[/quote]
,
be positive real numbers with 
.
and 
satisfy 
,
is an even number.
,
,
for which the equality holds.
triangle with 
.
,
point on AC, AB respectively such that 
,
.
,
lines intersect at point 
.
,
.![\[(a-b)^2 + a + b \geq 2\]](http://latex.artofproblemsolving.com/c/f/a/cfa01a5b54a72afe6746f547390e65dbbb82ea64.png)
and 
we define 
as the set of all products of an element of 
and 
then 
We call a set 
of positive integers good if there do not exist sets 
of positive integers, each with at least two elements and such that the sets 
and 
is good.
,
,
,
and 
are integers.)
baskets, each containing at least one candy. The total number of candies is 
.
consecutive non-empty baskets, while Sevinch can take all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least 
candies, regardless of the initial distribution of candies and Sevinch's actions.
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
,
,
at point 
.
,
,
,
,
can be divided into two groups, so that the angles in each group have a sum of 
.
be a function satisfying 
.
satisfies equations (A) and (B) below. 
for all integers 
.
.
such that 
,
is infinite.
and all triples 
be a point outside circle 
centered at 
.
.
intersects bigger 
arc at 
.
intersects 
at 
.
intersects small 
arc at 
.
is tangent to 
Y by
Let 
and 
be monic quadratic trinomials, and let 
and 
be the vertices of the parabolas 
and 
, respectively. Let 
denote the minimum value of the function 
. It is known that the differences 
and 
are equal positive numbers. Find the angle between the line 
and the 
-axis.
and 
be monic quadratic trinomials, and let 
and 
be the vertices of the parabolas 
and 
, respectively. Let 
denote the minimum value of the function 
. It is known that the differences 
and 
are equal positive numbers. Find the angle between the line 
and the 
-axis.This post has been edited 1 time. Last edited by egxa, Apr 18, 2025, 5:25 PM
