[/quote]
,
is said to be angelic if it has nonzero volume and satisfies ![\[ \begin{aligned} \angle BAC + \angle CAD + \angle DAB &= \angle ABC + \angle CBD + \angle DBA, \\ \angle ACB + \angle BCD + \angle DCA &= \angle ADB + \angle BDC + \angle CDA. \end{aligned} \]](http://latex.artofproblemsolving.com/a/2/0/a20a85e7b001bd7652905eca500639b1edc303d7.png)
Across all angelic tetrahedrons, what is the maximum number of distinct lengths that could appear in the set 
?
and have answers
such that: 
be an odd prime number. Prove that there exist sequence 
,
is not a perfect cube for any positive integer 
.
satisfies 
and 
.
lattice points with 
and 
coordinate no less than 
and no more than 
and walk 
feet north and turn 
degrees to the right.
feet east and turn 
feet south and turn 
feet to the east and turn 
feet south and turn 
and 
find the ordered pair 
.
such that
for all 
satisfying 
satisfying![\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]](http://latex.artofproblemsolving.com/a/e/6/ae65e3d53931fe9d60d0de76dc79540ccf44c26e.png)
and let 
denote the largest real number satisfying 
for all positive integers 
?
be an arbitrary point on the side 
of triangle 
.
and 
be the intersections of the lines through 
and 
,
be the intersection point of the circumcircle of 
with the circumcircle of 
be the midpoint of 
be the intersection point of line 
with the circumcircle of 
are collinear.
satisfying the condition![\[f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))\]](http://latex.artofproblemsolving.com/e/d/3/ed333c1123c4cb6a1e687dc572c5d847671caded.png)
be the set of positive integers. Find all unbounded functions 
such that![\[f(n + f(m) - 1) \mid f(n) + m - 1\]](http://latex.artofproblemsolving.com/3/4/a/34af2c3dae5a9e35efc7efd409739947c9e0e40b.png)
for all 
.
towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns.
questions.
Y by GA34-261
3. [Russia 1961]
Real numbers are written in an
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.
Real numbers are written in an
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.This post has been edited 2 times. Last edited by CatalanThinker, May 28, 2025, 5:43 AM
