[/quote]
,
:![\[
\left( \frac{a}{b+c} + \frac{b}{c+a} \right)
\left( \frac{b}{c+a} + \frac{c}{a+b} \right)
\left( \frac{c}{a+b} + \frac{a}{b+c} \right)
\geq \frac{(a+b+c)^2}{3(ab + bc + ca)}
\]](http://latex.artofproblemsolving.com/c/c/9/cc91c69e48bd016d5a35da221065415d665931ab.png)
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
be a triangle 
)
are the altitudes of the triangle. The bisector of 
intersects 
at 
.
be a point such that 
and 
.
passes through the midpoint of 
.
is circumcircle of an acute triangle 
is midpoint of arc 
and 
is incenter of triangle 
intersect 
and 
.
such that 
is parallel to 
.
.
.
and 
such that
such that for every 
and every prime 
the number 
is divisible by 
if and only if 
is divisible by 
be the lengths of the sides of a triangle. Prove that![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](http://latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
Prove that
,
,
be the roots of 
Find the numerical value of the expression:![\[\frac{\alpha + 2}{\alpha - 2} + \frac{\beta + 2}{\beta - 2} + \frac{\gamma + 2}{\gamma - 2}\]](http://latex.artofproblemsolving.com/b/5/9/b59f4515409422333e10857ea775a36e58490b5d.png)
be the roots of the polynomial 
.
Y by
Source: 2025 Nepal Mock TST Day 1 Problem 1
Prajit and Kritesh challenge each other with a marble game. In a bag, there are initially
red marbles and 
blue marbles. The rules of the game are as follows:
Move: In each turn, a player (either Prajit or Kritesh) removes two marbles from the bag.
If the two marbles are of the same color, they are both discarded and a red marble is added to the bag.
If the two marbles are of different colors, they are both discarded and a blue marble is added to the bag.
The game continues by repeating the above move.
Prove that no matter what sequence of moves is made, the process always terminates with exactly one marble left. In addition, find the possible colors of the marble remaining.
Prajit and Kritesh challenge each other with a marble game. In a bag, there are initially
red marbles and 
blue marbles. The rules of the game are as follows:Move: In each turn, a player (either Prajit or Kritesh) removes two marbles from the bag.
If the two marbles are of the same color, they are both discarded and a red marble is added to the bag.
If the two marbles are of different colors, they are both discarded and a blue marble is added to the bag.
The game continues by repeating the above move.
Prove that no matter what sequence of moves is made, the process always terminates with exactly one marble left. In addition, find the possible colors of the marble remaining.
at this point a player will remove the 2 remaining marbles and add one. The game then stops.
