All Questions ⁺⁰237911 Questions

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A number is chosen at random from among all 9-digit numbers that use each of the digits 1 through 9 exactly once. What is the probability that this number is a multiple of 2?

gnistory 6 hours ago

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Let ABC be a triangle with \angle B = 90^{\circ}. Suppose that point D lies on segment BC such that angle BAD = \angle CAD, and suppose that point E lies on segment AC such that angle EDA = 60^{\circ}. Given that AD = AC and that CE = 17, find BD.
<read more ..

HumanBemg 6 hours ago

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In the diagram below, AB = AC = 1, and arc BC is centered at A with \angle BAC = 60^\circ. Point D is on \overline{AB}, and point E is on arc BC so that BD = DE, and arc BE (centered at D) is tangent to \overline{AC}. Point F is on \overline{DB}, read more ..

HumanBemg 6 hours ago

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Two chords with length r are drawn on a circle of radius r. What is the maximum possible distance between their endpoints?

Jyrgernry 6 hours ago

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William writes the integers from 2 through 15 from left to right on a piece of paper, then does the following:

* He begins by circling the leftmost uncircled integer, then circles all its powers. Afterwards, he repeats this process until every read more ..

Jyrgernry 6 hours ago

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Siva starts at vertex A of square ABCD with side length 10. He walks towards vertex C, along the diagonal AC. He then walks towards B, then towards D. What is the length of his path?

cooIcooIcooI17 6 hours ago

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Mrs. Monstrous Millie wants to throw a party for a group of monsters and aliens. The monsters each have 5 heads with 7 ears on each head, and the aliens each have 5 heads with 3 ears on each head. As party favors, Mrs. Millie wants to give each guest a read more ..

cooIcooIcooI17 6 hours ago

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Given regular heptagon ABCDEFG, a circle can be drawn that is tangent to DC at C and to EF at F. What is radius of the circle if the side length of the heptagon is 1?

Jyrgernry 9 hours ago

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In the diagram, a semicircle is centered at $O,$ and $\overline{AB}$ is tangent to the semicircle at $B.$ Find $\angle BCD,$ in degrees.

Jyrgernry 9 hours ago

Apr 20, 2026

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The triangle with vertices at (1,0), (0,1), and (k,k) has area 1/2. Find k.

cooIcooIcooI17 Apr 20, 2026, 11:17 PM

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Distinct prime numbers p, q, and r satisfy the equation p(q + r) = 16. Which of the following prime numbers must either q or r be equal to?
2 3 5 7 11

cooIcooIcooI17 Apr 20, 2026, 11:17 PM

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In triangle ABC, let D be a point on side BC. Select all the true statements.
If AD is an altitude of triangle ABC, then AC > AD.
If AD is a median of triangle ABC, then BD > CD.
If AD is an angle bisector read more ..

Pythagorearn Apr 20, 2026, 11:16 PM

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All the sides of a triangle have integer length. The perimeter of the triangle is 10, and the triangle is scalene. How many such non-congruent triangles are there?

Pythagorearn Apr 20, 2026, 11:15 PM

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In triangle $XYZ,$ circles are drawn centered at $X$, $Y$, and $Z$, so that all pairs of circles are externally tangent. If $XY = 2,$ $XZ = 2,$ and $YZ = 2$, then find the sum of the areas of all three circles.

CInsbstnkng Apr 20, 2026

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In the diagram below, chords $\overline{AB}$ and $\overline{CD}$ are perpendicular, and meet at $X.$ Find the diameter of the circle.

CInsbstnkng Apr 20, 2026

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In equiangular octagaon EFGHIJKL, we know that EF = GH = IJ = KL = 1 and FG = HI = JK = LE = sqrt(2). Find the area of the octagon.

HumanBemg Apr 20, 2026

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Semicircles are drawn on diameters \overline{AB} and \overline{CD}, as shown below. Find AB.

HumanBemg Apr 20, 2026

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Given regular heptagon ABCDEFG, a circle can be drawn that is tangent to DC at C and to EF at F. What is radius of the circle if the side length of the heptagon is 1?

Pythagorearn Apr 20, 2026

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In the diagram, a semicircle is centered at $O,$ and $\overline{AB}$ is tangent to the semicircle at $B.$ Find $\angle BCD,$ in degrees.

Pythagorearn Apr 20, 2026

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The graph of y = p(x)/q(x) is shown below, where p(x) and q(x) are quadratic. (Assume that the grid lines are at integers.)
The horizontal asymptote is x = 0 and the only vertical asymptote is y = 0. Find p(3)/q(3)

BlackoutMiIkshake Apr 20, 2026

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Triangle ABC is rotated clockwise about A to triangle $AB'C',$ as shown in the diagram. We know the length of the path taken by vertex $B$ is $6.$ What is the length of the path taken by vertex $C?$

BlackoutMiIkshake Apr 20, 2026

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Two circles C_1 and C_2 are shown below. Find the radius of the circle that is externally tangent to both $C_1$ and $C_2,$ and tangent to the $x$-axis.

BlackoutMiIkshake Apr 20, 2026

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Triangle PQR is rotated clockwise about P to triangle PQ'R', as shown in the diagram. We know the length of the path taken by vertex Q is the same as the length of the path takesn by P. What is the length of the path taken by vertex R?

blackpanther Apr 20, 2026

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Line \ell is shown below. The line y = 2x + k has a distance of 3 from line y = 5x + k. Find all possible values of k.

blackpanther Apr 20, 2026

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