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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 2, 2025
0 replies
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Trash Trig Sum
P_Groudon   3
N 5 hours ago by P_Groudon
Find the smallest positive integer $n$ such that $$\sum_{k=0}^{298}\sin(k^2 + 2k + 2)\sin(2k + 2) = \frac{\cos(1) - \cos(n)}{2},$$where degrees are used.
3 replies
P_Groudon
Yesterday at 5:34 PM
P_Groudon
5 hours ago
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Geometry problem for enthusiasts
Raul_S_Baz   1
N Yesterday at 5:45 PM by Raul_S_Baz
IMAGE
1 reply
Raul_S_Baz
Yesterday at 4:51 PM
Raul_S_Baz
Yesterday at 5:45 PM
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Inequalities
sqing   3
N Yesterday at 4:24 PM by DAVROS
Let $a,b,c\ge \frac{1}{2}$ and $\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\le 1. $ Prove that
$$a+b+c\geq 2$$Let $a,b,c\ge \frac{1}{2}$ and $ \left(a+\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(a+\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\le \frac{9}{2}. $ Prove that
$$a^2+b^2+c^2\geq 1$$Let $a,b\ge \frac{1}{2}$ and $ \left( \frac{1}{a}-\frac{1}{b}+2\right)\left( \frac{1}{b}-\frac{1}{a}+2\right) \le \frac{20}{9}. $ Prove that
$$ a+b\geq 2$$Let $a,b\ge \frac{1}{2}$ and $a^2+b^2=1. $ Prove that
$$\left(\frac{2}{a}+\frac{1}{b}-1\right)\left(\frac{2}{a}-\frac{1}{b}+1\right)\ge \frac{13}{3}$$
3 replies
sqing
Mar 15, 2025
DAVROS
Yesterday at 4:24 PM
J
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Classic geometry problem
Raul_S_Baz   2
N Yesterday at 4:10 PM by Raul_S_Baz
IMAGE
2 replies
Raul_S_Baz
Mar 4, 2025
Raul_S_Baz
Yesterday at 4:10 PM
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No more topics!
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rhombus, line from bisector x circumcircle points (Puerto Rico TST 2024.7)
Equinox8   1
N Mar 15, 2025 by pooh123
Let $I$ be the incenter of $\triangle{ABC}$. Angle bisectors $BI$ and $CI$ intersect the circumcircle of $\triangle{ABC}$ at points $S$ and $T$, respectively ($S \neq B$ and $T \neq C$). Line $ST$ intersects sides $AB$ and $AC$ at points $K$ and $L$, respectively. Prove that quadrilateral $AKIL$ is a rhombus.

Note: A rhombus is a parallelogram with all sides equal.
1 reply
Equinox8
Mar 12, 2025
pooh123
Mar 15, 2025
J
rhombus, line from bisector x circumcircle points (Puerto Rico TST 2024.7)
G H J
Reveal topic tags
geometry
rhombus
angle bisector
circumcircle
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Equinox8
1730 posts
Equinox8
#1 h  Mar 12, 2025, 5:47 AM
Y by
Let $I$ be the incenter of $\triangle{ABC}$. Angle bisectors $BI$ and $CI$ intersect the circumcircle of $\triangle{ABC}$ at points $S$ and $T$, respectively ($S \neq B$ and $T \neq C$). Line $ST$ intersects sides $AB$ and $AC$ at points $K$ and $L$, respectively. Prove that quadrilateral $AKIL$ is a rhombus.

Note: A rhombus is a parallelogram with all sides equal.
This post has been edited 1 time. Last edited by Equinox8, Mar 12, 2025, 5:48 AM
Reason: \triangle no longer smooshed
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pooh123
2 posts
pooh123
#2 h  Mar 15, 2025, 1:38 PM
Y by
(Sorry for messy typing, not in latex)
Simple. A quick angle chasing shows that S and T are midpoints of arcs AC and AB respectively.
So ∠IBK=∠ABS=1/2 arc(AS)=1/2 arc (CS)=∠CTS=∠ITK, therefore B, I, K, T are concyclic.
Thus ∠IKB=∠ITB=∠BTC=∠BAC so IK||AC. Similarly, we also have IL||AB so AKIL is a parallelogram.
Since AI is the angle bisector of ∠BAC, we have AKIL is a rhombus.
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