Equations
by Jackson0423, Apr 22, 2025, 4:36 PM
Solve the system of equations
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](//latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
such that for any 
:
Woaah a lot of external tangents
by egxa, Apr 18, 2025, 5:14 PM
A quadrilateral 
with no parallel sides is inscribed in a circle 
. Circles 
are inscribed in triangles 
, respectively. Common external tangents are drawn between 
and 
, and 
, and 
, and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle 
. Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.
with no parallel sides is inscribed in a circle 
. Circles 
are inscribed in triangles 
, respectively. Common external tangents are drawn between 
and 
, and 
, and 
, and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle 
. Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.Calculate the distance of chess king!!
by egxa, Apr 18, 2025, 9:58 AM
A chess king was placed on a square of an 
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)spring break
by aaaaarush, Apr 7, 2025, 10:30 PM
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what r u guys doing for spring break??
i know pupitre is doing her bf
but anyways what is everyone else doing?
our schools spring break is 4/12-4/20 (i go to the same school as pupitre btw)
also i took aime in january and got my score back recently
i may or may not have received a four *sobbing in my room*
i know pupitre is doing her bf
but anyways what is everyone else doing?
our schools spring break is 4/12-4/20 (i go to the same school as pupitre btw)
also i took aime in january and got my score back recently
i may or may not have received a four *sobbing in my room*
and 
: Prove that :
such that 
for all 
As some nations like to say "Heavy theorems mostly do not help"
by Assassino9931, Dec 20, 2022, 12:02 AM
We say that a positive integer 
is lovely if there exist a positive integer 
and (not necessarily distinct) positive integers 
, 
, 
, 
such that 
and 
for 
.
a) Are there infinitely many lovely numbers?
b) Is there a lovely number, greater than
, which is a perfect square of an integer?
is lovely if there exist a positive integer 
and (not necessarily distinct) positive integers 
, 
, 
, 
such that 
and 
for 
.a) Are there infinitely many lovely numbers?
b) Is there a lovely number, greater than
, which is a perfect square of an integer?This post has been edited 1 time. Last edited by Assassino9931, Dec 20, 2022, 12:02 AM
Circumcircle excircle chaos
by CyclicISLscelesTrapezoid, Jul 12, 2022, 12:32 PM
Let 
be a triangle with circumcircle 
and let 
be the 
-excircle. Let 
and 
be the intersection points of and . Let 
and 
be the projections of onto the tangent lines to at and respectively. The tangent line at to the circumcircle of the triangle 
intersects the tangent line at to the circumcircle of the triangle 
at a point 
. Prove that 
.
be a triangle with circumcircle 
and let 
be the 
-excircle. Let 
and 
be the intersection points of and . Let 
and 
be the projections of onto the tangent lines to at and respectively. The tangent line at to the circumcircle of the triangle 
intersects the tangent line at to the circumcircle of the triangle 
at a point 
. Prove that 
.On existence of infinitely many positive integers satisfying
by shivangjindal, Apr 12, 2014, 12:09 PM
We denote the number of positive divisors of a positive integer 
by 
and the number of distinct prime divisors of by 
. Let be a positive integer. Prove that there exist infinitely many positive integers such that 
and 
does not divide 
for any positive integers 
satisfying 
.
by 
and the number of distinct prime divisors of by 
. Let be a positive integer. Prove that there exist infinitely many positive integers such that 
and 
does not divide 
for any positive integers 
satisfying 
.This post has been edited 1 time. Last edited by v_Enhance, May 11, 2016, 1:44 PM
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