All Questions ⁺⁰237703 Questions

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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 ..

BlackoutMiIkshake Oct 22, 2025

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Math

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

BlackoutMiIkshake Oct 22, 2025

Oct 21, 2025

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Math

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 ..

BlackoutMiIkshake Oct 21, 2025

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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 ..

BlackoutMiIkshake Oct 21, 2025

Oct 20, 2025

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Math

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?

BlackoutMiIkshake Oct 20, 2025

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Math

Three squares of a 5 \times 5 grid are colored. Two colorings are conisdered equivalent if one coloring can be rotated to form the other coloring, such as the two colorings below, and the grid can be reflected as well. We also want the three read more ..

BlackoutMiIkshake Oct 20, 2025

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+109

Math

Selma multiplies the number 9 the two-digit number \overline{AB}, and the three-digit number \overline{CDE}. She finds that the product of all three numbers is the five-digit number 41409. Compute the sum \overline{AB} + \overline{CDE}. read more ..

BlackoutMiIkshake Oct 20, 2025

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Math

Given a positive integer N, a multiplication decomposition of N is a sequence of positive integers (d_1, d_2, \dots, d_k) such that
* d_1 \neq 1,
* d_i divides d_{i + 1} for 1 \le i \le k - 1, and
* d_1 d_2 \dotsm d_k = N.
The last read more ..

BlackoutMiIkshake Oct 20, 2025

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+109

Math

A traffic light runs repeatedly through the following cycle: green for 40 seconds, then yellow for 5 seconds, then red for 35 seconds, which repeats indefinitely. Oliver picks a random time and watches the light for 75 seconds. What is the probability read more ..

BlackoutMiIkshake Oct 20, 2025

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Math

Consider a cube with side length 1, together with 14 congruent spheres each of radius r, with 8 of the spheres centered at the 8 vertices of the cube and the remaining $6$ spheres centered at the centers of the $6$ faces of the cube. Suppose $r$ is chosen read more ..

BlackoutMiIkshake Oct 20, 2025

Oct 18, 2025

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Math

What is the units digit of 1^1 + 2^2 + 3^4 + 4^8?

BlackoutMiIkshake Oct 18, 2025

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+109

Math

A square has two of its vertices at (12, 34) and (-98, 56),$ and another vertex at (a, b). What is the sum of all possible values of the area of the square?

BlackoutMiIkshake Oct 18, 2025

Oct 17, 2025

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Math

At Patty the panda's pizza parlor, the possible pizza toppings are pesto, pepperoni, peppers, and pineapple, and the possible desserts are pumpkin pie, powdered-sugar pretzels, and papaya popsicles. If Prajwal the porcupine, Petunia the peacock, and Pete read more ..

BlackoutMiIkshake Oct 17, 2025

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Math

In a rectangular prism, the surface area is 24 and the sum of the lengths of all twelve edges is 24. What is the volume of the prism?

BlackoutMiIkshake Oct 17, 2025

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Math

For how many prime numbers p less than 100 is the sum of p's digits less than or equal to 2?

BlackoutMiIkshake Oct 17, 2025

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Math

In the game Nerdle, a player tries to guess a true equation made up of $8$ symbols, each of which is either $+,$ $-,$ $*,$ $/,$ $=,$ or one of the digits from $0$ to $9.$ A square turns teal if the symbol is correct and in the correct location, magenta read more ..

BlackoutMiIkshake Oct 17, 2025

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Math

Define the importance of a point (x, y) on the coordinate plane to be x^2 + y^2. There exist some points with positive, integer $x$- and $y$-coordinates whose importances are 5. Find the area of the convex polygon determined by those points.

BlackoutMiIkshake Oct 17, 2025

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Math

A set of 16 positive integers has mean, median, and unique mode all equal to 5. What is the largest possible value of one of the numbers in the set?

BlackoutMiIkshake Oct 17, 2025

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Math

Haoyu gives Mingze and Yihan two different positive integers, and tells them that the product of their numbers is less than 1000. They are also aware that they have two different positive integers. They have the following conversation:
Mingze: read more ..

BlackoutMiIkshake Oct 17, 2025

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Math

Sunita and Mohammed play a game where they start with a pile of candy and take turns eating a prime number of pieces of candy from the pile. A player who leaves 0 or 1 candies in the pile after their move wins. If Sunita and Mohammed both play perfectly read more ..

BlackoutMiIkshake Oct 17, 2025

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Math

The shaded region below has area a+b\pi where a and b are positive integers. What is a+b? (You may assume the boundary of each region is made up of straight lines and circular arcs, and the squares of the grid have side length $1.$) read more ..

BlackoutMiIkshake Oct 17, 2025

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Math

Jason is driving from his home to his office, which is 20 miles away. He normally drives at a constant speed of 60 miles per hour for the whole trip. However today, 15 minutes into his drive, his car breaks down. He calls a taxi, and it arrives 20 minutes read more ..

BlackoutMiIkshake Oct 17, 2025

Oct 16, 2025

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Math

In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $CE =3$ and $DF = 3$, then what is $BD$?

cooIcooIcooI17 Oct 16, 2025

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