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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Yesterday at 3:18 PM
0 replies
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Olympiad Geometry problem-second time posting
kjhgyuio   6
N 4 minutes ago by ND_
Source: smo problem
In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD
6 replies
kjhgyuio
Yesterday at 1:03 AM
ND_
4 minutes ago
J
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Inspired by Ecrin_eren
sqing   1
N 34 minutes ago by lbh_qys
Source: Own
Let $ x ,y\geq 0 $ and $ x^2(y^2 + 9) + x^4y + 3y^2 \geq 27.$ Prove that
$$x^2 -x+ \frac{1}{2}y\geq 1$$$$x^2 -x+ \frac{1}{3}y\geq \frac{5}{8}$$$$x^2 -x+ y\geq 3-\sqrt 3$$
1 reply
sqing
an hour ago
lbh_qys
34 minutes ago
J
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The last nonzero digit of factorials
Tintarn   3
N 35 minutes ago by MyobDoesMath
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 2
For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$. The infinite sequence of these digits starts with $2,6,4,2,2$. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
3 replies
Tintarn
Mar 17, 2025
MyobDoesMath
35 minutes ago
J
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Problem inequality
inversionA007   10
N 42 minutes ago by Primeniyazidayi
Let $x>0, y>0, z>0$ and satisfy $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=3$. Prove that $ x^2+y^2+z^2-2 x y z \geq 1$.
10 replies
inversionA007
Jan 14, 2024
Primeniyazidayi
42 minutes ago
J
No more topics!
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Comparing Acute angles of Rhombus and Trapezoid
Iora   7
N Mar 31, 2025 by Math_01-person
Source: 2017 Azerbaijan Junior National Olympiad
A Rhombus and an Isosceles trapezoid that has same area is drawn in the same circle's outside. Compare their acute angles
(explain your answer)
7 replies
Iora
Apr 28, 2022
Math_01-person
Mar 31, 2025
J
Comparing Acute angles of Rhombus and Trapezoid
G H J
Reveal topic tags
geometry
rhombus
trapezoid
L
G H BBookmark kLocked kLocked NReply
Source: 2017 Azerbaijan Junior National Olympiad
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Iora
194 posts
Iora
#1 h  Apr 28, 2022, 2:09 PM • 1 Y
Y by ImSh95
A Rhombus and an Isosceles trapezoid that has same area is drawn in the same circle's outside. Compare their acute angles
(explain your answer)
This post has been edited 1 time. Last edited by Iora, May 3, 2022, 9:54 AM
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Telman
28 posts
Telman
#2 h  May 3, 2022, 6:57 AM • 3 Y
Y by ladyMS, Iora, ImSh95
The question is wrong it might be an isosceles trapezoid and a rhombus with same *area******
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Iora
194 posts
Iora
#3 h  May 3, 2022, 9:54 AM • 1 Y
Y by ImSh95
Telman wrote:
The question is wrong it might be an isosceles trapezoid and a rhombus with same *area******

Fixed :blush:
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Telman
28 posts
Telman
#4 h  May 3, 2022, 3:13 PM • 1 Y
Y by ImSh95
@above no problem
Hint to this problem :Each side of this trapezoid's length is equal to the side of the rhombus
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lian_the_noob12
173 posts
lian_the_noob12
#5 h  Mar 5, 2023, 7:22 PM
Y by
How does the circle effect here??
This post has been edited 2 times. Last edited by lian_the_noob12, Mar 5, 2023, 7:23 PM
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mathmax12
6001 posts
mathmax12
#6 h  Jul 31, 2023, 3:50 PM
Y by
Let a side length of the rhoumbus be $r$ now the area of the rhombus in terms of the angle is $r^2\cdot\cos(x)\cdot\sin(x)$

(let one of the angles be $2x$ and the other $180-2x$)
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Just1
7 posts
Just1
#7 h  Mar 11, 2025, 6:55 PM
Y by
Isn't it 2*r²*sin(x)*cos(x)?
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Math_01-person
1 post
Math_01-person
#8 h  Mar 31, 2025, 4:54 PM
Y by
Telman wrote:
The question is wrong it might be an isosceles trapezoid and a rhombus with same *area******
it is given that the areas are equal
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