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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.

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[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.
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[*]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!
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0 replies
jlacosta
Mar 2, 2025
0 replies
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Checkerboard
Ecrin_eren   1
N 2 minutes ago by Ecrin_eren
On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)
1 reply
Ecrin_eren
Today at 5:20 AM
Ecrin_eren
2 minutes ago
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BD tangent to (MDE) , rhombus ABCD with <DCB=60^o
parmenides51   1
N 25 minutes ago by vanstraelen
Source: 2021 Germany R4 10.6 https://artofproblemsolving.com/community/c3208025_
Let a rhombus $ABCD$ with $|\angle DCB| = 60^o$ be given . On the extension of the segment $\overline{CD}$ beyond $D$, a point $E$ is chosen arbitrarily. Let the line through $E$ and $A$ intersect the line $BC$ at the point $F$. Let $M$ be the intersection of the lines $BE$ and $DF$. Prove that the line $BD$ is tangent to the circumcircle of the triangle $MDE$.
1 reply
parmenides51
Oct 6, 2024
vanstraelen
25 minutes ago
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Geometry Problem #42
vankhea   2
N an hour ago by kaede_Arcadia
Source: Van Khea
Let $P$ be any point. Let $D, E, F$ be projection point from $P$ to $BC, CA, AB$. Circumcircle $(ABC)$ cuts circumcircle $(AEF), (BFD), (CDE)$ at $A_1, B_1, C_1$. Let $A_2, B_2, C_2$ be antipode of $A_1, B_1, C_1$ wrt $(AEF), (BFD), (CDE)$. Prove that $A_2, B_2, C_2, P$ are cyclic.
2 replies
vankhea
Sep 6, 2023
kaede_Arcadia
an hour ago
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divisibility
srnjbr   3
N an hour ago by srnjbr
Find all natural numbers n such that there exists a natural number l such that for every m members of the natural numbers the number m+m^2+...m^l is divisible by n.
3 replies
srnjbr
3 hours ago
srnjbr
an hour ago
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Inequalities
sqing   29
N Today at 1:20 PM by SomeonecoolLovesMaths
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
$$(a+8)(a^2-a+b+2)(b^2-b+5)\geq1331$$$$(a+10)(a^2-a+b+4)(b^2-b+7)\geq2197$$
29 replies
sqing
Mar 10, 2025
SomeonecoolLovesMaths
Today at 1:20 PM
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2019 Chile Classification / Qualifying NMO Juniors XXXI
parmenides51   6
N Today at 1:19 PM by bhontu
p1. Consider the sequence of positive integers $2, 3, 5, 6, 7, 8, 10, 11 ...$. which are not perfect squares. Calculate the $2019$-th term of the sequence.


p2. In a triangle $ABC$, let $D$ be the midpoint of side $BC$ and $E$ be the midpoint of segment $AD$. Lines $AC$ and $BE$ intersect at $F$. Show that $3AF = AC$.


p3. Find all positive integers $n$ such that $n! + 2019$ is a square perfect.


p4. In a party, there is a certain group of people, none of whom has more than $3$ friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?
6 replies
parmenides51
Oct 11, 2021
bhontu
Today at 1:19 PM
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Inequalities
sqing   12
N Today at 1:12 PM by sqing
Let $ a,b $ be real numbers such that $ a + b \geq |ab + 1|. $ Prove that$$ a^3 + b^3 \geq |a^3 b^3 + 1|$$Let $ a,b $ be real numbers such that $ 2(a + b ) \geq |ab + 1|. $ Prove that$$26( a^3 + b^3) \geq |a^3 b^3 + 1|$$Let $ a,b $ be real numbers such that $ 4(a + b) \geq 3|ab + 1|. $ Prove that$$148(a^3 + b^3) \geq27 |a^3 b^3 + 1|$$
12 replies
sqing
Mar 8, 2025
sqing
Today at 1:12 PM
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FB = BK , circumcircle and altitude related (In the World of Mathematics 516)
parmenides51   3
N Today at 12:09 PM by AshAuktober
Let $BT$ be the altitude and $H$ be the intersection point of the altitudes of triangle $ABC$. Point $N$ is symmetric to $H$ with respect to $BC$. The circumcircle of triangle $ATN$ intersects $BC$ at points $F$ and $K$. Prove that $FB = BK$.

(V. Starodub, Kyiv)
3 replies
parmenides51
Apr 19, 2020
AshAuktober
Today at 12:09 PM
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Polynomial with roots in geometric progression
red_dog   0
Today at 9:54 AM
Let $f\in\mathbb{C}[X], \ f=aX^3+bX^2+cX+d, \ a,b,c,d\in\mathbb{R}^*$ a polynomial whose roots $x_1,x_2,x_3$ are in geometric progression with ration $q\in(0,\infty)$. Find $S_n=x_1^n+x_2^n+x_3^n$.
0 replies
red_dog
Today at 9:54 AM
0 replies
J
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Good Functional equation question
vexploresmathysics   1
N Today at 9:30 AM by jasperE3
If f : R^+ --> R^+ satisfying f(f(x)/y ) = yf ( y ) + (f(x)). Then the value of α such that Sigma K = 1 to n [ 1 / f(K) ] = 420
1 reply
vexploresmathysics
Jul 1, 2024
jasperE3
Today at 9:30 AM
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Functional Equation
AnhQuang_67   2
N Today at 9:03 AM by jasperE3
Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying:
$$3f(\dfrac{x-1}{3x+2})-5f(\dfrac{1-x}{x-2})=\dfrac{8}{x-1}, \forall x \notin \{0,\dfrac{-2}{3},1,2\}$$
2 replies
AnhQuang_67
Jan 7, 2025
jasperE3
Today at 9:03 AM
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a+b+c=3 ine
jokehim   4
N Today at 8:26 AM by lbh_qys
Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\color{black}{\frac{a\left(b+c\right)}{bc+3}+\frac{b\left(c+a\right)}{ca+3}+\frac{c\left(a+b\right)}{ab+3}\le \frac{3}{2}.}$$Proposed by Phan Ngoc Chau
4 replies
jokehim
Mar 18, 2025
lbh_qys
Today at 8:26 AM
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IOQM P5 2024
SomeonecoolLovesMaths   13
N Today at 8:10 AM by quasar_lord
Let $a = \frac{x}{y} +\frac{y}{z} +\frac{z}{x}$, let $b = \frac{x}{z} +\frac{y}{x} +\frac{z}{y}$ and let $c = \left(\frac{x}{y} +\frac{y}{z} \right)\left(\frac{y}{z} +\frac{z}{x} \right)\left(\frac{z}{x} +\frac{x}{y} \right)$. The value of $|ab-c|$ is:
13 replies
SomeonecoolLovesMaths
Sep 8, 2024
quasar_lord
Today at 8:10 AM
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IOQM P4 2024
SomeonecoolLovesMaths   8
N Today at 8:04 AM by quasar_lord
Let $ABCD$ be a quadrilateral with $\angle ADC = 70^{\circ}$, $\angle ACD = 70^{\circ}$, $\angle ACB = 10^{\circ}$ and $\angle BAD = 110^{\circ}$. The measure of $\angle CAB$ (in degrees) is:
8 replies
SomeonecoolLovesMaths
Sep 8, 2024
quasar_lord
Today at 8:04 AM
J
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J
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A final homogeneous inequality
v_Enhance   1
N Jul 26, 2013 by sirknightingfail
Source: ELMO Shortlist 2013: Problem A9, by David Stoner
Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that
\[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\]Proposed by David Stoner
1 reply
v_Enhance
Jul 23, 2013
sirknightingfail
Jul 26, 2013
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A final homogeneous inequality
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Reveal topic tags
inequalities
function
logarithms
inequalities unsolved
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Source: ELMO Shortlist 2013: Problem A9, by David Stoner
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v_Enhance
6865 posts
v_Enhance
#1 h  Jul 23, 2013, 1:24 AM • 4 Y
Y by megarnie, HamstPan38825, Rounak_iitr, Adventure10
Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that
\[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\]Proposed by David Stoner
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sirknightingfail
260 posts
sirknightingfail
#2 h  Jul 26, 2013, 6:09 PM • 4 Y
Y by AFSA, megarnie, Adventure10, Mango247
solution
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