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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
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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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Trivial fun Equilateral
ItzsleepyXD   4
N 4 minutes ago by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P1
Let $ABC$ be a scalene triangle with point $P$ and $Q$ on the plane such that $\triangle BPC , \triangle CQB$ is an equilateral . Let $AB$ intersect $CP$ and $CQ$ at $X$ and $Z$ respectively and $AC$ intersect $BP$ and $BQ$ at $Y$ and $W$ respectively .
Prove that $XY\parallel ZW$
4 replies
ItzsleepyXD
Yesterday at 9:05 AM
cursed_tangent1434
4 minutes ago
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Geometry Proof
Jackson0423   2
N 8 minutes ago by aidan0626
In triangle \( \triangle ABC \), point \( P \) on \( AB \) satisfies \( DB = BC \) and \( \angle DCA = 30^\circ \).
Let \( X \) be the point where the perpendicular from \( B \) to line \( DC \) meets the angle bisector of \( \angle BCA \).
Then, the relation \( AD \cdot DC = BD \cdot AX \) holds.

Prove that \( \triangle ABC \) is an isosceles triangle.
2 replies
Jackson0423
Yesterday at 4:17 PM
aidan0626
8 minutes ago
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Do not try to case bash lol
ItzsleepyXD   2
N 10 minutes ago by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P3
Let $n,d\geqslant 6$ be a positive integer such that $d\mid 6^{n!}+1$ .
Prove that $d>2n+6$ .
2 replies
ItzsleepyXD
Yesterday at 9:08 AM
cursed_tangent1434
10 minutes ago
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N lines cutting each other in the plane
M.J.Espinas   4
N 20 minutes ago by Lemmas
Source: Iranian Math Olympiad(Second Round 2016)
Let $l_1,l_2,l_3,...,L_n$ be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let $A$ be the intersection point of lines $l_i,l_j$. We call $A$ an "Interior Point" if there are points $C,D$ on $l_i$ and $E,F$ on $l_j$ such that $A$ is between $C,D$ and $E,F$. Prove that there are at least $\frac{(n-2)(n-3)}{2}$ Interior points.($n>2$)
note: by point here we mean the points which are intersection point of two of $l_1,l_2,...,l_n$.
4 replies
M.J.Espinas
May 5, 2016
Lemmas
20 minutes ago
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Find the domain and range of $f(x)=2-|x-5|.$
Vulch   0
4 hours ago
Find the domain and range of $f(x)=2-|x-5|.$
0 replies
Vulch
4 hours ago
0 replies
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Inequalities
sqing   5
N 5 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
5 hours ago
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Amc 10 mock
Mathsboy100   4
N 5 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
5 hours ago
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Inequalities
sqing   0
6 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
6 hours ago
0 replies
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Purple Comet High School Math Meet 2024 P1
franklin2013   3
N 6 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
Yesterday at 11:20 PM
codegirl2013
6 hours ago
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Cool vieta sum
Kempu33334   0
Yesterday at 11:44 PM
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
Yesterday at 11:44 PM
0 replies
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Inequality, tougher than it looks
tom-nowy   2
N Yesterday at 11:28 PM 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
Apr 29, 2025
pooh123
Yesterday at 11:28 PM
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Stylish Numbers
pedronis   3
N Yesterday at 10:58 PM 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
Yesterday at 10:58 PM
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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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Inspired by Noname23
sqing   0
Mar 29, 2025
Source: Own
Let $ a,b>0 $ and $ a+b=\sqrt{10}. $ Prove that
$$ (a^4+b^2+1)(b^4+a^2+1)\ge \frac{1521}{16}$$$$ (a^4+2b^2+1)(b^4+2a^2+1) \ge \frac{2401}{16}$$$$ (a^4+b+1)(b^4+a+1)\ge \frac{881+116\sqrt{10}}{16}$$$$ (a^4+b^3+1)(b^4+a^3+1)\ge \frac{1091+290\sqrt{10}}{16}$$
0 replies
sqing
Mar 29, 2025
0 replies
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Inspired by Noname23
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sqing
41911 posts
sqing
#1 h  Mar 29, 2025, 3:34 AM
Y by
Let $ a,b>0 $ and $ a+b=\sqrt{10}. $ Prove that
$$ (a^4+b^2+1)(b^4+a^2+1)\ge \frac{1521}{16}$$$$ (a^4+2b^2+1)(b^4+2a^2+1) \ge \frac{2401}{16}$$$$ (a^4+b+1)(b^4+a+1)\ge \frac{881+116\sqrt{10}}{16}$$$$ (a^4+b^3+1)(b^4+a^3+1)\ge \frac{1091+290\sqrt{10}}{16}$$
This post has been edited 1 time. Last edited by sqing, Mar 29, 2025, 3:50 AM
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