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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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0 replies
jlacosta
Mar 2, 2025
0 replies
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Inequalities
sqing   0
33 minutes ago
Let $ a,b,c\geq 2 . $ Prove that
$$(a-1)(b^2-2)(c-1) - \frac{1}{2}abc\geq -2$$$$(a^2-2)(b-1)(c^2-2) - \frac{3}{2}abc\geq -6$$
0 replies
1 viewing
sqing
33 minutes ago
0 replies
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one intersting
teomihai   3
N 38 minutes ago by hexuhdecimal
Let $n $ it is fixed positiv number.
Find $card\{(x,y) | |x|+|y|<n \}$, where $x,y$ are integer numbers .
3 replies
teomihai
Mar 15, 2025
hexuhdecimal
38 minutes ago
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Discrete problem
ThanhNg   1
N an hour ago by RPCX
Prove that : From any 161 natural numbers, we can always find 81 numbers whose sum is divisible by 81.
1 reply
ThanhNg
Yesterday at 4:23 PM
RPCX
an hour ago
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Does there exist a function $f:\mathbb N^* \rightarrow \mathbb Z$ other than a p
kyotaro   0
3 hours ago
Does there exist a function $f:\mathbb N^* \rightarrow \mathbb Z$ other than a polynomial satisfying
$$f(a)-f(b) \mid a-b$$
0 replies
kyotaro
3 hours ago
0 replies
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Proof Question
MathRook7817   3
N 5 hours ago by Jack_w
Prove or disprove the assertion that when $1/n$ is written in decimal form with n being a natural number, that it's decimal either terminates or is eventually periodic.

I saw this question somewhere and I thinks it's interesting.
3 replies
MathRook7817
Today at 2:19 AM
Jack_w
5 hours ago
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Inequalities
lgx57   2
N 6 hours ago by sqing
Let $0 < a,b,c < 1$. Prove that

$$a(1-b)+b(1-c)+c(1-a)<1$$
2 replies
lgx57
Today at 7:43 AM
sqing
6 hours ago
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solve the system of equations
Havu   1
N 6 hours ago by Mathzeus1024
Solve the system of equations:
\[\begin{cases} 3x^2-2xy+3y^2+\dfrac{2}{x^2-2xy+y^2}=8\\ 2x+\dfrac{1}{x-y}=4 \end{cases}\]
1 reply
Havu
Today at 7:52 AM
Mathzeus1024
6 hours ago
J
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Inequalities
sqing   7
N Today at 8:12 AM by sqing
Let $a,b,c\ge \frac{1}{2}$ and $\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\le 1. $ Prove that
$$a+b+c\geq 2$$Let $a,b,c\ge \frac{1}{2}$ and $ \left(a+\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(a+\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\le \frac{9}{2}. $ Prove that
$$a^2+b^2+c^2\geq 1$$Let $a,b\ge \frac{1}{2}$ and $ \left( \frac{1}{a}-\frac{1}{b}+2\right)\left( \frac{1}{b}-\frac{1}{a}+2\right) \le \frac{20}{9}. $ Prove that
$$ a+b\geq 2$$Let $a,b\ge \frac{1}{2}$ and $a^2+b^2=1. $ Prove that
$$\left(\frac{2}{a}+\frac{1}{b}-1\right)\left(\frac{2}{a}-\frac{1}{b}+1\right)\ge \frac{13}{3}$$
7 replies
sqing
Mar 15, 2025
sqing
Today at 8:12 AM
J
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Quadratic Equation Problems
Saucepan_man02   7
N Today at 5:53 AM by soryn
P1) Let $\alpha < \beta < \gamma$ be the roots of $ax^3+bx^2+cx+d=0$ with $7a+3b>0, 3a+2b<0$ and $\alpha(\beta+1)+\beta(\gamma+1)+\gamma(1+\alpha)=0$.
If [.] denotes the greatest integer function, then the number of possible values of $[|3 \alpha |] + [|6 \beta|] + [|9 \gamma |]$ is equal to
7 replies
Saucepan_man02
Mar 3, 2025
soryn
Today at 5:53 AM
J
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Trash Trig Sum
P_Groudon   4
N Today at 5:41 AM by invisibleman
Find the smallest positive integer $n$ such that $$\sum_{k=0}^{298}\sin(k^2 + 2k + 2)\sin(2k + 2) = \frac{\cos(1) - \cos(n)}{2},$$where degrees are used.
4 replies
P_Groudon
Monday at 5:34 PM
invisibleman
Today at 5:41 AM
J
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Chessboard
Ecrin_eren   2
N Today at 5:24 AM by Ecrin_eren
On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)

2 replies
Ecrin_eren
Yesterday at 8:55 PM
Ecrin_eren
Today at 5:24 AM
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Inequality
BaCaPhe   1
N Today at 4:43 AM by lbh_qys
$\begin{array}{l} a,b,c \ge 1,ab + bc + ca \ge 4 + abc\\ {a^2} + {b^2} + {c^2} \ge 9? \end{array}$
I tried:
$\begin{array}{l} * {\text{In three numbers }}a,b,c \ge 1,{\text{ at least 2 numbers }} \ge {\text{2 or}} \le {\text{2}}\\ {\text{WLOG assume they are }}a{\text{ and }}b \Rightarrow \left( {a - 2} \right)\left( {b - 2} \right) \ge 0 \Leftrightarrow 4c + abc \ge 2ac + 2bc\\ \Leftrightarrow 2ac + 2bc \le 4c - 4 + 4 + abc \le 4c - 4 + ab + bc + ca \Leftrightarrow ac + bc \le 4c - 4 + ab \end{array}$
How do I process further?
1 reply
BaCaPhe
Today at 3:13 AM
lbh_qys
Today at 4:43 AM
J
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Inequalities
sqing   21
N Today at 4:36 AM by sqing
Let $ a,b>0 $ and $ \frac{1}{a}+\frac{1}{b}=1. $ Prove that
$$(a^2-a+1)(b^2-b+1) \geq 9$$$$ (a^2-a+b+1)(b^2-b+a+1) \geq 25$$Let $ a,b>0 $ and $ \frac{1}{a}+\frac{1}{b}=\frac{2}{3}. $ Prove that
$$(a+8)(a^2-a+b+2)(b^2-b+5)\geq1331$$$$(a+10)(a^2-a+b+4)(b^2-b+7)\geq2197$$
21 replies
sqing
Mar 10, 2025
sqing
Today at 4:36 AM
J
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Inequalities
sqing   6
N Today at 4:16 AM by sqing
Let $ a,b $ be real numbers such that $ a + b \geq |ab + 1|. $ Prove that$$ a^3 + b^3 \geq |a^3 b^3 + 1|$$Let $ a,b $ be real numbers such that $ 2(a + b ) \geq |ab + 1|. $ Prove that$$26( a^3 + b^3) \geq |a^3 b^3 + 1|$$Let $ a,b $ be real numbers such that $ 4(a + b) \geq 3|ab + 1|. $ Prove that$$148(a^3 + b^3) \geq27 |a^3 b^3 + 1|$$
6 replies
sqing
Mar 8, 2025
sqing
Today at 4:16 AM
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J
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Complex + Radical Evaluation
Saucepan_man02   1
N Mar 17, 2025 by MathRook7817
Evaluate: (without calculators)
$$ (\sqrt{6 - 2 \sqrt{5}} + i \sqrt{2 \sqrt{5} + 10})^5 + (\sqrt{6 - 2 \sqrt{5}} - i \sqrt{2 \sqrt{5} + 10})^5$$
1 reply
Saucepan_man02
Mar 17, 2025
MathRook7817
Mar 17, 2025
J
Complex + Radical Evaluation
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Reveal topic tags
complex numbers
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Saucepan_man02
1299 posts
Saucepan_man02
#1 h  Mar 17, 2025, 1:08 PM
Y by
Evaluate: (without calculators)
$$ (\sqrt{6 - 2 \sqrt{5}} + i \sqrt{2 \sqrt{5} + 10})^5 + (\sqrt{6 - 2 \sqrt{5}} - i \sqrt{2 \sqrt{5} + 10})^5$$
This post has been edited 1 time. Last edited by Saucepan_man02, Mar 17, 2025, 1:08 PM
Reason: EDIT
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MathRook7817
609 posts
MathRook7817
#2 h  Mar 17, 2025, 1:30 PM
Y by
I think you just find the magnitude of each one, then evaluate the angle, and then double it by 5 then evaluate the trig of it to find it

too lazy to evaluate it
This post has been edited 1 time. Last edited by MathRook7817, Mar 17, 2025, 1:31 PM
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