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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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Simple inequality
sqing   22
N 2 minutes ago by ND_
Source: JBMO 2011 Shortlist A3
$\boxed{\text{A3}}$If $a,b$ be positive real numbers, show that:$$ \displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}$$
22 replies
1 viewing
sqing
May 15, 2016
ND_
2 minutes ago
J
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greatest volume
hzbrl   0
7 minutes ago
Source: purple comet
A large sphere with radius 7 contains three smaller balls each with radius 3 . The three balls are each externally tangent to the other two balls and internally tangent to the large sphere. There are four right circular cones that can be inscribed in the large sphere in such a way that the bases of the cones are tangent to all three balls. Of these four cones, the one with the greatest volume has volume $n \pi$. Find $n$.
0 replies
hzbrl
7 minutes ago
0 replies
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Serbia national Olympiad Day 2 Problem 2
IgorM   19
N an hour ago by IndexLibrorumProhibitorum
Source: Serbia national Olympiad Day 2 Problem 2
Let $x,y,z$ be nonnegative positive integers.
Prove $\frac{x-y}{xy+2y+1}+\frac{y-z}{zy+2z+1}+\frac{z-x}{xz+2x+1}\ge 0$
19 replies
IgorM
Mar 28, 2015
IndexLibrorumProhibitorum
an hour ago
J
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find angle
TBazar   2
N an hour ago by sunken rock
Given $ABC$ triangle with $AC>BC$. We take $M$, $N$ point on AC, AB respectively such that $AM=BC$, $CM=BN$. $BM$, $AN$ lines intersect at point $K$. If $2\angle AKM=\angle ACB$, find $\angle ACB$
2 replies
TBazar
3 hours ago
sunken rock
an hour ago
J
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simple trapezoid
gggzul   3
N Today at 2:51 AM by imbadatmath1233
Let $ABCD$ be a trapezoid. By $x$ we denote the angle bisector of angle $X$ . Let $P=a\cap c$ and $Q=b\cap d$. Prove that $ABPQ$ is cyclic.
3 replies
gggzul
May 5, 2025
imbadatmath1233
Today at 2:51 AM
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Calculate the sidelength BC
MTA_2024   1
N Today at 2:44 AM by imbadatmath1233
Let $ABC$ be a triangle such that $AB=2AC$ and $\angle ABC =120°$. Let $D$ be the foot of the interior bissector of $\angle ABC$ (its intersection with $BC$).
If $AD=10$ calculate the sidelength $BC$.
1 reply
MTA_2024
Yesterday at 7:16 PM
imbadatmath1233
Today at 2:44 AM
J
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9 always remainder by 1-10 May Olympiad (Olimpiada de Mayo) 2019 L2 P1
parmenides51   2
N Today at 2:33 AM by imbadatmath1233
A positive integer is called piola if the $9$ is the remainder obtained by dividing it by $2, 3, 4, 5, 6, 7, 8, 9$ and $10$ and it's digits are all different and nonzero. How many piolas are there between $ 1$ and $100000$?
2 replies
parmenides51
Sep 24, 2021
imbadatmath1233
Today at 2:33 AM
J
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Area of triangle inside regular hexagon
smartvong   1
N Today at 2:17 AM by tom-nowy
The diagram shows a regular hexagon $ABCDEF$. $P$ is a point on the line segment $CD$, such that the areas of $\triangle PAF$ and $\triangle PEF$ are $64$ and $48$ respectively. Find the area of $\triangle PAB$.

IMAGE
1 reply
smartvong
Today at 1:40 AM
tom-nowy
Today at 2:17 AM
J
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Inequalities
sqing   2
N Today at 1:45 AM by sqing
Let $ a,b>0 $ and $\frac{a}{a^2+3}+ \frac{b}{b^2+ 3} \geq \frac{1}{2} . $ Prove that
$$a^2+ab+b^2\geq 3$$$$a^2-ab+b^2 \geq 1 $$Let $ a,b>0 $ and $\frac{a}{a^3+3}+ \frac{b}{b^3+ 3}\geq \frac{1}{2} . $ Prove that
$$a^3+ab+b^3 \geq 3$$$$ a^3-ab+b^3\geq 1 $$
2 replies
sqing
Yesterday at 12:59 PM
sqing
Today at 1:45 AM
J
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Unknown triangle area
smartvong   0
Today at 1:38 AM
The diagram shows a convex quadrilateral $ABCD$. The points $E$ and $F$ divide $AB$ into three equal parts while the points $G$ and $H$ divide $CD$ into three equal parts. The line segments $AH$ and $ED$ intersect at $I$. The line segments $CF$ and $BG$ intersect at $J$. Given that the areas of the triangles $AID$, $EHI$ and $FJG$ are $154$, $112$, and $99$ respectively, find the area of the triangle $BJC$.

IMAGE
0 replies
smartvong
Today at 1:38 AM
0 replies
J
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Prove that P,Q,R and S are cocyclic
MTA_2024   2
N Today at 1:09 AM by LearnMath_105
Let $ABC$ be an acute triangle. The perpendicular to $AC$ passing through $B$ cuts the circle of diameter $AC$ in $P$ and $Q$. The perpendicular to $AB$ passing through $C$ cuts circle of diameter $AB$ in $R$ and $S$.
Prove that $P$,$Q$,$R$ and $S$ are cocyclic.
2 replies
MTA_2024
Yesterday at 8:30 PM
LearnMath_105
Today at 1:09 AM
J
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Geometry
MTA_2024   5
N Yesterday at 11:58 PM by MathRook7817
Let $ABC$ be a triangle such that $AB=3$,$BC=5$ and $AC=6$.Let $D$ be a point on side $AC$ and $E$ one on side $BC$ so that the line $DE$ is tangent to the incircle of $\triangle ABC$ .
Evaluate the perimeter of triangle $\triangle CDE$.
5 replies
MTA_2024
Yesterday at 6:52 PM
MathRook7817
Yesterday at 11:58 PM
J
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Acute Angle Altitudes... say that ten times fast
Math-lover1   0
Yesterday at 11:30 PM
In acute triangle $ABC$, points $D$ and $E$ are the feet of the angle bisector and altitude from $A$, respectively. Suppose that $AC-AB=36$ and $DC-DB=24$. Compute $EC-EB$.
0 replies
Math-lover1
Yesterday at 11:30 PM
0 replies
J
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This shouldn't be a problem 15
derekli   3
N Yesterday at 8:37 PM by musicalpenguin
Hey guys I was practicing AIME and came across this problem which is definitely misplaced. It asks for the surface area of a plane within a cylinder which we can easily find out using a projection that is easy to find. I think this should be placed in problem 10 or below. What do you guys think?
3 replies
derekli
Tuesday at 2:15 PM
musicalpenguin
Yesterday at 8:37 PM
J
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J
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Inequality with three conditions
oVlad   3
N Apr 22, 2025 by sqing
Source: Romania EGMO TST 2019 Day 1 P3
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
3 replies
oVlad
Apr 21, 2025
sqing
Apr 22, 2025
J
Inequality with three conditions
G H J
Reveal topic tags
algebra
inequalities
L
G H BBookmark kLocked kLocked NReply
Source: Romania EGMO TST 2019 Day 1 P3
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oVlad
1746 posts
oVlad
#1 h  Apr 21, 2025, 1:48 PM
Y by
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
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Haris1
77 posts
Haris1
#2 h  Apr 21, 2025, 6:50 PM
Y by
Nice ineq,
$3(a+1)(b+1)(c+1)\geq (a+1)(b+1)(a+b)+(a+1)(a+c)(c+1)+(b+c)(b+1)(c+1)$
and using $(a+1)(b+1)(c+1)\geq (a+b)(b+c)(c+a)$ completes it.
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Quantum-Phantom
272 posts
Quantum-Phantom
#3 h  Apr 21, 2025, 11:24 PM
Y by
Because
\begin{align*}8abc+4-4\sum_{\rm cyc}a^2=&\left(1+\sum_{\rm cyc}a\right)\sum_{\rm cyc}(a+1-b-c)(b+1-c-a)\\&+\prod_{\rm cyc}(a+1-b-c)\ge0.\end{align*}
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sqing
42045 posts
sqing
#4 h  Apr 22, 2025, 4:24 AM
Y by
oVlad wrote:
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
Indian 2007
https://artofproblemsolving.com/community/c6h1754566p11450296
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