio of $OC$ to $OA$? Express your answer as a common fra... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A cone has a volume of $12288\\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. [asy]\n\nimport markers;\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has vertices with coordinates $A(2,3),$ $B(7,8),$ and $C(-4,6)$. The triangle is reflected about line $L$. The image points are $A'(2,-5),$ $B'(7,-10),$ and $C'(-4,-8)$. What is the equation of line $L$? [/INST]... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In rectangle $ABCD$, $P$ is a point on $BC$ so that $\\angle APD=90^{\\circ}$. $TS$ is perpendicular to $BC$ with $BP=PT$, as shown. $PD$ intersects $TS$ at $Q$. Point $R$ is on $CD$ such that $RA$ passes through $Q$. In $\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A line through $B$ cuts the area of $\\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line. [/INST] The line through $B$ that cuts the a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The vertices of a $3 - 4 - 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of these circles?\n\n[asy]unitsize(1cm);\ndraw(Circle((1.8,2.4),1),linewidth(0... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Square $ABCD$ has side length $1$ unit. Points $E$ and $F$ are on sides $AB$ and $CB$, respectively, with $AE = CF$. When the square is folded along the lines $DE$ and $DF$, sides $AD$ and $CD$ coincide and lie on diagonal $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is on side $EG$ of the triangle, as shown. The triangle's altitude from $F$ to side $EG$ is 7 inches, and $EG = 10 \\text{ inches}$. The leng... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $A$, $B$, $C$, and $D$ are points on a circle, and segments $\\overline{AC}$ and $\\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP < DP.$\n\n[asy]\n\nunitsize(0.6 inch);\n\ndraw... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A rectangular prism measures 10-inches by 20-inches by 10-inches. What is the length, in inches, of the diagonal connecting point A and point B? Express your answer in simplest radical form. [asy]\nunitsize(0.75cm);\ndefaultpe... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Square $ABCD$ has area $200$. Point $E$ lies on side $\\overline{BC}$. Points $F$ and $G$ are the midpoints of $\\overline{AE}$ and $\\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has area $34$, what is the are... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The hypotenuse of a right triangle whose legs are consecutive whole numbers is 29 units. What is the sum of the lengths of the two legs? [/INST] Let's call the length of the shorter leg $x$. Then the length of the longer leg i... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter of $BC$ meets side $AC$ at $D.$ If $AD = 1$ and $BD = 4,$ then what is $CD$? [/INST] We might try sketching a diagram: [asy]\npair... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Suppose $\\cos R = \\frac{4}{9}$ in the diagram below. What is $QS$?\n\n\n\n[asy]\n\npair Q,R,S;\n\nS = (0,0);\n\nQ = (sqrt(65),0);\n\nR = (sqrt(65),-4);\n\ndraw(S--Q--R--S);\n\ndraw(rightanglemark(S,Q,R,13));\n\nlabel(\"$S$\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Three faces of a right rectangular prism have areas of 48, 49 and 50 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number. [/INST] If the length, width, and height of t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circles centered at $A$ and $B$ each have radius 2, as shown. Point $O$ is the midpoint of $\\overline{AB}$, and $OA=2\\sqrt{2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is $BD$? [/INST] Let $\\overline{CH}$ be an altitude of $\\triangle ABC$. ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A whole block of modeling clay is a right rectangular prism six inches by two inches by one inch. How many whole blocks need to be unwrapped to mold a cylindrical sculpture seven inches high and four inches in diameter? [/INST... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Isosceles triangle $ABE$ of area 100 square inches is cut by $\\overline{CD}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 75 square inches. If the altitude of triangle $ABE$ from ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A bridge is built by suspending a plank of wood between two triangular wedges with equal heights, as in the following diagram: [asy]\nimport olympiad;\nimport math;\n\n// Draw triangles\npair A = (0, 1);\npair B = (-cos(1.3962... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the degree measure of angle $LOQ$ when polygon $\\allowbreak LMNOPQ$ is a regular hexagon? [asy]\ndraw((-2,0)--(-1,1.73205081)--(1,1.73205081)--(2,0)--(1,-1.73205081)--(-1,-1.73205081)--cycle);\ndraw((-1,-1.73205081)--... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The line $y = b-x$ with $0 < b < 4$ intersects the $y$-axis at $P$ and the line $x=4$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 9:25, what is the value of $b$? Express the answer as a ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two vertices of an obtuse triangle are $(6,4)$ and $(0,0)$. The third vertex is located on the negative branch of the $x$-axis. What are the coordinates of the third vertex if the area of the triangle is 30 square units? [/INS... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has vertices at $A(5,8)$, $B(3,-2)$, and $C(6,1)$. The point $D$ with coordinates $(m,n)$ is chosen inside the triangle so that the three small triangles $ABD$, $ACD$ and $BCD$ all have equal areas. What is the ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In right triangle $DEF$, we have $\\angle D = 25^\\circ$, $\\angle E = 90^\\circ$, and $EF = 9$. Find $DE$ to the nearest tenth. You may use a calculator for this problem. [/INST] We start with a diagram:\n\n[asy]\npair D,EE... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two circles of radius 2 are centered at $(2,0)$ and at $(0,2)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\\pi$. [/INST] The two circles i... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The measures of the interior angles of a convex hexagon form an increasing arithmetic sequence. How many such sequences are possible if the hexagon is not equiangular and all of the angle degree measures are positive integers ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 4 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest r... | custom_source | 5 |
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