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Geometry Finale: Incircles and concurrency
lminsl 174
N
by LitleCabage0639
Source: IMO 2019 Problem 6
Let 
be the incentre of acute triangle 
with 
. The incircle 
of is tangent to sides 
, and 
at 
and 
, respectively. The line through 
perpendicular to 
meets at 
. Line 
meets again at 
. The circumcircles of triangle 
and 
meet again at 
.
Prove that lines
and 
meet on the line through 
perpendicular to 
.
Proposed by Anant Mudgal, India
be the incentre of acute triangle 
with 
. The incircle 
of is tangent to sides 
, and 
at 
and 
, respectively. The line through 
perpendicular to 
meets at 
. Line 
meets again at 
. The circumcircles of triangle 
and 
meet again at 
.Prove that lines
and 
meet on the line through 
perpendicular to 
.Proposed by Anant Mudgal, India
174 replies
,
cubefree divisibility
DottedCaculator 61
N
by sansgankrsngupta
Source: 2021 ISL N1
Find all positive integers 
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
61 replies
+1 w
Cauchy and multiplicative function over a field extension
miiirz30 5
N
by AshAuktober
Source: 2025 Euler Olympiad, Round 2
Find all functions ![$f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$](//latex.artofproblemsolving.com/c/a/1/ca172b912bf5bf7178cbbd5745fbc0a6a8b27503.png)
such that for all ![$x, y \in \mathbb{Q}[\sqrt{2}]$](//latex.artofproblemsolving.com/9/f/e/9fede0b2fd22836372b8afc488ccd8a14db5d0c2.png)
,
where ![$\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$](//latex.artofproblemsolving.com/f/a/c/fac8bf68761534a139d570bf66b1aa5dfb59e8db.png)
.
Proposed by Stijn Cambie, Belgium
![$f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$](http://latex.artofproblemsolving.com/c/a/1/ca172b912bf5bf7178cbbd5745fbc0a6a8b27503.png)
such that for all ![$x, y \in \mathbb{Q}[\sqrt{2}]$](http://latex.artofproblemsolving.com/9/f/e/9fede0b2fd22836372b8afc488ccd8a14db5d0c2.png)
,
where ![$\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$](http://latex.artofproblemsolving.com/f/a/c/fac8bf68761534a139d570bf66b1aa5dfb59e8db.png)
.Proposed by Stijn Cambie, Belgium
5 replies
Circle
bilarev 2
N
by Blackbeam999
Let 
be a circle, and 
are parallel chords and 
is a
line from C, that intersects in its middle point 
and 
. 
is the middle of 
. Prove that 
be a circle, and 
are parallel chords and 
is aline from C, that intersects
in its middle point 
and 
. 
is the middle of 
. Prove that 
2 replies
Show that 2 triangles are bilogic
kosmonauten3114 0
Source: My own
Given a scalene acute triangle 
with orthic triangle 
, let 
, 
, 
be points such that 
. Let 
be the trilinear polar of the polar conjugate of wrt , and define 
and 
cyclically. Let 
be the triangle bounded by , , .
Show that and are bilogic.
with orthic triangle 
, let 
, 
, 
be points such that 
. Let 
be the trilinear polar of the polar conjugate of wrt , and define 
and 
cyclically. Let 
be the triangle bounded by , , .Show that
and are bilogic.
0 replies
Interesting functions with iterations over integers
miiirz30 0
Source: 2025 Euler Olympiad, Round 2
For any subset 
, a function 
is called interesting if the following two conditions hold:
1. There is no element
such that 
.
2. For every, we have 
(where 
denotes the -th iteration of 
).
Prove that:
a) There exist infinitely many interesting functions
.
b) There exist infinitely many positive integers
for which there is no interesting function
Proposed by Giorgi Kekenadze, Georgia
, a function 
is called interesting if the following two conditions hold:1. There is no element
such that 
.2. For every
, we have 
(where 
denotes the -th iteration of 
).Prove that:
a) There exist infinitely many interesting functions
.b) There exist infinitely many positive integers
for which there is no interesting function
Proposed by Giorgi Kekenadze, Georgia
0 replies
Moving stones on an infinite row
miiirz30 0
Source: 2025 Euler Olympiad, Round 2
We are given an infinite row of cells extending infinitely in both directions. Some cells contain one or more stones. The total number of stones is finite. At each move, the player performs one of the following three operations:
1. Take three stones from some cell, and add one stone to the cells located one cell to the left and one cell to the right, each skipping one cell in between.
2. Take two stones from some cell, and add one stone to the cell one cell to the left, skipping one cell and one stone to the adjacent cell to the right.
3. Take one stone from each of two adjacent cells, and add one stone to the cell to the right of these two cells.
The process ends when no moves are possible. Prove that the process always terminates and the final distribution of stones does not depend on the choices of moves made by the player.
IMAGE
Proposed by Luka Tsulaia, Georgia
1. Take three stones from some cell, and add one stone to the cells located one cell to the left and one cell to the right, each skipping one cell in between.
2. Take two stones from some cell, and add one stone to the cell one cell to the left, skipping one cell and one stone to the adjacent cell to the right.
3. Take one stone from each of two adjacent cells, and add one stone to the cell to the right of these two cells.
The process ends when no moves are possible. Prove that the process always terminates and the final distribution of stones does not depend on the choices of moves made by the player.
IMAGE
Proposed by Luka Tsulaia, Georgia
0 replies
Alice, Bob and 6 boxes
Anulick 1
N
by RPCX
Source: SMMC 2024, B1
Alice has six boxes labelled 1 through 6. She secretly chooses exactly two of the boxes and places a coin inside each. Bob is trying to guess which two boxes contain the coins. Each time Bob guesses, he does so by tapping exactly two of the boxes. Alice then responds by telling him the total number of coins inside the two boxes that he tapped. Bob successfully finds the two coins when Alice responds with the number 2.
What is the smallest positive integer such that Bob can always find the two coins in at most guesses?
What is the smallest positive integer
such that Bob can always find the two coins in at most guesses?
1 reply
Prove the statement
Butterfly 12
N
by oty
Given an infinite sequence ![$\{x_n\} \subseteq [0,1]$](//latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
, there exists some constant 
, for any 
, among the sequence 
and 
could be chosen to satisfy 
and 
.
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
, there exists some constant 
, for any 
, among the sequence 
and 
could be chosen to satisfy 
and 
.
12 replies
