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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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stat moment
fruitmonster97   4
N 41 minutes ago by BS2012
Define xiooix deviation of a set as the average positive difference from each element and the average of said set. Prove that the xiooix deviation of a set is less than or equal to standard deviation.
4 replies
fruitmonster97
3 hours ago
BS2012
41 minutes ago
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Expected number of flips
Bread10   7
N 2 hours ago by Pengu14
An unfair coin has a $\frac{4}{7}$ probability of coming up heads and $\frac{3}{7}$ probability of coming up tails. The expected number of flips necessary to first see the sequence $HHTHTHHT$ in that consecutive order can be written as $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. Find the number of factors of $n$.

$\textbf{(A)}~40\qquad\textbf{(B)}~42\qquad\textbf{(C)}~44\qquad\textbf{(D)}~45\qquad\textbf{(E)}~48$
7 replies
Bread10
3 hours ago
Pengu14
2 hours ago
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Help with Competitive Geometry?
REACHAW   3
N 2 hours ago by REACHAW
Hi everyone,
I'm struggling a lot with geometry. I've found algebra, number theory, and even calculus to be relatively intuitive. However, when I took geometry, I found it very challenging. I stumbled my way through the class and can do the basic 'textbook' geometry problems, but still struggle a lot with geometry in competitive math. I find myself consistently skipping the geometry problems during contests (even the easier/first ones).

It's difficult for me to see the solution path. I can do the simpler textbook tasks (eg. find congruent triangles) but not more complex ones (eg. draw these two lines to form similar triangles).

Do you have any advice, resources, or techniques I should try?
3 replies
REACHAW
Monday at 11:51 PM
REACHAW
2 hours ago
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fractional part
Ecrin_eren   3
N 3 hours ago by rchokler
{x^2}+{x}=0.64

How many positive real values of x satisfy this equation?
3 replies
Ecrin_eren
Apr 13, 2025
rchokler
3 hours ago
J
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Angle oriented geometry
Problems_eater   0
3 hours ago
Let $A, B, C,D$ be four distinct points in the plane.
Which of the following statements, expressed using oriented angles, are always true?

1.If lines $AB$ and $CD$ are distinct and parallel, then
the oriented angle $ABC$ is equal to the oriented angle DCB.

2.If $B$ lies on the segment $AC$, then
the oriented angle $DBA$ plus the oriented angle $DBC $equals $180°$.

3.If the oriented angle$ ABC$ plus the oriented angle $BCD$ equals 0°, then
lines $AB $and $CD$ are parallel.

4.If the oriented angle $ABC$ plus the oriented angle $BCD$ equals $180°$, then
lines $AB$ and $CD$are parallel.
0 replies
Problems_eater
3 hours ago
0 replies
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how many quadrilaterals ?
Ecrin_eren   6
N Yesterday at 5:31 PM by mathprodigy2011
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"
6 replies
Ecrin_eren
Apr 13, 2025
mathprodigy2011
Yesterday at 5:31 PM
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Plane geometry problem with inequalities
ReticulatedPython   3
N Yesterday at 2:48 PM by vanstraelen
Let $A$ and $B$ be points on a plane such that $AB=1.$ Let $P$ be a point on that plane such that $$\frac{AP^2+BP^2}{(AP)(BP)}=3.$$Prove that $$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$
Source: Own
3 replies
ReticulatedPython
Apr 10, 2025
vanstraelen
Yesterday at 2:48 PM
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Inequalities
sqing   1
N Yesterday at 1:55 PM by sqing
Let $ a,b $ be reals such that $ a^2-ab+b^2 =3$ . Prove that
$$ \frac{13}{ 10 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq \frac{1}{ 2 }$$$$ \frac{6}{ 5 }>\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq \frac{1}{ 5 }$$$$ \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq \frac{1}{ 14 }$$
1 reply
sqing
Yesterday at 8:59 AM
sqing
Yesterday at 1:55 PM
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idk12345678 Math Contest
idk12345678   21
N Yesterday at 1:25 PM by idk12345678
Welcome to the 1st idk12345678 Math Contest.
You have 4 hours. You do not have to prove your answers.
Post \signup username to sign up. Post your answers in a hide tag and I will tell you your score.*


The contest is attached to the post

Clarifications

*I mightve done them wrong feel free to ask about an answer
21 replies
idk12345678
Apr 10, 2025
idk12345678
Yesterday at 1:25 PM
J
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purple comet math competition question
AVY2024   4
N Yesterday at 1:02 PM by K1mchi_
Given that (1 + tan 1)(1 + tan 2). . .(1 + tan 45) = 2n, find n
4 replies
AVY2024
Yesterday at 11:00 AM
K1mchi_
Yesterday at 1:02 PM
J
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Inequalities
sqing   25
N Yesterday at 12:06 PM by sqing
Let $ a,b,c,d>0 $ and $(a+c)(b+d)=ac+\frac{3}{2}bd.$ Prove that
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq \frac{20-\sqrt{10}}{3}$$Let $ a,b,c,d>0 $ and $(a+c)(b+d)=ac+\frac{4}{3}bd.$ Prove that
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq \frac{21-\sqrt{6}}{3}$$
25 replies
sqing
Dec 3, 2024
sqing
Yesterday at 12:06 PM
J
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Polynomials
CuriousBabu   3
N Yesterday at 11:40 AM by osszhangbanvan
\[ \frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0 \]
Find the number of real solutions.
3 replies
CuriousBabu
Monday at 4:09 PM
osszhangbanvan
Yesterday at 11:40 AM
J
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Combination
aria123   0
Yesterday at 10:59 AM
Prove that three squares of side length $4$ cannot completely cover a square of side length $5$, if the three smaller squares do not overlap in their interiors (i.e., they may touch at edges or corners, but no part of one lies over another).
0 replies
aria123
Yesterday at 10:59 AM
0 replies
J
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Geo Mock #6
Bluesoul   3
N Yesterday at 3:16 AM by dudade
Consider triangle $ABC$ with $AB=5, BC=8, AC=7$, denote the incenter of the triangle as $I$. Extend $BI$ to meet the circumcircle of $\triangle{AIC}$ at $Q\neq I$, find the length of $QC$.
3 replies
Bluesoul
Apr 1, 2025
dudade
Yesterday at 3:16 AM
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no of rectangles by vertices of a cube (HOMC 2018 ind. sen1)
parmenides51   4
N Aug 3, 2020 by heheXD1
How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle).

A. $6$ B. $8$ C. $12$ D. $18$ E. $16$
4 replies
parmenides51
Feb 2, 2020
heheXD1
Aug 3, 2020
J
no of rectangles by vertices of a cube (HOMC 2018 ind. sen1)
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Reveal topic tags
geometry
rectangle
cube
3D geometry
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parmenides51
30630 posts
parmenides51
#1 h  Feb 2, 2020, 5:52 PM • 1 Y
Y by Adventure10
How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle).

A. $6$ B. $8$ C. $12$ D. $18$ E. $16$
Z K Y
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brainiacmaniac31
2170 posts
brainiacmaniac31
#2 h  Feb 2, 2020, 5:56 PM • 1 Y
Y by Adventure10
There are $6$ square faces, and $6$ rectangles which bisect the cube and are formed by $2$ edges and two face diagonals, for an answer of $\boxed{12}$.
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heheXD1
429 posts
heheXD1
#3 h  Aug 2, 2020, 11:48 PM
Y by
nothing .
This post has been edited 1 time. Last edited by heheXD1, Aug 3, 2020, 12:15 AM
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Inconsistent
1455 posts
Inconsistent
#4 h  Aug 2, 2020, 11:51 PM
Y by
Umm... @above do you have another example of a rectangle? I'm pretty sure it's 12.
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heheXD1
429 posts
heheXD1
#5 h  Aug 3, 2020, 12:15 AM
Y by
Oh I think I over counted earlier.
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