[/quote]
,
be an integer, 
a real number, and 
be positive real numbers such 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)
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)
be positive integers such that 
,
,
are perfect squares. Prove that 
is also a perfect square.
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)
with 
.
be the midpoint of side 
,
and 
be points on segments 
and 
,
.
be the circumcircle of 
,
the circumcircle of 
.
to 
,
and 
,
intersect 
,
intersect 
.
is tangent to both 
be an acute scalene triangle. Denote by 
the circle with diameter 
,
be the contact points of the tangents from 
and 
lie on opposite sides of 
and 
and 
lie on opposite sides of 
.
be the circle with diameter 
,
,
the circle with diameter 
.
are concurrent.
were selected 
pairwise distinct points, abscissas of which are integer numbers from the segment ![$[0; 100000]$](http://latex.artofproblemsolving.com/0/5/e/05e1d2855ef6a6dfaf9b24f2713455ab1017a30d.png)
.
,
,
such that areas of triangles 
and 
are equals.
and 
such that 
and 
such that 
and on side 
and 
such that 
Let 
and 
are concurrent.
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
,
,
at point 
.
,
,
,
,
can be divided into two groups, so that the angles in each group have a sum of 
.
and 
of the acute-angled triangle 
,
intersect at point 
,
.
is satisfied. Point 
is taken on the circumcircle of 
.
and 
.
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles 
.
.
.
and 
at 
.
be an acute triangle with 
be the center of its circumcircle. Let 
be the reflection of 
at 
,
intersects the line through 
be a point on the ray 
,
.
.
Y by
Let 
be the set of all lines in the plane and 
be a set of 
points in the plane. For a line 
let 
be the number of distinct points among the orthogonal projections of the points from on 
. Find the maximum possible number of distinct values of (this quantity is computed for any line ) as varies.
be the set of all lines in the plane and 
be a set of 
points in the plane. For a line 
let 
be the number of distinct points among the orthogonal projections of the points from on 
. Find the maximum possible number of distinct values of (this quantity is computed for any line ) as varies.This post has been edited 1 time. Last edited by Assassino9931, Apr 27, 2025, 10:18 PM
.![[asy]
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![[asy]
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[/asy]](http://latex.artofproblemsolving.com/b/0/0/b00767e3d62432eb03c82ad9f2932593bc00ebe7.png)
![[asy]
draw((-1,1)--(2,4),green);
draw((-1,0)--(3,4),green);
draw((-1,-1)--(4,4),green);
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draw((1,-1)--(5,3),green);
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dot((0,0));
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[/asy]](http://latex.artofproblemsolving.com/7/6/a/76a5a455bbcfec1c8dc6064634b7b589ff8af6be.png)
![[asy]
draw((-1,1)--(1,-1),green);
draw((-1,2)--(2,-1),green);
draw((-1,3)--(3,-1),green);
draw((0,3)--(4,-1),green);
draw((0,4)--(4,0),green);
draw((1,4)--(5,0),green);
draw((2,4)--(5,1),green);
dot((0,0));
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dot((2,3));
dot((3,3));
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![[asy]
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draw((-1,1)--(4,3.5),green);
draw((-1,1.5)--(3,3.5),green);
draw((0,2.5)--(2,3.5),green);
dot((0,0));
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dot((2,0));
dot((3,0));
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dot((1,1));
dot((2,1));
dot((3,1));
dot((4,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((4,2));
dot((1,3));
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dot((3,3));
[/asy]](http://latex.artofproblemsolving.com/e/b/e/ebeb5a7ecc83b65fa3bfd0efc8630690677658f8.png)
to 
is possible. Then either 
to 
is possible. If 
is possible then all the points lie inside some 
affine box, which is less than 
is the next smallest, which means that there cannot be more than 
affine box that contains all the points and transform it into a regular array. Then note that the best possible slopes are 
and 
and if we want 
to be possible, the points must be in the form of the construction, and then that leaves no room for 
to be possible.
