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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
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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Fibonacci...?
Jackson0423   1
N 9 minutes ago by KAME06
The sequence \( F \) is defined by \( F_0 = F_1 = 2025 \) and for all positive integers \( n \geq 2 \), \( F_n = F_{n-1} + F_{n-2} \). Show that for every positive integer \( k \), there exists a suitable positive integer \( j \) such that \( F_j \) is a multiple of \( k \).
1 reply
Jackson0423
35 minutes ago
KAME06
9 minutes ago
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4 concyclic points
buzzychaoz   18
N 34 minutes ago by bjump
Source: Japan Mathematical Olympiad Finals 2015 Q4
Scalene triangle $ABC$ has circumcircle $\Gamma$ and incenter $I$. The incircle of triangle $ABC$ touches side $AB,AC$ at $D,E$ respectively. Circumcircle of triangle $BEI$ intersects $\Gamma$ again at $P$ distinct from $B$, circumcircle of triangle $CDI$ intersects $\Gamma$ again at $Q$ distinct from $C$. Prove that the $4$ points $D,E,P,Q$ are concyclic.
18 replies
1 viewing
buzzychaoz
Apr 1, 2016
bjump
34 minutes ago
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angles in triangle
AndrewTom   33
N 37 minutes ago by zuat.e
Source: BrMO 2012/13 Round 2
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.
33 replies
AndrewTom
Feb 1, 2013
zuat.e
37 minutes ago
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Geometry Proof
Jackson0423   0
41 minutes ago
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.
0 replies
Jackson0423
41 minutes ago
0 replies
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Algebraic Manipulation
Darealzolt   1
N 2 hours ago by Soupboy0
Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$.
1 reply
Darealzolt
4 hours ago
Soupboy0
2 hours ago
J
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BrUMO 2025 Team Round Problem 13
lpieleanu   1
N 2 hours ago by vanstraelen
Let $\omega$ be a circle, and let a line $\ell$ intersect $\omega$ at two points, $P$ and $Q.$ Circles $\omega_1$ and $\omega_2$ are internally tangent to $\omega$ at points $X$ and $Y,$ respectively, and both are tangent to $\ell$ at a common point $D.$ Similarly, circles $\omega_3$ and $\omega_4$ are externally tangent to $\omega$ at $X$ and $Y,$ respectively, and are tangent to $\ell$ at points $E$ and $F,$ respectively.

Given that the radius of $\omega$ is $13,$ the segment $\overline{PQ}$ has a length of $24,$ and $YD=YE,$ find the length of segment $\overline{YF}.$
1 reply
lpieleanu
Apr 27, 2025
vanstraelen
2 hours ago
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Inequlities
sqing   33
N 3 hours ago by sqing
Let $ a,b,c\geq 0 $ and $ a^2+ab+bc+ca=3 .$ Prove that$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$$$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$$
33 replies
sqing
Jul 19, 2024
sqing
3 hours ago
J
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Very tasteful inequality
tom-nowy   1
N 3 hours ago by sqing
Let $a,b,c \in (-1,1)$. Prove that $$(a+b+c)^2+3>(ab+bc+ca)^2+3(abc)^2.$$
1 reply
tom-nowy
Today at 10:47 AM
sqing
3 hours ago
J
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Inequalities
sqing   8
N 3 hours ago by sqing
Let $x\in(-1,1). $ Prove that
$$ \dfrac{1}{\sqrt{1-x^2}} + \dfrac{1}{2+ x^2} \geq \dfrac{3}{2}$$$$ \dfrac{2}{\sqrt{1-x^2}} + \dfrac{1}{1+x^2} \geq 3$$
8 replies
sqing
Apr 26, 2025
sqing
3 hours ago
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đề hsg toán
akquysimpgenyabikho   1
N 5 hours ago by Lankou
làm ơn giúp tôi giải đề hsg

1 reply
akquysimpgenyabikho
Apr 27, 2025
Lankou
5 hours ago
J
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Inequalities
sqing   2
N Today at 10:05 AM by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
2 replies
sqing
Jul 12, 2024
sqing
Today at 10:05 AM
J
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9 Physical or online
wimpykid   0
Today at 6:49 AM
Do you think the AoPS print books or the online books are better?

0 replies
wimpykid
Today at 6:49 AM
0 replies
J
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Three variables inequality
Headhunter   6
N Today at 6:08 AM by lbh_qys
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
6 replies
Headhunter
Apr 20, 2025
lbh_qys
Today at 6:08 AM
J
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Sequence
lgx57   8
N Today at 5:08 AM by Vivaandax
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
8 replies
lgx57
Apr 27, 2025
Vivaandax
Today at 5:08 AM
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J
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Estonian Math Competitions 2006/2007
STARS   0
Jul 29, 2008
Source: Finalround Problem 9
A circle passing through the endpoints of the leg AB of an isosceles triangle ABC
intersects the base BC in point P. A line tangent to the circle in point B intersects the circumcircle of ABC in point Q. Prove that P lies on line AQ if and only if AQ and BC are perpendicular.
0 replies
STARS
Jul 29, 2008
0 replies
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Estonian Math Competitions 2006/2007
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Reveal topic tags
geometry
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geometry unsolved
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G H BBookmark kLocked kLocked NReply
Source: Finalround Problem 9
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STARS
84 posts
STARS
#1 h  Jul 29, 2008, 8:57 AM • 1 Y
Y by Adventure10
A circle passing through the endpoints of the leg AB of an isosceles triangle ABC
intersects the base BC in point P. A line tangent to the circle in point B intersects the circumcircle of ABC in point Q. Prove that P lies on line AQ if and only if AQ and BC are perpendicular.
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