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!
geometry
B
No tags match your search
M
G
Topic
First Poster
Last Poster
H
one cyclic formed by two cyclic
CrazyInMath   10
N a few seconds ago by InterLoop
Source: EGMO 2025/3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
10 replies
+4 w
CrazyInMath
3 hours ago
InterLoop
a few seconds ago
J
H
Problem of power of 2
nhathhuyyp5c   0
2 hours ago
Let $a,b,c$ be odd positive integers such that $a>b$ and both $a^2+c$ and $b^2+c$ are powers of 2. Prove that $ac>2b^2.$
0 replies
nhathhuyyp5c
2 hours ago
0 replies
J
H
geometry parabola problem
smalkaram_3549   10
N 2 hours ago by ReticulatedPython
How would you solve this without using calculus?
10 replies
smalkaram_3549
Friday at 9:52 PM
ReticulatedPython
2 hours ago
J
H
number of contestants
Ecrin_eren   0
2 hours ago




In a mathematics competition with 10 questions, every group of 3 questions is solved by more than half of the participants. No participant has solved any 7 questions. Exactly one participant has solved 6 questions. What could be the number of participants?
1994
2000
2013
2048
None of the above
0 replies
Ecrin_eren
2 hours ago
0 replies
J
H
Inequalities
sqing   9
N 3 hours ago by sqing
Let $ a,b,c,d\geq 0 ,a-b+d=21 $ and $ a+3b+4c=101 $. Prove that
$$ - \frac{1681}{3}\leq ab - cd \leq 820$$$$ - \frac{16564}{9}\leq ac -bd \leq 420$$$$ - \frac{10201}{48}\leq ad- bc \leq\frac{1681}{3}$$
9 replies
sqing
Apr 11, 2025
sqing
3 hours ago
J
H
Cyclic quadrilateral from midpoints and perpendicular from center
Kizaruno   0
3 hours ago
Let triangle $ABC$ be inscribed in a circle with center $O$. A line $d$ intersects sides $AB$ and $AC$ at points $E$ and $D$, respectively. Let $M$, $N$, and $P$ be the midpoints of segments $BD$, $CE$, and $DE$, respectively. Let $Q$ be the foot of the perpendicular from $O$ to line $DE$.

Prove that the four points $M$, $N$, $P$, and $Q$ lie on a circle.
0 replies
Kizaruno
3 hours ago
0 replies
J
H
Inequalities
sqing   1
N 4 hours ago by sqing
Let $ a,b,c\in [0,1] $ . Prove that
$$(a+b+c)\left(\frac{1}{a^2+3}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{19}{4}$$$$(a+b+c)\left(\frac{1}{a^2+ 4}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{23}{5}$$$$(a+b+c)\left(\frac{1}{a^2+ \frac{5}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{34}{7}$$$$(a+b+c)\left(\frac{1}{a^2+ \frac{7}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{14}{3}$$
1 reply
sqing
Yesterday at 3:33 AM
sqing
4 hours ago
J
H
Inequalities
sqing   0
4 hours ago
Let $ a,b\geq 0 $ and $\frac{a+1}{a^2+b}+\frac{b+3}{b^2+a}=1. $ Prove that
$$a+3b-ab\leq 9$$$$a+b-ab\leq 2+\sqrt{5}$$$$a+3b+ab\leq \frac{13+5\sqrt{17}}{2}$$
0 replies
sqing
4 hours ago
0 replies
J
H
inequality
Ecrin_eren   1
N 5 hours ago by sqing
Let a, b, c be positive real numbers. What is the minimum value of the following expression?

(18ca + 12bc + ab)² / (36abc² + 2ab²c + 3a²bc)
1 reply
Ecrin_eren
5 hours ago
sqing
5 hours ago
J
H
A rather difficult question
BeautifulMath0926   0
5 hours ago
I got a difficult equation for users to solve:
Find all functions f: R to R, so that to all real numbers x and y,
1+f(x)f(y)=f(x+y)+f(xy)+xy(x+y-2) holds.
0 replies
BeautifulMath0926
5 hours ago
0 replies
J
H
Inequalities
sqing   10
N 5 hours ago by sqing
Let $ a,b,c $ be real numbers so that $ a+2b+3c=2 $ and $ 2ab+6bc+3ca =1. $ Show that
$$-\frac{1}{6} \leq ab-bc+ ca\leq \frac{1}{2}$$$$\frac{5-\sqrt{61}}{9} \leq a-b+c\leq \frac{5+\sqrt{61}}{9} $$
10 replies
sqing
Apr 9, 2025
sqing
5 hours ago
J
H
J
H
The incircle problem
danil_e   1
N Dec 1, 2023 by ancamagelqueme
Given triangle \(ABC\) inscribed in circle \((O)\) and circumscribed about circle \((I)\). A circle passing through \(B\) and \(C\) is tangent to \((I)\) at \(N_a\) and intersects \(AB\) and \(AC\) at \(A_c\) and \(A_b\) respectively. Similarly, define \(B_c, B_a, N_b\) and \(C_b, C_a, N_c\) correspondingly. Let \(XYZ\) be the triangle formed by the radical axis of circles \((N_aBC)\), \((N_bCA)\), \((N_cBA)\) (as shown in the figure); \(MNP\) is the triangle formed by the intersection of lines \(A_cC_a\), \(B_cC_b\), \(A_bB_a\) (as shown in the figure).

a) Prove that: Triangle \(XYZ\) and triangle \(MNP\) are in perspective axially.
b) Let \(S\) be the point of concurrence of \(XM\), \(YN\), \(ZP\). Prove that: \(S\), \(I\), \(O\) are collinear.
1 reply
danil_e
Dec 1, 2023
ancamagelqueme
Dec 1, 2023
J
The incircle problem
G H J
Reveal topic tags
geometry
incircle
L
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
danil_e
25 posts
danil_e
#1 h  Dec 1, 2023, 4:23 PM
Y by
Given triangle \(ABC\) inscribed in circle \((O)\) and circumscribed about circle \((I)\). A circle passing through \(B\) and \(C\) is tangent to \((I)\) at \(N_a\) and intersects \(AB\) and \(AC\) at \(A_c\) and \(A_b\) respectively. Similarly, define \(B_c, B_a, N_b\) and \(C_b, C_a, N_c\) correspondingly. Let \(XYZ\) be the triangle formed by the radical axis of circles \((N_aBC)\), \((N_bCA)\), \((N_cBA)\) (as shown in the figure); \(MNP\) is the triangle formed by the intersection of lines \(A_cC_a\), \(B_cC_b\), \(A_bB_a\) (as shown in the figure).

a) Prove that: Triangle \(XYZ\) and triangle \(MNP\) are in perspective axially.
b) Let \(S\) be the point of concurrence of \(XM\), \(YN\), \(ZP\). Prove that: \(S\), \(I\), \(O\) are collinear.
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ancamagelqueme
104 posts
ancamagelqueme
#2 h  Dec 1, 2023, 11:44 PM
Y by
The point $S$ is NOT aligned with $I=X_1$ and $O=X_3$. If it is on the line $X_{57}X_{2346}$.

The barycentric coordinates of $S$, with respect to $ABC$, are given by the triangle center function

f(a,b,c)= a (a+b-c) (a-b+c)(a(b+c)-(b-c)^2)((b-c)^6 (b+c)^2 (b^2-3 b c+c^2)-2 (b-c)^4 (4 b^5-b^4 c-11 b^3 c^2-11 b^2 c^3-b c^4+4 c^5) a+(b-c)^2 (27 b^6+4 b^5 c-39 b^4 c^2-112 b^3 c^3-39 b^2 c^4+4 b c^5+27 c^6) a^2-2 (b-c)^2 (24 b^5+55 b^4 c+97 b^3 c^2+97 b^2 c^3+55 b c^4+24 c^5) a^3+2 (21 b^6+56 b^5 c+141 b^4 c^2+140 b^3 c^3+141 b^2 c^4+56 b c^5+21 c^6) a^4-2 b c (49 b^3+167 b^2 c+167 b c^2+49 c^3) a^5-2 (21 b^4+7 b^3 c-34 b^2 c^2+7 b c^3+21 c^4) a^6+2 (24 b^3+35 b^2 c+35 b c^2+24 c^3) a^7+(-27 b^2-41 b c-27 c^2) a^8+8 (b+c) a^9-a^10)
Z K Y
N Quick Reply
G
H
=
a