G
Topic
First Poster
Last Poster
Inequality with three conditions
oVlad 2
N
by Quantum-Phantom
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 
2 replies
GCD Functional Equation
pinetree1 61
N
by ihategeo_1969
Source: USA TSTST 2019 Problem 7
Let 
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and 
such that 
for all integers .
Ankan Bhattacharya
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and 
such that 
for all integers .Ankan Bhattacharya
61 replies
An easy FE
oVlad 3
N
by jasperE3
Source: Romania EGMO TST 2017 Day 1 P3
Determine all functions 
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](//latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
3 replies
Interesting F.E
Jackson0423 12
N
by jasperE3
Show that there does not exist a function
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](//latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,
![\[
f(x + y^2) \geq f(x) + y.
\]](//latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
~Korea 2017 P7
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
~Korea 2017 P7
12 replies
p^3 divides (a + b)^p - a^p - b^p
62861 49
N
by Ilikeminecraft
Source: USA January TST for IMO 2017, Problem 3
Prove that there are infinitely many triples 
of positive integers with 
prime, 
, and 
, such that 
is a multiple of 
.
Noam Elkies
of positive integers with 
prime, 
, and 
, such that 
is a multiple of 
.Noam Elkies
49 replies
1 viewing
3D geometry theorem
KAME06 0
Let 
a point in the space and 
the centroid of a tetrahedron 
. Prove that:
a point in the space and 
the centroid of a tetrahedron 
. Prove that:
0 replies
Funny easy transcendental geo
qwerty123456asdfgzxcvb 1
N
by golue3120
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.
1 reply
demonic monic polynomial problem
iStud 0
Source: Monthly Contest KTOM April P4 Essay
(a) Let 
be a monic polynomial so that there exists another real coefficients 
that satisfy
![\[P(x^2-2)=P(x)Q(x)\]](//latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from
(b) For arbitrary polynomial that satisfies (a), determine whether should have real coefficients or not.
be a monic polynomial so that there exists another real coefficients 
that satisfy![\[P(x^2-2)=P(x)Q(x)\]](http://latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from (b) For arbitrary polynomial
that satisfies (a), determine whether should have real coefficients or not.
0 replies
fun set problem
iStud 0
Source: Monthly Contest KTOM April P2 Essay
Given a set 
with exactly 9 elements that is subset of 
. Prove that there exist two subsets 
and 
that satisfy the following:
- and are non-empty subsets from ,
- the sum of all elements in each of and are equal, and
-
is an empty subset.
with exactly 9 elements that is subset of 
. Prove that there exist two subsets 
and 
that satisfy the following:-
and are non-empty subsets from ,- the sum of all elements in each of
and are equal, and-
is an empty subset.
0 replies
