All-Russian Olympiad
by ABCD1728, May 15, 2025, 7:00 AM
This post has been edited 1 time. Last edited by ABCD1728, 4 hours ago
Reason: wrong spelling
Reason: wrong spelling
inequality
by danilorj, May 14, 2025, 9:08 PM
Let 
be nonnegative real numbers such that 
. Prove that
![\[
\frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1,
\]](//latex.artofproblemsolving.com/5/b/d/5bd3349071e075519bd986c845c500125b7d46f8.png)
and determine all such triples 
where the equality holds.
be nonnegative real numbers such that 
. Prove that![\[
\frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1,
\]](http://latex.artofproblemsolving.com/5/b/d/5bd3349071e075519bd986c845c500125b7d46f8.png)
and determine all such triples 
where the equality holds.D1032 : A general result on polynomial 2
by Dattier, May 14, 2025, 5:19 PM
Let ![$P \in \mathbb Q[x,y]$](//latex.artofproblemsolving.com/a/3/8/a384cbf38fd3a7f812b22baaeb0860e7f6eb26c0.png)
with 
and ![$\forall (a,b) \in \mathbb Z^2 \cap [0,d]^2, P(a,b) \in \mathbb Z$](//latex.artofproblemsolving.com/a/6/5/a65ecb4bb2d9b5e6096f322c2de8bd11b945ed6a.png)
.
Is it true that
?
![$P \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/a/3/8/a384cbf38fd3a7f812b22baaeb0860e7f6eb26c0.png)
with 
and ![$\forall (a,b) \in \mathbb Z^2 \cap [0,d]^2, P(a,b) \in \mathbb Z$](http://latex.artofproblemsolving.com/a/6/5/a65ecb4bb2d9b5e6096f322c2de8bd11b945ed6a.png)
.Is it true that
?Ah, easy one
by irregular22104, May 14, 2025, 4:01 PM
In the number series 
every next number (from the fifth number) is the unit number of the sum of the four numbers preceding it. Is there any cases that we get the numbers 
and 
in this series?
every next number (from the fifth number) is the unit number of the sum of the four numbers preceding it. Is there any cases that we get the numbers 
and 
in this series?greatest volume
by hzbrl, May 8, 2025, 9:56 AM
A large sphere with radius 7 contains three smaller balls each with radius 3 . The three balls are each externally tangent to the other two balls and internally tangent to the large sphere. There are four right circular cones that can be inscribed in the large sphere in such a way that the bases of the cones are tangent to all three balls. Of these four cones, the one with the greatest volume has volume 
. Find 
.
. Find 
.Functional Equation!
by EthanWYX2009, Mar 29, 2025, 10:48 AM
Find all functions 
such that 
is unbounded and
![\[2f(m)f(n)-f(n-m)-1\]](//latex.artofproblemsolving.com/0/f/7/0f77b5addbf8f5e603eb1b6923cdac795cd132e9.png)
is a perfect square for all integer 
such that 
is unbounded and![\[2f(m)f(n)-f(n-m)-1\]](http://latex.artofproblemsolving.com/0/f/7/0f77b5addbf8f5e603eb1b6923cdac795cd132e9.png)
is a perfect square for all integer 
mods with a twist
by sketchydealer05, Apr 16, 2023, 10:03 PM
We are given a positive integer 
. For each positive integer 
, we define its twist 
as follows: write as 
, where 
are non-negative integers and 
, then 
. For the positive integer , consider the infinite sequence 
where 
and 
is the twist of 
for each positive integer 
.
Prove that this sequence contains
if and only if the remainder when is divided by 
is either or 
.
. For each positive integer 
, we define its twist 
as follows: write as 
, where 
are non-negative integers and 
, then 
. For the positive integer , consider the infinite sequence 
where 
and 
is the twist of 
for each positive integer 
.Prove that this sequence contains
if and only if the remainder when is divided by 
is either or 
.angle relations in a convex ABCD given, double segment wanted
by parmenides51, Sep 19, 2018, 8:54 PM
In convex quadrilateral 
, the diagonals 
and 
meet at the point 
. We know that 
and 
. If we have 
prove that 
.
Proposed by Iman Maghsoudi
, the diagonals 
and 
meet at the point 
. We know that 
and 
. If we have 
prove that 
.Proposed by Iman Maghsoudi
This post has been edited 1 time. Last edited by parmenides51, Sep 20, 2018, 9:22 AM
Reason: Proposed
Reason: Proposed
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