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< NA'T = < ADT wanted, starting with a right triangle, symmetric, projections
parmenides51 3
N
by tilya_TASh
Source: JBMO Shortlist 2018 G2
Let 
be a right angled triangle with 
and 
its altitude. We draw parallel lines from 
to the vertical sides of the triangle and we call 
their points of intersection with 
and 
respectively. The parallel line from 
to 
intersects the line at the point 
. Let 
be the symmetric of 
with respect to the line and 
the projections of 
onto and respectively. If 
is the point of intersection of the lines 
and 
, prove that 
.
be a right angled triangle with 
and 
its altitude. We draw parallel lines from 
to the vertical sides of the triangle and we call 
their points of intersection with 
and 
respectively. The parallel line from 
to 
intersects the line at the point 
. Let 
be the symmetric of 
with respect to the line and 
the projections of 
onto and respectively. If 
is the point of intersection of the lines 
and 
, prove that 
.
3 replies
x(x - y) = 8y - 7 in NxN
parmenides51 4
N
by Namisgood
Source: JBMO 2008 Shortlist N1
Find all the positive integers 
and 
that satisfy the equation 
and 
that satisfy the equation 
4 replies
IMO Genre Predictions
ohiorizzler1434 3
N
by NicoN9
Everybody, with IMO upcoming, what are you predictions for the problem genres?
Personally I predict:
Personally I predict:
3 replies
Insects walk
Giahuytls2326 0
Source: somewhere in the internet
A 100 × 100 chessboard is divided into unit squares. Each square has an arrow pointing up, down, left, or right. The board square is surrounded by a wall, except to the right of the top right corner square. An insect is placed in one of the squares.
Every second, the insect moves one unit in the direction of the arrow in its square. As the insect moves, the arrow of the square it just left rotates 90° clockwise.
If the specified movement cannot be performed, then the insect will not move for that second, but the arrow in the square it is standing on will still rotate. Is it possible that the insect never leaves the board?
Every second, the insect moves one unit in the direction of the arrow in its square. As the insect moves, the arrow of the square it just left rotates 90° clockwise.
If the specified movement cannot be performed, then the insect will not move for that second, but the arrow in the square it is standing on will still rotate. Is it possible that the insect never leaves the board?
0 replies
Equal sum of digits
Fudicuehfosonrcjeong 0
Is it true that for any two positive integers a, b there exists a positive integer k such that s(ka)=s(kb), where s(n) is sum of digits in base 10?
0 replies
Common tangent of mixtilinear incircles
CyclicISLscelesTrapezoid 3
N
by Ilikeminecraft
Source: MOP 2020/1Z
Let 
be a quadrilateral inscribed in circle 
. Circles 
and 
are drawn internally tangent to , such that is tangent to 
and 
while is tangent to 
and 
. Prove that we can draw a line parallel to 
which is simultaneously tangent to both and .
be a quadrilateral inscribed in circle 
. Circles 
and 
are drawn internally tangent to , such that is tangent to 
and 
while is tangent to 
and 
. Prove that we can draw a line parallel to 
which is simultaneously tangent to both and .
3 replies
1 viewing
Construct
Pomegranat 2
N
by Blackbeam999
Source: idk
Let 
be a prime number. Prove that there exists a natural number 
such that
![\[
p \mid m^n - n.
\]](//latex.artofproblemsolving.com/3/a/3/3a38b299f0328d176286d65e02b92e9918e8109d.png)
be a prime number. Prove that there exists a natural number 
such that![\[
p \mid m^n - n.
\]](http://latex.artofproblemsolving.com/3/a/3/3a38b299f0328d176286d65e02b92e9918e8109d.png)
2 replies
Almost Squarefree Integers
oVlad 4
N
by HeshTarg
Source: Romania Junior TST 2025 Day 1 P1
A positive integer 
is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
the ratio 
is an integer.
is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
the ratio 
is an integer.
4 replies
A nice and easy gem off of StackExchange
NamelyOrange 2
N
by Royal_mhyasd
Source: https://math.stackexchange.com/questions/3818796/
Define 
as the set of all numbers of the form 
for some nonnegative 
and 
. Find (with proof) all pairs 
such that 
and 
.
Rephrased: Solve
over 
, and prove that your solution(s) is/are the only one(s).
as the set of all numbers of the form 
for some nonnegative 
and 
. Find (with proof) all pairs 
such that 
and 
.Rephrased: Solve
over 
, and prove that your solution(s) is/are the only one(s).
2 replies
