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.
be an equilateral triangle, 
be a point inside it, and 
be the intersections of 
with the sides of . If 
are the midpoints of 
, 
, 
, show that there is a triangle with sides 
, 
and 
.
+1 w
such that 
![\[f(x)+f(\lfloor x\rfloor)f(\{x\})=x,\]](http://latex.artofproblemsolving.com/6/4/1/6411452c9f27ac7a8a25141927ed2f34e810e6dd.png)
and draw its graph. Find all 
for which the equation 
has solutions for any 
.
be a fixed integer and 
.
and 
, the number![\[\sum\limits_{k=1}^n \dfrac{|a-r\omega^k|^2}{|a|^2+r^2}\]](http://latex.artofproblemsolving.com/e/e/3/ee3b0568b9a758cf22c23f96dbb7e7afe3601964.png)
is an integer. Interpet this result geometrically.
be two integers and 
be a positive integer such that 
. Show that ![\[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\]](http://latex.artofproblemsolving.com/e/5/f/e5f9486f980870ff72858b57f7b50cd2aa9d142a.png)
for 
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .
consisting of 
consecutive non-negative integers can be partitioned in two disjoint sets 
and 
each containing 
elements so that 
for 
of all sequences of integers 
such that 
and let 
. Show that![\[\sum_{s\in A} M(s)=(n+1)^{k+1}-(1^k+2^k+\ldots +(n+1)^k).\]](http://latex.artofproblemsolving.com/7/6/e/76e7fce0da6bb9537b73d8699f7a21a6b8cb61fa.png)
with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each 
along a great circle of .
be a positive integer not divisible by 3. Prove that there are infinitely many positive integers 
such that 
and 
are divisible by , where 
is the sum of digits of 
.
be a prime number, and 
be a sequence defined by 
and 
, 
. Show that one of the first 
terms of the sequence is not prime. we have![\[\max(|\sin x|,|\sin (x+1)|)>\frac13.\]](http://latex.artofproblemsolving.com/e/d/e/ede9b524370c457fe41038cca5663e3c9bdaf320.png)
of lattice points with positive coordinates in the plane. For each point 
from , we draw a segment between it and each of the points in the set ![\[S(P)=\{(a+b,c)\mid c\in\mathbb{Z}, \, c>a+b\}.\]](http://latex.artofproblemsolving.com/3/d/0/3d091c92ad9bf8644c59b77a1ad0209c41765093.png)
Show that there is no colouring of the points in with a finite number of colours such that every two points joined by a segment are coloured with different colours.
in the plane of a square 
for which ![\[\max (PA, PC)=\frac1{\sqrt2}(PB+PD).\]](http://latex.artofproblemsolving.com/3/9/8/3984b1970837a85af0bac93fd711843630a1fd99.png)
, where is a prime number. Show that if is an even number, then the polynomial ![\[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]](http://latex.artofproblemsolving.com/5/d/f/5df80f885e63852e4f53a6145dafd18d63dc3526.png)
is divisible by 
.
- are sides of triangle 
. It is known, that if we increase any one side by 
, we get new in case of a) and in case b)
,to have acute angle implies that 
thus the sides are greater than 
or equal,it s easy to verify that 
satisfy the condition so the minimal area is 