G
Topic
First Poster
Last Poster
Projections on lateral faces of pyramid are coplanar
Miquel-point 0
Source: Romanian NMO 2025 8.4
From a point 
inside a square 
we raise a segment 
perpendicular to the plane of the square. Show that the projections of on the planes 
, 
, 
and 
are coplanar if and only if ![$O\in [AC]\cup [BD]$](//latex.artofproblemsolving.com/6/6/6/666df8f2afa87f15c79f9c96d9d1cea17a2a7af0.png)
.
inside a square 
we raise a segment 
perpendicular to the plane of the square. Show that the projections of on the planes 
, 
, 
and 
are coplanar if and only if ![$O\in [AC]\cup [BD]$](http://latex.artofproblemsolving.com/6/6/6/666df8f2afa87f15c79f9c96d9d1cea17a2a7af0.png)
.0 replies
NT equation
EthanWYX2009 3
N
by pavel kozlov
Source: 2025 TST T11
Let 
. Proof that
![\[
(2^x - 1)(5^x - 1) = y^n
\]](//latex.artofproblemsolving.com/f/8/f/f8f013a96863bc5f39aa23edb98eed9aec54a17f.png)
have no positive integer solution 
.
. Proof that![\[
(2^x - 1)(5^x - 1) = y^n
\]](http://latex.artofproblemsolving.com/f/8/f/f8f013a96863bc5f39aa23edb98eed9aec54a17f.png)
have no positive integer solution 
.
3 replies
math olympiads
Lirimath 1
N
by maromex
Let a,b,c be real numbers such that a^2(b+c)+b^2(c+a)+c^2(a+b)=3(a+b+c-1) and a+b+c differnet by 0.Prove that ab+bc+ca=3 if and only if abc=1
1 reply
math olympiad
Lirimath 2
N
by maromex
Let a,b,c be positive real numbers such that a+b+c=3abc.Prove that
a^2+b^2+c^2+3>=2(ab+bc+ca).
a^2+b^2+c^2+3>=2(ab+bc+ca).
2 replies
Interesting F.E
Jackson0423 9
N
by Sedro
Show that there does not exist a function
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](//latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,
![\[
f(x^2 + y) \geq f(x) + y.
\]](//latex.artofproblemsolving.com/c/7/c/c7cd7c9a4d4372afc23e4503bb1c05fc7d07e788.png)
~Korea 2017 P7
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x^2 + y) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/c/7/c/c7cd7c9a4d4372afc23e4503bb1c05fc7d07e788.png)
~Korea 2017 P7
9 replies
1 viewing
Three-player money transfer game with unique winner per round
rilarfer 1
N
by Lankou
Source: ASJTNic 2005
Ana, Bárbara, and Cecilia play a game with the following rules:
[list]
[*] In each round, exactly one player wins.
[*] The two losing players each give half of their current money to the winner.
[/list]
The game proceeds as follows:
[list=1]
[*] Ana wins the first round.
[*] Bárbara wins the second round.
[*] Cecilia wins the third round.
[/list]
At the end of the game, the players have the following amounts:
[list]
[*] Ana: C$35
[*] Bárbara: C$75
[*] Cecilia: C$150
[/list]
How much money did each of them have at the beginning?
[list]
[*] In each round, exactly one player wins.
[*] The two losing players each give half of their current money to the winner.
[/list]
The game proceeds as follows:
[list=1]
[*] Ana wins the first round.
[*] Bárbara wins the second round.
[*] Cecilia wins the third round.
[/list]
At the end of the game, the players have the following amounts:
[list]
[*] Ana: C$35
[*] Bárbara: C$75
[*] Cecilia: C$150
[/list]
How much money did each of them have at the beginning?
1 reply
Find all integer solutions to an exponential equation involving powers of 2 and
rilarfer 2
N
by teomihai
Source: ASJTNic 2005
Find all integer pairs 
such that:
such that:
2 replies
Winning strategy in a two-player subtraction game starting with 65 tokens
rilarfer 1
N
by CHESSR1DER
Source: ASJTNic 2005
Juan and Pedro play the following game:
[list]
[*] There are initially 65 tokens.
[*] The players alternate turns, starting with Juan.
[*] On each turn, a player may remove between 1 and 7 tokens.
[*] The player who removes the last token wins.
[/list]
Describe and justify a strategy that guarantees Juan a win.
[list]
[*] There are initially 65 tokens.
[*] The players alternate turns, starting with Juan.
[*] On each turn, a player may remove between 1 and 7 tokens.
[*] The player who removes the last token wins.
[/list]
Describe and justify a strategy that guarantees Juan a win.
1 reply
Radius of circle tangent to two equal circles and a common line
rilarfer 1
N
by Lankou
Source: ASJTNic 2005
Two circles of radius 2 are tangent to each other and to a straight line. A third circle is placed so that it is tangent to both of the other circles and also tangent to the same straight line.
What is the radius of the third circle?
IMAGE
What is the radius of the third circle?
IMAGE
1 reply
Four-variable FE mod n
TheUltimate123 2
N
by cosmicgenius
Source: PRELMO 2023/3 (http://tinyurl.com/PRELMO)
Let 
be a positive integer, and let 
denote the integers modulo . Determine the number of functions 
satisfying 
for all 
.
be a positive integer, and let 
denote the integers modulo . Determine the number of functions 
satisfying 
for all 
.
2 replies
