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Last Poster
weird conditions in geo
Davdav1232 1
N
by NO_SQUARES
Source: Israel TST 7 2025 p1
Let 
be an isosceles triangle with 
. Let 
be a point on 
. Let 
be a point inside the triangle such that 
and
![\[
CL \cdot BD = BL \cdot CD.
\]](//latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)
Prove that the circumcenter of triangle 
lies on line 
.
be an isosceles triangle with 
. Let 
be a point on 
. Let 
be a point inside the triangle such that 
and![\[
CL \cdot BD = BL \cdot CD.
\]](http://latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)
Prove that the circumcenter of triangle 
lies on line 
.
1 reply
Functional equation on R
rope0811 15
N
by ezpotd
Source: IMO ShortList 2003, algebra problem 2
Find all nondecreasing functions 
such that
(i)
(ii)
for all real numbers 
such that 
.
Proposed by A. Di Pisquale & D. Matthews, Australia
such that(i)
(ii)
for all real numbers 
such that 
.Proposed by A. Di Pisquale & D. Matthews, Australia
15 replies
all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng 34
N
by LenaEnjoyer
Source: Balkan MO 2025 P3
Find all functions 
such that for all 
,
![\[f(x+yf(x))+y = xy + f(x+y).\]](//latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
Proposed by Giannis Galamatis, Greece
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
Proposed by Giannis Galamatis, Greece
34 replies
Miklos Schweitzer 1971_7
ehsan2004 1
N
by pi_quadrat_sechstel
Let 
be an integer, let 
be a set of 
elements, and let 
, be distinct subsets of of size at least 
such that ![\[ A_i \cap A_j \not= \emptyset, A_i \cap A_k \not= \emptyset, A_j \cap A_k \not= \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not= \emptyset \ .\]](//latex.artofproblemsolving.com/c/1/1/c113f5cdce00dfec6714c382a8e1bc74f188f025.png)
Show that 
.
P. Erdos
be an integer, let 
be a set of 
elements, and let 
, be distinct subsets of of size at least 
such that ![\[ A_i \cap A_j \not= \emptyset, A_i \cap A_k \not= \emptyset, A_j \cap A_k \not= \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not= \emptyset \ .\]](http://latex.artofproblemsolving.com/c/1/1/c113f5cdce00dfec6714c382a8e1bc74f188f025.png)
Show that 
.P. Erdos
1 reply
Functional equation with a twist (it's number theory)
Davdav1232 0
Source: Israel TST 8 2025 p2
Prove that for all primes 
such that 
or 
, there exist integers
![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](//latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that
![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](//latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
such that 
or 
, there exist integers![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](http://latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
0 replies
Grid combi with T-tetrominos
Davdav1232 0
Source: Israel TST 8 2025 p1
Let 
denote the maximum number of 
-tetrominoes that can be placed on an 
board such that each -tetromino covers at least one cell that is not covered by any other -tetromino.
Find the smallest real number
such that
![\[
f(N) \leq cN^2
\]](//latex.artofproblemsolving.com/7/4/8/748109c3d63203f9a67a167434e10a909f38e83f.png)
for all positive integers 
.
denote the maximum number of 
-tetrominoes that can be placed on an 
board such that each -tetromino covers at least one cell that is not covered by any other -tetromino.Find the smallest real number
such that![\[
f(N) \leq cN^2
\]](http://latex.artofproblemsolving.com/7/4/8/748109c3d63203f9a67a167434e10a909f38e83f.png)
for all positive integers 
.
0 replies
forced vertices in graphs
Davdav1232 0
Source: Israel TST 7 2025 p2
Let 
be a graph colored using 
colors. We say that a vertex is forced if it has neighbors in all the other 
colors.
Prove that for any
-regular graph , there exists a coloring using 
colors such that at least 
of the colors have a forced vertex of that color.
Note: The graph coloring must be valid, this means no
vertices of the same color may be adjacent.
be a graph colored using 
colors. We say that a vertex is forced if it has neighbors in all the other 
colors.Prove that for any
-regular graph , there exists a coloring using 
colors such that at least 
of the colors have a forced vertex of that color.Note: The graph coloring must be valid, this means no
vertices of the same color may be adjacent.
0 replies
Can this sequence be bounded?
darij grinberg 70
N
by ezpotd
Source: German pre-TST 2005, problem 4, ISL 2004, algebra problem 2
Let 
, 
, 
, ... be an infinite sequence of real numbers satisfying the equation 
for all 
, where and are two different positive reals.
Can this sequence, , , ... be bounded?
Proposed by Mihai Bălună, Romania
, 
, 
, ... be an infinite sequence of real numbers satisfying the equation 
for all 
, where and are two different positive reals.Can this sequence
, , , ... be bounded?Proposed by Mihai Bălună, Romania
70 replies
find angle
TBazar 4
N
by vanstraelen
Given 
triangle with 
. We take 
, 
point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point 
. If 
, find 
triangle with 
. We take 
, 
point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point 
. If 
, find 
4 replies
Polys with int coefficients
adihaya 4
N
by sangsidhya
Source: 2012 INMO (India National Olympiad), Problem #3
Define a sequence 
of functions by 
for 
. Prove that each 
is a polynomial with integer coefficients.
of functions by 
for 
. Prove that each 
is a polynomial with integer coefficients.
4 replies
