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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Inequalities
hn111009   0
15 minutes ago
Let $a,b,c>0$ satisfied $a^2+b^2+c^2=9.$ Find the minimum of $$P=\dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}.$$
0 replies
hn111009
15 minutes ago
0 replies
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Olympiad
sasu1ke   3
N 40 minutes ago by sasu1ke
IMAGE
3 replies
sasu1ke
2 hours ago
sasu1ke
40 minutes ago
J
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How to judge a number is prime or not?
mingzhehu   1
N 2 hours ago by scrabbler94
A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N

For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime
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mingzhehu
Yesterday at 2:45 PM
scrabbler94
2 hours ago
J
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Eazy equation clap
giangtruong13   0
Yesterday at 4:03 PM
Find all $x,y,z$ satisfy that: $$\frac{x}{y+z}=2x-1; \frac{y}{x+z}=3y-1;\frac{z}{x+y}=5x-1$$
0 replies
giangtruong13
Yesterday at 4:03 PM
0 replies
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area wanted, joining midpoints with vertices in square 2019 Portugal p1
parmenides51   1
N Oct 11, 2020 by vanstraelen
In a square of side $10$ cm , the vertices are joined to the midpoints on the opposite sides, as shown in the figure. How much does the area of the colored region measure?
IMAGE
1 reply
parmenides51
Oct 11, 2020
vanstraelen
Oct 11, 2020
J
area wanted, joining midpoints with vertices in square 2019 Portugal p1
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Reveal topic tags
geometry
square
area
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parmenides51
30629 posts
parmenides51
#1 h  Oct 11, 2020, 5:22 AM
Y by
In a square of side $10$ cm , the vertices are joined to the midpoints on the opposite sides, as shown in the figure. How much does the area of the colored region measure?
https://1.bp.blogspot.com/-bHrc1Nu0PQI/X4KaJysLAcI/AAAAAAAAMk0/LLGv1fotQO0Tk1AXqQymG_nNdpyWcbjyACLcBGAsYHQ/s109/2019%2BPortugal%2Bp1.png
This post has been edited 3 times. Last edited by parmenides51, Oct 11, 2020, 9:04 PM
Reason: title typo
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vanstraelen
8949 posts
vanstraelen
#3 h  Oct 11, 2020, 10:52 AM
Y by
Given $A(0,0),B(10,0),C(10,10),D(0,10)$; midpoints $E(5,0),F(10,5)$.

White triangle bottom left $A(0,0),E(5,0),T(4,2)$; area $=5$ cm$^{2}$.
$8$ white triangles, area $=40$ cm$^{2}$.
Area colored region $=60$ cm$^{2}$.
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