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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
Italian WinterCamps test07 Problem4
mattilgale 89
N
by cj13609517288
Source: ISL 2006, G3, VAIMO 2007/5
Let 
be a convex pentagon such that
![\[ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.
\]](//latex.artofproblemsolving.com/d/3/d/d3d9a82f4318190298a8f008d417a364e03f1fca.png)
The diagonals 
and 
meet at 
. Prove that the line 
bisects the side 
.
Proposed by Zuming Feng, USA
be a convex pentagon such that![\[ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.
\]](http://latex.artofproblemsolving.com/d/3/d/d3d9a82f4318190298a8f008d417a364e03f1fca.png)
The diagonals 
and 
meet at 
. Prove that the line 
bisects the side 
.Proposed by Zuming Feng, USA
89 replies
Simple triangle geometry [a fixed point]
darij grinberg 49
N
by cj13609517288
Source: German TST 2004, IMO ShortList 2003, geometry problem 2
Three distinct points 
, 
, and 
are fixed on a line in this order. Let 
be a circle passing through and whose center does not lie on the line 
. Denote by the intersection of the tangents to at and . Suppose meets the segment 
at 
. Prove that the intersection of the bisector of 
and the line does not depend on the choice of .
, 
, and 
are fixed on a line in this order. Let 
be a circle passing through and whose center does not lie on the line 
. Denote by the intersection of the tangents to at and . Suppose meets the segment 
at 
. Prove that the intersection of the bisector of 
and the line does not depend on the choice of .
49 replies
Kosovo MO 2010 Problem 5
Com10atorics 19
N
by CM1910
Source: Kosovo MO 2010 Problem 5
Let 
be positive real numbers such that 
. Prove that
.
be positive real numbers such that 
. Prove that
.
19 replies
Hard combi
EeEApO 1
N
by EeEApO
In a quiz competition, there are a total of 
questions, each with 
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?
Now, suppose my spouse and I move into a new home. Every year, we have twins. Starting at the age of
, each of our twin children also begins to have twins every year. If this pattern continues, how many years will it take for my family to grow large enough to have the required number of members to guarantee winning the quiz gift?
questions, each with 
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?Now, suppose my spouse and I move into a new home. Every year, we have twins. Starting at the age of
, each of our twin children also begins to have twins every year. If this pattern continues, how many years will it take for my family to grow large enough to have the required number of members to guarantee winning the quiz gift?
1 reply
