[/quote]
,
and 
.
table is colored black or white. Every cell on table border is black. It is known, that in every 
square there are cells of two colors. Prove, that exist 
(use in a problem for junior) to practice. Can anyone share to me?
be a non-isosceles triangle with orthocenter 
,
.
be midpoints of 
and 
be reflections of 
.
have the same centroid.
be a quadrilateral such that 
and 
are acute and 
.
,
.
.
,
be real numbers such that 
Prove that
Let 
Prove that
,
.
at 
.
at 
.
be the midpoint of 
,
meet 
.
at 
.
is tangent to 
be positive real numbers satisfying 
.![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
and 
, then we can deduce 
\le 2 $ ?
be the maximal number of the vertices of a convex lattice polygon with exactly 
lattice points in the interior. Show that:
for 
.
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
Y by
A car is driving along a straight highway at a speed of 
km per hour. Not far from the highway there is a parallel to him a 
-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than 
km per hour. Not far from the highway there is a parallel to him a 
-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than 
This post has been edited 1 time. Last edited by parmenides51, Apr 26, 2021, 1:05 PM
