[/quote]
,
be the set of integers. Determine all functions 
such that, for all integers 
and 
,
Proposed by Liam Baker, South Africa
be a positive integer. Determine all positive integers 
that satisfy the following condition:
of divisors of ![\[
\frac{1}{d_1} + \frac{1}{d_2} + \cdots + \frac{1}{d_k} > N,
\]](http://latex.artofproblemsolving.com/2/c/2/2c20e94d7b7d5d5e9bcc32d67c375e29cdef9ae1.png)
there exists a subset of the fractions 
whose sum is exactly 
be a non-isosceles triangle with incircle 
)
the points where 
,
at 
.
and 
meet 
at 
and 
,
,
,
are concurrent at a point.
and 
be the midpoints of 
be the orthocenter of triangle 
.
are collinear.
such that the following inequality holds:![\[
x^k y^k z^k (x^2 + y^2 + z^2) \leq 3
\]](http://latex.artofproblemsolving.com/1/1/4/114213c088a60405d584cf25d20b8e82cc07c9ac.png)
for all positive real numbers 
satisfying the condition 
![\[
x^k y^k z^k (x^2 + y^2 + z^2) \leq xy+yz+zx
\]](http://latex.artofproblemsolving.com/2/1/f/21f7e38350bd3856d283f0f826819b4d96898c79.png)
for all positive real numbers 
and circle 
are externally tangent at point 
.
on circle 
and 
.
is extended to intersect circle 
.
is taken on the arc 
of circle 
is extended to intersect the angle bisector of 
at point 
.
and the circle passing through points 
intersect at another point 
.
.
be positive integer such that ![\[ \text{LCM}(x+2, y+2)+\text{LCM}(x, y)=2\text{LCM}(x+1, y+1).\]](http://latex.artofproblemsolving.com/7/3/3/7338e4d6058ededa9b8cd9fcc945ae362db4bad4.png)
Prove that 
is divisible by 
.
be the circumcircle of triangle 
,
and 
.
and 
intersect at 
.
intersects line 
at 
.
intersects line 
.
with diameter 
intersects line 
at points 
and 
,
at points 
and 
.
,
,
is equal to the sum of the other two.
,
be the set of vertices of 
be the set of triangles with vertices all in 
of 
be the set of triangles of this type.
be the set of triangles of this type.
,
the area of ![\[\sum_{T \in \mathcal A} S_T \geq \sum_{T \in \mathcal B} S_T.\]](http://latex.artofproblemsolving.com/5/e/d/5ed261e1e812466ea3e78f3b681f6bbdb7ebccd1.png)
.
such that:![\[ f(xy + f(x))+f(y) = xf(y)+f(x+y) \]](http://latex.artofproblemsolving.com/6/0/4/604d965a38e7df743b97369340946ef8d1c56e5e.png)
for all real numbers 
with 
is satisfied that :
is odd
![\[
S = 2^0 x_0^2 + 2^1 x_1^2 + \dots + 2^n x_n^2,
\]](http://latex.artofproblemsolving.com/7/b/8/7b818fccfea8f1dcb9f80bde51dba224f91c3713.png)
where 
are nonnegative integers such that 
.
,
.
,
,
,
,
,
Determine the minimum possible value of![\[\sum_{I \subseteq S} (-1)^{n-|I|} Y_I^{k},\]](http://latex.artofproblemsolving.com/6/e/f/6ef80d82ee22435b9b2994ec60790860621d9aea.png)
where 
denotes the number of elements in the finite set 
.
and 
and 
.
and 
meet at 
and intersect 
and 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
Y by
Let n>1 be odd. A row of n spaces is initially empty. Alice and Bob alternate moves (Alice first); on each turn a player may either
1. Place a stone in any empty space, or
2. Remove a stone from a non-empty space S, then (if they exist) place stones in the nearest empty spaces immediately to the left and to the right of S.
Furthermore, no move may produce a position that has appeared earlier. The player loses when they cannot make a legal move.
Assuming optimal play, which move(s) can Alice make on her first turn?
1. Place a stone in any empty space, or
2. Remove a stone from a non-empty space S, then (if they exist) place stones in the nearest empty spaces immediately to the left and to the right of S.
Furthermore, no move may produce a position that has appeared earlier. The player loses when they cannot make a legal move.
Assuming optimal play, which move(s) can Alice make on her first turn?
