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.
is clearly not true. Even a 1st grader can see that. And what does "a number not a product" mean? A product is a number? Literally makes 0 sense.
.
as 
? What's wrong with them? Even a 1st grader can see that's so easy to simplify. Not to mention them acting all high and mighty since they memorized the cosine addition formula... so annoying.
if you do go around, acting like some kind of god again, "anticodon." Everyone will worship you finally for being able to copy from Google correctly!...
+2 w
people are playing a game at Cheesy's math casino, where is a prime number. Let 
be a positive integer. A subset of length 
from the set of integers from 
to inclusive is randomly chosen, with an equal probability (
and is fixed). The winner of Cheesy's game is person 
, if the sum of the chosen numbers are congruent to 
for 
., find all values of such that no person will sue Cheesy for creating unfair games (i.e. all the winning outcomes are equally likely)., let 
be the set of positive integers that do not exceed and are coprime to . Define 
as the smallest positive integer that allows to be partitioned into disjoint subsets, each forming an arithmetic progression.
satisfying 
, 
, and 
.
be a quadrilateral and let 
be a line. Let intersect the lines 
at points 
respectively. Given that these six points on are in the order 
, show that the circles with diameter 
are coaxal.
be a positive integer greater or equal to 
and let 
, 
, ..., 
be a sequence of non-negative real numbers. Find the maximum value of 
in terms of .
positive real numbers such that 
Prove that![\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+(k+5)(a+b+c) \geqslant 3(k+6),\]](http://latex.artofproblemsolving.com/3/d/a/3da7c98ff14a70b4bbbb6504bfb2c397fd148e45.png)
for all ![$0 \leqslant k \leqslant k_0 = \frac{3\big(\sqrt[3]{2}+\sqrt[3]{4}\big)-7}{2}.$](http://latex.artofproblemsolving.com/7/7/5/77526925e38a958cb4167555e8fbc1056506c522.png)
![$P\in \mathbb{R}[x] $](http://latex.artofproblemsolving.com/1/1/9/1194a1382b5ae7a9945cf8d9a6281954cf8cdb3d.png)
with even-degree
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
be an acute triangle incribed in a circle 
.Let 
be the midpoint of 
.Let 
and 
be altitudes from 
and 
of triangle , respectively, and let them intersect at 
.Let 
be the intersection point of tangents to the circle at points 
.Prove that 
and 
are concurrent.
or![$$a=b=\frac{\sqrt[3]{4}+2\sqrt[3]{2}-2}{3},\,c=\frac{2\big(1+\sqrt[3]{2}\big)}{3},$$](http://latex.artofproblemsolving.com/c/b/d/cbd0471249c6ac97348f9748ee55aa5e56af03ca.png)
or any cyclic permutation.
so 
.
and the number of blue marbles is 
.
,
the number of total marbles. Testing the options, we see that 
is the total number of marbles.
red marbles, 
blue marbles, and 
total marbles. It's given that 
and 
Now, we have
so 
implying that 
Adding these ratios up, we can see that the number of marbles in the jar must be a multiple of 7. The only multiple of 7 in the choices is 
so the answer must be 
