be a convex pentagon such that![\[ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.
\]](http://latex.artofproblemsolving.com/d/3/d/d3d9a82f4318190298a8f008d417a364e03f1fca.png)
The diagonals 
and 
meet at 
.
bisects the side 
.
,
,
are fixed on a line in this order. Let 
be a circle passing through 
.
at 
.
and the line 
be positive real numbers such that 
.
.
q
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?
,
.
Let 
.
Let 
.
Let 
such that
Proposed by Ngo Van Trang, Vietnam
s.t. 
and 
be positiv integer numbers.
,
and 
are the feet of the angle bisector and altitude from 
and 
.
.
be a positive integer.Prove that there is a polynomial 
,![$$a^{2n+1}+b^{2n+1}+c^{2n+1}=abc[P(a,b)+P(b,c)+P(a,c)]$$](http://latex.artofproblemsolving.com/5/2/0/520813bebb2f297928208f7b7b2eea76d24043ac.png)
.
.
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
