Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Stay ahead of learning milestones! Enroll in a class over the summer!
Pythagorean Theorem
B
No tags match your search
M
G
Topic
First Poster
Last Poster
H
Trigonometry article for geometry
xytunghoanh   0
an hour ago
Does anyone have any articles on using trigonometry to prove geometry problems (Law of Sines, Ceva's Theorem in trigonometric form,..) that they can share with me?
Thanks!
0 replies
1 viewing
xytunghoanh
an hour ago
0 replies
J
H
centroid lies outside of triangle (not clickbait)
Scilyse   1
N an hour ago by LoloChen
Source: 数之谜 January (CHN TST Mock) Problem 5
Let $P$ be a convex polygon with centroid $G$, and let $\mathcal P$ be the set of vertices of $P$. Let $\mathcal X$ be the set of triangles with vertices all in $\mathcal P$. We sort the elements $\triangle ABC$ of $\mathcal X$ into the following three types:
[list]
[*] (Type 1) $G$ lies in the strict interior of $\triangle ABC$; let $\mathcal A$ be the set of triangles of this type.
[*] (Type 2) $G$ lies in the strict exterior of $\triangle ABC$; let $\mathcal B$ be the set of triangles of this type.
[*] (Type 3) $G$ lies on the boundary of $\triangle ABC$.
[/list]
For any triangle $T$, denote by $S_T$ the area of $T$. Prove that \[\sum_{T \in \mathcal A} S_T \geq \sum_{T \in \mathcal B} S_T.\]
1 reply
1 viewing
Scilyse
Jan 26, 2025
LoloChen
an hour ago
J
H
Function equation
LeDuonggg   1
N an hour ago by luutrongphuc
Find all functions $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$ , such that for all $x,y>0$:
\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]
1 reply
LeDuonggg
Yesterday at 2:59 PM
luutrongphuc
an hour ago
J
H
4 lines concurrent
Zavyk09   6
N an hour ago by hectorleo123
Source: Homework
Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$. $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$.
6 replies
Zavyk09
Apr 9, 2025
hectorleo123
an hour ago
J
H
No More than √㏑x㏑㏑x Digits
EthanWYX2009   4
N 2 hours ago by tom-nowy
Source: 2024 April 谜之竞赛-3
Let $f(x)\in\mathbb Z[x]$ have positive integer leading coefficient. Show that there exists infinte positive integer $x,$ such that the number of digit that doesn'r equal to $9$ is no more than $\mathcal O(\sqrt{\ln x\ln\ln x}).$

Created by Chunji Wang, Zhenyu Dong
4 replies
EthanWYX2009
Mar 24, 2025
tom-nowy
2 hours ago
J
H
Old hard problem
ItzsleepyXD   1
N 2 hours ago by ItzsleepyXD
Source: IDK
Let $ABC$ be a triangle and let $O$ be its circumcenter and $I$ its incenter.
Let $P$ be the radical center of its three mixtilinears and let $Q$ be the isogonal conjugate of $P$.
Let $G$ be the Gergonne point of the triangle $ABC$.
Prove that line $QG$ is parallel with line $OI$ .
1 reply
ItzsleepyXD
Apr 25, 2025
ItzsleepyXD
2 hours ago
J
H
Existence of a solution of a diophantine equation
syk0526   5
N 2 hours ago by cursed_tangent1434
Source: North Korea Team Selection Test 2013 #6
Show that $ x^3 + x+ a^2 = y^2 $ has at least one pair of positive integer solution $ (x,y) $ for each positive integer $ a $.
5 replies
syk0526
May 17, 2014
cursed_tangent1434
2 hours ago
J
H
Inequality with 3 variables
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq 0 ,a^3b^3+b^3c^3+c^3a^3+2abc\geq 1 . $ Prove that$$a+b+c\geq 2 $$Let $ a,b,c\geq 0 ,a^3b^3+b^3c^3+c^3a^3+6abc\geq 9 . $ Prove that$$a+b+c\geq 2\sqrt 3 $$Let $ a,b,c\geq 0 ,a^3b+b^3c+c^3a+6abc\geq 9 . $ Prove that$$a+b+c\geq 3 $$Let $ a,b,c\geq 0 ,a^3b+b^3c+c^3a+3abc\geq 3 . $ Prove that$$a+b+c\geq \frac{4}{\sqrt 3} $$
0 replies
sqing
2 hours ago
0 replies
J
H
Inequality with 3 variables and a special condition
Nuran2010   5
N 2 hours ago by sqing
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$.
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$.

Determine the equality case.
5 replies
Nuran2010
Apr 29, 2025
sqing
2 hours ago
J
H
Chain of floors
Assassino9931   0
3 hours ago
Source: Vojtech Jarnik IMC 2025, Category I, P2
Determine all real numbers $x>1$ such that
\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]for any positive integer $n$.
0 replies
Assassino9931
3 hours ago
0 replies
J
a