conversations listlengths 2 2 | source stringclasses 1
value | score int64 5 5 |
|---|---|---|
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Square $ABCD$ has sides of length 2. Set $S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A right circular cone has a base with radius $600$ and height $200\\sqrt{7}.$ A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is $125$, and crawls along the surface of the cone to ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$and $CD=4.$ Draw circles of radius 3 centered at $A$ and $B,$ and circles of radius 2 centered at $C$ and $D.$ A circle contained within the trapezoi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $ABCDE$ be a convex pentagon with $AB \\parallel CE, BC \\parallel AD, AC \\parallel DE, \\angle ABC=120^\\circ, AB=3, BC=5,$ and $DE = 15.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: $ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ lies in the cartesian plane and has an area of $70$. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20),$ respectively, and the coordinates of $A$ are $(p,q).$ The line containing the median to side $BC$ ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \\sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In quadrilateral $ABCD,\\ BC=8,\\ CD=12,\\ AD=10,$ and $m\\angle A= m\\angle B = 60^\\circ.$ Given that $AB = p + \\sqrt{q},$ where $p$ and $q$ are positive integers, find $p+q.$\n [/INST] [asy]draw((0,0)--(20.87,0)--(15.87,8.... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O,$ and that the ratio of the volume of $O$ to that of $C$ is $\\frac mn,$ where $m$ and $n$ are relatively prime int... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Square $ABCD$ has center $O,\\ AB=900,\\ E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\\angle EOF =45^\\circ,$ and $EF=400.$ Given that $BF=p+q\\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is external... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In quadrilateral $ABCD$, $\\angle B$ is a right angle, diagonal $\\overline{AC}$ is perpendicular to $\\overline{CD}$, $AB=18$, $BC=21$, and $CD=14$. Find the perimeter of $ABCD$.\n [/INST] From the problem statement, we const... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the volume, in cubic inches, of a rectangular box, whose faces have areas of $24$ square inches, $16$ square inches and $6$ square inches? [/INST] If $l$, $w$, and $h$ represent the dimensions of the rectangular box, w... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A sphere is inscribed in a cube. Given that one edge of the cube is 6 inches, how many cubic inches are in the volume of the inscribed sphere? Express your answer in terms of $\\pi$. [/INST] [asy]\nsize(70);\ndraw(Circle((6,6)... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region $\\mathcal{R}$ be the union of the eight circular regions. Line $l,$ with slope 3, divides $\\mathcal{R}$ into two regio... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\\overline{BC}$ and $\\overline{CD},$ respectively, so that $\\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABC... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$.[asy]unitsize(0.2 cm); pair A, B, C, D, E, F; A = (0,13); B = (13,13); C = (13,0); D = (... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entire... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $ABCD$ be an isosceles trapezoid with $\\overline{AD}||\\overline{BC}$ whose angle at the longer base $\\overline{AD}$ is $\\dfrac{\\pi}{3}$. The diagonals have length $10\\sqrt {21}$, and point $E$ is at distances $10\\sq... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\overline{AB}$ be a diameter of circle $\\omega$. Extend $\\overline{AB}$ through $A$ to $C$. Point $T$ lies on $\\omega$ so that line $CT$ is tangent to $\\omega$. Point $P$ is the foot of the perpendicular from $A$ to ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A square piece of paper has sides of length $100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\\sqrt{17}$ from the corner, and they meet on the... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The base of a triangular piece of paper $ABC$ is $12\\text{ cm}$ long. The paper is folded down over the base, with the crease $DE$ parallel to the base of the paper. The area of the triangle that projects below the base is $1... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $AB = AC = 100$, and $BC = 56$. Circle $P$ has radius $16$ and is tangent to $\\overline{AC}$ and $\\overline{BC}$. Circle $Q$ is externally tangent to $P$ and is tangent to $\\overline{AB}$ and $\\overline{... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\\angle A = 37^\\circ$, $\\angle D = 53^\\circ$, and $M$ and $N$ be the midpoints of $\\overline{BC}$ and $\\overline{AD}... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangl... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In right $\\triangle ABC$ with hypotenuse $\\overline{AB}$, $AC = 12$, $BC = 35$, and $\\overline{CD}$ is the altitude to $\\overline{AB}$. Let $\\omega$ be the circle having $\\overline{CD}$ as a diameter. Let $I$ be a point ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $AB = 10$, $BC = 14$, and $CA = 16$. Let $D$ be a point in the interior of $\\overline{BC}$. Let points $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD$, respectively. The circumcircles of t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In parallelogram $ABCD$, point $M$ is on $\\overline{AB}$ so that $\\frac {AM}{AB} = \\frac {17}{1000}$ and point $N$ is on $\\overline{AD}$ so that $\\frac {AN}{AD} = \\frac {17}{2009}$. Let $P$ be the point of intersection o... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\\overline{AC}$ and $\\overline{AB}$ respectively so that $AK = CK$, and $\\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the poin... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\\frac{3}5$. Poin... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In rectangle $ABCD$, $AB=100$. Let $E$ be the midpoint of $\\overline{AD}$. Given that line $AC$ and line $BE$ are perpendicular, find the greatest integer less than $AD$.\n [/INST] [asy] pair A=(0,10), B=(0,0), C=(14,0), D=(1... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Equilateral triangle $T$ is inscribed in circle $A$, which has radius $10$. Circle $B$ with radius $3$ is internally tangent to circle $A$ at one vertex of $T$. Circles $C$ and $D$, both with radius $2$, are internally tangent... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\triangle PQR$ be a triangle in the plane, and let $S$ be a point outside the plane of $\\triangle PQR$, so that $SPQR$ is a pyramid whose faces are all triangles.\n\nSuppose that every edge of $SPQR$ has length $18$ or ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \\le 10$ and $3y - x \\ge 15$. When $\\mathcal{R}$ is revolved around the line whose equation is $3y - x =... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Rectangle $ABCD$ and a semicircle with diameter $AB$ are coplanar and have nonoverlapping interiors. Let $\\mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $\\ell$ meets the semicircle, segment... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle{ABC}$ with $AB = 12$, $BC = 13$, and $AC = 15$, let $M$ be a point on $\\overline{AC}$ such that the incircles of $\\triangle{ABM}$ and $\\triangle{BCM}$ have equal radii. Then $\\frac{AM}{CM} = \\frac{p}{q}$, w... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8: 7$. Find the minimum possible value of their common perimeter.\n [... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ with right angle at $C$, $\\angle BAC < 45^\\circ$ and $AB = 4$. Point $P$ on $\\overline{AB}$ is chosen such that $\\angle APC = 2\\angle ACP$ and $CP = 1$. The ratio $\\frac{AP}{BP}$ can be represented in the ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $ABCDEF$ be a regular hexagon. Let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $AF$, respectively. The segments $\\overline{AH}$, $\\overline{BI}$, $\\overline{CJ}$, $\\over... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $AB=125$, $AC=117$ and $BC=120$. The angle bisector of angle $A$ intersects $\\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\\overline{AC}$ at point $K$. Let $M$ and $N$ be the... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\\overline{AC}$ with $V$ on $\\overline{AW}$, points $X$ and $Y$ are on $\\overline{BC}$ with $X$ on $\\overline{CY}$, and points $Z$ and $U$ a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A circle with center $O$ has radius 25. Chord $\\overline{AB}$ of length 30 and chord $\\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: On an old-fashioned bicycle the front wheel has a radius of $2.5$ feet and the back wheel has a radius of $4$ inches. If there is no slippage, how many revolutions will the back wheel make while the front wheel makes $100$ re... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 12$ and $\\angle O_{1}PO_{2} = 120^{\\circ}$, then $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: On square $ABCD$, point $E$ lies on side $AD$ and point $F$ lies on side $BC$, so that $BE=EF=FD=30$. Find the area of the square $ABCD$.\n [/INST] Drawing the square and examining the given lengths,[asy] size(2inch, 2inch); c... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.\n [/INST] The average angle in an 18-gon is $160^\\circ$... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In triangle $ABC$, $AB=20$ and $AC=11$. The angle bisector of $\\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ is inscribed in circle $\\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\\overline{BC}$ at $D$ and circle $\\omega$ at a second point $E$. Let $\\gamma$ be the circle with diamete... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A rectangular box has width $12$ inches, length $16$ inches, and height $\\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle RED$, $\\measuredangle DRE=75^{\\circ}$ and $\\measuredangle RED=45^{\\circ}$. $RD=1$. Let $M$ be the midpoint of segment $\\overline{RD}$. Point $C$ lies on side $\\overline{ED}$ such that $\\overline{RC}\\perp... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle{ABC}, AB=10, \\angle{A}=30^\\circ$ , and $\\angle{C=45^\\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\\perp{BC}$, $\\angle{BAD}=\\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of th... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19.$ Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line through $P$ and $Q$ intersects the circle at $X... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circle $C$ with radius 2 has diameter $\\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\\overline{AB}$. Th... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\\angle B$ and $\\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\\triangle ABC$.\n [/INST] Le... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: With all angles measured in degrees, the product $\\prod_{k=1}^{45} \\csc^2(2k-1)^\\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.\n [/INST] Let $x = \\cos 1^\\circ + i \\sin 1^\\circ$. Then from the ide... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\\overline{AB}$, vertex $Q$ on $\\overline{AC}$, and vertices $R$ and $S$ on $\\overline{BC}$. In terms of the side lengt... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Centered at each lattice point in the coordinate plane are a circle radius $\\frac{1}{10}$ and a square with sides of length $\\frac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0,0)$ to $(10... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Equilateral $\\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\\triangle ABC$ and are on opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\\triangle PA... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\\sqrt{n}$, where $m$ and $n$ are positive integer... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\triangle ABC$ have side lengths $AB=30$, $BC=32$, and $AC=34$. Point $X$ lies in the interior of $\\overline{BC}$, and points $I_1$ and $I_2$ are the incenters of $\\triangle ABX$ and $\\triangle ACX$, respectively. Fin... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle ABC$, the sides have integer lengths and $AB=AC$. Circle $\\omega$ has its center at the incenter of $\\triangle ABC$. An excircle of $\\triangle ABC$ is a circle in the exterior of $\\triangle ABC$ that is tang... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \\neq C$ is a point of intersection of the circumcircles of $\\triangle ACD$ and $\\triangle EBC$ sat... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In convex quadrilateral $KLMN$ side $\\overline{MN}$ is perpendicular to diagonal $\\overline{KM}$, side $\\overline{KL}$ is perpendicular to diagonal $\\overline{LN}$, $MN = 65$, and $KL = 28$. The line through $L$ perpendicu... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\\triangle ABC$ and $\\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the inter... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1.$ Point $P$ lies inside the circle so that the region bounded by $\\overline{PA_1},\\overline{PA_2},$ and the minor arc $\\widehat{A_1A_2}$ of the ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\\overline{AB}$ and from $B$ to $\\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\\triangle ABC$ in two distin... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\\ell_A,\\ell_B$, and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$, and $\\overline{AB}$, respectively, such that the intersections of $\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$.\n [/INST] Assume the incircle... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A triangle has sides of length 5 and 6 units. The length of the third side is $x$ units, where $x$ is an integer. What is the largest possible perimeter of the triangle? [/INST] If a triangle has sides of length 5 and 6 units,... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\triangle ABC$ be an acute scalene triangle with circumcircle $\\omega$. The tangents to $\\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppos... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\\left(\\frac58,... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangles $\\triangle ABC$ and $\\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$ wh... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere w... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In square $ABCD$, points $E$ and $H$ lie on $\\overline{AB}$ and $\\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\\overline{BC}$ and $\\overline{CD}$, respectively, and points $I$ and $J$ lie on $\\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Line segment $\\overline{AB}$ is a diameter of a circle with $AB = 24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\\triangle ABC$ traces out... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\\omega_1$, $\\omega_2$, and... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In $\\triangle{ABC}$ with side lengths $AB = 13$, $AC = 12$, and $BC = 5$, let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcirc... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Lisa, a child with strange requirements for her projects, is making a rectangular cardboard box with square bases. She wants the height of the box to be 3 units greater than the side of the square bases. What should the height... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\\triangle{ABP}$, $\\triangle{BCP}$, $\\triangle{CDP}$, and $\\triangle{DAP}$ are the vertices of a convex... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\\triangle ABC \\sim \\triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\\triangle ABC \\sim \\triangle CEB$ and the area of $\\tria... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The diameter $AB$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and line $ED$ is perpendicular to line $AD$. Segment $AE$ intersects the circle at a ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\\dfrac{1}{2}$ is $\\dfrac{a-b\\pi}{c}$... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible?\n$\\textb... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $ABCDE$ be a pentagon inscribed in a circle such that $AB = CD = 3$, $BC = DE = 10$, and $AE= 14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\\frac{m}{n}$, where $m$ and $n$ are relatively prime posit... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?\n$\\textbf{(A)}\\ \\sqrt{15} \\q... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $R$ be a unit square region and $n \\geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many poi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\\triangle ABC$ parallel to $\\overline{BC}$ intersects $\\overline{AB}$ at $M$ and $\\overline{AC}$ at $N.$ What is the peri... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A triangle has three different integer side lengths and a perimeter of 20 units. What is the maximum length of any one side? [/INST] A triangle with sides of 9,8 and 3 will satisfy these conditions. This will have longest sid... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $A_0=(0,0)$. Distinct points $A_1,A_2,\\dots$ lie on the $x$-axis, and distinct points $B_1,B_2,\\dots$ lie on the graph of $y=\\sqrt{x}$. For every positive integer $n,\\ A_{n-1}B_nA_n$ is an equilateral triangle. What is... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Triangle $ABC$ has $\\angle C = 60^{\\circ}$ and $BC = 4$. Point $D$ is the midpoint of $BC$. What is the largest possible value of $\\tan{\\angle BAD}$?\n$\\mathrm{(A)}\\ \\frac{\\sqrt{3}}{6}\\qquad\\mathrm{(B)}\\ \\frac{\\sq... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the c... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Which of the following could NOT be the lengths of the external diagonals of a right regular prism [a \"box\"]? (An $\\textit{external diagonal}$ is a diagonal of one of the rectangular faces of the box.)\n$\\text{(A) }\\{4,5,... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: [asy] draw(circle((4,1),1),black+linewidth(.75)); draw((0,0)--(8,0)--(8,6)--cycle,black+linewidth(.75)); MP(\"A\",(0,0),SW);MP(\"B\",(8,0),SE);MP(\"C\",(8,6),NE);MP(\"P\",(4,1),NW); MP(\"8\",(4,0),S);MP(\"6\",(8,3),E);MP(\"10\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: [asy] draw((0,0)--(1,sqrt(3)),black+linewidth(.75),EndArrow); draw((0,0)--(1,-sqrt(3)),black+linewidth(.75),EndArrow); draw((0,0)--(1,0),dashed+black+linewidth(.75)); dot((1,0)); MP(\"P\",(1,0),E); [/asy]\nLet $S$ be the set o... | custom_source | 5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.