cheryyunl/rover_test · Datasets at Hugging Face

id stringlengths 9 19	reasoning_image imagewidth (px) 200 4.26k ⌀	example_image imagewidth (px) 2.19k 2.19k ⌀	image2 imagewidth (px) 937 1.95k ⌀	prompt stringlengths 57 928	problem_type stringclasses 5 values	answer stringlengths 1 176
logical_111		Not supported with pagination yet	Not supported with pagination yet	As shown, two concentric circles have center $O$. In the larger circle, chord $AC$ is tangent to the smaller circle at $B$. Through $B$, draw the line perpendicular to $AO$, meeting the larger circle at $E$ and $F$, with foot $G$ on $AO$. Segment $CF$ is a diameter of the larger circle. (1) Prove that $AB=BC$. (2) Find the ratio $AE:AB:BE$.	logical	<1>AB=BC</1><2>6:3:1</2>
physical_56			Not supported with pagination yet	Given this robotics scene (image resolution: 1752x1138 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Put the coke next to the sprite into the white box with the right arm Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[1127, 332], [994, 411], [896, 439], [850, 553], [868, 697], [827, 537], [961, 561], [1034, 684], [1075, 775], [1007, 952]]
logical_44		Not supported with pagination yet	Not supported with pagination yet	In the figure, $\triangle ABC$ is an isosceles right triangle with right angle at $B$. Point $P$ is outside $\triangle ABC$. Segment $BP$ is rotated clockwise by $90^\circ$ about $B$ to $BP'$. It is given that $\angle AP'B=135^\circ$ and $P'A:P'C=1:3$. Find the value of $PB:P'A$.	logical	1:2
perception_next_44	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	right
physical_84		Not supported with pagination yet	Not supported with pagination yet	In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which container will catch the most liquid? (Choose from: purple container, red container)	physical	red container
physical_14			Not supported with pagination yet	Given this robotics scene (image resolution: 2258x1442 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Turn the teapot handle to the left Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[1081, 473], [1107, 546], [1119, 633], [1081, 706], [1034, 759], [959, 821], [890, 849], [833, 890], [804, 935], [753, 1003]]
physical_79		Not supported with pagination yet	Not supported with pagination yet	In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Will both the brown container and the purple container catch falling liquids? (Answer: Yes/No)	physical	Yes
logical_78		Not supported with pagination yet	Not supported with pagination yet	In triangle $ABC$, $BP$ is the angle bisector of $\angle B$, and $AP$ is perpendicular to $BP$ at $P$. The area of $\triangle PBC$ is $12$, and the area of $\triangle APB$ is twice the area of $\triangle APC$. Find the area of $\triangle APB$.	logical	4
physical_17			Not supported with pagination yet	Given this robotics scene (image resolution: 2560x1336 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Insert the flowers into the white vase Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[1590, 850], [1606, 919], [1636, 1007], [1524, 906], [1489, 822], [1411, 751], [1327, 714], [1258, 697], [1146, 696], [1022, 695]]
logical_106		Not supported with pagination yet	Not supported with pagination yet	In right triangle $ABC$, $\angle C=90^\circ$, $\angle A=30^\circ$. Point $O$ lies on hypotenuse $AB$. The circle with center $O$ and radius $2$ passes through $B$, is tangent to $AC$ at $D$, and intersects $BC$ at $E$. Find the area of the shaded region enclosed by segments $CD$, $CE$, and $DE$.\n	logical	\frac{3\sqrt{3}}{2}-\frac{2\pi}{3}
physical_73		Not supported with pagination yet	Not supported with pagination yet	In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which containers will catch the falling liquids? (List all from: brown container, white container, yellow container)	physical	brown container, white container
logical_49		Not supported with pagination yet	Not supported with pagination yet	In right triangle $ABC$, $ riangle ABC$ has $ riangle C=90^\\circ$, $ riangle A=30^\\circ$, and $BC=2$. A circle centered at $C$ has radius $1$. Point $P$ lies on hypotenuse $AB$. Through $P$, draw a tangent $PQ$ to the circle (with $Q$ the point of tangency). What is the minimum value of $PQ$?	logical	2
logical_66		Not supported with pagination yet	Not supported with pagination yet	As shown, points B, C, D lie on circle O. Through C, draw line AC parallel to BD, meeting the extension of OB at A. Connect CD. Given $\angle CDB = \angle OBD = 30^\circ$ and $DB = 6\sqrt{3}\,\text{cm}$. (1) What is the positional relationship between line AC and circle O? Explain why. (2) Find the area of the shaded region bounded by chord CD, chord BD, and arc BC. (Leave the result in terms of $\pi$.)	logical	<1>tangent</1><2>6\pi</2>
logical_131		Not supported with pagination yet	Not supported with pagination yet	In the figure, $AB$ is the diameter of semicircle $O$. Lines $AD$ and $BC$ are tangent to circle $O$ at $A$ and $B$, respectively. Line $CD$ is tangent to circle $O$ at $E$. Lines $AD$ and $CD$ intersect at $D$, and lines $BC$ and $CD$ intersect at $C$. Connect $OD$ and $OC$. Among the following statements, which are correct? ① $OD^{2}=DE\cdot CD$; ② $AD+BC=CD$; ③ $OD=OC$; ④ $S_{\text{trapezoid }ABCD}=\dfrac{1}{2}\,CD\cdot OA$; ⑤ $\angle DOC=90^\circ$. Fill in the indices of all correct statements.	logical	①②⑤
perception_61		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	A
perception_next_80	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	right
physical_47			Not supported with pagination yet	Given this robotics scene (image resolution: 2496x1334 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Move apple to "Denver Nuggets" Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[1950, 449], [1799, 447], [1618, 476], [1481, 523], [1357, 591], [1264, 719], [1479, 719], [1652, 777], [1775, 828], [1874, 896]]
perception_67		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	B
perception_next_91	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	right
perception_84		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	B
logical_135		Not supported with pagination yet	Not supported with pagination yet	As shown, C is a moving point on ray OE. AB is a chord through C. Line DA intersects OE at D. Consider the following three statements: (1) DA is a tangent to circle O. (2) DA = DC. (3) OD is perpendicular to OB. Using any two as conditions and the remaining one as the conclusion, write a true implication in the form “○○$\Rightarrow$○”, and provide a proof.	logical	①②⇒③
logical_93		Not supported with pagination yet	Not supported with pagination yet	In the figure, P is a point outside circle O. The tangents PA and PB touch circle O at A and B, respectively. Given $PA=4\text{ cm}$ and $\angle P=40^\circ$. Point C is any point on segment AB. Through C, draw the tangent to circle O, which meets PA and PB at D and E, respectively. Find: (1) the perimeter of $\triangle PDE$; (2) the measure of $\angle DOE$.	logical	<1>8</1><2>70^\circ</2>
physical_108		Not supported with pagination yet	Not supported with pagination yet	In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which containers will catch the falling liquids? (List all from: cyan container, white container, blue container)	physical	cyan container
physical_67		Not supported with pagination yet	Not supported with pagination yet	How many cubes on the tabletop will the car collide with? List them.	physical	red cube, yellow cube, purple cube, blue cube
perception_next_105	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	left
logical_90		Not supported with pagination yet	Not supported with pagination yet	As shown, a circle with center $A$ and radius $1.2$ moves so that $A$ lies on the line $y=\tfrac{3}{4}x-3$. Let $OP$ be a tangent from the origin $O$ to the circle, with point of tangency $P$. What is the minimum possible length of $OP$?	logical	\frac{63}{25}
perception_89		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	C
physical_70		Not supported with pagination yet	Not supported with pagination yet	In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which containers will catch the falling liquids? (List all from: black container, yellow container, brown container)	physical	black container, brown container
physical_60			Not supported with pagination yet	Given this robotics scene (image resolution: 2124x1142 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Pick up the apple to the left of the orange and place it on the first shelf of the cabinet Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[722, 420], [542, 492], [408, 554], [434, 690], [461, 850], [506, 647], [701, 578], [911, 549], [1222, 577], [1431, 624]]
perception_38		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	C
perception_next_87	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	right
logical_140		Not supported with pagination yet	Not supported with pagination yet	In triangle $ABC$, $AB=AC=4$ and $\tan B=\dfrac{\sqrt{7}}{3}$. Point $D$ lies on $BC$ with $CD=1$. Fold $\triangle ABD$ along line $AD$ to obtain $\triangle AED$, where point $B$ maps to $E$. Line $DE$ meets side $AC$ at $F$. Find the length of $EF$.	logical	\frac{64}{15}
logical_102		Not supported with pagination yet	Not supported with pagination yet	In the circle with center $O$, $AB$ is a diameter, and points $C$ and $D$ lie on the circle. Given $\angle CAB=40^\circ$, find the measure of $\angle D$ (the inscribed angle at point $D$ that subtends arc $AB$).	logical	50
logical_122		Not supported with pagination yet	Not supported with pagination yet	In right triangle $ABC$, $\angle ACB=90^{\circ}$ and $\angle A=30^{\circ}$. Point $D$ lies on $AB$. Circle $O$ has diameter $BD$ and is tangent to $AC$ at $E$. Extend $DE$ to meet the extension of $BC$ at $F$. (1) Prove that $\triangle BDF$ is equilateral. (2) Connect $AF$ and $DC$. Given $BC=3$, outline a method to find the area of quadrilateral $AFCD$.	logical	<1>Equilateral</1><2>Use AB=2BC and AC=\sqrt{3}\,BC to get AB, AC; from AO=2OE deduce AD, DB, DH; from BF=BD get CF; then area(AFCD)=area(\triangle AFC)+area(\triangle DHC)</2>
perception_57		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	A
perception_50		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	B
perception_next_23	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	right
physical_82		Not supported with pagination yet	Not supported with pagination yet	In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which container will catch the most liquid? (Choose from: purple container, pink container, grey container, brown container)	physical	purple container
logical_126		Not supported with pagination yet	Not supported with pagination yet	In the figure, circle $O$ has radius $OC$ perpendicular to diameter $AB$. Point $P$ moves on segment $OB$ (excluding $O$ and $B$). The extension of $CP$ meets the circle again at $D$. On the extension of $OB$ beyond $B$, take point $E$ so that $ED=EP$. (1) Show that $ED$ is a tangent to circle $O$. (2) When $OC=2$ and $ED=2\sqrt{3}$, find $\tan\angle E$ and the area $S$ of the shaded region (leave irrational numbers in exact form).	logical	<1>\frac{\sqrt{3}}{3}</1><2>2\sqrt{3}-\frac{2\pi}{3}</2>
logical_80		Not supported with pagination yet	Not supported with pagination yet	Given circle $O$ with diameter $AB$. Point $C$ lies on circle $O$ (distinct from $A,B$). Extend $BC$ beyond $C$ to point $D$ such that $BC=CD$. Line $DA$ meets the circle again at $E$. Through $C$, draw the tangent to circle $O$, which meets $AD$ at $F$. (I) If $\angle DBA=60^{\circ}$, show that $E$ is the midpoint of $AD$. (II) If $CF=\dfrac{1}{2}R$, where $R$ is the radius of circle $O$, find $\angle DBA$.	logical	<1>E is the midpoint of AD</1><2>60^\circ</2>
physical_11			Not supported with pagination yet	Given this robotics scene (image resolution: 2208x1432 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Place the orange on the white plate Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[1652, 374], [1737, 506], [1791, 588], [1824, 729], [1849, 846], [1865, 958], [1585, 852], [1406, 851], [1230, 881], [1081, 1051]]
logical_147		Not supported with pagination yet	Not supported with pagination yet	In the figure, AB is the diameter of semicircle O. Points C and D divide the semicircle into three equal arcs. If chord $CD=6$, find the area of the shaded region.	logical	6\pi
logical_5		Not supported with pagination yet	Not supported with pagination yet	Two circles are centered at the origin, as shown. The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle. If $QR=3$, what is the value of $k$?	logical	7
perception_next_111	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	left
physical_46			Not supported with pagination yet	Given this robotics scene (image resolution: 2550x1276 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Move banana to "Germany" Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[2235, 768], [2138, 842], [2045, 926], [1985, 984], [1804, 1047], [1824, 957], [1909, 874], [1870, 798], [1806, 787], [1712, 745]]
logical_75		Not supported with pagination yet	Not supported with pagination yet	In rhombus $ABCD$, point $E$ lies on diagonal $AC$ and $DE\perp BE$. (1) Are triangles $\triangle ADE$ and $\triangle ABE$ congruent? (2) If $\angle DAB=60^\circ$ and $AD=2\sqrt{3}$, find the length of $DE$.	logical	<1>Yes</1><2>3</2>
physical_69		Not supported with pagination yet	Not supported with pagination yet	A playground contains obstacles of various colors and poses, along with randomly arranged pits. Soft balls are launched from various initial positions. These balls move dynamically, bouncing and undergoing permanent deformation, potentially colliding with obstacles and falling into pits. Will the brown ball finally drop into the right pit? A. Yes B. No C. Can not answer D. Uncertain.	physical	B
physical_23			Not supported with pagination yet	Given this robotics scene (image resolution: 1460x1080 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Open drawer with distractor on the top Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[695, 550], [698, 593], [702, 654], [707, 712], [728, 751], [741, 798], [806, 809], [862, 817], [920, 833], [982, 860]]
physical_86		Not supported with pagination yet	Not supported with pagination yet	In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which containers will catch the falling liquids? (List all from: purple container, green container, pink container)	physical	green container
perception_70		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	B
physical_6			Not supported with pagination yet	Given this robotics scene (image resolution: 2236x1394 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Stack the pink block on the blue block Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[1394, 383], [1448, 456], [1472, 550], [1467, 724], [1375, 571], [1264, 519], [1173, 510], [1103, 524], [1069, 606], [1074, 728]]
logical_107		Not supported with pagination yet	Not supported with pagination yet	In the figure, $AB$ is a diameter of circle $O$. Line $CD$ is tangent to circle $O$ at $D$. The extension of $AB$ meets $CD$ at $C$. If $\angle CAD=32^\circ$, find $\angle C$.	logical	26^\circ
perception_100		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	B
perception_next_72	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	right
logical_76		Not supported with pagination yet	Not supported with pagination yet	In the circle with center $O$ and radius $5$, triangle $ABC$ is inscribed with $AB=AC=4$. Let $BD$ be a diameter of the circle. Find the area of quadrilateral $ABCD$.	logical	44\!/5
logical_63		Not supported with pagination yet	Not supported with pagination yet	In triangle $ABC$, $\angle C=90^\circ$. Point $D$ is the midpoint of $BC$, and $AD=BC$. Find $\sin\angle BAD$.	logical	3/2
perception_4		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	B
logical_86		Not supported with pagination yet	Not supported with pagination yet	In the figure, $AB$ is a diameter of circle $O$. Point $C$ lies on the circle. The tangent to circle $O$ at $C$ meets the extension of $AB$ at $D$, and $\angle D = 30^\circ$. (1) Find $\angle A$. (2) Through $C$, draw $CF \perp AB$, with foot $E$ on $AB$, and $CF$ meets the circle again at $F$. Given $CF = \dfrac{4}{3}$, find the length of arc $BC$. (Leave the answer in terms of $\pi$.)	logical	<1>30</1><2>\frac{4\pi}{3}</2>
perception_22		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	D
perception_72		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	D
perception_69		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	C
perception_49		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	C
physical_80		Not supported with pagination yet	Not supported with pagination yet	In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which container will catch the most liquid? (Choose from: black container, brown container)	physical	brown container
logical_9		Not supported with pagination yet	Not supported with pagination yet	Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. A circle is tangent to the sides $AB$ and $AC$ at $X$ and $Y$ respectively, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on the side $BC$. Given that $AB = 6$, find the area of the portion of the circle that lies outside the triangle.	logical	$\pi - 2$
perception_1		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	B
perception_46		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	D
physical_99		Not supported with pagination yet	Not supported with pagination yet	In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which container will catch the most liquid? (Choose from: pink container, cyan container, blue container)	physical	blue container
physical_83		Not supported with pagination yet	Not supported with pagination yet	In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which container will catch the most liquid? (Choose from: cyan container, brown container, orange container)	physical	cyan container
logical_17		Not supported with pagination yet	Not supported with pagination yet	In the adjoining figure, , $CD$ is the diameter of a semicircle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semicircle, and $B$ is the point of intersection (distinct from $E$) of line segment $AE$ with the semicircle. If length $AB$ equals length $OD$, and the measure of $\angle EOD$ is $45^\circ$, then find the measure of $\angle BAO$, in degrees.	logical	$15^\circ$
physical_49			Not supported with pagination yet	Given this robotics scene (image resolution: 2434x1262 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Move the blue objects together Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[1474, 445], [1474, 545], [1467, 662], [1393, 759], [1307, 826], [1180, 897], [1227, 776], [1160, 702], [1046, 657], [908, 682]]
logical_26		Not supported with pagination yet	Not supported with pagination yet	In triangle $ABC,$ the side lengths are $a = 7,$ $b = 9,$ and $c = 4.$ The incenter $I$ is shown in the diagram below. The position vector of the incenter can be expressed as a weighted average of the position vectors of the vertices: \[\overrightarrow{I} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C},\] where $x, y,$ and $z$ are constants such that $x + y + z = 1.$ Find the ordered triple $(x,y,z).$	logical	$\left( \frac{7}{20}, \frac{9}{20}, \frac{1}{5} \right)$
perception_next_85	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	right
physical_53			Not supported with pagination yet	Given this robotics scene (image resolution: 1998x1146 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Place land animal onto plate with left robot arm Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[622, 614], [735, 706], [841, 754], [960, 787], [1039, 885], [1041, 761], [1152, 686], [1299, 809], [1428, 873], [1437, 1026]]
physical_105	Not supported with pagination yet	Not supported with pagination yet	Not supported with pagination yet	An array of both movable and fixed pulleys, along with anchor points, is arranged on a wall. Ropes are connected to pulleys, loads, or anchor points and can be wound around the pulleys. These loads, with varying masses, interact with other forces, creating distinct motion patterns. The model's primary objective is to identify tension distributions within this rope system. Additionally, it aims to recognize correlations or constraints among moving objects, such as the coordinated movement of loads and the rotation of pulleys on a single rope. Furthermore, the model is expected to infer numerical relationships between the loads' masses. If we want the gray sphere to move up, what can we do? (Select all answers that satisfy the conditions.) A. Increase the mass of the yellow sphere B. Decrease the mass of the yellow sphere C. Increase the mass of the green cube D. Decrease the mass of the gray sphere	physical	C, D
perception_34		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	D
logical_47		Not supported with pagination yet	Not supported with pagination yet	In isosceles triangle $ABC$ with $AB=AC$, $AD$ is the altitude to base $BC$. Given $AD:DC=1:3$. Rotate triangle $ADC$ about point $D$ to obtain triangle $DEF$ such that points $A$ and $C$ correspond to $E$ and $F$ respectively, and $EF$ coincides with line $AB$. Let $AC$ and $DF$ intersect at $O$. Find the ratio $S_{\triangle AOE}:S_{\triangle DOE}$.	logical	\frac{32}{45}
physical_91		Not supported with pagination yet	Not supported with pagination yet	A playground contains obstacles of various colors and poses, along with randomly arranged pits. Soft balls are launched from various initial positions. These balls move dynamically, bouncing and undergoing permanent deformation, potentially colliding with obstacles and falling into pits. Where will the orange ball, green ball, and black ball finally drop?	physical	Orange ball: left pit; Green ball: right pit; Black ball: right pit
logical_31		Not supported with pagination yet	Not supported with pagination yet	Starting from the green dot, find the path to point C. List the points you pass through in order (not including C).	logical	Point E.
logical_11		Not supported with pagination yet	Not supported with pagination yet	The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.	logical	$\frac{5\sqrt{5}}{3}$
perception_59		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	B
logical_99		Not supported with pagination yet	Not supported with pagination yet	In the circle with center $O$, $AB$ is a diameter. Points $C$ and $D$ lie on the circle. Given $BC=8$ and $\cos\angle D=\dfrac{2}{3}$, find the length of $AB$.	logical	12
physical_51			Not supported with pagination yet	Given this robotics scene (image resolution: 1600x1322 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Move the pan to "30" Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[424, 305], [570, 360], [684, 392], [784, 438], [906, 484], [1084, 555], [1035, 457], [880, 526], [781, 586], [635, 698]]
logical_128		Not supported with pagination yet	Not supported with pagination yet	In circle $O$, diameter $CD$ passes through the midpoint $G$ of chord $EF$. Given $ \angle OEF = 34^\circ$, find the measure of $\angle DCF$.	logical	28
perception_56		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	B
logical_69		Not supported with pagination yet	Not supported with pagination yet	In circle $O$, $AB$ is a diameter. Chord $CD$ is perpendicular to $AB$ at point $E$. Given $\angle BAD = 30^{\circ}$ and $AE = 2$, find the length of $CD$.	logical	4\sqrt{3}
logical_36		Not supported with pagination yet	Not supported with pagination yet	Starting from the green dot, find the path to point C. List the points you pass through in order (not including C).	logical	Point C.
perception_75		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	D
logical_29		Not supported with pagination yet	Not supported with pagination yet	What is the area in square units of the quadrilateral XYZW shown below?	logical	The area of the quadrilateral XYZW is 2304 square units.
logical_145		Not supported with pagination yet	Not supported with pagination yet	In the figure, $BA$ is a diameter of circle $O$. Points $C$ and $E$ lie on circle $O$. The extensions of $BC$ and $BE$ meet line $AD$ at points $D$ and $F$, respectively. Given $BA^{2}=BC\cdot BD$, answer the following: (I) Is line $AD$ tangent to circle $O$ at $A$? (II) What is the value of $\angle D+\angle CEF$?	logical	<1>tangent</1><2>180^\circ</2>
physical_50			Not supported with pagination yet	Given this robotics scene (image resolution: 2450x1372 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Pick the object that is different from other objects Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]	embodied	[[1365, 506], [1312, 617], [1239, 724], [1162, 793], [1054, 861], [916, 941], [929, 863], [923, 743], [858, 687], [747, 722]]
logical_101		Not supported with pagination yet	Not supported with pagination yet	In the figure, $AB$ is a diameter of circle $O$, and $AC$ is a chord. Point $D$ is the midpoint of chord $BC$. Line $DE$ is perpendicular to $AC$ at $E$. Given $DE=6$ cm and $CE=2$ cm. (1) Is $DE$ a tangent to circle $O$? (2) Find the length of chord $AC$. (3) Find the length of diameter $AB$.	logical	<1>Yes</1><2>16</2><3>20</3>
perception_78		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	D
perception_next_115	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	left
perception_next_114	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	left
logical_62		Not supported with pagination yet	Not supported with pagination yet	In square $ABCD$ and square $CEFG$, $AD=6$, $CE=2\sqrt{2}$. Point $F$ lies on $CD$. Draw $DE$, draw $BG$ and extend it to meet $CD$ at $M$, and $DE$ at $H$. Find the length of $BH$.	logical	18/5
perception_next_38	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	left
perception_77		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	B
logical_15		Not supported with pagination yet	Not supported with pagination yet	In $\triangle ABC$ lines $CE$ and $AD$ are drawn so that $\dfrac{CD}{DB}=\dfrac{3}{1}$ and $\dfrac{AE}{EB}=\dfrac{3}{2}$. Let $r=\dfrac{CP}{PE}$ where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals?	logical	The final calculated value for r is 5.
perception_next_119	Not supported with pagination yet	Not supported with pagination yet		Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.	perception_next	right
perception_102		Not supported with pagination yet	Not supported with pagination yet	This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.	perception	C

logical_111

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As shown, two concentric circles have center $O$. In the larger circle, chord $AC$ is tangent to the smaller circle at $B$. Through $B$, draw the line perpendicular to $AO$, meeting the larger circle at $E$ and $F$, with foot $G$ on $AO$. Segment $CF$ is a diameter of the larger circle. (1) Prove that $AB=BC$. (2) Find the ratio $AE:AB:BE$.

logical

<1>AB=BC</1><2>6:3:1</2>

physical_56

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Given this robotics scene (image resolution: 1752x1138 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Put the coke next to the sprite into the white box with the right arm Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[1127, 332], [994, 411], [896, 439], [850, 553], [868, 697], [827, 537], [961, 561], [1034, 684], [1075, 775], [1007, 952]]

logical_44

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In the figure, $\triangle ABC$ is an isosceles right triangle with right angle at $B$. Point $P$ is outside $\triangle ABC$. Segment $BP$ is rotated clockwise by $90^\circ$ about $B$ to $BP'$. It is given that $\angle AP'B=135^\circ$ and $P'A:P'C=1:3$. Find the value of $PB:P'A$.

logical

1:2

perception_next_44

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

right

physical_84

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In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which container will catch the most liquid? (Choose from: purple container, red container)

physical

red container

physical_14

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Given this robotics scene (image resolution: 2258x1442 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Turn the teapot handle to the left Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[1081, 473], [1107, 546], [1119, 633], [1081, 706], [1034, 759], [959, 821], [890, 849], [833, 890], [804, 935], [753, 1003]]

physical_79

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In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Will both the brown container and the purple container catch falling liquids? (Answer: Yes/No)

physical

Yes

logical_78

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In triangle $ABC$, $BP$ is the angle bisector of $\angle B$, and $AP$ is perpendicular to $BP$ at $P$. The area of $\triangle PBC$ is $12$, and the area of $\triangle APB$ is twice the area of $\triangle APC$. Find the area of $\triangle APB$.

logical

4

physical_17

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Given this robotics scene (image resolution: 2560x1336 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Insert the flowers into the white vase Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[1590, 850], [1606, 919], [1636, 1007], [1524, 906], [1489, 822], [1411, 751], [1327, 714], [1258, 697], [1146, 696], [1022, 695]]

logical_106

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In right triangle $ABC$, $\angle C=90^\circ$, $\angle A=30^\circ$. Point $O$ lies on hypotenuse $AB$. The circle with center $O$ and radius $2$ passes through $B$, is tangent to $AC$ at $D$, and intersects $BC$ at $E$. Find the area of the shaded region enclosed by segments $CD$, $CE$, and $DE$.\n

logical

\frac{3\sqrt{3}}{2}-\frac{2\pi}{3}

physical_73

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In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which containers will catch the falling liquids? (List all from: brown container, white container, yellow container)

physical

brown container, white container

logical_49

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In right triangle $ABC$, $ riangle ABC$ has $ riangle C=90^\\circ$, $ riangle A=30^\\circ$, and $BC=2$. A circle centered at $C$ has radius $1$. Point $P$ lies on hypotenuse $AB$. Through $P$, draw a tangent $PQ$ to the circle (with $Q$ the point of tangency). What is the minimum value of $PQ$?

logical

2

logical_66

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As shown, points B, C, D lie on circle O. Through C, draw line AC parallel to BD, meeting the extension of OB at A. Connect CD. Given $\angle CDB = \angle OBD = 30^\circ$ and $DB = 6\sqrt{3}\,\text{cm}$. (1) What is the positional relationship between line AC and circle O? Explain why. (2) Find the area of the shaded region bounded by chord CD, chord BD, and arc BC. (Leave the result in terms of $\pi$.)

logical

<1>tangent</1><2>6\pi</2>

logical_131

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In the figure, $AB$ is the diameter of semicircle $O$. Lines $AD$ and $BC$ are tangent to circle $O$ at $A$ and $B$, respectively. Line $CD$ is tangent to circle $O$ at $E$. Lines $AD$ and $CD$ intersect at $D$, and lines $BC$ and $CD$ intersect at $C$. Connect $OD$ and $OC$. Among the following statements, which are correct? ① $OD^{2}=DE\cdot CD$; ② $AD+BC=CD$; ③ $OD=OC$; ④ $S_{\text{trapezoid }ABCD}=\dfrac{1}{2}\,CD\cdot OA$; ⑤ $\angle DOC=90^\circ$. Fill in the indices of all correct statements.

logical

①②⑤

perception_61

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

A

perception_next_80

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

right

physical_47

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Given this robotics scene (image resolution: 2496x1334 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Move apple to "Denver Nuggets" Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[1950, 449], [1799, 447], [1618, 476], [1481, 523], [1357, 591], [1264, 719], [1479, 719], [1652, 777], [1775, 828], [1874, 896]]

perception_67

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

B

perception_next_91

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

right

perception_84

Not supported with pagination yet

This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

B

logical_135

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As shown, C is a moving point on ray OE. AB is a chord through C. Line DA intersects OE at D. Consider the following three statements: (1) DA is a tangent to circle O. (2) DA = DC. (3) OD is perpendicular to OB. Using any two as conditions and the remaining one as the conclusion, write a true implication in the form “○○$\Rightarrow$○”, and provide a proof.

logical

①②⇒③

logical_93

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In the figure, P is a point outside circle O. The tangents PA and PB touch circle O at A and B, respectively. Given $PA=4\text{ cm}$ and $\angle P=40^\circ$. Point C is any point on segment AB. Through C, draw the tangent to circle O, which meets PA and PB at D and E, respectively. Find: (1) the perimeter of $\triangle PDE$; (2) the measure of $\angle DOE$.

logical

<1>8</1><2>70^\circ</2>

physical_108

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In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which containers will catch the falling liquids? (List all from: cyan container, white container, blue container)

physical

cyan container

physical_67

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How many cubes on the tabletop will the car collide with? List them.

physical

red cube, yellow cube, purple cube, blue cube

perception_next_105

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

left

logical_90

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As shown, a circle with center $A$ and radius $1.2$ moves so that $A$ lies on the line $y=\tfrac{3}{4}x-3$. Let $OP$ be a tangent from the origin $O$ to the circle, with point of tangency $P$. What is the minimum possible length of $OP$?

logical

\frac{63}{25}

perception_89

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

C

physical_70

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In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which containers will catch the falling liquids? (List all from: black container, yellow container, brown container)

physical

black container, brown container

physical_60

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Given this robotics scene (image resolution: 2124x1142 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Pick up the apple to the left of the orange and place it on the first shelf of the cabinet Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[722, 420], [542, 492], [408, 554], [434, 690], [461, 850], [506, 647], [701, 578], [911, 549], [1222, 577], [1431, 624]]

perception_38

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

C

perception_next_87

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

right

logical_140

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In triangle $ABC$, $AB=AC=4$ and $\tan B=\dfrac{\sqrt{7}}{3}$. Point $D$ lies on $BC$ with $CD=1$. Fold $\triangle ABD$ along line $AD$ to obtain $\triangle AED$, where point $B$ maps to $E$. Line $DE$ meets side $AC$ at $F$. Find the length of $EF$.

logical

\frac{64}{15}

logical_102

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In the circle with center $O$, $AB$ is a diameter, and points $C$ and $D$ lie on the circle. Given $\angle CAB=40^\circ$, find the measure of $\angle D$ (the inscribed angle at point $D$ that subtends arc $AB$).

logical

50

logical_122

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In right triangle $ABC$, $\angle ACB=90^{\circ}$ and $\angle A=30^{\circ}$. Point $D$ lies on $AB$. Circle $O$ has diameter $BD$ and is tangent to $AC$ at $E$. Extend $DE$ to meet the extension of $BC$ at $F$. (1) Prove that $\triangle BDF$ is equilateral. (2) Connect $AF$ and $DC$. Given $BC=3$, outline a method to find the area of quadrilateral $AFCD$.

logical

<1>Equilateral</1><2>Use AB=2BC and AC=\sqrt{3}\,BC to get AB, AC; from AO=2OE deduce AD, DB, DH; from BF=BD get CF; then area(AFCD)=area(\triangle AFC)+area(\triangle DHC)</2>

perception_57

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

A

perception_50

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

B

perception_next_23

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

right

physical_82

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In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which container will catch the most liquid? (Choose from: purple container, pink container, grey container, brown container)

physical

purple container

logical_126

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In the figure, circle $O$ has radius $OC$ perpendicular to diameter $AB$. Point $P$ moves on segment $OB$ (excluding $O$ and $B$). The extension of $CP$ meets the circle again at $D$. On the extension of $OB$ beyond $B$, take point $E$ so that $ED=EP$. (1) Show that $ED$ is a tangent to circle $O$. (2) When $OC=2$ and $ED=2\sqrt{3}$, find $\tan\angle E$ and the area $S$ of the shaded region (leave irrational numbers in exact form).

logical

<1>\frac{\sqrt{3}}{3}</1><2>2\sqrt{3}-\frac{2\pi}{3}</2>

logical_80

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Given circle $O$ with diameter $AB$. Point $C$ lies on circle $O$ (distinct from $A,B$). Extend $BC$ beyond $C$ to point $D$ such that $BC=CD$. Line $DA$ meets the circle again at $E$. Through $C$, draw the tangent to circle $O$, which meets $AD$ at $F$. (I) If $\angle DBA=60^{\circ}$, show that $E$ is the midpoint of $AD$. (II) If $CF=\dfrac{1}{2}R$, where $R$ is the radius of circle $O$, find $\angle DBA$.

logical

<1>E is the midpoint of AD</1><2>60^\circ</2>

physical_11

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Given this robotics scene (image resolution: 2208x1432 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Place the orange on the white plate Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[1652, 374], [1737, 506], [1791, 588], [1824, 729], [1849, 846], [1865, 958], [1585, 852], [1406, 851], [1230, 881], [1081, 1051]]

logical_147

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In the figure, AB is the diameter of semicircle O. Points C and D divide the semicircle into three equal arcs. If chord $CD=6$, find the area of the shaded region.

logical

6\pi

logical_5

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Two circles are centered at the origin, as shown. The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle. If $QR=3$, what is the value of $k$?

logical

7

perception_next_111

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

left

physical_46

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Given this robotics scene (image resolution: 2550x1276 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Move banana to "Germany" Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[2235, 768], [2138, 842], [2045, 926], [1985, 984], [1804, 1047], [1824, 957], [1909, 874], [1870, 798], [1806, 787], [1712, 745]]

logical_75

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In rhombus $ABCD$, point $E$ lies on diagonal $AC$ and $DE\perp BE$. (1) Are triangles $\triangle ADE$ and $\triangle ABE$ congruent? (2) If $\angle DAB=60^\circ$ and $AD=2\sqrt{3}$, find the length of $DE$.

logical

<1>Yes</1><2>3</2>

physical_69

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A playground contains obstacles of various colors and poses, along with randomly arranged pits. Soft balls are launched from various initial positions. These balls move dynamically, bouncing and undergoing permanent deformation, potentially colliding with obstacles and falling into pits. Will the brown ball finally drop into the right pit? A. Yes B. No C. Can not answer D. Uncertain.

physical

B

physical_23

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Given this robotics scene (image resolution: 1460x1080 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Open drawer with distractor on the top Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[695, 550], [698, 593], [702, 654], [707, 712], [728, 751], [741, 798], [806, 809], [862, 817], [920, 833], [982, 860]]

physical_86

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In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which containers will catch the falling liquids? (List all from: purple container, green container, pink container)

physical

green container

perception_70

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

B

physical_6

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Given this robotics scene (image resolution: 2236x1394 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Stack the pink block on the blue block Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[1394, 383], [1448, 456], [1472, 550], [1467, 724], [1375, 571], [1264, 519], [1173, 510], [1103, 524], [1069, 606], [1074, 728]]

logical_107

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In the figure, $AB$ is a diameter of circle $O$. Line $CD$ is tangent to circle $O$ at $D$. The extension of $AB$ meets $CD$ at $C$. If $\angle CAD=32^\circ$, find $\angle C$.

logical

26^\circ

perception_100

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

B

perception_next_72

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

right

logical_76

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In the circle with center $O$ and radius $5$, triangle $ABC$ is inscribed with $AB=AC=4$. Let $BD$ be a diameter of the circle. Find the area of quadrilateral $ABCD$.

logical

44\!/5

logical_63

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In triangle $ABC$, $\angle C=90^\circ$. Point $D$ is the midpoint of $BC$, and $AD=BC$. Find $\sin\angle BAD$.

logical

3/2

perception_4

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

B

logical_86

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In the figure, $AB$ is a diameter of circle $O$. Point $C$ lies on the circle. The tangent to circle $O$ at $C$ meets the extension of $AB$ at $D$, and $\angle D = 30^\circ$. (1) Find $\angle A$. (2) Through $C$, draw $CF \perp AB$, with foot $E$ on $AB$, and $CF$ meets the circle again at $F$. Given $CF = \dfrac{4}{3}$, find the length of arc $BC$. (Leave the answer in terms of $\pi$.)

logical

<1>30</1><2>\frac{4\pi}{3}</2>

perception_22

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

D

perception_72

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

D

perception_69

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

C

perception_49

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

C

physical_80

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In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which container will catch the most liquid? (Choose from: black container, brown container)

physical

brown container

logical_9

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Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. A circle is tangent to the sides $AB$ and $AC$ at $X$ and $Y$ respectively, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on the side $BC$. Given that $AB = 6$, find the area of the portion of the circle that lies outside the triangle.

logical

$\pi - 2$

perception_1

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

B

perception_46

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

D

physical_99

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In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which container will catch the most liquid? (Choose from: pink container, cyan container, blue container)

physical

blue container

physical_83

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In this device, different colored liquids are released from emitters at the top. Gravity pulls them down fixed ramps that change their flow direction. Please focus on exploring the physical properties and dynamic trajectories of these liquids. In the image, liquid can be seen falling from some emitters at the top. Which container will catch the most liquid? (Choose from: cyan container, brown container, orange container)

physical

cyan container

logical_17

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In the adjoining figure, , $CD$ is the diameter of a semicircle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semicircle, and $B$ is the point of intersection (distinct from $E$) of line segment $AE$ with the semicircle. If length $AB$ equals length $OD$, and the measure of $\angle EOD$ is $45^\circ$, then find the measure of $\angle BAO$, in degrees.

logical

$15^\circ$

physical_49

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Given this robotics scene (image resolution: 2434x1262 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Move the blue objects together Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[1474, 445], [1474, 545], [1467, 662], [1393, 759], [1307, 826], [1180, 897], [1227, 776], [1160, 702], [1046, 657], [908, 682]]

logical_26

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In triangle $ABC,$ the side lengths are $a = 7,$ $b = 9,$ and $c = 4.$ The incenter $I$ is shown in the diagram below. The position vector of the incenter can be expressed as a weighted average of the position vectors of the vertices: \[\overrightarrow{I} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C},\] where $x, y,$ and $z$ are constants such that $x + y + z = 1.$ Find the ordered triple $(x,y,z).$

logical

$\left( \frac{7}{20}, \frac{9}{20}, \frac{1}{5} \right)$

perception_next_85

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

right

physical_53

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Given this robotics scene (image resolution: 1998x1146 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Place land animal onto plate with left robot arm Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[622, 614], [735, 706], [841, 754], [960, 787], [1039, 885], [1041, 761], [1152, 686], [1299, 809], [1428, 873], [1437, 1026]]

physical_105

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An array of both movable and fixed pulleys, along with anchor points, is arranged on a wall. Ropes are connected to pulleys, loads, or anchor points and can be wound around the pulleys. These loads, with varying masses, interact with other forces, creating distinct motion patterns. The model's primary objective is to identify tension distributions within this rope system. Additionally, it aims to recognize correlations or constraints among moving objects, such as the coordinated movement of loads and the rotation of pulleys on a single rope. Furthermore, the model is expected to infer numerical relationships between the loads' masses. If we want the gray sphere to move up, what can we do? (Select all answers that satisfy the conditions.) A. Increase the mass of the yellow sphere B. Decrease the mass of the yellow sphere C. Increase the mass of the green cube D. Decrease the mass of the gray sphere

physical

C, D

perception_34

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

D

logical_47

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In isosceles triangle $ABC$ with $AB=AC$, $AD$ is the altitude to base $BC$. Given $AD:DC=1:3$. Rotate triangle $ADC$ about point $D$ to obtain triangle $DEF$ such that points $A$ and $C$ correspond to $E$ and $F$ respectively, and $EF$ coincides with line $AB$. Let $AC$ and $DF$ intersect at $O$. Find the ratio $S_{\triangle AOE}:S_{\triangle DOE}$.

logical

\frac{32}{45}

physical_91

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A playground contains obstacles of various colors and poses, along with randomly arranged pits. Soft balls are launched from various initial positions. These balls move dynamically, bouncing and undergoing permanent deformation, potentially colliding with obstacles and falling into pits. Where will the orange ball, green ball, and black ball finally drop?

physical

Orange ball: left pit; Green ball: right pit; Black ball: right pit

logical_31

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Starting from the green dot, find the path to point C. List the points you pass through in order (not including C).

logical

Point E.

logical_11

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The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.

logical

$\frac{5\sqrt{5}}{3}$

perception_59

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

B

logical_99

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In the circle with center $O$, $AB$ is a diameter. Points $C$ and $D$ lie on the circle. Given $BC=8$ and $\cos\angle D=\dfrac{2}{3}$, find the length of $AB$.

logical

12

physical_51

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Given this robotics scene (image resolution: 1600x1322 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Move the pan to "30" Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[424, 305], [570, 360], [684, 392], [784, 438], [906, 484], [1084, 555], [1035, 457], [880, 526], [781, 586], [635, 698]]

logical_128

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In circle $O$, diameter $CD$ passes through the midpoint $G$ of chord $EF$. Given $ \angle OEF = 34^\circ$, find the measure of $\angle DCF$.

logical

28

perception_56

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

B

logical_69

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In circle $O$, $AB$ is a diameter. Chord $CD$ is perpendicular to $AB$ at point $E$. Given $\angle BAD = 30^{\circ}$ and $AE = 2$, find the length of $CD$.

logical

4\sqrt{3}

logical_36

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Starting from the green dot, find the path to point C. List the points you pass through in order (not including C).

logical

Point C.

perception_75

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

D

logical_29

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What is the area in square units of the quadrilateral XYZW shown below?

logical

The area of the quadrilateral XYZW is 2304 square units.

logical_145

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In the figure, $BA$ is a diameter of circle $O$. Points $C$ and $E$ lie on circle $O$. The extensions of $BC$ and $BE$ meet line $AD$ at points $D$ and $F$, respectively. Given $BA^{2}=BC\cdot BD$, answer the following: (I) Is line $AD$ tangent to circle $O$ at $A$? (II) What is the value of $\angle D+\angle CEF$?

logical

<1>tangent</1><2>180^\circ</2>

physical_50

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Given this robotics scene (image resolution: 2450x1372 pixels), generate a 10-point end-effector trajectory to complete the task. TASK: Pick the object that is different from other objects Output format: A list of 10 (x, y) pixel coordinates representing the trajectory, where (0, 0) is the top-left corner. Example: [[x1, y1], [x2, y2], ..., [x10, y10]]

embodied

[[1365, 506], [1312, 617], [1239, 724], [1162, 793], [1054, 861], [916, 941], [929, 863], [923, 743], [858, 687], [747, 722]]

logical_101

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In the figure, $AB$ is a diameter of circle $O$, and $AC$ is a chord. Point $D$ is the midpoint of chord $BC$. Line $DE$ is perpendicular to $AC$ at $E$. Given $DE=6$ cm and $CE=2$ cm. (1) Is $DE$ a tangent to circle $O$? (2) Find the length of chord $AC$. (3) Find the length of diameter $AB$.

logical

<1>Yes</1><2>16</2><3>20</3>

perception_78

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

D

perception_next_115

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

left

perception_next_114

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

left

logical_62

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In square $ABCD$ and square $CEFG$, $AD=6$, $CE=2\sqrt{2}$. Point $F$ lies on $CD$. Draw $DE$, draw $BG$ and extend it to meet $CD$ at $M$, and $DE$ at $H$. Find the length of $BH$.

logical

18/5

perception_next_38

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

left

perception_77

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

B

logical_15

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In $\triangle ABC$ lines $CE$ and $AD$ are drawn so that $\dfrac{CD}{DB}=\dfrac{3}{1}$ and $\dfrac{AE}{EB}=\dfrac{3}{2}$. Let $r=\dfrac{CP}{PE}$ where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals?

logical

The final calculated value for r is 5.

perception_next_119

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Determine if camera moves clockwise (left) or counter-clockwise (right) from Image 1 to Image 2.

perception_next

right

perception_102

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This is a visual puzzle. A gray box is covering part of the image. Please select from options A, B, C, or D which piece correctly fills the missing area.

perception

C