[/quote]
,
be a triangle with circumcircle 
.
and 
intersect at 
.
intersects the line 
at 
and 
is the midpoint of 
.
and 
intersect on 
![\[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\]](http://latex.artofproblemsolving.com/b/9/d/b9d355433e896e8b97f28c2e4235140fd2348e77.png)
correct to four decimal places.
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)
,
be positive real numbers. Prove that![\[\frac{a^4 + 3}{b} + \frac{b^4 + 3}{c} + \frac{c^4 + 3}{a} \ge 12.\]](http://latex.artofproblemsolving.com/1/f/d/1fdfd64c9aafa8741ad1594db46cc16a403b1a99.png)
When does equality hold?
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 
islands of the Arctic Ocean. Every bear sometimes swims from one island to another. It turned out that every bear made at least one swim in a year, but no two bears made equal swams. At the same time, exactly one swim was made between each two islands 
:
and 
such that 
.
be a convex polyhedron with the following properties:
edges.
.
colors such that for each color 
and any two distinct vertices 
,
to 
all of whose edges have color 
such that
divides 
for every 
,
we have![\[
f\bigl(f(a)+kb\bigr)\;=\;f\bigl(a + k\,f(b)\bigr).
\]](http://latex.artofproblemsolving.com/c/f/f/cff41352422c8765978b8fe6dc966ba934a9df49.png)
be an acute scalene triangle. Denote by 
the circle with diameter 
,
be the contact points of the tangents from 
lie on opposite sides of 
and 
lie on opposite sides of 
.
be the circle with diameter 
,
,
the circle with diameter 
.
are concurrent.
Y by
The numbers 
are written in a row in that exact order. Igor and Ruslan take turns inserting the signs 
between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by 
, Igor wins; otherwise, Ruslan wins. Which player has a winning strategy regardless of the opponent’s moves?
are written in a row in that exact order. Igor and Ruslan take turns inserting the signs 
between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by 
, Igor wins; otherwise, Ruslan wins. Which player has a winning strategy regardless of the opponent’s moves?This post has been edited 1 time. Last edited by egxa, Apr 18, 2025, 5:24 PM
