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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 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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Inequalities
sqing   2
N 3 minutes ago by sqing
Let $ a,b,c> 0 $ and $ \frac{a}{a^2+ab+c}+\frac{b}{b^2+bc+a}+\frac{c}{c^2+ca+b} \geq 1$. Prove that
$$ a+b+c\leq 3 $$
2 replies
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sqing
Yesterday at 3:52 AM
sqing
3 minutes ago
J
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Inequalities
sqing   0
28 minutes ago
Let $a,b$ be real numbers such that $ a^2+b^2+a^3 +b^3=4 . $ Prove that
$$a+b \leq 2$$Let $a,b$ be real numbers such that $a+b + a^2+b^2+a^3 +b^3=6 . $ Prove that
$$a+b \leq 2$$
0 replies
sqing
28 minutes ago
0 replies
J
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that statement is true
pennypc123456789   3
N an hour ago by sqing
we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8 $ , then we can deduce $a^2+b^2$ \le 2 $ ?
3 replies
pennypc123456789
Mar 23, 2025
sqing
an hour ago
J
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Distance vs time swimming problem
smalkaram_3549   1
N 2 hours ago by Lankou
How should I approach a problem where we deal with velocities becoming negative and stuff. I know that they both travel 3 Lengths of the pool before meeting a second time.
1 reply
smalkaram_3549
Today at 2:57 AM
Lankou
2 hours ago
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No more topics!
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2 altitudes tangent to circumcircle 2009 Estonia Open Senior 2.4
parmenides51   1
N Oct 3, 2020 by natmath
a) An altitude of a triangle is also a tangent to its circumcircle. Prove that some angle of the triangle is larger than $90^o$ but smaller than $135^o$.
b) Some two altitudes of the triangle are both tangents to its circumcircle. Find the angles of the triangle.
1 reply
parmenides51
Oct 3, 2020
natmath
Oct 3, 2020
J
2 altitudes tangent to circumcircle 2009 Estonia Open Senior 2.4
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Reveal topic tags
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altitude
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parmenides51
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parmenides51
#1 h  Oct 3, 2020, 9:46 PM
Y by
a) An altitude of a triangle is also a tangent to its circumcircle. Prove that some angle of the triangle is larger than $90^o$ but smaller than $135^o$.
b) Some two altitudes of the triangle are both tangents to its circumcircle. Find the angles of the triangle.
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natmath
8219 posts
natmath
#2 h  Oct 3, 2020, 11:20 PM
Y by
A) So pick an arbitrary point, $A$, on the circle and draw the tangent. Now pick a line perpendicular to the tangent, the intersections of this line and the circle will be $B,C$. We need to prove that one of the arcs is between 180 and 270. If $C$ is further away from $A$, then it is clear that major arc $AC$ follows these conditions.

B) It's pretty clear that the triangle must be isoceles. Say the congruent angles are $A$ and $B$ on a circle. Then $C$ is on the midpoint of minor arc $AB$. And the line through $B$ and $C$ is perpendicular to the tangent at $A$ (vice versa for $B$ because of symmetry).
After some angle chasing, we get the angle between the tangents at $A$ and $B$ is $60^\circ$, so the angles are $120,30,30$
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