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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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Inequalities
sqing   4
N 2 hours ago by sqing
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
$$\left(\frac{2}{a}+\frac{1}{b}-1\right)\left(\frac{2}{a}-\frac{1}{b}+1\right)\ge \frac{13}{3}$$
4 replies
sqing
Mar 15, 2025
sqing
2 hours ago
J
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Binomial Summation
Saucepan_man02   0
4 hours ago
Given that: $l_n = \sum_{r=1}^{n-1} \frac{2r^2-r(n-2)+1}{(n-r) \binom{n}{r}}$, find $l_{15}$.
0 replies
Saucepan_man02
4 hours ago
0 replies
J
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Inequality with integers and indices
Michael Niland   5
N 4 hours ago by ohiorizzler1434
Without using a calculator, show that $2^{\frac{1}{2}}$ $< 3^{\frac{1}{3}}$, and for a positive integer $n\geq3$, prove that

$n^{\frac{1}{n}}$ $>(n+1)^{\frac{1}{n+1}}$.
5 replies
Michael Niland
Yesterday at 6:52 AM
ohiorizzler1434
4 hours ago
J
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Quadratic Equation Problems
Saucepan_man02   4
N 5 hours ago by Saucepan_man02
P1) Let $\alpha < \beta < \gamma$ be the roots of $ax^3+bx^2+cx+d=0$ with $7a+3b>0, 3a+2b<0$ and $\alpha(\beta+1)+\beta(\gamma+1)+\gamma(1+\alpha)=0$.
If [.] denotes the greatest integer function, then the number of possible values of $[|3 \alpha |] + [|6 \beta|] + [|9 \gamma |]$ is equal to
4 replies
Saucepan_man02
Mar 3, 2025
Saucepan_man02
5 hours ago
J
No more topics!
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Inequality with roots of cubic
Michael Niland   0
Yesterday at 7:23 AM
The equation $x^3+px+q =0$ has real roots $ a_1 ,a_2 , a_3 $ where $a_1 \leq a_2 \leq a_3$.

Similarly the equation $x^3 +rx +s=0$ has real roots $ b_1, b_2, b_3 $ where $b_1 \leq b_2,\leq b_3$.

Prove that if $ \frac{a_1}{b_1} \leq \frac{a_2}{b_2} \leq \frac{a_3}{b_3}$ (s non zero), then $(\frac{p}{r})^3 =(\frac{q}{s})^2$
0 replies
Michael Niland
Yesterday at 7:23 AM
0 replies
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Inequality with roots of cubic
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Michael Niland
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Michael Niland
#1 h  Yesterday at 7:23 AM
Y by
The equation $x^3+px+q =0$ has real roots $ a_1 ,a_2 , a_3 $ where $a_1 \leq a_2 \leq a_3$.

Similarly the equation $x^3 +rx +s=0$ has real roots $ b_1, b_2, b_3 $ where $b_1 \leq b_2,\leq b_3$.

Prove that if $ \frac{a_1}{b_1} \leq \frac{a_2}{b_2} \leq \frac{a_3}{b_3}$ (s non zero), then $(\frac{p}{r})^3 =(\frac{q}{s})^2$
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