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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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[*]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   2
N a minute ago by hn111009
Let $a,b,c>0$ satisfied $a^2+b^2+c^2=9.$ Find the minimum of $$P=\dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}.$$
2 replies
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hn111009
an hour ago
hn111009
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Olympiad
sasu1ke   3
N 2 hours ago by sasu1ke
IMAGE
3 replies
sasu1ke
3 hours ago
sasu1ke
2 hours ago
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How to judge a number is prime or not?
mingzhehu   1
N 3 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
1 reply
mingzhehu
Yesterday at 2:45 PM
scrabbler94
3 hours ago
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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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4-digit evens with 0,1,2,3,4,5 (Puerto Rico TST 2024.6)
Equinox8   3
N Mar 29, 2025 by wisewigglyjaguar
Find the sum of all $4$-digit even numbers that can be written using the digits $0, 1, 2, 3, 4,$ and $5$. Digits can be repeated in a number.
3 replies
Equinox8
Mar 24, 2025
wisewigglyjaguar
Mar 29, 2025
J
4-digit evens with 0,1,2,3,4,5 (Puerto Rico TST 2024.6)
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Equinox8
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Equinox8
#1 h  Mar 24, 2025, 3:45 PM
Y by
Find the sum of all $4$-digit even numbers that can be written using the digits $0, 1, 2, 3, 4,$ and $5$. Digits can be repeated in a number.
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lpieleanu
2875 posts
lpieleanu
#2 h  Mar 29, 2025, 6:29 PM
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Note that the expected value of such a number is $$3 \cdot 1000 + \frac{5}{2} \cdot 100 + \frac{5}{2} \cdot 10 + 2 = 3277,$$so the answer is $$3277 \cdot 5 \cdot 6 \cdot 6 \cdot 3 = \boxed{1769580}.$$$\square$
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ethan2011
262 posts
ethan2011
#3 h  Mar 29, 2025, 10:37 PM
Y by
Ain't no way this was on the TST.
The expected value of the number is
$\frac{1+2+3+4+5}{5}\cdot 1000+\frac{0+1+2+3+4+5}{6}\cdot 100+\frac{0+1+2+3+4+5}{6}\cdot 10+\frac{0+2+4}{3}=3277$.
Since the total amount of possible 4-digit evens is $5\cdot 6\cdot 6\cdot 3=540$, our answer is $540\cdot 3277=\boxed{1769580}$
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wisewigglyjaguar
548 posts
wisewigglyjaguar
#4 h  Mar 29, 2025, 11:59 PM
Y by
Bruh, Using the method of expected value makes the problem so easy, if you use any other method, then it's complicated.
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