AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
by 
by 
inches (
and are integers). He paints the entire surface of the block, cuts the block into inch cubes, and notices that exactly half the cubes are completely unpainted. Prove that the number of essentially different blocks with this property is finite. (Do not attempt to enumerate them.)
South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude 
S.In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances?
board, there is initially a tree of height 
. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:
. Determine the largest 
such that the gardener can ensure there are eventually majestic trees on the board, no matter how the lumberjack plays.
such that the equation
be a positive integer and 
be a polynomial with integer coefficients such that 
.
is divisible by 
.
such that for every positive real 
, there exists an integer 
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
be a positive integer and let 
be a strictly increasing sequence of positive real numbers with sum equal to 2. Let 
be a subset of 
such that the value of![\[
\left|1-\sum_{i \in X} a_{i}\right|
\]](http://latex.artofproblemsolving.com/2/5/1/2515e074de1782f38453e1fce8ab7c2ec7cfe4c0.png)
is minimised. Prove that there exists a strictly increasing sequence of positive real numbers 
with sum equal to 2 such that![\[
\sum_{i \in X} b_{i}=1.
\]](http://latex.artofproblemsolving.com/3/7/4/37410820fc4c546f11d57212cfce53a7ec4941e3.png)
be a scalene triangle with circumcircle 
, and suppose the incircle of touches 
at 
. The angle bisector of 
meets and at 
and 
. The circumcircle of 
intersects the 
-excircle at 
, 
, and at 
. Prove that line 
passes through either or .
be positive real numbers such that : 
. Prove that : 
and 
be relatively prime positive integers, even. For each positive integer 
, let 
be chosen so that
is a minimum. Prove that
![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
be relatively prime nonconstant polynomials. Show that there can be at most three real numbers 
such that 
is the square of a polynomial.
be a permutation of the integers 
Call 
a big integer if 
for all 
Find the mean number of big integers over all permutations on the first postive integers.
be a linear transformation on linear space 
satisfying:
but 
, and 
is the eigensubspace of eigenvalue . Prove that can be decomposed to 
-cyclic subspace's direct sum.
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
