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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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[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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0 replies
jlacosta
Mar 2, 2025
0 replies
J
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Very interesting inequality
sqing   0
4 minutes ago
Source: Own
Let $ a,b,c\geq 2 . $ Prove that
$$(a-1)(b^2-2)(c^3-3)- \frac{5}{2}abc\geq -10$$$$(a-\frac{3}{2})(b^2-2)(c^3-3)- \frac{5}{2}abc\geq -15$$$$(a-\frac{3}{2})(b^2-\frac{3}{2})(c^3-3)- \frac{25}{8}abc\geq - \frac{155}{8}$$$$(a-\frac{3}{2})(b^2-\frac{3}{2})(c^3-3)- 3abc\geq - \frac{363}{20}$$$$(a-\frac{3}{2})(b^2-\frac{3}{2})(c^3-\frac{5}{2})- \frac{55}{16}abc\geq - \frac{341}{16}$$
0 replies
sqing
4 minutes ago
0 replies
J
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inequality
senku23   3
N 13 minutes ago by SunnyEvan
Let x,y,z in R+ prove that 8(x^3+y^3+z^3)2≥9(x^2+yz)(y^2+zx)(z^2+xy).
3 replies
senku23
3 hours ago
SunnyEvan
13 minutes ago
J
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Cool Number Theory
Fermat_Fanatic108   1
N 33 minutes ago by Fermat_Fanatic108
For an integer with 5 digits $n=abcde$ (where $a, b, c, d, e$ are the digits and $a\neq 0$) we define the \textit{permutation sum} as the value $$bcdea+cdeab+deabc+eabcd$$For example the permutation sum of 20253 is $$02532+25320+53202+32025=113079$$Let $m$ and $n$ be two fivedigit integers with the same permutation sum.
Prove that $m=n$.
1 reply
Fermat_Fanatic108
34 minutes ago
Fermat_Fanatic108
33 minutes ago
J
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ratio chasing inside a triangle, segment trisecting
parmenides51   10
N 35 minutes ago by sangsidhya
Source: CRMO 2012 Region 2 p5
Let $ABC$ be a triangle. Let $D, E$ be a points on the segment $BC$ such that $BD =DE = EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine $BP:PQ$.
10 replies
parmenides51
Sep 30, 2018
sangsidhya
35 minutes ago
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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
J
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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
J
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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
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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
J
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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
J
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J
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Problem 2
blug   1
N Mar 15, 2025 by aidan0626
Source: Polish Junior Math Olympiad Finals 2025
A party is attended by boys and girls. Each person attending the party knows exactly 3 boys and exactly 7 girls among the other people. Prove that the number of all the people attending the party is divisible by 20.
1 reply
blug
Mar 15, 2025
aidan0626
Mar 15, 2025
J
Problem 2
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Source: Polish Junior Math Olympiad Finals 2025
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blug
61 posts
blug
#1 h  Mar 15, 2025, 7:54 PM
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A party is attended by boys and girls. Each person attending the party knows exactly 3 boys and exactly 7 girls among the other people. Prove that the number of all the people attending the party is divisible by 20.
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1745 posts
aidan0626
#3 h  Mar 15, 2025, 9:42 PM
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