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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.

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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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a^2 + b^2 + c^2 \le 38 HOMC 2014 S Q13
parmenides51   5
N 13 minutes ago by sqing
Let $a, b,c$ satises the conditions $\begin{cases} 5 \ge a \ge b \ge c \ge 0 \\ a + b \le 8 \\ a + b + c = 10 \end{cases}$
Prove that $a^2 + b^2 + c^2 \le 38$
5 replies
parmenides51
Sep 10, 2019
sqing
13 minutes ago
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Ask mininum
TangenT-maTh-   0
18 minutes ago
Find the mininum value of function$f(x)=\cos^2 x-4\cos x-2\sqrt{3}\sin x$
0 replies
1 viewing
TangenT-maTh-
18 minutes ago
0 replies
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Inequalities
sqing   19
N 40 minutes ago by sqing
Let $ a,b>0 $ and $ \frac{1}{a}+\frac{1}{b}=1. $ Prove that
$$(a^2-a+1)(b^2-b+1) \geq 9$$$$ (a^2-a+b+1)(b^2-b+a+1) \geq 25$$Let $ a,b>0 $ and $ \frac{1}{a}+\frac{1}{b}=\frac{2}{3}. $ Prove that
$$(a+8)(a^2-a+b+2)(b^2-b+5)\geq1331$$$$(a+10)(a^2-a+b+4)(b^2-b+7)\geq2197$$
19 replies
sqing
Mar 10, 2025
sqing
40 minutes ago
J
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Inequalities
nhathhuyyp5c   1
N an hour ago by Noname23
Let $a,b,c$ be reals such that $0<a\leq b\leq c\leq 2$ and $\frac{1}{b}+\frac{2}{c}\geq 2$ and $\frac{1}{a}+\frac{1}{b}+\frac{2}{c}\geq 3$. Prove that $$a^3+b^3+c^3\leq 10$$
1 reply
nhathhuyyp5c
Mar 10, 2025
Noname23
an hour ago
J
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Geometry problem
Bibi_math   0
2 hours ago
Can someone solve this problem?
Let ABC be a triangle. Let C1 be point on midle on AB and B1 be point on midle on side AC. Let T is intersection of AA1 and BB1. Prove that АC1TB1 is chord quadrilateral if and only if the lenght of BC is AT multiplied with the root of 3.

Sorry, my english is not good enough.
0 replies
Bibi_math
2 hours ago
0 replies
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Progressions
dsrmurthy2011   1
N 2 hours ago by vanstraelen
If a, b, c are in H. P
Then √bc/√b+√c , √ac/√c+√a , √ab/ √a+√b are in which progression. Prove it
1 reply
dsrmurthy2011
Yesterday at 6:34 PM
vanstraelen
2 hours ago
J
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Can you tell the steps to solve this problem?
sky1983   2
N 3 hours ago by vanstraelen
How to solve this problem?
2 replies
sky1983
4 hours ago
vanstraelen
3 hours ago
J
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number theory
IOQMaspirant   1
N 4 hours ago by SomeonecoolLovesMaths
gcd(a,b) = 1
ax + by = gcd(a,b) by beizouts lemma
is there any bound in x or y or I can just do the following
a(a+b) + b(a+b) = gcd(a,b)
(a^2 + b^2) + 2(ab) = gcd(a,b)
gcd(a^2 + b^2, ab ) = gcd(a,b) [ gcd(a^2 + b^2, ab) = x(a^2 + b^2) + y(ab)]

if there is something wrong in this please point it out and please help me clear my doubts about beizouts lemma as I am reading from a book so maybe I have misinterpreted something
am I using beizouts lemma properly?
if no then how to use it
1 reply
IOQMaspirant
Today at 4:23 AM
SomeonecoolLovesMaths
4 hours ago
J
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Inequalities
sqing   1
N Today at 5:31 AM by sqing
Let $ a, b,c\geq 0 $ and $a^2+b+ c=1.$ Prove that
$$ \frac{a} {\frac65-a^2}+\frac{b} {\frac65-b^2}+\frac{c} {\frac65-c^2} \geq \frac{20} {19}$$$$ \frac{a} {\frac32-a^2}+\frac{b} {\frac32-b^2}+\frac{c} {\frac32-c^2} \geq \frac{4} {5}$$$$ \frac{a} {\frac72-a^2}+\frac{b} {\frac72-b^2}+\frac{c} {\frac72-c^2} \geq \frac{4} {13}$$$$ \frac{a} {\frac{18}{5}-a^2}+\frac{b} {\frac{18}{5}-b^2}+\frac{c} {\frac{18}{5}-c^2} \geq \frac{20} {67}$$
1 reply
sqing
Mar 9, 2025
sqing
Today at 5:31 AM
J
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reals x,z; x^2+5xz+z^2=7, x^2z+xz^2=2, x+z/=2; (6xz)^2? (Puerto Rico TST 2024.8)
Equinox8   2
N Today at 3:04 AM by rchokler
Let $x$ and $z$ be real numbers such that $$x^2+5xz+z^2=7$$$$x^2z+xz^2=2$$
If $x+z\neq2$, determine the value of $(6xz)^2$.
2 replies
Equinox8
Yesterday at 5:57 AM
rchokler
Today at 3:04 AM
J
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Inequalities
sqing   4
N Today at 1:32 AM by sqing
Let $ a,b $ be reals such that $ a^2+b^2=1. $ Prove that
$$\sqrt{ a+1}+ \sqrt{ b+1}+k \sqrt{ 2ab+1} \leq \sqrt{2}(k+ \sqrt{ 2+\sqrt{2}})$$Where $ k>0. $
4 replies
sqing
Mar 9, 2025
sqing
Today at 1:32 AM
J
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Mathematica
Wiselady   6
N Today at 1:06 AM by Wiselady
Let ABC be any triangle. Let side BC pass through point C to point D such that CD=AC. Let P be the second intersection point of the circumscribed circle ACD with the circle with BC as the diameter. Let BP and AC meet at point E and let CP and AB meet at point F. Prove that D, E, F lie on the same straight line.
6 replies
Wiselady
Yesterday at 2:23 PM
Wiselady
Today at 1:06 AM
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can i see yo dih and can u make it super veiny
hexuhdecimal   0
Yesterday at 11:41 PM
Let $\triangle{ABC}$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $M$ be the reflection of $O$ over $AC$. Given that $\angle{BAH} = 30^{\circ}$ and the circumradius of $\triangle{ABC}$ is $10$, the area of triangle $\triangle{AOM}$ can be written as $a\sqrt{b}$. Find $a+b$.
0 replies
hexuhdecimal
Yesterday at 11:41 PM
0 replies
J
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Diophantine
Musashi123   4
N Yesterday at 11:24 PM by hashbrown2009
Find positive interger x,y such that
y^2+y=x^3+x
4 replies
Musashi123
Mar 11, 2025
hashbrown2009
Yesterday at 11:24 PM
J
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J
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Inequalities
sqing   1
N Today at 5:31 AM by sqing
Let $ a, b,c\geq 0 $ and $a^2+b+ c=1.$ Prove that
$$ \frac{a} {\frac65-a^2}+\frac{b} {\frac65-b^2}+\frac{c} {\frac65-c^2} \geq \frac{20} {19}$$$$ \frac{a} {\frac32-a^2}+\frac{b} {\frac32-b^2}+\frac{c} {\frac32-c^2} \geq \frac{4} {5}$$$$ \frac{a} {\frac72-a^2}+\frac{b} {\frac72-b^2}+\frac{c} {\frac72-c^2} \geq \frac{4} {13}$$$$ \frac{a} {\frac{18}{5}-a^2}+\frac{b} {\frac{18}{5}-b^2}+\frac{c} {\frac{18}{5}-c^2} \geq \frac{20} {67}$$
1 reply
sqing
Mar 9, 2025
sqing
Today at 5:31 AM
J
Inequalities
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Reveal topic tags
inequalities
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G H BBookmark kLocked kLocked NReply
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sqing
40978 posts
sqing
#1 h  Mar 9, 2025, 2:24 PM
Y by
Let $ a, b,c\geq 0 $ and $a^2+b+ c=1.$ Prove that
$$ \frac{a} {\frac65-a^2}+\frac{b} {\frac65-b^2}+\frac{c} {\frac65-c^2} \geq \frac{20} {19}$$$$ \frac{a} {\frac32-a^2}+\frac{b} {\frac32-b^2}+\frac{c} {\frac32-c^2} \geq \frac{4} {5}$$$$ \frac{a} {\frac72-a^2}+\frac{b} {\frac72-b^2}+\frac{c} {\frac72-c^2} \geq \frac{4} {13}$$$$ \frac{a} {\frac{18}{5}-a^2}+\frac{b} {\frac{18}{5}-b^2}+\frac{c} {\frac{18}{5}-c^2} \geq \frac{20} {67}$$
Z K Y
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sqing
40978 posts
sqing
#2 h  Today at 5:31 AM
Y by
Let $a,b> 0$ and $a^2+b^2+a+b+ab=5 .$ Prove that
$$\frac{a+2}{b+3}+\frac{b+2}{a+b+2} +\frac{a+b+1}{a+3} \ge \frac{9}{4}$$
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