AoPS Academy
+8 w
be an integer. Prove that there exists a set 
of 
positive integers satisfying the following property: For every 
the set can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality 
.
, 
, 
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
, the number
is an integer. Prove that there is a positive integer 
such that 
for all 
.
be a positive integer. On the table, we have 
ornaments in different colours, not necessarily of each colour. Prove that we can hang the ornaments on Christmas trees in such a way that there are exactly ornaments on each tree and the ornaments on every tree are of at most 
different colours.
be a fixed positive integer and consider an 
grid of real numbers. Determine the greatest possible number of cells 
in the grid such that the entry in is both strictly greater than the average of 's column and strictly less than the average of 's row.
such that: 
and 
. Prove that
Equality holds when 
Equality holds when 
Equality holds when 
, and 
. The spheres with centers , and touch the plane at 
, and 
, respectively. Prove that 
of the pizza is covered by both toppings.
is a positive integer. For an -topping pizza, determine the probability, in terms of , that some region of the pizza with non-zero area is covered by all toppings.
liter and is initially filled with tea of % concentration. I want to serve tea to my guests, but they insist that the tea must have at least a % tea concentration to be considered good, such that 
. I can add as much water as needed, as long as the teapot’s capacity isn’t exceeded, and I can pour out tea in arbitrarily small amounts—allowing me to continuously adjust the concentration by topping it off with water. Using an optimal strategy, what is the maximum total volume of good tea (i.e. tea with at least % concentration) that I can serve to my guests?
is the midpoint of 
and 
is the midpoint of 
. Suppose 
is similar to 
. Find the length of .
points, pick points at an axis-aligned 
rectangle 
and 
. Now, for the inductive step, assume that there is a 
rectangle that contains all 
points. Then, shift this rectangle by 
. Consider the angles of rectangular strips of width that cover two points in the rectangle. Suppose the maximum angle is 
. Then, we can take 
. The maximum angle of a rectangular strip of width that covers two points of this new set of 
points is at most 
. For 
, starts at 
, so the maximum angle for the first 
points is at most 
. Otherwise, for 
, the tangent is at most
. Then, the maximum value of 
is at most 
for 
points, 
for 
points, 
for 
points, 
for 
points, 
for 
points, 
for 
points, and 
for 
points. Then, the last points are placed in a rectangle with 
-coordinate at most 
. Therefore, all 
points fit inside the 
square, and no rectangular strip of width covers at most two points of .
then prove that 
Equality holds when 
or 
or 
satisfying the following conditions:
for all 
Define the set ![\[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]](http://latex.artofproblemsolving.com/d/e/b/deb50830b9fe10c86ea169c044c1b26f515b2514.png)
then 
，
satisfy for 