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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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Olympiad
sasu1ke   0
4 minutes ago
IMAGE
0 replies
sasu1ke
4 minutes ago
0 replies
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How to judge a number is prime or not?
mingzhehu   1
N 43 minutes 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
1 reply
mingzhehu
Today at 2:45 PM
scrabbler94
43 minutes ago
J
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Eazy equation clap
giangtruong13   0
Today 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
Today at 4:03 PM
0 replies
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inequality
revol_ufiaw   3
N Today at 2:55 PM by MS_asdfgzxcvb
Prove that that for any real $x \ge 0$ and natural number $n$,
$$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$$
3 replies
revol_ufiaw
Today at 2:05 PM
MS_asdfgzxcvb
Today at 2:55 PM
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trisect a diagonal on 1x4 rectangle, ruler only (2010 Lomonosov Tournament)
parmenides51   2
N Apr 22, 2021 by jasperE3
A diagonal of a $1\times 4$ rectangle is drawn on checkered paper. Show how, using only the ruler without divisions, divide this segment into three equal parts.
2 replies
parmenides51
Apr 21, 2021
jasperE3
Apr 22, 2021
J
trisect a diagonal on 1x4 rectangle, ruler only (2010 Lomonosov Tournament)
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Reveal topic tags
geometry
rectangle
Ruler
Lomonosov Tournament
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parmenides51
30629 posts
parmenides51
#1 h  Apr 21, 2021, 12:09 PM
Y by
A diagonal of a $1\times 4$ rectangle is drawn on checkered paper. Show how, using only the ruler without divisions, divide this segment into three equal parts.
This post has been edited 1 time. Last edited by parmenides51, May 20, 2022, 2:51 PM
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Hellboy1234
134 posts
Hellboy1234
#2 h  Apr 21, 2021, 12:59 PM
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nice problem :coolspeak:
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jasperE3
11165 posts
jasperE3
#3 h  Apr 22, 2021, 3:52 AM
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Let the rectangle have vertices on $(0,0),(0,1),(4,0),(4,1)$ on the Cartesian plane, and the diagonal be between $(0,0)$ and $(4,1)$. The first trisection point is the intersection of the line through $(0,-2)$ and $(4,5)$ and the line between $(0,5)$ and $(2,-2)$. The second trisection point is the intersection of the line between $(0,4)$ and $(4,-1)$ and the line between $(2,-2)$ and $(3,2)$.
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