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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
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
1 viewing
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
two tangent circles
KPBY0507 3
N
by Sanjana42
Source: FKMO 2021 Problem 5
The incenter and -excenter of 
is 
and 
. The foot from 
to 
is 
and 
. The intersection of 
and 
is . The circumcenter of 
is .
Show that the circumcircle of
is tangent to the -excircle if is on the incircle of .
-excenter of 
is 
and 
. The foot from 
to 
is 
and 
. The intersection of 
and 
is . The circumcenter of 
is .Show that the circumcircle of
is tangent to the -excircle if is on the incircle of .
3 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
