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A Projection Theorem
buratinogigle 2
N
by wh0nix
Source: VN Math Olympiad For High School Students P1 - 2025
In triangle 
, prove that
![\[ a = b\cos C + c\cos B. \]](//latex.artofproblemsolving.com/7/e/d/7ed699f18ede191e5b89bc79d7f0592294347818.png)
, prove that![\[ a = b\cos C + c\cos B. \]](http://latex.artofproblemsolving.com/7/e/d/7ed699f18ede191e5b89bc79d7f0592294347818.png)
2 replies
Turbo's en route to visit each cell of the board
Lukaluce 18
N
by yyhloveu1314
Source: EGMO 2025 P5
Let 
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.
Proposed by Melek Güngör, Turkey
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.Proposed by Melek Güngör, Turkey
18 replies
Perhaps a classic with parameter
mihaig 1
N
by LLriyue
Find the largest positive constant 
such that
for all reals 
satisfying 
such that
for all reals 
satisfying 
1 reply
Connected graph with k edges
orl 26
N
by Maximilian113
Source: IMO 1991, Day 2, Problem 4, IMO ShortList 1991, Problem 10 (USA 5)
Suppose 
is a connected graph with 
edges. Prove that it is possible to label the edges 
in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.
Note: Graph-Definition. A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices
belongs to at most one edge. The graph 
is connected if for each pair of distinct vertices 
there is some sequence of vertices 
such that each pair 
is joined by an edge of 
.
is a connected graph with 
edges. Prove that it is possible to label the edges 
in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.Note: Graph-Definition. A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices
belongs to at most one edge. The graph 
is connected if for each pair of distinct vertices 
there is some sequence of vertices 
such that each pair 
is joined by an edge of 
.
26 replies
and 
. Prove that
Where 
Pls solve this FE
ItzsleepyXD 2
N
by ItzsleepyXD
Source: My friend
Let 
be the set of real numbers. Determine all functions 
that satisfy the equation![\[f(x^2f(x+y))=f(xyf(x))+xf(x)^2\]](//latex.artofproblemsolving.com/b/4/b/b4b1e22e937394550f13cfdab142ca5bf4a26aea.png)
for all real numbers 
and 
.
be the set of real numbers. Determine all functions 
that satisfy the equation![\[f(x^2f(x+y))=f(xyf(x))+xf(x)^2\]](http://latex.artofproblemsolving.com/b/4/b/b4b1e22e937394550f13cfdab142ca5bf4a26aea.png)
for all real numbers 
and 
.
2 replies
The old one is gone.
EeEeRUT 3
N
by ItzsleepyXD
Source: EGMO 2025 P2
An infinite increasing sequence 
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
Show that there exists an infinite sequence
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .Show that there exists an infinite sequence
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
3 replies
Interesting inequalities
sqing 4
N
by sqing
Source: Own
Let 
be reals such that 
. Prove that
Let be reals such that 
. Prove that
be reals such that 
. Prove that
Let be reals such that 
. Prove that
4 replies
Ant wanna come to A
Rohit-2006 3
N
by Rohit-2006
An insect starts from 
and in 
steps and has to reach again. But in between one of the s steps and can't go . Find probability. For example 
is valid but 
is not valid.
*Too many edits, my brain had gone to a trip
and in 
steps and has to reach again. But in between one of the s steps and can't go . Find probability. For example 
is valid but 
is not valid.*Too many edits, my brain had gone to a trip
3 replies
BMO Shortlist 2021 A5
Lukaluce 16
N
by Sedro
Source: BMO Shortlist 2021
Find all functions 
such that
holds for all 
.
Proposed by Nikola Velov, North Macedonia
such that
holds for all 
.Proposed by Nikola Velov, North Macedonia
16 replies
