G
Topic
First Poster
Last Poster
NEPAL TST 2025 DAY 2
Tony_stark0094 8
N
by cursed_tangent1434
Consider an acute triangle 
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
Define
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
If
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.Define
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .If
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
8 replies
Interesting inequalities
sqing 4
N
by sqing
Source: Own
Let 
and 
. Prove that
Where 
and 
. Prove that
Where 
4 replies
1 viewing
NEPAL TST DAY 2 PROBLEM 2
Tony_stark0094 6
N
by cursed_tangent1434
Kritesh manages traffic on a 
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the grid.
Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove
cars from the grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.
What is the minimum value of that guarantees that Kritesh's job is possible?
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the grid.Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove
cars from the grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.What is the minimum value of
that guarantees that Kritesh's job is possible?
6 replies
NEPAL TST DAY-2 PROBLEM 1
Tony_stark0094 9
N
by cursed_tangent1434
Let the sequence 
be defined by
![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](//latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that
![\[
a_n^{2025} >n^{2024}
\]](//latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
be defined by![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](http://latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that![\[
a_n^{2025} >n^{2024}
\]](http://latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
9 replies
and 
Prove that
Where 
Weird Inequality Problem
Omerking 4
N
by sqing
Following inequality is given:
Find the range of values that can be taken by :
Where
are positive reals.
Find the range of values that can be taken by :
Where
are positive reals.
4 replies
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
