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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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Inequalities
sqing   6
N 2 minutes ago by lbh_qys
Let $x,y\ge 0$ such that $ 13(x^3+y^3) \leq 125(1+xy)$. Prove that
$$ k(x+y)-xy\leq 5(2k-5)$$Where $k\geq 5.6797. $
$$ 6(x+y)-xy\leq 35$$
6 replies
sqing
Apr 20, 2025
lbh_qys
2 minutes ago
J
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Inequalities
sqing   0
11 minutes ago
Let $ a,b \in [0 ,1] . $ Prove that
$$\frac{a}{ 1-ab+b }+\frac{b }{ 1-ab+a } \leq 2$$$$\frac{a}{ 1-ab+b }+\frac{b }{ 1-ab+a }+\frac{ab }{1+ab }\leq \frac{5}{2}$$$$\frac{a}{ 1+ab+b }+\frac{b }{ 1+ab+a } +\frac{ab }{1+ab }\leq \frac{7}{6 }$$
0 replies
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sqing
11 minutes ago
0 replies
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Inequalities from SXTX
sqing   20
N 38 minutes ago by sqing
T702. Let $ a,b,c>0 $ and $ a+2b+3c=\sqrt{13}. $ Prove that $$ \sqrt{a^2+1} +2\sqrt{b^2+1} +3\sqrt{c^2+1} \geq 7$$S
T703. Let $ a,b $ be real numbers such that $ a+b\neq 0. $. Find the minimum of $ a^2+b^2+(\frac{1-ab}{a+b} )^2.$
T704. Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that $$ \frac{a^2+7}{(c+a)(a+b)} + \frac{b^2+7}{(a+b)(b+c)} +\frac{c^2+7}{(b+c)(c+a)} \geq 6$$S
20 replies
sqing
Feb 18, 2025
sqing
38 minutes ago
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Theory of Equations
P162008   3
N 2 hours ago by P162008
Let $a,b,c,d$ and $e\in [-2,2]$ such that $\sum_{cyc} a = 0, \sum_{cyc} a^3 = 0, \sum_{cyc} a^5 = 10.$ Find the value of $\sum_{cyc} a^2.$
3 replies
P162008
Wednesday at 11:27 AM
P162008
2 hours ago
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Three 3-digit numbers
miiirz30   5
N Apr 16, 2025 by fruitmonster97
Leonard wrote three 3-digit numbers on the board whose sum is $1000$. All of the nine digits are different. Determine which digit does not appear on the board.

Proposed by Giorgi Arabidze, Georgia
5 replies
miiirz30
Mar 31, 2025
fruitmonster97
Apr 16, 2025
J
Three 3-digit numbers
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miiirz30
21 posts
miiirz30
#1 h  Mar 31, 2025, 5:38 PM
Y by
Leonard wrote three 3-digit numbers on the board whose sum is $1000$. All of the nine digits are different. Determine which digit does not appear on the board.

Proposed by Giorgi Arabidze, Georgia
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jasperE3
11248 posts
jasperE3
#2 h  Mar 31, 2025, 5:48 PM
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Solution
This post has been edited 1 time. Last edited by jasperE3, Mar 31, 2025, 5:50 PM
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megarnie
5596 posts
megarnie
#4 h  Apr 1, 2025, 12:30 AM
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The sum is $1 \pmod 9$, and $\left( \sum_{i=0}^9 i \right) - 1 \equiv \boxed{8} \pmod 9$.
This post has been edited 1 time. Last edited by megarnie, Apr 1, 2025, 12:30 AM
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henryli3333
10 posts
henryli3333
#5 h  Apr 1, 2025, 4:05 AM
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that makes so much sense thanks jasper
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sansgankrsngupta
131 posts
sansgankrsngupta
#6 h  Apr 16, 2025, 11:10 AM
Y by
OG! LOL: This question is wrong there are nou such numbers
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fruitmonster97
2481 posts
fruitmonster97
#7 h  Apr 16, 2025, 12:42 PM
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To further on above, let the numbers be $abc, def, ghi.$ Then $a+d+g=9,b+e+h=9,c+d+i=10.$ Thus, the number missing is $45-9-9-10=\boxed{17}$ :P
This post has been edited 1 time. Last edited by fruitmonster97, Apr 16, 2025, 12:43 PM
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