id stringlengths 1 4 | media images listlengths 1 1 | messages stringlengths 84 1.58k | answer stringclasses 371
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2259 | [{"role": "user", "question": "In the right triangle shown the sum of the distances $ BM$ and $ MA$ is equal to the sum of the distances $ BC$ and $ CA$. If $ MB = x$, $ CB = h$, and $ CA = d$, then $ x$ equals:\n<image1>", "hint": "", "options": {"A": "$\\frac{hd}{2h + d}$", "B": "$d - h$", "C": "$\\frac{1}{2}d$", "D... | A | mcq | 5 | metric geometry - length | |
226 | [{"role": "user", "question": "The triangle pictured is right-angled. $M$ is the midoint of the hypotenuse $\\mathrm{AB}$ and $\\angle \\mathrm{BCA}=90^{\\circ}$. How big is $\\angle \\mathrm{BMC}$?\n<image1>", "hint": "", "options": {"A": "$105^{\\circ}$", "B": "$108^{\\circ}$", "C": "$110^{\\circ}$", "D": "$120^{\\ci... | D | mcq | 4 | metric geometry - angle | |
2260 | [{"role": "user", "question": "Given triangle $ PQR$ with $ \\overline{RS}$ bisecting $ \\angle R$, $ PQ$ extended to $ D$ and $ \\angle n$ a right angle, then:\n<image1>", "hint": "", "options": {"A": "$\\angle m = \\frac{1}{2}(\\angle p - \\angle q)$", "B": "$\\angle m = \\frac{1}{2}(\\angle p + \\angle q)$", "C": "$... | B | mcq | 4 | metric geometry - angle | |
2261 | [{"role": "user", "question": "In the diagram, if points $ A$, $ B$ and $ C$ are points of tangency, then $ x$ equals:\n<image1>", "hint": "", "options": {"A": "$\\frac{3}{16}\"$", "B": "$\\frac{1}{8}\"$", "C": "$\\frac{1}{32}\"$", "D": "$\\frac{3}{32}\"$", "E": "$\\frac{1}{16}\"$"}, "choices": []}] | E | mcq | 5 | metric geometry - length | |
2262 | [{"role": "user", "question": "In the figure, $ PA$ is tangent to semicircle $ SAR$; $ PB$ is tangent to semicircle $ RBT$; $ SRT$ is a straight line; the arcs are indicated in the figure. Angle $ APB$ is measured by:\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{2}(a - b)$", "B": "$\\frac{1}{2}(a + b)$", "C": ... | E | mcq | 4 | metric geometry - angle | |
2263 | [{"role": "user", "question": "In circle $ O$ chord $ AB$ is produced so that $ BC$ equals a radius of the circle. $ CO$ is drawn and extended to $ D$. $ AO$ is drawn. Which of the following expresses the relationship between $ x$ and $ y$?\n\n<image1>", "hint": "", "options": {"A": "$x=3y \\\\$", "B": "$x=2y \\\\$",... | A | mcq | 4 | metric geometry - angle | |
2264 | [{"role": "user", "question": "In the figure $ \\overline{AB} = \\overline{AC}$, angle $ BAD = 30^{\\circ}$, and $ \\overline{AE} = \\overline{AD}$.\n<image1>Then angle $ CDE$ equals:", "hint": "", "options": {"A": "$7\\frac{1}{2}^{\\circ}$", "B": "$10^{\\circ}$", "C": "$12\\frac{1}{2}^{\\circ}$", "D": "$15^{\\circ}$",... | D | mcq | 4 | metric geometry - angle | |
2265 | [{"role": "user", "question": "Circle $ O$ has diameters $ AB$ and $ CD$ perpendicular to each other. $ AM$ is any chord intersecting $ CD$ at $ P$. Then $ AP\\cdot AM$ is equal to:\n<image1>", "hint": "", "options": {"A": "$AO\\cdot OB$", "B": "$AO\\cdot AB$", "C": "$CP\\cdot CD$", "D": "$CP\\cdot PD$", "E": "$CO\\cdo... | B | mcq | 5 | metric geometry - length | |
2266 | [{"role": "user", "question": "In right triangle $ ABC$, $ BC = 5$, $ AC = 12$, and $ AM = x$; $ \\overline{MN} \\perp \\overline{AC}$, $ \\overline{NP} \\perp \\overline{BC}$; $ N$ is on $ AB$. If $ y = MN + NP$, one-half the perimeter of rectangle $ MCPN$, then:\n<image1>", "hint": "", "options": {"A": "$y = \\frac{1... | C | mcq | 5 | metric geometry - length | |
2267 | [{"role": "user", "question": "In triangle $ ABC$, $ AC = CD$ and $ \\angle CAB - \\angle ABC = 30^\\circ$. Then $ \\angle BAD$ is:\n<image1>", "hint": "", "options": {"A": "$30^\\circ$", "B": "$20^\\circ$", "C": "$22\\frac{1}{2}^\\circ$", "D": "$10^\\circ$", "E": "$15^\\circ$"}, "choices": []}] | E | mcq | 4 | metric geometry - angle | |
2268 | [{"role": "user", "question": "In circle $ O$, the midpoint of radius $ OX$ is $ Q$; at $ Q$, $ \\overline{AB} \\perp \\overline{XY}$. The semi-circle with $ \\overline{AB}$ as diameter intersects $ \\overline{XY}$ in $ M$. Line $ \\overline{AM}$ intersects circle $ O$ in $ C$, and line $ \\overline{BM}$ intersects cir... | A | mcq | 5 | metric geometry - length | |
2269 | [{"role": "user", "question": "Let $ ABC$ be an equilateral triangle inscribed in circle $ O$. $ M$ is a point on arc $ BC$. Lines $ \\overline{AM}$, $ \\overline{BM}$, and $ \\overline{CM}$ are drawn. Then $ AM$ is:\n<image1>", "hint": "", "options": {"A": "$\\text{equal to }{BM + CM}$", "B": "$\\text{less than }{BM +... | A | mcq | 5 | metric geometry - length | |
227 | [{"role": "user", "question": "In the figure the square has side length 2. The semi-circles pass through the midpoint of the square and have their centres on the corners of the square. The grey circles have their centres on the sides of the square and touch the semi-circles. How big is the total area of the grey parts?... | A | mcq | 5 | metric geometry - area | |
2270 | [{"role": "user", "question": "The parallel sides of a trapezoid are $ 3$ and $ 9$. The non-parallel sides are $ 4$ and $ 6$. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is:\n<image1>", "hint": "", "options": {"... | C | mcq | 5 | metric geometry - length | |
2271 | [{"role": "user", "question": "In the accompanying figure $ \\overline{CE}$ and $ \\overline{DE}$ are equal chords of a circle with center $ O$. Arc $ AB$ is a quarter-circle. Then the ratio of the area of triangle $ CED$ to the area of triangle $ AOB$ is:\n<image1>", "hint": "", "options": {"A": "$\\sqrt{2} : 1$", "B"... | E | mcq | 5 | metric geometry - area | |
2272 | [{"role": "user", "question": "In a general triangle $ ADE$ (as shown) lines $ \\overline{EB}$ and $ \\overline{EC}$ are drawn. Which of the following angle relations is true?\n<image1>", "hint": "", "options": {"A": "$x + z = a + b$", "B": "$y + z = a + b$", "C": "$m + x = w + n \\\\$", "D": "$x + z + n = w + c + m$"... | E | mcq | 4 | metric geometry - angle | |
2273 | [{"role": "user", "question": "$ ABCD$ is a rectangle (see the accompanying diagram) with $ P$ any point on $ \\overline{AB}$. $ \\overline{PS} \\perp \\overline{BD}$ and $ \\overline{PR} \\perp \\overline{AC}$. $ \\overline{AF} \\perp \\overline{BD}$ and $ \\overline{PQ} \\perp \\overline{AF}$. Then $ PR + PS$ is equa... | D | mcq | 5 | metric geometry - length | |
2274 | [{"role": "user", "question": "In this diagram a scheme is indicated for associating all the points of segment $ \\overline{AB}$ with those of segment $ \\overline{A'B'}$, and reciprocally. To described this association scheme analytically, let $ x$ be the distance from a point $ P$ on $ \\overline{AB}$ to $ D$ and let... | C | mcq | 5 | metric geometry - length | |
2275 | [{"role": "user", "question": "In this figure the center of the circle is $O$. $AB \\perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then:\n<image1>", "hint": "", "options": {"A": "$AP^2 = PB \\times AB$", "B": "$AP \\times DO = PB \\times AD$", "C": "$AB^2 = AD \\times DE$", "D"... | A | mcq | 5 | metric geometry - length | |
2276 | [{"role": "user", "question": "In this diagram $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which is inscribed equilateral triangle $DEF$. Designate angle $BFD$ by $a$, angle $ADE$ by $b$, and angle $FEC$ by $c$. Then:\n<image1>", "hint": "", "options": {"A": "$b=\\frac{a+c}{2}$", "B": "$b=\\fra... | D | mcq | 4 | metric geometry - angle | |
2277 | [{"role": "user", "question": "Chord $EF$ is the perpendicular bisector of chord $BC$, intersecting it in $M$. Between $B$ and $M$ point $U$ is taken, and $EU$ extended meets the circle in $A$. Then, for any selection of $U$, as described, triangle $EUM$ is similar to triangle:\n\n<image1>", "hint": "", "options": {"A"... | A | mcq | 5 | metric geometry - area | |
2278 | [{"role": "user", "question": "Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of triangle $CEF$ is $200$ square inches. Then the number of inches in $BE$ is:\n\n<image1>", "hint": "", "option... | 12 | numeric | 5 | metric geometry - length | |
2279 | [{"role": "user", "question": "Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 32$ and $GF = 24$, then $BE$ equals:\n\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 16 | numeric | 5 | metric geometry - length | |
228 | [{"role": "user", "question": "The chord $A B$ touches the smaller of the two concentric circles. The length $A B=$ 16. How big is the area of the grey part?\n<image1>", "hint": "", "options": {"A": "$32 \\pi$", "B": "$63 \\pi$", "C": "$64 \\pi$", "D": "$32 \\pi^{2}$", "E": "It depends on the radius of the circles."}, ... | C | mcq | 5 | metric geometry - area | |
2280 | [{"role": "user", "question": "In triangle $ABC$ lines $CE$ and $AD$ are drawn so that\n\n$\\frac{CD}{DB}=\\frac{3}{1}$ and $\\frac{AE}{EB}=\\frac{3}{2}$. Let $r=\\frac{CP}{PE}$\n\nwhere $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals:\n\n<image1>", "hint": "", "options": {"A": "$3$", "B": "$\\frac{3}{2... | D | mcq | 5 | metric geometry - length | |
2281 | [{"role": "user", "question": "In this figure $\\angle RFS = \\angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\\frac{1}{2}$ inches. The length of $RS$, in inches, is:\n\n<image1>", "hint": "", "options": {"A": "$\\text{undetermined}$", "B": "$4$", "C": "$5\\frac{1}{2}$", "D": "$6$", "E": "$6\\fra... | E | mcq | 5 | metric geometry - length | |
2282 | [{"role": "user", "question": "$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches.\n<image1>\nThen $PB$, in inches, equals:", "hint": "", "options": {"A": "$2\\sqrt{3}$", "B": "$3\\sqrt{2}$", "C": "$3\\sqrt{3}$", "D": "$4\\sqrt{2}$", "E": "$2$"}, "choices": []}] | B | mcq | 5 | metric geometry - length | |
2283 | [{"role": "user", "question": "In this figure the radius of the circle is equal to the altitude of the equilateral triangle $ABC$. The circle is made to roll along the side $AB$, remaining tangent to it at a variable point $T$ and intersecting lines $AC$ and $BC$ in variable points $M$ and $N$, respectively. Let $n$ be... | E | mcq | 4 | metric geometry - angle | |
2284 | [{"role": "user", "question": "The magnitudes of the sides of triangle $ABC$ are $a$, $b$, and $c$, as shown, with $c\\le b\\le a$. Through interior point $P$ and the vertices $A$, $B$, $C$, lines are drawn meeting the opposite sides in $A'$, $B'$, $C'$, respectively.\n<image1>\nLet $s=AA'+BB'+CC'$. Then, for all posit... | A | mcq | 5 | metric geometry - length | |
2285 | [{"role": "user", "question": "An \"n-pointed star\" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\\cdots,k,\\cdots,n$, $n\\geq 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong t... | E | mcq | 4 | metric geometry - angle | |
2286 | [{"role": "user", "question": "Triangle $ABC$ is inscribed in a circle with center $O'$. A circle with center $O$ is inscribed in triangle $ABC$. $AO$ is drawn, and extended to intersect the larger circle in $D$.\n<image1>\n Then, we must have:", "hint": "", "options": {"A": "$CD=BD=O'D$", "B": "$AO=CO=OD$", "C": "$CD... | D | mcq | 5 | metric geometry - length | |
2287 | [{"role": "user", "question": "<image1>\nIn this figure $AB$ is a diameter of a circle, centered at $O$, with radius $a$. A chord $AD$ is drawn and extended to meet the tangent to the circle at $B$ in point $C$. Point $E$ is taken on $AC$ so that $AE=DC$. Denoting the distances of $E$ from the tangent through $A$ a... | A | mcq | 5 | metric geometry - length | |
2288 | [{"role": "user", "question": "<image1>\n\nIn this diagram semi-circles are constructed on diameters $\\overline{AB}$, $\\overline{AC}$, and $\\overline{CB}$, so that they are mutually tangent. If $\\overline{CD} \\bot \\overline{AB}$, then the ratio of the shaded area to the area of a circle with $\\overline{CD}$ as ... | D | mcq | 5 | metric geometry - area | |
2289 | [{"role": "user", "question": "In this diagram, not drawn to scale, figures $\\text{I}$ and $\\text{III}$ are equilateral triangular regions with respective areas of $32\\sqrt{3}$ and $8\\sqrt{3}$ square inches. Figure $\\text{II}$ is a square region with area $32$ sq. in. Let the length of segment $AD$ be decreased by... | D | mcq | 5 | metric geometry - area | |
229 | [{"role": "user", "question": "The big equilateral triangle consists of 36 small equilateral triangles which each have an area of $1 \\mathrm{~cm}^{2}$. Determine the area of $A B C$.\n<image1>", "hint": "", "options": {"A": "$11 \\mathrm{~cm}^{2}$", "B": "$12 \\mathrm{~cm}^{2}$", "C": "$13 \\mathrm{~cm}^{2}$", "D": "$... | A | mcq | 5 | combinatorial geometry | |
2290 | [{"role": "user", "question": "In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD, O, G, H, J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF$... | D | mcq | 5 | metric geometry - area | |
2291 | [{"role": "user", "question": "<image1>\n\nA parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at a point $5$ inches from the center of $M$ is:", "hint": "", "options": {"A": "$1$", "B": "$15$", "C": "$15\\frac{1}{3}$", "D": "$15\\frac{1}{2}$", "E": "$15\\frac{3}{... | B | mcq | 5 | metric geometry - length | |
2292 | [{"role": "user", "question": "Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$ and $v=2xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is:\n<image1>\n<image2>\n<image3>\n<image4>\n<image5>", "hint": "", "opti... | D | mcq | 4 | analytic geometry | |
2293 | [{"role": "user", "question": "In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice the measure of angle $B$, and the measures of segments $AB$ and $CD$ are $a$ and $b$ respectively.\n<image1>\nThen the measure of $AB$ is equal to", "hint": "", "options": {"A": "$\\frac{1}{... | E | mcq | 4 | metric geometry - angle | |
2294 | [{"role": "user", "question": "Points $A,B,Q,D,$ and $C$ lie on the circle shown and the measures of arcs $\\widehat{BQ}$ and $\\widehat{QD}$ are $42^\\circ$ and $38^\\circ$ respectively.\n<image1>\nThe sum of the measures of angles $P$ and $Q$ is", "hint": "", "options": {"A": "$80^\\circ$", "B": "$62^\\circ$", "C": "... | C | mcq | 4 | metric geometry - angle | |
2295 | [{"role": "user", "question": "<image1>\n\nPascal's triangle is an array of positive integers(See figure), in which the first row is $1$, the second row is two $1$'s, each row begins and ends with $1$, and the $k^\\text{th}$ number in any row when it is not $1$, is the sum of the $k^\\text{th}$ and $(k-1)^\\text{th}$ n... | D | mcq | 5 | combinatorics | |
2296 | [{"role": "user", "question": "<image1>\n\nIn triangle $ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the midpoints of $BF$. The point $E$ divides side $BC$ in the ratio", "hint": "", "options": {"A": "1:4", "B": "1:3", "C": "2:5", "D": "4:... | B | mcq | 5 | metric geometry - length | |
2297 | [{"role": "user", "question": "<image1>\n\nQuadrilateral $ABCD$ is inscribed in a circle with side $AD$, a diameter of length $4$. If sides $AB$ and $BC$ each have length $1$, then side $CD$ has length", "hint": "", "options": {"A": "$\\frac{7}{2}$", "B": "$\\frac{5\\sqrt{2}}{2}$", "C": "$\\sqrt{11}$", "D": "$\\sqrt{13... | A | mcq | 5 | metric geometry - length | |
2298 | [{"role": "user", "question": "<image1>\n\nInside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$ respectively. The ... | C | mcq | 5 | metric geometry - length | |
2299 | [{"role": "user", "question": "<image1>\n\nIf the sum of the measures in degrees of angles $A,~B,~C,~D,~E$ and $F$ in the figure above is $90n$, then $n$ is equal to", "hint": "", "options": {}, "choices": []}] | 4 | numeric | 4 | metric geometry - angle | |
23 | [{"role": "user", "question": "Tom bought a chocolate heart (see the picture) to Mary on her birthday.\n<image1>\nHow many grams did the chocolate weigh, if each square weighs 10 grams?", "hint": "", "options": {}, "choices": []}] | 140 | numeric | 1 | combinatorial geometry | |
230 | [{"role": "user", "question": "Which of the following graphs represents the solution set of $(x-|x|)^{2}+(y-|y|)^{2}=4$?\n<image1>", "hint": "", "options": {"A": "A", "B": "B", "C": "C", "D": "D", "E": "E"}, "choices": []}] | A | mcq | 4 | analytic geometry | |
2300 | [{"role": "user", "question": "<image1>\n\nThe radius of the smallest circle containing the symmetric figure composed of the $3$ unit squares shown above is", "hint": "", "options": {"A": "$\\sqrt{2}$", "B": "$\\sqrt{1.25}$", "C": "$1.25$", "D": "$\\frac{5\\sqrt{17}}{16}$", "E": "$\\text{None of these}$"}, "choices": [... | D | mcq | 5 | metric geometry - length | |
2301 | [{"role": "user", "question": "<image1>\n\nIn the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to", "hint": "", "options": {"A": "3x+2", "B": "3x+1", "C"... | E | mcq | 5 | metric geometry - length | |
2302 | [{"role": "user", "question": "<image1>\n\nA rectangular piece of paper $6$ inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease $L$ in terms of angle $\\theta$ is", "hint": "", "options": {"A": "$3\\sec ^2\\theta\\csc\\theta$", "B": "$6\\sin\\theta\\s... | A | mcq | 5 | transformation geometry | |
2303 | [{"role": "user", "question": "<image1>\n\nChords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is", "hint": "", "options": {"A": "$4\\sqrt{5}$", "B": "$... | B | mcq | 5 | metric geometry - length | |
2304 | [{"role": "user", "question": "<image1>\n\nEquilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ retur... | D | mcq | 5 | metric geometry - length | |
2305 | [{"role": "user", "question": "A circle with a circumscribed and an inscribed square centered at the origin $ O$ of a rectangular coordinate system with positive $ x$ and $ y$ axes $ OX$ and $ OY$ is shown in each figure $ I$ to $ IV$ below.\n<image1>\n\nThe inequalities\n\\[ |x| + |y| \\leq \\sqrt{2(x^2 + y^2)} \\leq ... | B | mcq | 4 | analytic geometry | |
2306 | [{"role": "user", "question": "In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long.\n<image1>\nOf the three equations\n\\[ \\textbf{I.}\\ d-s=1, \\qquad \\... | E | mcq | 5 | metric geometry - length | |
2307 | [{"role": "user", "question": "In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is\n<image1>", "hint": "", "options": {"A": "$2\\sqrt{3}-3$", "B": "$1-\\frac{\\sqrt{3}}{3}$", "C": "$\\frac{\\sqrt{3}}{4}$", ... | A | mcq | 5 | metric geometry - area | |
2308 | [{"role": "user", "question": "In the adjoining figure $ TP$ and $ T'Q$ are parallel tangents to a circle of radius $ r$, with $ T$ and $ T'$ the points of tangency. $ PT''Q$ is a third tangent with $ T''$ as point of tangency. If $ TP=4$ and $ T'Q=9$ then $ r$ is\n\n<image1>", "hint": "", "options": {"A": "$25/6$", "B... | B | mcq | 5 | metric geometry - length | |
2309 | [{"role": "user", "question": "In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogra... | C | mcq | 5 | metric geometry - area | |
231 | [{"role": "user", "question": "A strip of paper is folded three times as shown. Determine $\\beta$ if $\\alpha=70^{\\circ}$.\n<image1>", "hint": "", "options": {"A": "$140^{\\circ}$", "B": "$130^{\\circ}$", "C": "$120^{\\circ}$", "D": "$110^{\\circ}$", "E": "$100^{\\circ}$"}, "choices": []}] | C | mcq | 5 | transformation geometry | |
2310 | [{"role": "user", "question": "In the adjoining figure triangle $ ABC$ is such that $ AB = 4$ and $ AC = 8$. If $ M$ is the midpoint of $ BC$ and $ AM = 3$, what is the length of $ BC$?\n<image1>", "hint": "", "options": {"A": "$2\\sqrt{26}$", "B": "$2\\sqrt{31}$", "C": "$9$", "D": "$4+2\\sqrt{13}$", "E": "$\\text{not... | B | mcq | 5 | metric geometry - length | |
2311 | [{"role": "user", "question": "In the adjoining figure $AB$ and $BC$ are adjacent sides of square $ABCD$; $M$ is the midpoint of $AB$; $N$ is the midpoint of $BC$; and $AN$ and $CM$ intersect at $O$. The ratio of the area of $AOCD$ to the area of $ABCD$ is\n<image1>", "hint": "", "options": {"A": "$\\frac{5}{6}$", "B":... | C | mcq | 5 | metric geometry - area | |
2312 | [{"role": "user", "question": "In triangle $ABC$ shown in the adjoining figure, $M$ is the midpoint of side $BC$, $AB=12$ and $AC=16$. Points $E$ and $F$ are taken on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE=2AF$ then $\\frac{EG}{GF}$ equals\n<image1>", "hint": "", "options": {"A": ... | A | mcq | 5 | metric geometry - length | |
2313 | [{"role": "user", "question": "<image1>\n\nIn the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC=DC=3$, $OD=2$, and $AB=6$, then the radius of the circle is", "hint": "", "options": {"A": "$3+\\sqrt{3}$", "B": "$... | E | mcq | 5 | metric geometry - length | |
2314 | [{"role": "user", "question": "<image1>\n\nIn the adjoining figure, circle $\\mathit{K}$ has diameter $\\mathit{AB}$; cirlce $\\mathit{L}$ is tangent to circle $\\mathit{K}$ and to $\\mathit{AB}$ at the center of circle $\\mathit{K}$; and circle $\\mathit{M}$ tangent to circle $\\mathit{K}$, to circle $\\mathit{L}$ and... | C | mcq | 5 | metric geometry - area | |
2315 | [{"role": "user", "question": "<image1>\n\nIn the adjoining figure, every point of circle $\\mathit{O'}$ is exterior to circle $\\mathit{O}$. Let $\\mathit{P}$ and $\\mathit{Q}$ be the points of intersection of an internal common tangent with the two external common tangents. Then the length of $PQ$ is", "hint": "", "o... | C | mcq | 5 | metric geometry - length | |
2316 | [{"role": "user", "question": "<image1>\n\nIn triangle $ABC$, $AB=AC$ and $\\measuredangle A=80^\\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\\measuredangle EDF$ equals", "hint": "", "options": {"A": "$30^\\circ$", "B": "$40^\\circ$", "C": "$50^\\ci... | C | mcq | 4 | metric geometry - angle | |
2317 | [{"role": "user", "question": "<image1>\n\nIn the adjoining figure $\\measuredangle E=40^\\circ$ and arc $AB$, arc $BC$, and arc $CD$ all have equal length. Find the measure of $\\measuredangle ACD$.", "hint": "", "options": {"A": "$10^\\circ$", "B": "$15^\\circ$", "C": "$20^\\circ$", "D": "$\\left(\\frac{45}{2}\\right... | B | mcq | 4 | metric geometry - angle | |
2318 | [{"role": "user", "question": "<image1>\n\nEach of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is", "hint": "", "options": {"A": "$36+9\\sqrt{2}$", "B... | D | mcq | 5 | metric geometry - length | |
2319 | [{"role": "user", "question": "<image1>\n\nIf $a,b,$ and $d$ are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon (see adjoining figure), then", "hint": "", "options": {"A": "$d=a+b$", "B": "$d^2=a^2+b^2$", "C": "$d^2=a^2+ab+b^2$", "D": "$b=\\frac{a+d}{2}$", "E": "$b... | A | mcq | 5 | metric geometry - length | |
232 | [{"role": "user", "question": "A barcode as pictured is made up of alternate black and white stripes. The code always starts and ends with a black stripee. Each strip (black or white) has the width 1 or 2 and the total width of the barcode is 12. How many different barcodes if this kind are there if one reads from left... | 116 | numeric | 5 | combinatorics | |
2320 | [{"role": "user", "question": "The following four statements, and only these are found on a card:\n<image1>\n(Assume each statement is either true or false.) Among them the number of false statements is exactly", "hint": "", "options": {}, "choices": []}] | 3 | numeric | 1 | logic | |
2321 | [{"role": "user", "question": "<image1>\n\nVertex $E$ of equilateral triangle $ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\\sqrt{1+\\sqrt{3}}$ then the area of $\\triangle ABF$ is", "hint": "", "options": {"A": "$1$", "B": "$... | C | mcq | 5 | metric geometry - area | |
2322 | [{"role": "user", "question": "<image1>\nIn $\\triangle ABC$, $AB = 10~ AC = 8$ and $BC = 6$. Circle $P$ is the circle with smallest radius which passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$ , of circle $P$ with sides $AC$ and $BC$, respectively. The length... | B | mcq | 5 | metric geometry - length | |
2323 | [{"role": "user", "question": "<image1>\n\nIf $\\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, then the measure of $\\measuredangle A_{44}A_{45}A_{43}$ equals", "hint": "", "options": {"A": "$30^\\circ$", "B": "$45^\\circ$", "C": "$60^\\circ... | E | mcq | 4 | metric geometry - angle | |
2324 | [{"role": "user", "question": "<image1>\n\nIf rectangle $ABCD$ has area $72$ square meters and $E$ and $G$ are the midpoints of sides $AD$ and $CD$, respectively, then the area of rectangle $DEFG$ in square meters is", "hint": "", "options": {}, "choices": []}] | 18 | numeric | 5 | metric geometry - area | |
2325 | [{"role": "user", "question": "<image1>\n\nIn the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD$. What is the measure of $\\measuredangle AED$ in degrees?", "hint": "", "options": {}, "choices": []}] | 15 | numeric | 4 | metric geometry - angle | |
2326 | [{"role": "user", "question": "<image1>\n\nIn the adjoining figure, $CD$ is the diameter of a semi-circle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semi-circle, and $B$ is the point of intersection (distinct from $E$ ) of line segment $AE$ with the semi-circle. If length $... | B | mcq | 4 | metric geometry - angle | |
2327 | [{"role": "user", "question": "<image1>\n\nPoints $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$ , respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, so that points $A$ and $... | C | mcq | 5 | transformation geometry | |
2328 | [{"role": "user", "question": "The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.\n<image1>", "hint": "", "options": {"A": "$\\frac{1}{2}$", "B": "$\\frac{3... | C | mcq | 3 | solid geometry | |
2329 | [{"role": "user", "question": "Circles with centers $A ,~ B,$ and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$.\n<image1>\nIf $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A... | D | mcq | 5 | metric geometry - length | |
233 | [{"role": "user", "question": "The picture on the right shows a tile pattern. The side length of the bigger tiles is a and of the smaller ones b. The dotted lines (horizontal and tilted) include an angle of $30^{\\circ}$. How big is the ratio a:b?\n<image1>", "hint": "", "options": {"A": "$(2 \\cdot \\sqrt{3}): 1$", "B... | B | mcq | 4 | metric geometry - angle | |
2330 | [{"role": "user", "question": "<image1>\n\nIn $\\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ and $\\measuredangle BAC = 60^\\circ$, $\\measuredangle ABC = 100^\\circ$, $\\measuredangle ACB = 20^\\circ$ and $\\measuredangle DEC = 80^\\circ$, then the area of $\\t... | B | mcq | 5 | metric geometry - area | |
2331 | [{"role": "user", "question": "In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares.\n<image1>\nThe measure of $\\angle GDA$ is", "hint": "", "options": {"A": "$90^\\circ$", "B": "$105^\\circ$", "C": "$120^\\circ$", "D": "$135^\\circ$", "E": "$150^\\circ$"}, "choices": []}] | C | mcq | 4 | metric geometry - angle | |
2332 | [{"role": "user", "question": "If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\\overline{AQ}$, and $\\measuredangle QPC = 60^\\circ$, then the length of $PQ$ divided by the length of $AQ$ is\n<image1>", "hint": "", "options": {"A": "$\\frac{\\sqrt{3}}{2}$", "B": "$\\frac{\\sqrt{3}}{3}$", "C": "$\\f... | B | mcq | 5 | metric geometry - length | |
2333 | [{"role": "user", "question": "Sides $AB,BC,CD$ and $DA$ of convex polygon $ABCD$ have lengths 3,4,12, and 13, respectively, and $\\measuredangle CBA$ is a right angle. The area of the quadrilateral is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 36 | numeric | 5 | metric geometry - area | |
2334 | [{"role": "user", "question": "In triangle $ABC$, $\\measuredangle CBA=72^\\circ$, $E$ is the midpoint of side $AC$, and $D$ is a point on side $BC$ such that $2BD=DC$; $AD$ and $BE$ intersect at $F$. The ratio of the area of triangle $BDF$ to the area of quadrilateral $FDCE$ is\n\n<image1>", "hint": "", "options": {"A... | A | mcq | 5 | metric geometry - area | |
2335 | [{"role": "user", "question": "In $\\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\\angle BAC$, $BN\\perp AN$ and $\\theta$ is the measure of $\\angle BAC$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then length $MN$ equals\n\n<image1>", "hint": "", "options": {"A": "$2$", "B": ... | B | mcq | 5 | metric geometry - length | |
2336 | [{"role": "user", "question": "<image1>\n\nEquilateral $ \\triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $ T$ and tangent to sides $ AB$ and $ AC$ at points $ P$ and $ Q$. If side $ BC$ has length $ 12$, then segment $ PQ$ has length", "hint": "", "options": {"A": ... | C | mcq | 5 | metric geometry - length | |
2337 | [{"role": "user", "question": "<image1>\n\nIn triangle $ ABC$ in the adjoining figure, $ AD$ and $ AE$ trisect $ \\angle BAC$. The lengths of $ BD$, $ DE$ and $ EC$ are $ 2$, $ 3$, and $ 6$, respectively. The length of the shortest side of $ \\triangle ABC$ is", "hint": "", "options": {"A": "$2\\sqrt{10}$", "B": "$11$"... | A | mcq | 5 | metric geometry - length | |
2338 | [{"role": "user", "question": "In the adjoining figure triangle $ ABC$ is inscribed in a circle. Point $ D$ lies on $ \\stackrel{\\frown}{AC}$ with $ \\stackrel{\\frown}{DC} = 30^\\circ$, and point $ G$ lies on $ \\stackrel{\\frown}{BA}$ with $ \\stackrel{\\frown}{BG}\\, > \\, \\stackrel{\\frown}{GA}$. Side $ AB$ and s... | C | mcq | 5 | metric geometry - area | |
2339 | [{"role": "user", "question": "In the adjoining diagram, $BO$ bisects $\\angle CBA$, $CO$ bisects $\\angle ACB$, and $MN$ is parallel to $BC$. If $AB=12$, $BC=24$, and $AC=18$, then the perimeter of $\\triangle AMN$ is\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 30 | numeric | 5 | metric geometry - length | |
234 | [{"role": "user", "question": "In the picture on the right a number should be written next to each point. The sum of the numbers on the corners of each side of the hexagon should be equal. Two numbers have already been inserted. Which number should be in the place marked '$x$'?\n<image1>", "hint": "", "options": {}, "c... | 1 | numeric | 5 | algebra | |
2340 | [{"role": "user", "question": "In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB$, $BC$, and $CD$ are diameters of circle $O$, $N$, and $P$, respectively. Circles $O$, $N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points... | C | mcq | 5 | metric geometry - length | |
2341 | [{"role": "user", "question": "In the adjoining figure of a rectangular solid, $\\angle DHG=45^\\circ$ and $\\angle FHB=60^\\circ$. Find the cosine of $\\angle BHD$.\n\n<image1>", "hint": "", "options": {"A": "$\\frac{\\sqrt{3}}{6}$", "B": "$\\frac{\\sqrt{2}}{6}$", "C": "$\\frac{\\sqrt{6}}{3}$", "D": "$\\frac{\\sqrt{6}... | D | mcq | 3 | solid geometry | |
2342 | [{"role": "user", "question": "In the adjoining figure, the triangle $ABC$ is a right triangle with $\\angle BCA=90^\\circ$. Median $CM$ is perpendicular to median $BN$, and side $BC=s$. The length of $BN$ is\n\n<image1>", "hint": "", "options": {"A": "$s\\sqrt{2}$", "B": "$\\frac{3}{2}s\\sqrt{2}$", "C": "$2s\\sqrt{2}$... | E | mcq | 5 | metric geometry - length | |
2343 | [{"role": "user", "question": "In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If $AG=2$, $GF=13$, $FC=1$, and $HJ=7$, then $DE$ equals\n\n<image1>", "hint": "", "options": {"A": "$2\\sqrt{22}$", "B": "$7\\sqrt{3}$", "C": "$9$", "D": "$10$", "E": "$13$"}, "choices": []}] | A | mcq | 5 | metric geometry - length | |
2344 | [{"role": "user", "question": "The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. For variety, at each intersection where he has... | D | mcq | 5 | combinatorics | |
2345 | [{"role": "user", "question": "In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, and sides $BC$ and $ED$. Each side has length of 1. Also, $\\measuredangle FAB = \\measuredangle BCD = 60^\\circ$. The area of the figure is\n\n<image1>", "hint": "", "options": {"A": "$\\frac{\\... | D | mcq | 5 | metric geometry - area | |
2346 | [{"role": "user", "question": "In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$ ($L_1$ is the line that is above the circles and $L_2$ is the line that goes under the circles). If the radius of the largest circle is 18 and that of the smallest one is 8, ... | 12 | numeric | 5 | metric geometry - length | |
2347 | [{"role": "user", "question": "Triangle $\\triangle ABC$ in the figure has area $10$. Points $D$, $E$ and $F$, all distinct from $A$, $B$ and $C$, are on sides $AB$, $BC$ and $CA$ respectively, and $AD = 2$, $DB = 3$. If triangle $\\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is\n\n<image1... | C | mcq | 5 | metric geometry - area | |
2348 | [{"role": "user", "question": "Distinct points $A$ and $B$ are on a semicircle with diameter $MN$ and center $C$. The point $P$ is on $CN$ and $\\angle CAP = \\angle CBP = 10^{\\circ}$. If $\\stackrel{\\frown}{MA} = 40^{\\circ}$, then $\\stackrel{\\frown}{BN}$ equals\n\n<image1>", "hint": "", "options": {"A": "$10^{\... | C | mcq | 4 | metric geometry - angle |
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