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Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
Inspired by Czech-Polish-Slovak 2024
sqing 1
N
by sqing
Source: Own
Let 
Prove that
Let 
Prove that
Let 
Prove that![$$ a+b^2+c\geq \frac{135 \sqrt[3]{90}-2}{2}$$](//latex.artofproblemsolving.com/8/6/6/866623c65f54d6147a43b1b241c2cba96863b624.png)
Prove that
Let 
Prove that
Let 
Prove that![$$ a+b^2+c\geq \frac{135 \sqrt[3]{90}-2}{2}$$](http://latex.artofproblemsolving.com/8/6/6/866623c65f54d6147a43b1b241c2cba96863b624.png)
1 reply
FE i created on bijective function with x≠y
benjaminchew13 8
N
by benjaminchew13
Source: own (probably)
Find all bijective functions 
such that 
for all 
such that 
.
such that 
for all 
such that 
.
8 replies
1 viewing
Sum of divisors
Kimchiks926 3
N
by math-olympiad-clown
Source: Baltic Way 2022, Problem 17
Let 
be a positive integer such that the sum of its positive divisors is at least 
. Prove that has at least 
distinct prime factors.
be a positive integer such that the sum of its positive divisors is at least 
. Prove that has at least 
distinct prime factors.
3 replies
Find the number of interesting numbers
WakeUp 13
N
by mathematical-forest
Source: China TST 2011 - Quiz 1 - D1 - P3
A positive integer is known as an interesting number if satisfies
![\[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \]](//latex.artofproblemsolving.com/0/c/a/0ca11fa76313b65fd76a3cceee90955960e44ef6.png)
for all
.
Find the number of interesting numbers.
is known as an interesting number if satisfies![\[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \]](http://latex.artofproblemsolving.com/0/c/a/0ca11fa76313b65fd76a3cceee90955960e44ef6.png)
for all
.Find the number of interesting numbers.
13 replies
A complex FE from Iran
mojyla222 7
N
by mathematical-forest
Source: Iran 2024 3rd round algebra exam P2
A surjective function 
is given. Find all functions 
such that for all 
we have
Proposed by Mojtaba Zare, Amirabbas Mohammadi
is given. Find all functions 
such that for all 
we have
Proposed by Mojtaba Zare, Amirabbas Mohammadi
7 replies
interesting geometry config (3/3)
Royal_mhyasd 1
N
by Royal_mhyasd
Let 
be an acute triangle, 
its orthocenter and 
the center of its nine point circle. Let 
be a point on the parallel through 
to 
such that 
and and 
are on different sides of 
and 
a point on the parallel through 
to 
such that 
and and are on different sides of . If 
and 
are the reflections of over and respectively, 
and 
are the intersections of 
and 
respectively with the circumcircle of , prove that the intersection of lines 
and 
lies on 
.
final problem for this "points on parallels forming strange angles with the orthocenter" config, for now. personally i think its pretty cool :D
be an acute triangle, 
its orthocenter and 
the center of its nine point circle. Let 
be a point on the parallel through 
to 
such that 
and and 
are on different sides of 
and 
a point on the parallel through 
to 
such that 
and and are on different sides of . If 
and 
are the reflections of over and respectively, 
and 
are the intersections of 
and 
respectively with the circumcircle of , prove that the intersection of lines 
and 
lies on 
.final problem for this "points on parallels forming strange angles with the orthocenter" config, for now. personally i think its pretty cool :D
1 reply
interesting geo config (2/3)
Royal_mhyasd 4
N
by Royal_mhyasd
Source: own
Let be an acute triangle and its orthocenter. Let be a point on the parallel through to such that 
. Define and 
as points on the parallels through to and through to similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
be an acute triangle and its orthocenter. Let be a point on the parallel through to such that 
. Define and 
as points on the parallels through to and through to similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
4 replies
Polyline with increasing links
NO_SQUARES 1
N
by Noirshade
Source: 239 MO 2025 10-11 p1
There are 
points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?
points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?
1 reply
Find the perfect squares
Johann Peter Dirichlet 5
N
by ririgggg
Source: Problem 1, Brazil MO 1993
The sequence 
is defined by 
. Find all terms which are perfect squares.
is defined by 
. Find all terms which are perfect squares.
5 replies
