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basically INAMO 2010/6
iStud 0
Source: Monthly Contest KTOM April P1 Essay
Call 
kawaii if it satisfies 
(
is the number of positive factors of , while 
is the number of integers not more than that are relatively prime with ). Find all that is kawaii.
kawaii if it satisfies 
(
is the number of positive factors of , while 
is the number of integers not more than that are relatively prime with ). Find all that is kawaii.
0 replies
trolling geometry problem
iStud 0
Source: Monthly Contest KTOM April P3 Essay
Given a cyclic quadrilateral 
with 
and 
. Lines 
and 
intersect at 
, and lines 
and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let 
and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to 
if and only if 
divides into two triangles with equal area.
with 
and 
. Lines 
and 
intersect at 
, and lines 
and 
intersect at 
. Let 
be the midpoints of sides 
, respectively. Let 
and 
be points on segment 
and 
, respectively, so that 
is the angle bisector of 
and 
is the angle bisector of 
. Prove that 
is parallel to 
if and only if 
divides into two triangles with equal area.
0 replies
My hardest algebra ever created (only one solve in the contest)
mshtand1 6
N
by mshtand1
Source: Ukraine IMO TST P9
Find all functions 
for which, for all 
, the following identity holds:
![\[
f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x}
\]](//latex.artofproblemsolving.com/9/0/c/90c180110402e1a32b70edb2b0a03a28727457d1.png)
Proposed by Mykhailo Shtandenko
for which, for all 
, the following identity holds:![\[
f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x}
\]](http://latex.artofproblemsolving.com/9/0/c/90c180110402e1a32b70edb2b0a03a28727457d1.png)
Proposed by Mykhailo Shtandenko
6 replies
Killer NT that nobody solved (also my hardest NT ever created)
mshtand1 4
N
by mshtand1
Source: Ukraine IMO 2025 TST P8
A positive integer number 
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where
![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](//latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
Proposed by Arsenii Nikolaev and Mykhailo Shtandenko
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
Proposed by Arsenii Nikolaev and Mykhailo Shtandenko
4 replies
Advanced topics in Inequalities
va2010 22
N
by Primeniyazidayi
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
22 replies
Funny easy transcendental geo
qwerty123456asdfgzxcvb 0
Let 
be a logarithmic spiral centered at the origin (ie curve satisfying for any point on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
Prove that for a point
on the spiral, the polar of wrt. is tangent to the spiral.
be a logarithmic spiral centered at the origin (ie curve satisfying for any point on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.Prove that for a point
on the spiral, the polar of wrt. is tangent to the spiral.
0 replies
Nice problem about a trapezoid
manlio 1
N
by kiyoras_2001
Have you a nice solution for this problem?
Thank you very much
Thank you very much
1 reply
product of the first n terms
FireBreathers 5
N
by ihategeo_1969
Does there exist an infinite sequence of positive integers 
such that every positive integer appears exactly once and the product of the first terms is a perfect 
power ?
such that every positive integer appears exactly once and the product of the first terms is a perfect 
power ?
5 replies
Inequality with three conditions
oVlad 1
N
by Haris1
Source: Romania EGMO TST 2019 Day 1 P3
Let 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](//latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
1 reply
Paint and Optimize: A Grid Strategy Problem
mojyla222 2
N
by sami1618
Source: Iran 2025 second round p2
Ali and Shayan are playing a turn-based game on an infinite grid. Initially, all cells are white. Ali starts the game, and in the first turn, he colors one unit square black. In the following turns, each player must color a white square that shares at least one side with a black square. The game continues for exactly 2808 turns, after which each player has made 1404 moves. Let 
be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape by playing optimally. (The perimeter of shape is defined as the total length of the boundary segments between a black and a white cell.)
What are the possible values of the perimeter of, assuming both players play optimally?
be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape by playing optimally. (The perimeter of shape is defined as the total length of the boundary segments between a black and a white cell.)What are the possible values of the perimeter of
, assuming both players play optimally?
2 replies
