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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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help me ..
exoticc   2
N 3 hours ago by sodiumaka
Find all pairs of functions (f,g) : R->R that satisfy:
f(1)=2025;
g(f(x+y))+2x+y-1=f(x)+(2x+y)g(y) , ∀x,y∈R
2 replies
exoticc
Today at 9:26 AM
sodiumaka
3 hours ago
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Inequalities
sqing   20
N 4 hours ago by ytChen
Let $ a,b,c\geq 0 ,a+b+c\leq 3. $ Prove that
$$a^2+b^2+c^2+ab +2ca+2bc + abc \leq \frac{251}{27}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{2}{5}abc \leq \frac{4861}{540}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{7}{20}abc \leq \frac{2381411}{26460}$$
20 replies
sqing
May 21, 2025
ytChen
4 hours ago
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How do I prove this? - Sets and symmetric difference
smadadi1000   0
4 hours ago
For sets A, B and C, prove that (A Δ B) Δ C = (A Δ C) Δ (A \ B).

The textbook - Proofs by Jay Cummings - had this definition for symmetric difference:
A Δ B = ( A U B)/(A ∩ B)

this is exercise 3.37 (e).
0 replies
smadadi1000
4 hours ago
0 replies
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21st PMO National Orals #9
yes45   0
4 hours ago
In square $ABCD$, $P$ and $Q$ are points on sides $CD$ and $BC$, respectively, such that $\angle{APQ} = 90^\circ$. If $AP = 4$ and $PQ = 3$, find the area of $ABCD$.

Answer Confirmation
Solution
0 replies
yes45
4 hours ago
0 replies
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Base 9 Repunits = Triangular Numbers
MCrawford   4
N May 9, 2006 by Iron Fist
Prove that every positive integer that can be written using only 1 as each digit in base 9 is a triangular number:
\begin{eqnarray*} 1_9 &=& 1 \\ 11_9 &=& 1 + 2 + 3 + 4 \\ 111_9 &=& 1 + 2 + \cdots + 13 \\ 1111_9 &=& 1 + 2 + \cdots + 40 \\ & \vdots & \end{eqnarray*}
4 replies
MCrawford
May 8, 2006
Iron Fist
May 9, 2006
J
Base 9 Repunits = Triangular Numbers
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MCrawford
6325 posts
MCrawford
#1 h  May 8, 2006, 9:24 PM • 1 Y
Y by Adventure10
Prove that every positive integer that can be written using only 1 as each digit in base 9 is a triangular number:
\begin{eqnarray*} 1_9 &=& 1 \\ 11_9 &=& 1 + 2 + 3 + 4 \\ 111_9 &=& 1 + 2 + \cdots + 13 \\ 1111_9 &=& 1 + 2 + \cdots + 40 \\ & \vdots & \end{eqnarray*}
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chess64
4794 posts
chess64
#2 h  May 8, 2006, 10:00 PM • 3 Y
Y by Adventure10, Adventure10, Mango247
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towersfreak2006
3012 posts
towersfreak2006
#3 h  May 8, 2006, 10:06 PM • 2 Y
Y by Adventure10, Mango247
MCrawford wrote:
Prove that every positive integer that can be written using only 1 as each digit in base 9 is a triangular number:
\begin{eqnarray*} 1_9 &=& 1 \\ 11_9 &=& 1 + 2 + 3 + 4 \\ 111_9 &=& 1 + 2 + \cdots + 13 \\ 1111_9 &=& 1 + 2 + \cdots + 40 \\ & \vdots & \end{eqnarray*}
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maniacman384
282 posts
maniacman384
#4 h  May 9, 2006, 12:16 AM • 2 Y
Y by Adventure10, Mango247
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Iron Fist
98 posts
Iron Fist
#5 h  May 9, 2006, 12:20 AM • 2 Y
Y by Adventure10, Mango247
maniacman384 wrote:
I just noticed something, any number that starts with a 1 and has the rest of its digits as zeros is a triangular number.

Last time I checked, 1000 was a triangle number, not in base 10 nor base 9...
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