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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
J
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Combinatorial Game
Cats_on_a_computer   2
N 2 minutes ago by Cats_on_a_computer

Let n>1 be odd. A row of n spaces is initially empty. Alice and Bob alternate moves (Alice first); on each turn a player may either
1. Place a stone in any empty space, or
2. Remove a stone from a non-empty space S, then (if they exist) place stones in the nearest empty spaces immediately to the left and to the right of S.

Furthermore, no move may produce a position that has appeared earlier. The player loses when they cannot make a legal move.
Assuming optimal play, which move(s) can Alice make on her first turn?
2 replies
Cats_on_a_computer
16 minutes ago
Cats_on_a_computer
2 minutes ago
J
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2-var inequality
sqing   4
N 6 minutes ago by sqing
Source: Own
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^2}{b^2}+\frac{1}{a^2}-a^2\geq 1$$$$ \frac{a^3}{b^3}+\frac{1}{a^3}-a^3\geq 1$$
4 replies
sqing
an hour ago
sqing
6 minutes ago
J
H
A game of digits and seventh powers
v_Enhance   28
N 12 minutes ago by blueprimes
Source: Taiwan 2014 TST3 Quiz 1, P2
Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6B_6A_7}$). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.
28 replies
v_Enhance
Jul 18, 2014
blueprimes
12 minutes ago
J
H
n containers distribute gas to a car for a single loop along the track
parmenides51   16
N 22 minutes ago by v_Enhance
Source: Spanish Mathematical Olympiad 1997 P6
The exact quantity of gas needed for a car to complete a single loop around a track is distributed among $n$ containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas, can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.
16 replies
+1 w
parmenides51
Jul 31, 2018
v_Enhance
22 minutes ago
J
No more topics!
H
J
H
nice system of equations
outback   4
N Apr 23, 2025 by Raj_singh1432
Solve in positive numbers the system

$ x_1+\frac{1}{x_2}=4, x_2+\frac{1}{x_3}=1, x_3+\frac{1}{x_4}=4, ..., x_{99}+\frac{1}{x_{100}}=4, x_{100}+\frac{1}{x_1}=1$
4 replies
outback
Oct 8, 2008
Raj_singh1432
Apr 23, 2025
J
nice system of equations
G H J
Reveal topic tags
inequalities
L
G H BBookmark kLocked kLocked NReply
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outback
293 posts
outback
#1 h  Oct 8, 2008, 2:41 PM • 2 Y
Y by Adventure10, Mango247
Solve in positive numbers the system

$ x_1+\frac{1}{x_2}=4, x_2+\frac{1}{x_3}=1, x_3+\frac{1}{x_4}=4, ..., x_{99}+\frac{1}{x_{100}}=4, x_{100}+\frac{1}{x_1}=1$
Z K Y
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Bovy11
151 posts
Bovy11
#2 h  Oct 8, 2008, 7:14 PM • 3 Y
Y by Adventure10, Mango247, ehuseyinyigit
outback wrote:
Solve in positive numbers the system

$ x_1 + \frac {1}{x_2} = 4, x_2 + \frac {1}{x_3} = 1, x_3 + \frac {1}{x_4} = 4, ..., x_{99} + \frac {1}{x_{100}} = 4, x_{100} + \frac {1}{x_1} = 1$

solution
This post has been edited 2 times. Last edited by Bovy11, Oct 9, 2008, 1:33 PM
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t0rajir0u
12167 posts
t0rajir0u
#3 h  Oct 8, 2008, 7:37 PM • 2 Y
Y by Adventure10, Mango247
Was $ x_{100} + \frac{1}{x_1} = 1$ a typo?
Z K Y
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outback
293 posts
outback
#4 h  Oct 9, 2008, 12:32 AM • 2 Y
Y by Adventure10, Mango247
t0rajir0u wrote:
Was $ x_{100} + \frac {1}{x_1} = 1$ a typo?

Why do you think it is?
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Raj_singh1432
3 posts
Raj_singh1432
#5 h  Apr 23, 2025, 4:57 PM
Y by
t0rajir0u wrote:
Was $ x_{100} + \frac{1}{x_1} = 1$ a typo?

No
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