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< NA'T = < ADT wanted, starting with a right triangle, symmetric, projections
parmenides51   3
N 2 hours ago by tilya_TASh
Source: JBMO Shortlist 2018 G2
Let $ABC$ be a right angled triangle with $\angle A = 90^o$ and $AD$ its altitude. We draw parallel lines from $D$ to the vertical sides of the triangle and we call $E, Z$ their points of intersection with $AB$ and $AC$ respectively. The parallel line from $C$ to $EZ$ intersects the line $AB$ at the point $N$. Let $A' $ be the symmetric of $A$ with respect to the line $EZ$ and $I, K$ the projections of $A'$ onto $AB$ and $AC$ respectively. If $T$ is the point of intersection of the lines $IK$ and $DE$, prove that $\angle NA'T = \angle  ADT$.
3 replies
parmenides51
Jul 22, 2019
tilya_TASh
2 hours ago
x(x - y) = 8y - 7 in NxN
parmenides51   4
N 2 hours ago by Namisgood
Source: JBMO 2008 Shortlist N1
Find all the positive integers $x$ and $y$ that satisfy the equation $x(x - y) = 8y - 7$
4 replies
parmenides51
Oct 14, 2017
Namisgood
2 hours ago
IMO Genre Predictions
ohiorizzler1434   3
N 2 hours ago by NicoN9
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
3 replies
ohiorizzler1434
3 hours ago
NicoN9
2 hours ago
Insects walk
Giahuytls2326   0
2 hours ago
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?
0 replies
Giahuytls2326
2 hours ago
0 replies
Equal sum of digits
Fudicuehfosonrcjeong   0
3 hours ago
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
Fudicuehfosonrcjeong
3 hours ago
0 replies
Common tangent of mixtilinear incircles
CyclicISLscelesTrapezoid   3
N 3 hours ago by Ilikeminecraft
Source: MOP 2020/1Z
Let $ABCD$ be a quadrilateral inscribed in circle $\Omega$. Circles $\omega_A$ and $\omega_D$ are drawn internally tangent to $\Omega$, such that $\omega_A$ is tangent to $\overline{AB}$ and $\overline{AC}$ while $\omega_D$ is tangent to $\overline{DB}$ and $\overline{DC}$. Prove that we can draw a line parallel to $\overline{AD}$ which is simultaneously tangent to both $\omega_A$ and $\omega_D$.
3 replies
1 viewing
CyclicISLscelesTrapezoid
Jan 6, 2023
Ilikeminecraft
3 hours ago
Construct
Pomegranat   2
N 3 hours ago by Blackbeam999
Source: idk
Let \( p \) be a prime number. Prove that there exists a natural number \( n \) such that
\[
p \mid m^n - n.
\]
2 replies
Pomegranat
Apr 30, 2025
Blackbeam999
3 hours ago
square root problem
kjhgyuio   2
N 3 hours ago by wh0nix
........
2 replies
kjhgyuio
5 hours ago
wh0nix
3 hours ago
Almost Squarefree Integers
oVlad   4
N 4 hours ago by HeshTarg
Source: Romania Junior TST 2025 Day 1 P1
A positive integer $n\geqslant 3$ is almost squarefree if there exists a prime number $p\equiv 1\bmod 3$ such that $p^2\mid n$ and $n/p$ is squarefree. Prove that for any almost squarefree positive integer $n$ the ratio $2\sigma(n)/d(n)$ is an integer.
4 replies
oVlad
Apr 12, 2025
HeshTarg
4 hours ago
A nice and easy gem off of StackExchange
NamelyOrange   2
N 5 hours ago by Royal_mhyasd
Source: https://math.stackexchange.com/questions/3818796/
Define $S$ as the set of all numbers of the form $2^i5^j$ for some nonnegative $i$ and $j$. Find (with proof) all pairs $(m,n)$ such that $m,n\in S$ and $m-n=1$.


Rephrased: Solve $2^a5^b-2^c5^d=1$ over $(\mathbb{N}_0)^4$, and prove that your solution(s) is/are the only one(s).
2 replies
NamelyOrange
Yesterday at 8:13 PM
Royal_mhyasd
5 hours ago
a