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Brazilian Locus
kraDracsO 16
N
by Giant_PT
Source: IberoAmerican, Day 2, P4
Let 
and 
be two fixed points in the plane. For each point 
of the plane, outside of the line 
, let 
be the barycenter of the triangle 
. Determine the locus of points such that 
.
Note: The locus is the set of all points of the plane that satisfies the property.
and 
be two fixed points in the plane. For each point 
of the plane, outside of the line 
, let 
be the barycenter of the triangle 
. Determine the locus of points such that 
.Note: The locus is the set of all points of the plane that satisfies the property.
16 replies
Disconnected Tree Subsets
AwesomeYRY 25
N
by john0512
Source: TSTST 2021/5
Let 
be a tree on 
vertices with exactly 
leaves. Suppose that there exists a subset of at least 
vertices of , no two of which are adjacent. Show that the longest path in contains an even number of edges.
Vincent Huang
be a tree on 
vertices with exactly 
leaves. Suppose that there exists a subset of at least 
vertices of , no two of which are adjacent. Show that the longest path in contains an even number of edges. Vincent Huang
25 replies
Concurrency of tangent touchpoint lines on thales circles
MathMystic33 1
N
by Giant_PT
Source: 2024 Macedonian Team Selection Test P4
Let 
be an acute scalene triangle. Denote by 
the circle with diameter , and let 
be the contact points of the tangents from to , chosen so that and 
lie on opposite sides of 
and and 
lie on opposite sides of 
. Similarly, let 
be the circle with diameter 
, with tangents from touching at 
, and 
the circle with diameter , with tangents from touching at 
.
Prove that the lines
are concurrent.
be an acute scalene triangle. Denote by 
the circle with diameter , and let 
be the contact points of the tangents from to , chosen so that and 
lie on opposite sides of 
and and 
lie on opposite sides of 
. Similarly, let 
be the circle with diameter 
, with tangents from touching at 
, and 
the circle with diameter , with tangents from touching at 
.Prove that the lines
are concurrent.
1 reply
Balkan MO 2022/1 is reborn
Assassino9931 8
N
by Giant_PT
Source: Bulgaria EGMO TST 2023 Day 1, Problem 1
Let be a triangle with circumcircle . The tangents at and intersect at . The circumcircle of triangle 
intersects the line 
at 
and 
is the midpoint of 
. Prove that the lines 
and 
intersect on .
be a triangle with circumcircle . The tangents at and intersect at . The circumcircle of triangle 
intersects the line 
at 
and 
is the midpoint of 
. Prove that the lines 
and 
intersect on .
8 replies
USAMO 1985 #2
Mrdavid445 6
N
by anticodon
Determine each real root of ![\[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\]](//latex.artofproblemsolving.com/b/9/d/b9d355433e896e8b97f28c2e4235140fd2348e77.png)
correct to four decimal places.
![\[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\]](http://latex.artofproblemsolving.com/b/9/d/b9d355433e896e8b97f28c2e4235140fd2348e77.png)
correct to four decimal places.
6 replies
Inequality with rational function
MathMystic33 3
N
by ariopro1387
Source: Macedonian Mathematical Olympiad 2025 Problem 2
Let 
be an integer, 
a real number, and 
be positive real numbers such that 
. Prove that:
![\[
\frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n.
\]](//latex.artofproblemsolving.com/0/8/d/08de0c74a4e36b50a64d17875d3fd93eeb5b52de.png)
When does equality hold?
be an integer, 
a real number, and 
be positive real numbers such that 
. Prove that:![\[
\frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n.
\]](http://latex.artofproblemsolving.com/0/8/d/08de0c74a4e36b50a64d17875d3fd93eeb5b52de.png)
When does equality hold?
3 replies
k A cyclic weighted inequality
MathMystic33 2
N
by grupyorum
Source: 2024 Macedonian Team Selection Test P2
Let 
be positive real numbers. Prove that there exists a cyclic permutation 
of 
such that for all positive real numbers 
the following holds:
![\[
\frac{a}{x\,a + y\,b + z\,c}
\;+\;
\frac{b}{x\,b + y\,c + z\,a}
\;+\;
\frac{c}{x\,c + y\,a + z\,b}
\;\ge\;
\frac{3}{x + y + z}.
\]](//latex.artofproblemsolving.com/1/5/2/15261b3f712c2cc8ffbeb92a9711e06ad0992b5d.png)
be positive real numbers. Prove that there exists a cyclic permutation 
of 
such that for all positive real numbers 
the following holds:![\[
\frac{a}{x\,a + y\,b + z\,c}
\;+\;
\frac{b}{x\,b + y\,c + z\,a}
\;+\;
\frac{c}{x\,c + y\,a + z\,b}
\;\ge\;
\frac{3}{x + y + z}.
\]](http://latex.artofproblemsolving.com/1/5/2/15261b3f712c2cc8ffbeb92a9711e06ad0992b5d.png)
2 replies
k Perfect squares imply GCD is a perfect square
MathMystic33 1
N
by grupyorum
Source: 2024 Macedonian Team Selection Test P6
Let 
be positive integers such that 
, 
, and 
are perfect squares. Prove that 
is also a perfect square.
be positive integers such that 
, 
, and 
are perfect squares. Prove that 
is also a perfect square.
1 reply
Divisibility condition with primes
MathMystic33 1
N
by grupyorum
Source: 2024 Macedonian Team Selection Test P1
Let 
be distinct primes and let 
be nonnegative integers. Define
![\[
m \;=\;
\frac12
\Bigl(\prod_{i=2}^k p_i^{a_i}\Bigr)
\Bigl(\prod_{i=1}^k(p_i+1)\;+\;\sum_{i=1}^k(p_i-1)\Bigr),
\]](//latex.artofproblemsolving.com/7/9/a/79af85375957986298d37c00d288b6d402f40106.png)
![\[
n \;=\;
\frac12
\Bigl(\prod_{i=2}^k p_i^{a_i}\Bigr)
\Bigl(\prod_{i=1}^k(p_i+1)\;-\;\sum_{i=1}^k(p_i-1)\Bigr).
\]](//latex.artofproblemsolving.com/e/2/3/e23799ff03400d5eade3731bb83456fcd702a5e1.png)
Prove that
![\[
p^2-1 \;\bigm|\; p\,m \;-\; n.
\]](//latex.artofproblemsolving.com/3/7/e/37e4535b59ef183908730946213429e9ebc05dd7.png)
be distinct primes and let 
be nonnegative integers. Define![\[
m \;=\;
\frac12
\Bigl(\prod_{i=2}^k p_i^{a_i}\Bigr)
\Bigl(\prod_{i=1}^k(p_i+1)\;+\;\sum_{i=1}^k(p_i-1)\Bigr),
\]](http://latex.artofproblemsolving.com/7/9/a/79af85375957986298d37c00d288b6d402f40106.png)
![\[
n \;=\;
\frac12
\Bigl(\prod_{i=2}^k p_i^{a_i}\Bigr)
\Bigl(\prod_{i=1}^k(p_i+1)\;-\;\sum_{i=1}^k(p_i-1)\Bigr).
\]](http://latex.artofproblemsolving.com/e/2/3/e23799ff03400d5eade3731bb83456fcd702a5e1.png)
Prove that![\[
p^2-1 \;\bigm|\; p\,m \;-\; n.
\]](http://latex.artofproblemsolving.com/3/7/e/37e4535b59ef183908730946213429e9ebc05dd7.png)
1 reply
