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.
candies is 
. If 
, the probability of getting 
candies is 
times the probability of getting 
candies plus 
the probability of getting 
candies. What is the probability that Spencer gets 
candies when he plays the game? Express your answer as a common fraction
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
be a positive integer and let 
be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of around the circle such that the product of any two neighbors is of the form 
for some positive integer 
.
be a fixed triangle, and let 
, 
, 
be the midpoints of sides 
, 
, 
, respectively. Let 
be a variable point on the circumcircle. Let lines 
, 
, 
meet the circumcircle again at 
, 
, 
, respectively. Assume that the points 
, 
, 
, , , are distinct, and lines 
, 
, 
form a triangle. Prove that the area of this triangle does not depend on .
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
be an acute-angled triangle with circumcircle 
. A circle 
is internally tangent to at 
and also tangent to 
at 
. Let 
and 
intersect at 
and 
respectively. Let 
and 
be points on line such that 
is the midpoint of 
and 
is the midpoint of 
. Lines 
and 
meet at 
and intersect again at 
and 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
,
,. . . ,
is 
if for every index 
that holds 
, there exist at least 
terms of the serie such that they are all less or equal to .
, where ,,. . . , is an serie.
be a triangle with 
. The angle bisectors of 
and 
meet the sides 
and 
at 
and 
, respectively. Let 
be the incentre of triangle 
. Suppose that 
. Find all possible values of .
be a fixed positive integer. There are 
people, numbered 
, participating in a tennis tournament. For any two positive integers 
with 
, player has a higher skill level than player 
. Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among courts, numbered 
.
, the winner of court moves to court 
and the loser of court stays on court ; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court . such that, regardless of the initial pairing, the players 
all change courts immediately after the th round. be positive reals. Prove that![\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]](http://latex.artofproblemsolving.com/6/9/3/6937791380a8642edca3e773dc72dbf3f0e0f206.png)
, how many digits does 
have?
not 
has the most factors?
has the most factors?
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
.
,
lies on diagonal 
.
.
has 
,
,
.
.
to 
have the digit 
and are divisible by 
now, we have covered all the numbers in the 7 ones place. (27, if I counted right) now, lets find numbers with tens digit 7 that do NOT have 7 in its one place, since we already counted the _77 in the first case.
this totals up to 25 numbers, so the total is
so 
?
without a calculator.
for 
:
:
.
and 
where the rest has been omitted because of the floor function. Now 
and multiplying by 
yields 
grid is made. 
unit squares are created by separating the side with length 
in 
in 
and 
,
rectangles can be created.
grid, each unit square is created by dividing the sides of lengths 
and 
.
horizontal lines and 
vertical lines:
that can be formed, we multiply the number of ways to choose the horizontal lines and the ways to choose the vertical lines:
satisfy 
,
and 
are positive integers, what is the probability that 
is divisible by 
?
.
,
represents a distinct digit from 
.
is our answer. (By python, 
does not work and surprisingly, 
is the only permutation number that is divisible by 
8907