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In triangle ABC, the angle bisector of angle BAC meets BC at D, such that AD = AB. Line segment AD is extended to E, such that angle DBE = angle BAD = 17 degrees. Find angle ABD.

CInsbstnkng Jul 27, 2025

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Recall that a partition of a positive integer n means a way of writing n as the sum of some positive integers, where the order of the parts does not matter. For example, there are five partitions of 4:
4
3 + 1
2 + 2 read more ..

CInsbstnkng Jul 27, 2025

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Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of triangle ABC?
A is at (0,0), B is at (1,0), and C is at (1,1).

CInsbstnkng Jul 27, 2025

Jul 26, 2025

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In the complex plane, z, z2, z3, form, in some order, three of the vertices of a non-degenerate square. Enter all possible areas of the square, separated by commas.

Clnsbstnkng Jul 26, 2025

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In the diagram below, \overline{AD} and \overline{BE} are angle bisectors of \angle BAC and \angle ABC, respectively, and they intersect at T. We know that BD=12, AE=8, and BF=3+AE. Find AB.

CInsbstnkng Jul 26, 2025

Jul 25, 2025

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Compute the smallest positive integer n such that n+i, (n+i)2, and (n+i)3 are the vertices of a triangle in the complex plane whose area is greater than 2015.

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Clnsbstnkng Jul 25, 2025

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In the diagram, \overline{AD} is an altitude, \overline{BE} is a median, and \overline{AD}, \overline{BE}, and \overline{CF} are concurrent. If AE = 11, BE = 9, and AD = 6\sqrt{2}, what is AF?

CInsbstnkng Jul 25, 2025

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How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?

CInsbstnkng Jul 25, 2025

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How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?

CInsbstnkng Jul 25, 2025

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Number Theory

Karlanna places 600 marbles into m total boxes such that each box contains an equal number of marbles. There is more than one box, and each box contains more than one marble. For how many values of m can this be done?

bqimath Jul 25, 2025

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Right triangles ABC, ADE, EFG, and GHC are all similar. Also, $[ADE] = 1,$ $[EFG] = 1,$ and $[GHC] = 1.$ Find the area of the red region.

CInsbstnkng Jul 25, 2025

Jul 24, 2025

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Let f(x) be a polynomial such that f(0) = 4, f(1) = 8, and f(2) = -6. Find the remainder when f(x) is divided by x.

CInsbstnkng Jul 24, 2025

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The polynomial
g(x) = x^3 - x^2 - (m^2 + m + 18) x + 2m^2 - 14m - 6
is divisible by x - 5 and all of its zeroes are integers. Find all possible values of m.

CInsbstnkng Jul 24, 2025

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Let z be a complex number such that |z|=1.
Find the maximum value of |1+z|+|1-z+z2|.

Clnsbstnkng Jul 24, 2025

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Find the number of ways of arranging one A, two Bs, three Cs, and four Ds, so that no two Bs are next to each other, no two Cs are next to each other, and no two Ds are next to each other.

CInsbstnkng Jul 24, 2025

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A permutation of the numbers (1,2,3,\dots,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3) is a permutation of (1,2,3,4,5).
Find the number of permutations (x_1, x_2, \dots, x_8) read more ..

CInsbstnkng Jul 24, 2025

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Find the number of sequences (a_1, a_2, a_3, \dots, a_8) such that:
* a_i \in \{1, 2, 3, 4, 5, 6, 7, 8\} for all 1 \le i \le 8.
* Every number1, 2, 3, 4, 5, 6, 7, 8 appears at least once in the sequence.

CInsbstnkng Jul 24, 2025

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A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits (spades, hearts, diamonds, and clubs), such that there is exactly one card for any given rank and suit.
You are dealt a hand of read more ..

CInsbstnkng Jul 24, 2025

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