Middle School Math
Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8
Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8
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Middle School Math
Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8
Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8
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Another System
worthawholebean 3
N
2 hours ago
by P162008
Source: HMMT 2008 Guts Problem 33
Let
,
,
be nonzero real numbers such that
and
. Find the value of
.






3 replies
Inequality with three conditions
oVlad 2
N
2 hours ago
by Quantum-Phantom
Source: Romania EGMO TST 2019 Day 1 P3
Let
be non-negative real numbers such that
Prove 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)

2 replies

GCD Functional Equation
pinetree1 61
N
2 hours ago
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








Ankan Bhattacharya
61 replies
An easy FE
oVlad 3
N
3 hours ago
by jasperE3
Source: Romania EGMO TST 2017 Day 1 P3
Determine all functions
such that
for any real numbers
and

![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)


3 replies
1 viewing
Interesting F.E
Jackson0423 12
N
3 hours ago
by jasperE3
Show that there does not exist a function
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)

![\[
f(x + y^2) \geq f(x) + y.
\]](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
3 hours ago
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






Noam Elkies
49 replies
3D geometry theorem
KAME06 0
3 hours ago
Let
a point in the space and
the centroid of a tetrahedron
. Prove that:




0 replies
Funny easy transcendental geo
qwerty123456asdfgzxcvb 1
N
3 hours ago
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.






Prove that for a point



1 reply
demonic monic polynomial problem
iStud 0
4 hours ago
Source: Monthly Contest KTOM April P4 Essay
(a) Let
be a monic polynomial so that there exists another real coefficients
that satisfy
Determine all complex roots that are possible from 
(b) For arbitrary polynomial
that satisfies (a), determine whether
should have real coefficients or not.


![\[P(x^2-2)=P(x)Q(x)\]](http://latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)

(b) For arbitrary polynomial


0 replies
