G
Topic
First Poster
Last Poster
Dou Fang Geometry in Taiwan TST
Li4 9
N
by WLOGQED1729
Source: 2025 Taiwan TST Round 3 Mock P2
Let 
and 
be the incircle and circumcircle of the acute triangle 
, respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on 
. Extend 
and 
, intersecting at 
and 
, respectively. Prove that 
and 
intersects on .
Proposed by kyou46, Li4, Revolilol.
and 
be the incircle and circumcircle of the acute triangle 
, respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on 
. Extend 
and 
, intersecting at 
and 
, respectively. Prove that 
and 
intersects on .Proposed by kyou46, Li4, Revolilol.
9 replies
A4 BMO SHL 2024
mihaig 0
Source: Someone known
Let 
be real numbers such that 
Prove
Prove that if 
is a positive constant, then
is not always true
be real numbers such that 
Prove
Prove that if 
is a positive constant, then
is not always true0 replies
.
Equal segments in a cyclic quadrilateral
a_507_bc 4
N
by AylyGayypow009
Source: Greece JBMO TST 2023 P2
Consider a cyclic quadrilateral 
in which 
and 
. Let 
be a point on the side 
and 
a point on the line such that 
. Prove that 
.
in which 
and 
. Let 
be a point on the side 
and 
a point on the line such that 
. Prove that 
.
4 replies
satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)
Geometry
AlexCenteno2007 0
Source: NCA
Let ABC be an acute triangle. The altitudes from B and C intersect the sides AC and AB at E and F, respectively. The internal bisector of ∠A intersects BE and CF at T and S, respectively. The circles with diameters AT and AS intersect the circumcircle of ABC at X and Y, respectively. Prove that XY, EF, and BC meet at the exsimilicenter of BTX and CSY
0 replies
Prove that
Let 
Prove that
Based on IMO 2024 P2
Miquel-point 1
N
by MathLuis
Source: KoMaL B. 5461
Prove that for any positive integers 
, 
, 
and 
there exists infinitely many positive integers 
for which 
and 
are not relatively primes.
Proposed by Géza Kós
, 
, 
and 
there exists infinitely many positive integers 
for which 
and 
are not relatively primes.Proposed by Géza Kós
1 reply
egmo 2018 p4
microsoft_office_word 29
N
by math-olympiad-clown
Source: EGMO 2018 P4
A domino is a 
or 
tile.
Let
be an integer. Dominoes are placed on an 
board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some 
such that each row and each column has a value of 
. Prove that a balanced configuration exists for every , and find the minimum number of dominoes needed in such a configuration.
or 
tile.Let
be an integer. Dominoes are placed on an 
board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some 
such that each row and each column has a value of 
. Prove that a balanced configuration exists for every , and find the minimum number of dominoes needed in such a configuration.
29 replies
Tangents to a cyclic quadrilateral
v_Enhance 24
N
by hectorleo123
Source: ELMO Shortlist 2013: Problem G9, by Allen Liu
Let be a cyclic quadrilateral inscribed in circle whose diagonals meet at . Lines 
and 
meet at . Segment 
intersects at . Lines 
and meet at 
, and lines 
and meet at 
. Prove that 
and concur with the tangent to at .
Proposed by Allen Liu
be a cyclic quadrilateral inscribed in circle whose diagonals meet at . Lines 
and 
meet at . Segment 
intersects at . Lines 
and meet at 
, and lines 
and meet at 
. Prove that 
and concur with the tangent to at .Proposed by Allen Liu
24 replies
