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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)!

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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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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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Inspired by giangtruong13
sqing   3
N 7 minutes ago by kokcio
Source: Own
Let $ a,b,c,d\geq 0 ,a-b+d=21 $ and $ a+3b+4c=101 $. Prove that
$$ 61\leq a+b+2c+d\leq \frac{265}{3}$$$$- \frac{2121}{2}\leq ab+bc-2cd+da\leq \frac{14045}{12}$$$$\frac{519506-7471\sqrt{7471}}{27}\leq ab+bc-2cd+3da\leq 33620$$
3 replies
sqing
Yesterday at 2:57 AM
kokcio
7 minutes ago
J
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Inequality with a,b,c
GeoMorocco   1
N 17 minutes ago by Natrium
Source: Morocco Training
Let $ a,b,c $ be positive real numbers such that : $ ab+bc+ca=3 $ . Prove that : $$\frac{\sqrt{1+a^2}}{1+ab}+\frac{\sqrt{1+b^2}}{1+bc}+\frac{\sqrt{1+c^2}}{1+ca}\ge \sqrt{\frac{3(a+b+c)}{2}}$$
1 reply
GeoMorocco
Yesterday at 10:05 PM
Natrium
17 minutes ago
J
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NEPAL TST 2025 DAY 2
Tony_stark0094   2
N 17 minutes ago by ThatApollo777
Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$.

If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.
2 replies
Tony_stark0094
Today at 8:40 AM
ThatApollo777
17 minutes ago
J
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IMO Shortlist 2013, Number Theory #1
lyukhson   149
N 23 minutes ago by SSS_123
Source: IMO Shortlist 2013, Number Theory #1
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
149 replies
lyukhson
Jul 10, 2014
SSS_123
23 minutes ago
J
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anglechasing, orthocenter , 2 angle bisectors and 3 midpoints related
parmenides51   0
Nov 22, 2020
Source: 2020 Bulgaria JBMO TST 2.2
Let $\vartriangle ABC$ be an acute triangle with orthocenter $H$. The angle bisectors of angles $CAH$ and $CBH$ intersect at point $Q$. The midpoint of the segments $AB$ and $CH$ are $M,N$ respectively.
a) Prove that the point $Q$ lies on the segment $MN$.
b) If the point $Q$ is the midpoint of the segment $MN$, then find $\angle ACB$
0 replies
parmenides51
Nov 22, 2020
0 replies
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anglechasing, orthocenter , 2 angle bisectors and 3 midpoints related
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Reveal topic tags
geometry
angle bisector
orthocenter
angles
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Source: 2020 Bulgaria JBMO TST 2.2
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parmenides51
30630 posts
parmenides51
#1 h  Nov 22, 2020, 12:18 AM • 1 Y
Y by Mango247
Let $\vartriangle ABC$ be an acute triangle with orthocenter $H$. The angle bisectors of angles $CAH$ and $CBH$ intersect at point $Q$. The midpoint of the segments $AB$ and $CH$ are $M,N$ respectively.
a) Prove that the point $Q$ lies on the segment $MN$.
b) If the point $Q$ is the midpoint of the segment $MN$, then find $\angle ACB$
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