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for which the number 
is a perfect square.
such that the centroid of the triangle lies on the circle inscribed in the triangle. Prove that 
we can find positive integers 
such that 
. Show that in any such equation there must be two of the 
numbers with a common divisor (
).
is a triangle. Show that 
be the common chord of two circles 
and 
. A line through 
intersects at 
and at 
. Let 
be a point on the segment , different from 
and . The line 
intersect at 
and 
. The line 
intersects at 
and 
.
lie on the circumference of a circle 
. is 
, prove that 
is perpendicular to 
.
inhabitants of an island, where is even, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that two necklaces have a marble of the same type if and only if their owners are friends.
different types of stones. types.
for which the equation 
has positive integer solutions.
, consider the numbers 
is even. for which 
and 
are primes.
be the sequence of Fibonacci numbers: 
, 
and 
, for every 
.
, divides 
.
such that 
, ![$2 \cdot \sqrt3 \cdot \sqrt[3]{4} ...\sqrt[n-1]{n} > n$](http://latex.artofproblemsolving.com/8/8/b/88b20da824a8641e1dea5d3ae6b8a075a063e9e5.png)
, ![$2 \cdot \sqrt3 \cdot \sqrt[3]{4} ,,, \sqrt[n-1]{p} > n$](http://latex.artofproblemsolving.com/7/8/d/78d0c378c006705c7bf6334e6b526be2342c5e9f.png)
![$2 \cdot \sqrt3 \cdot \sqrt[3]{4} \ \ldots \sqrt[n-1]{n} > n$](http://latex.artofproblemsolving.com/e/c/7/ec759e53df7e3969d099e129074718e960ea8c7f.png)
and also 
., when 
, it is obvious. Assume that for 
, the inequality holds, then when 
, ![\[2 \cdot \sqrt3 \cdot \sqrt[3]{4} \ \ldots \sqrt[n-1]{n} \cdot \sqrt[n]{n+1} > n \sqrt[n]{n+1} >n+1 \iff n+1 > \left(1+\frac{1}{n}\right)^n\]](http://latex.artofproblemsolving.com/5/f/6/5f6bfcab1bdf7ed14a8b4707d5c68daeaaa907ee.png)
which is true since we know the RHS converges to 
when approaches infinity but LHS will grow larger and so bigger than RHS. 
