AoPS Academy
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
be defined as follows:
, and for all positive integers 
,![\[
a_n = a_{\left\lfloor \frac{n}{3} \right\rfloor} + a_{\left\lfloor \frac{n}{2} \right\rfloor}.
\]](http://latex.artofproblemsolving.com/8/0/0/8005effb2b72a59ce55f49a5204169ec8ffa3b60.png)
Find the sum of all values 
for which there exists a unique positive integer such that 
.
, points 
are constructed as follows: Points 
and 
lie on the extension of side 
such that 
. Points 
and 
lie on the extension of side 
such that 
. Points 
and 
lie on the extension of side 
such that 
..![\[
\angle ACE + \angle FBD + \angle EAC = 180^\circ.
\]](http://latex.artofproblemsolving.com/c/b/8/cb88e7d6ec129d50bde13e6a08ee16b275f42042.png)
with legs 
and 
is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?
be non-negative real numbers such that 
. Find the maximum and minimum values of the expression![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
be a permutation of 
such that there are exactly 
ordered pairs of integers 
satisfying 
and 
. How many possible exist?
students, 
students play cricket, 
students play hockey and 
students play basketball. 
students play both basketball and hockey, 
students play both cricket and basketball, 
students play both cricket and hockey, and 
students play all three: basketball, hockey, and cricket. Find the number of students who do not play any game.
be reals such that 
. Prove that
Let be reals such that 
. Prove that
,
,
, 
, 
is not greater than 
. red balls and 
white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...
unit. The area of the hexagon is ...
and are the roots of the cubic equation 
. The value of 
is ...
so that 
and 
is ...
, 
, 
, 
has an average of 
and a median of 
. The largest number of such data is ...
are integers greater than 
which are four consecutive quarters of an arithmetic row with 
. If 
and 
are squares of two consecutive natural numbers, then the smallest value of 
is ..., with 
, 
and 
. The points 
and 
lies on the line segment 
. with 
and 
. The measure of the angle 
is ...
meet 
for each natural number 
. The value of 
is ....
provided that the difference of any two numbers at least 
is ...
which satisfies 
are as many as ... with 
and 
. Point lies on the side so that 
. Suppose 
is a point on the side extension 
so that 
is perpendicular to . The point lies on the ray such that 
and 
. The large angle 
is ...
consists of integers with the following properties: For every three different members of there are two of them whose sum is a member of . The largest value of is ....
with 
positive reals is ....
with 
is ....
square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of 
coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of so that the game never ends for any initial square selection is ....
, the angle - bisector 
(
) and the median 
(
) intersect at point 
. Lines and 
intesect at point . The parallel through 
to 
intersects at point 
. Prove that 
, the angle - bisector () and the median () intersect at point . Lines and intesect at point . The parallel through to intersects at point . Prove that
is median, 
, thus 
. From angle bisector theorem, 
and 
. From 
we get 
. 
. From 
we get 
; with 
we are done.
and 
.
such that 
.
Where 
Write only the answers to the questions given.
to be more exact:
in years 2002-08 time was 90' for part A and 120' for part B
