Y by
Prove that if a convex quadrilateral 
satisfies the equation
![\[
(AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2,
\]](//latex.artofproblemsolving.com/7/d/a/7daf2d6ac00609b5252b2b8b05d96a5ca0952a94.png)
then must be a rectangle.
satisfies the equation![\[
(AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2,
\]](http://latex.artofproblemsolving.com/7/d/a/7daf2d6ac00609b5252b2b8b05d96a5ca0952a94.png)
then must be a rectangle.G
Topic
First Poster
Last Poster
3D geometry theorem
KAME06 1
N
by mathuz
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:
1 reply
trolling geometry problem
iStud 2
N
by mathuz
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.
2 replies
GCD of a sequence
oVlad 7
N
by grupyorum
Source: Romania EGMO TST 2017 Day 1 P2
Determine all pairs 
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
7 replies
Another System
worthawholebean 3
N
by P162008
Source: HMMT 2008 Guts Problem 33
Let 
, 
, 
be nonzero real numbers such that 
and 
. Find the value of
.
, 
, 
be nonzero real numbers such that 
and 
. Find the value of
.
3 replies
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
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
A Characterization of Rectangles
G
H
G
H
Source: VN Math Olympiad For High School Students P8 - 2025
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
![\[
AB^2 + CD^2 + AD^2 + BC^2 \geq AC^2 + BD^2,
\]](http://latex.artofproblemsolving.com/a/4/6/a46763d7a72ba44005eade1bffca31e28077efa9.png)
and according to Ptolemy's inequality,![\[
AB \cdot CD + AD \cdot BC \geq AC \cdot BD.
\]](http://latex.artofproblemsolving.com/7/7/c/77c004953365450da97733dd26d294da97b51cdb.png)
Consequently,![\[
(AB+CD)^2 + (AD+BC)^2 \geq (AC+BD)^2.
\]](http://latex.artofproblemsolving.com/8/6/8/8683f36a6457c49d3f6eaff66efe8ffa67105a70.png)
Equality holds if and only if 
is a parallelogram inscribed in a circle, that is, a rectangle.N Quick Reply
G
H
