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Multiple of power of two.
dgrozev 4
N
by Assassino9931
Source: Bulgarian TST, 2020, p2
Given two odd natural numbers 
prove that for each 
there exists 
such that either 
or 
is multiple of 
prove that for each 
there exists 
such that either 
or 
is multiple of 
4 replies
Parallelity and equal angles given, wanted an angle equality
BarisKoyuncu 6
N
by expsaggaf
Source: 2022 Turkey JBMO TST P4
Given a convex quadrilateral 
such that 
. The lines 
and 
intersect at a point 
and the line passing through which is parallel to 
, intersects 
at 
. Prove that
such that 
. The lines 
and 
intersect at a point 
and the line passing through which is parallel to 
, intersects 
at 
. Prove that
6 replies
2020 IGO Intermediate P3
turko.arias 13
N
by fe.
Source: 7th Iranian Geometry Olympiad (Intermediate) P3
In acute-angled triangle 
(
), point 
is the orthocenter and point 
is the midpoint of the segment . The median 
intersects the circumcircle of triangle at 
. The line 
intersects the perpendicular bisector of at 
and the circumcircle of the triangle again at 
. Point 
lies on circle 
, passing through 
and , such that 
is a trapezoid (
). Prove that 
and 
meet on .
Proposed by Alireza Dadgarnia
(
), point 
is the orthocenter and point 
is the midpoint of the segment . The median 
intersects the circumcircle of triangle at 
. The line 
intersects the perpendicular bisector of at 
and the circumcircle of the triangle again at 
. Point 
lies on circle 
, passing through 
and , such that 
is a trapezoid (
). Prove that 
and 
meet on .Proposed by Alireza Dadgarnia
13 replies
Functional equation involving decimal-place count
saulgodman 0
Source: Own
Let
![\[
S = \left\{\, x \in \mathbb{Q} : x \text{ has a finite decimal expansion} \,\right\}.
\]](http://latex.artofproblemsolving.com/1/c/e/1ce81426f5d169a9b4c254ef91a1fe34250504b4.png)
For each 
, define![\[
d(x) = \text{the number of digits after the decimal point in the (reduced) decimal form of } x.
\]](http://latex.artofproblemsolving.com/8/1/9/819d07d26274a051a776fcf9f6d7c668ab8d0dab.png)
Find all functions 
such that, whenever both 
and 
,![\[
f(x) + f(y) = f(x+y) + d(x y).
\]](http://latex.artofproblemsolving.com/5/c/a/5ca142ba21a23dc21db6e3c712bc0a76d561647c.png)
0 replies
Did you talk to Noga Alon?
pohoatza 36
N
by ezpotd
Source: IMO Shortlist 2006, Combinatorics 3, AIMO 2007, TST 6, P2
Let 
be a finite set of points in the plane such that no three of them are on a line. For each convex polygon 
whose vertices are in , let 
be the number of vertices of , and let 
be the number of points of which are outside . A line segment, a point, and the empty set are considered as convex polygons of 
, 
, and 
vertices respectively. Prove that for every real number 
![\[\sum_{P}{x^{a(P)}(1 - x)^{b(P)}} = 1,\]](//latex.artofproblemsolving.com/3/2/7/3279b2917f21c8f3c2b819e39a32528ad18e4b4b.png)
where the sum is taken over all convex polygons with vertices in .
Alternative formulation:
Let
be a finite point set in the plane and no three points are collinear. A subset 
of will be called round if its elements is the set of vertices of a convex 
gon 
For each round subset let 
be the number of points from which are exterior from the convex gon Subsets with 
and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset of construct the polynomial
![\[ P_A(x) = x^{|A|}(1 - x)^{r(A)}.
\]](//latex.artofproblemsolving.com/e/b/c/ebc6f8fa2cc303e1062c5a95d4f65c9d2691d0ca.png)
Show that the sum of polynomials for all round subsets is exactly the polynomial
Proposed by Federico Ardila, Colombia
be a finite set of points in the plane such that no three of them are on a line. For each convex polygon 
whose vertices are in , let 
be the number of vertices of , and let 
be the number of points of which are outside . A line segment, a point, and the empty set are considered as convex polygons of 
, 
, and 
vertices respectively. Prove that for every real number 
![\[\sum_{P}{x^{a(P)}(1 - x)^{b(P)}} = 1,\]](http://latex.artofproblemsolving.com/3/2/7/3279b2917f21c8f3c2b819e39a32528ad18e4b4b.png)
where the sum is taken over all convex polygons with vertices in .Alternative formulation:
Let
be a finite point set in the plane and no three points are collinear. A subset 
of will be called round if its elements is the set of vertices of a convex 
gon 
For each round subset let 
be the number of points from which are exterior from the convex gon Subsets with 
and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset of construct the polynomial![\[ P_A(x) = x^{|A|}(1 - x)^{r(A)}.
\]](http://latex.artofproblemsolving.com/e/b/c/ebc6f8fa2cc303e1062c5a95d4f65c9d2691d0ca.png)
Show that the sum of polynomials for all round subsets is exactly the polynomial
Proposed by Federico Ardila, Colombia
36 replies
sqrt(n) or n+p (Generalized 2017 IMO/1)
vincentwant 2
N
by vincentwant
Let 
be an odd prime. Define 
over the positive integers as follows:
Let be chosen such that there exists an ordered pair of positive integers 
where 
such that 
. Prove that there exists at least three integers 
such that 
and 
is a perfect square.
be an odd prime. Define 
over the positive integers as follows:
Let
be chosen such that there exists an ordered pair of positive integers 
where 
such that 
. Prove that there exists at least three integers 
such that 
and 
is a perfect square.
2 replies
Aslı tries to make the amount of stones at every unit square is equal
AlperenINAN 1
N
by expsaggaf
Source: Turkey JBMO TST 2025 P2
Let 
be a positive integer. Aslı and Zehra are playing a game on an 
grid. Initially, 
stones are placed on some of the unit squares of this grid.
On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square.
For which values of, regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?
be a positive integer. Aslı and Zehra are playing a game on an 
grid. Initially, 
stones are placed on some of the unit squares of this grid.On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square.
For which values of
, regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?
1 reply
Arrange positive divisors of n in rectangular table!
cjquines0 44
N
by ezpotd
Source: 2016 IMO Shortlist C2
Find all positive integers for which all positive divisors of can be put into the cells of a rectangular table under the following constraints:
[list]
[*]each cell contains a distinct divisor;
[*]the sums of all rows are equal; and
[*]the sums of all columns are equal.
[/list]
for which all positive divisors of can be put into the cells of a rectangular table under the following constraints:[list]
[*]each cell contains a distinct divisor;
[*]the sums of all rows are equal; and
[*]the sums of all columns are equal.
[/list]
44 replies
Problem 1: Triangle triviality
ZetaX 135
N
by mathnerd_101
Source: IMO 2006, 1. day
Let be triangle with incenter 
. A point in the interior of the triangle satisfies ![\[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\]](//latex.artofproblemsolving.com/d/6/2/d622094b95a1f317063cf2f02a8b66b31751638f.png)
Show that 
, and that equality holds if and only if 
.
be triangle with incenter 
. A point in the interior of the triangle satisfies ![\[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\]](http://latex.artofproblemsolving.com/d/6/2/d622094b95a1f317063cf2f02a8b66b31751638f.png)
Show that 
, and that equality holds if and only if 
.
135 replies
Swapping string consisting a,b,c
MarkBcc168 45
N
by ezpotd
Source: IMO Shortlist 2017 C2
Let be a positive integer. Define a chameleon to be any sequence of 
letters, with exactly occurrences of each of the letters 
and 
. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon , there exists a chameleon 
such that cannot be changed to using fewer than 
swaps.
be a positive integer. Define a chameleon to be any sequence of 
letters, with exactly occurrences of each of the letters 
and 
. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon , there exists a chameleon 
such that cannot be changed to using fewer than 
swaps.
45 replies
