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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 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$
Vulch   1
N an hour ago by aidan0626
Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$
1 reply
Vulch
2 hours ago
aidan0626
an hour ago
J
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Find the domain and range of $f(x)=\frac{4-x}{x-4}.$
Vulch   1
N 2 hours ago by aidan0626
Find the domain and range of $f(x)=\frac{4-x}{x-4}.$
1 reply
Vulch
2 hours ago
aidan0626
2 hours ago
J
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Find the domain and range of $f(x)=2-|x-5|.$
Vulch   0
2 hours ago
Find the domain and range of $f(x)=2-|x-5|.$
0 replies
Vulch
2 hours ago
0 replies
J
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Inequalities
sqing   5
N 3 hours ago by pooh123
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
5 replies
sqing
Jul 12, 2024
pooh123
3 hours ago
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Amc 10 mock
Mathsboy100   4
N 3 hours ago by pooh123
let \[\lfloor x \rfloor\]denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which \[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]
4 replies
Mathsboy100
Oct 9, 2024
pooh123
3 hours ago
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Inequalities
sqing   0
4 hours ago
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq 4\left(\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{32}{9}\left(\frac{a+b}{b+c}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left( \frac{a+b}{b+c}+ \frac{b+c}{c+a}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}\right)$$
0 replies
sqing
4 hours ago
0 replies
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Purple Comet High School Math Meet 2024 P1
franklin2013   3
N 4 hours ago by codegirl2013
Joe ate one half of a fifth of a pizza. Gale ate one third of a quarter of that pizza. The difference in the amounts that the two ate was $\frac{1}{n}$ of the pizza, where $n$ is a positive integer. Find $n$.
3 replies
franklin2013
5 hours ago
codegirl2013
4 hours ago
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Cool vieta sum
Kempu33334   0
4 hours ago
Let the roots of \[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]be $r_1, r_2, \dots, r_{108}$. Find \[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]without Newton Sums.
0 replies
Kempu33334
4 hours ago
0 replies
J
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Inequality, tougher than it looks
tom-nowy   2
N 4 hours ago by pooh123
Prove that for $a,b \in \mathbb{R}$
$$ 2(a^2+1)(b^2+1) \geq 3(a+b). $$Is there an elegant way to prove this?
2 replies
tom-nowy
Tuesday at 3:51 AM
pooh123
4 hours ago
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Stylish Numbers
pedronis   3
N 5 hours ago by pedronis
A positive even integer $n$ is called stylish if the set $\{1, 2, \ldots, n\}$ can be partitioned into $\frac{n}{2}$ pairs such that the sum of the elements in each pair is a power of $3$. For example, $6$ is stylish because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned as $\{1,2\}, \{3,6\}, \{4,5\}$, with sums $3$, $9$, and $9$ respectively. Determine the number of stylish numbers less than $3^{2025}$.
3 replies
pedronis
Apr 13, 2025
pedronis
5 hours ago
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Transformation of a cross product when multiplied by matrix A
Math-lover1   2
N Yesterday at 9:23 PM by Math-lover1
I was working through AoPS Volume 2 and this statement from Chapter 11: Cross Products/Determinants confused me.
[quote=AoPS Volume 2]A quick comparison of $|\underline{A}|$ to the cross product $(\underline{A}\vec{i}) \times (\underline{A}\vec{j})$ reveals that a negative determinant [of $\underline{A}$] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
I understand that this is true for the unit vectors $\vec{i} = (1 \ 0)$ and $\vec{j} = (0 \ 1)$, but am confused on how to prove this statement for general vectors $\vec{v}$ and $\vec{w}$ although its supposed to be a quick comparison.

How do I prove this statement easily with any two 2D vectors?
2 replies
Math-lover1
Tuesday at 10:29 PM
Math-lover1
Yesterday at 9:23 PM
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unfair coin, points winning 2024 TMC AIME Mock #9
parmenides51   5
N Yesterday at 8:44 PM by Math-lover1
Krithik has an unfair coin with a $\frac13$ chance of landing heads when flipped. Krithik is playing a game where he starts with $1$ point. Every turn, he flips the coin, and if it lands heads, he gains $1$ point, and if it lands tails, he loses $1$ point. However, after the turn, if he has a negative number of points, his point counter resets to $1$. Krithik wins when he earns $8$ points. Find the expected number of turns until Krithik wins.
5 replies
parmenides51
Apr 26, 2025
Math-lover1
Yesterday at 8:44 PM
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BrUMO 2025 Team Round Problem 15
lpieleanu   1
N Yesterday at 8:01 PM by vanstraelen
Let $\triangle{ABC}$ be an isosceles triangle with $AB=AC.$ Let $D$ be a point on the circumcircle of $\triangle{ABC}$ on minor arc $AB.$ Let $\overline{AD}$ intersect the extension of $\overline{BC}$ at $E.$ Let $F$ be the midpoint of segment $AC,$ and let $G$ be the intersection of $\overline{EF}$ and $\overline{AB}.$ Let the extension of $\overline{DG}$ intersect $\overline{AC}$ and the circumcircle of $\triangle{ABC}$ at $H$ and $I,$ respectively. Given that $DG=3, GH=5,$ and $HI=1,$ compute the length of $\overline{AE}.$
1 reply
lpieleanu
Apr 27, 2025
vanstraelen
Yesterday at 8:01 PM
J
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trigonometric functions
VivaanKam   9
N Yesterday at 7:12 PM by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
9 replies
VivaanKam
Tuesday at 8:29 PM
aok
Yesterday at 7:12 PM
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max area of regular hexagon inside equliateral (HOMC 2018 ind. sen2)
parmenides51   1
N Feb 2, 2020 by brainiacmaniac31
What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side $3$?

A. $3\sqrt7$ B. $\frac{3 \sqrt3}{2}$ C. $2\sqrt5$ D. $\frac{3\sqrt3}{8}$ E. $3\sqrt5$
1 reply
parmenides51
Feb 2, 2020
brainiacmaniac31
Feb 2, 2020
J
max area of regular hexagon inside equliateral (HOMC 2018 ind. sen2)
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Reveal topic tags
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parmenides51
30650 posts
parmenides51
#1 h  Feb 2, 2020, 5:54 PM • 2 Y
Y by Adventure10, abcabcpqrpqr
What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side $3$?

A. $3\sqrt7$ B. $\frac{3 \sqrt3}{2}$ C. $2\sqrt5$ D. $\frac{3\sqrt3}{8}$ E. $3\sqrt5$
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brainiacmaniac31
2170 posts
brainiacmaniac31
#2 h  Feb 2, 2020, 6:01 PM • 2 Y
Y by Adventure10, abcabcpqrpqr
[asy] import geometry; import olympiad; pair a,b,c; a=dir(90); b=dir(-30); c=dir(210); draw(a--b--c--a^^(2a+b)/3--(2a+c)/3^^(a+2b)/3--(a+2c)/3^^(b+2c)/3--(b+2a)/3^^(2b+a)/3--(2b+c)/3^^(c+2a)/3--(c+2b)/3^^(2c+a)/3--(2c+b)/3); [/asy]
Tile the triangle like so; the area of the largest hexagon is clearly the central $6$ triangles, which is $\frac{2}{3}$ of the area of the triangle. Thus, the answer is $\frac{2}{3}\left(\frac{9\sqrt3}{4}\right)=\boxed{\frac{3\sqrt3}{2}}$.
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