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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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jlacosta
Apr 2, 2025
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Angles, similar triangles, geometry problem
smalkaram_3549   0
2 minutes ago
Completely stuck on this problem.
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smalkaram_3549
2 minutes ago
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New geometry problem
titaniumfalcon   3
N 3 hours ago by mathprodigy2011
Post any solutions you have, with explanation or proof if possible, good luck!
3 replies
titaniumfalcon
Thursday at 10:40 PM
mathprodigy2011
3 hours ago
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School Math Problem
math_cool123   5
N 3 hours ago by math_cool123
Find all ordered pairs of nonzero integers $(a, b)$ that satisfy $$(a^2+b)(a+b^2)=(a-b)^3.$$
5 replies
math_cool123
Apr 2, 2025
math_cool123
3 hours ago
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KSEA NMSC Mock Contest Group B (Last Problem)
Shiyul   3
N 4 hours ago by Giant_PT
Let $a_n$ be a sequence defined by $a_n = a^2 + 1$. Then the product of four consecutive terms in $a_n$ can be written as the product of two terms in $a_n$. Find $p + q$ if $(a_(11))(a_(12))(a_(13))(a_(14)) = (a_p)(a_q)$.
3 replies
Shiyul
Yesterday at 3:09 PM
Giant_PT
4 hours ago
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polyhedron wit 14 faces, area wanted (Formula of Unity 2020 1st gr10)
parmenides51   1
N May 7, 2020 by natmath
The surface of a wooden $1$ m$^3$ cube is painted. From each vertex, a pyramid is cut off, as a result, a polyhedron with $14$ faces is obtained. Each painted face is a rectangle, and each unpainted face is an equilateral triangle (the triangles are not necessarily equal). Find the total area of the painted faces of the polyhedron if it is\sqrt3 times less than the total area of the unpainted ones.

(A. Tesler )
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parmenides51
May 6, 2020
natmath
May 7, 2020
J
polyhedron wit 14 faces, area wanted (Formula of Unity 2020 1st gr10)
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parmenides51
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parmenides51
#1 h  May 6, 2020, 11:30 PM • 1 Y
Y by Mango247
The surface of a wooden $1$ m$^3$ cube is painted. From each vertex, a pyramid is cut off, as a result, a polyhedron with $14$ faces is obtained. Each painted face is a rectangle, and each unpainted face is an equilateral triangle (the triangles are not necessarily equal). Find the total area of the painted faces of the polyhedron if it is\sqrt3 times less than the total area of the unpainted ones.

(A. Tesler )
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natmath
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natmath
#2 h  May 7, 2020, 12:44 AM
Y by
There are 2 sets of 4 vertices that are all pairwise endpoints of a face diagonal. The 4 pyramid cut from a set must be congruent in order to make a rectangle (because rectangles have opposite sides congruent). Call these two side lengths (of the equilateral triangles) a and b. We know that $a+b=\sqrt{2}$
The total area of the painted faces is 8ab and the total area of the painted faces is $\sqrt{3}(a^2+b^2)$. So we know that $a^2+b^2=8ab$

From our first equation $a^2+2ab+b^2=2$, so $10ab=2$; $8ab=\boxed{\frac{8}{5}}$
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