holds 
.
with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
.
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
such that the equation 
has an integer solution 
.
Adit Aggarwal, India.
Y by Adventure10
Q1) Let 
be 3 distinct mumbers from {1,2,3,4,,5,6}. Show that 7 divides 
.
Q2) Show that there is a natural number
such that the number 
ends exacly in 2009 zeros.
Q3) Let a,b,c be positive integers with no common factor and satisfy the conditions
. Prove that a+b is a square.
Q4) Suppose that
, where 
. Prove that a is divisible by 23 for any positive integer n.
Q5) Prove that
is divisible by 42 for any positive integer m.
Q6) Suppose that 4 real numbers a,b,c,d satisfy the conditions
Find the set of all possible values the number M=ab+cd can take
Q7) Let a,b,c,d be positive integers such that a+b+c+d=99.Find the smallest and the greatest values of the following product P=abcd.
Q8) Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three times less than 1004.
Q9) Give an acute-angled triangle ABC with area S, let points A',B',C' be located as follows: A' is the point where altitude from A on BC meets the outwards facing semicirle drawn on BX as diameter.Points B',C' are located similarly. Evaluate the sum
.
Q10) Prove that
,where d is diameter of the inscribed cricle of 
.
Q11) Let A={1,2, . . . ,100} and B is a subset of A having 48 elements. Show that B has two distint elements
and 
whose sum is divisible by 11.
be 3 distinct mumbers from {1,2,3,4,,5,6}. Show that 7 divides 
.Q2) Show that there is a natural number
such that the number 
ends exacly in 2009 zeros.Q3) Let a,b,c be positive integers with no common factor and satisfy the conditions
. Prove that a+b is a square.Q4) Suppose that
, where 
. Prove that a is divisible by 23 for any positive integer n.Q5) Prove that
is divisible by 42 for any positive integer m.Q6) Suppose that 4 real numbers a,b,c,d satisfy the conditions
Find the set of all possible values the number M=ab+cd can take
Q7) Let a,b,c,d be positive integers such that a+b+c+d=99.Find the smallest and the greatest values of the following product P=abcd.
Q8) Find all the pairs of the positive integers such that the product of the numbers of any pair plus the half of one of the numbers plus one third of the other number is three times less than 1004.
Q9) Give an acute-angled triangle ABC with area S, let points A',B',C' be located as follows: A' is the point where altitude from A on BC meets the outwards facing semicirle drawn on BX as diameter.Points B',C' are located similarly. Evaluate the sum
.Q10) Prove that
,where d is diameter of the inscribed cricle of 
.Q11) Let A={1,2, . . . ,100} and B is a subset of A having 48 elements. Show that B has two distint elements
and 
whose sum is divisible by 11.
with the area 
. Let 
be located as follows: 
is the point where altitude
on 
meets the outwards facing semicirle drawn on ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
as diameter. Points 
and 
are located similarly.![$T=[BCA_1]^2+[CAB_1]^2+[ABC_1]^2$](http://latex.artofproblemsolving.com/c/b/f/cbf63b73acd4959c04538c0d447e051516a3fd56.png)
, where denoted ![$[XYZ]$](http://latex.artofproblemsolving.com/c/7/6/c7608a3e97e44fc5bcf0cd62a1f7dfc0a0a2e7d4.png)
- the area of ![$4\cdot \sum\sin A\cos B\cos C=2\cdot \sum\sin A\left[\cos (B+C)+\cos (B-C)\right]=$](http://latex.artofproblemsolving.com/f/4/6/f46411773081f30cabc9df6ebfff1f5c5964f90c.png)
![$2\cdot \sum\sin A\left[-\cos A+\cos (B-C)\right]=$](http://latex.artofproblemsolving.com/1/2/0/120d35382ab946fbceb8202916b3acb82ecb0e77.png)
![$\sum \left[-\sin 2A+2\sin (B+C)\cos (B-C)\right]=$](http://latex.artofproblemsolving.com/e/3/0/e30f439cadb9cc95cbebff8352fb5dbd2355a834.png)
.
. So 
![$[BCA_1]^2=\left(\frac 12\cdot BC\cdot DA_1\right)^2=$](http://latex.artofproblemsolving.com/2/0/e/20e4047abe7bd7a93df385a945d8f2848eca6c8f.png)
![$\sum [BCA_1]^2=\frac {abc}{4}\cdot\sum a\cdot\cos B\cos C=$](http://latex.artofproblemsolving.com/7/c/d/7cdb7c672f76e1fbbd15a386d968b96c3117321c.png)
![$\boxed{\sum [BCA_1]^2=2R^2S\cdot \frac {S}{2R^2}=S^2=[ABC]^2}$](http://latex.artofproblemsolving.com/7/5/f/75fb9c79561c2b67d6c6992342c57e7671b06d76.png)
. Otherwise, 
.
,
.
has exactly 2008 zeros and the number 
has exactly 2010 zeros, and since the function 
,
,
But since 
,
,
,
,
,
,
.
![$\sqrt[4]{abcd} \leq \frac{99}{4} \Leftrightarrow abcd \leq \frac{99^4}{16}$](http://latex.artofproblemsolving.com/6/a/e/6aef3fe61b96c49f5afdac20505556b4a869feb3.png)
but this is assuming 
,
and so maximum will be at 
,
(I offer no proof, it is something very intuitive, I know that wouldn't probably suffice at olympiad. It is easy to prove for 
variables, not sure how to do it for 
variables), and so maximum is 
.
,
and so minimum is 
.
?
and 
divids both 
divides all three of 
and 
s
(
rather than 
from the way the problem was stated.
and 
the orthocenter of ![$\frac{[A'BC]^2}{[ABC]^2}=\frac{A'D^2}{AD^2}$](http://latex.artofproblemsolving.com/e/3/b/e3b81f3113755c887c912afa389dc1e865933205.png)
,
,
,![$\frac{[A'BC]^2}{[ABC]^2}=\frac{DH\cdot AD}{AD^2}=\frac{DH}{AD}=\frac{[BCH]}{[ABC]}$](http://latex.artofproblemsolving.com/5/4/4/544aa54683c42e9d35c7b5143752383e6c3efa62.png)
.![$\frac{\Sigma{[BCH]}}{[ABC]}=\frac{[ABC]}{[ABC]}=1$](http://latex.artofproblemsolving.com/f/1/1/f11cd513b58867ecb83cf48ccdbd5fed00f19ffd.png)
.
.
and 
because of factor 
.
,
,
,
are clear again from factor 
,
.
,
.
,
,
and so 
.
.
,
in order for the sum 
to be divisible by 
.
elements of 
which give a remainder of 
or 
when divisible by 
when divisible by 
,![$area [ABC]^{2}$](http://latex.artofproblemsolving.com/9/c/0/9c060357345d784b54636ffebf466e2541828657.png)
,
and 
satisfy given system of equations.
