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.
, the sequence 
of real numbers is defined as follows. Let 
, and for 
, let![\[a_n = \sum_{i=1}^{n-1} (a_i)^{n-i+1}.\]](http://latex.artofproblemsolving.com/8/3/0/83062000c09953ccdf171f90d90e169642630302.png)
Find all positive real numbers such that 
for all positive integers 
.
, 
is on the angle bisector of 
, 
and 
is on 
. to 
and 
is equal.
of (not necessarily distinct) integers has the following properties: 
for all integers 
, and ![\[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\]](http://latex.artofproblemsolving.com/d/1/c/d1cdc8a3d6b56870cf2c22f2a37924813f5152b0.png)
for all integers 
. Prove that all integers 
occur in the sequence (that is, for all , there exists with 
).
denote the set of integers considered modulo 
(hence has elements). Find all positive integers for which there exists a bijective function 
, such that the 101 functions![\[g(x), \quad g(x) + x, \quad g(x) + 2x, \quad \dots, \quad g(x) + 100x\]](http://latex.artofproblemsolving.com/4/0/1/401bc720ecfa2188e1735d623c161205b84f043b.png)
are all bijections on .
is irreducible for every natural number 
.
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
be an acute-angled triangle with circumcircle 
and circumcentre 
. Points 
and 
lie on such that 
and 
. Let 
meet 
at 
, and 
meet 
at 
.
and 
have an intersection lie on line 
.
and be positive integers such that 
. Define 
for 
. Prove that if all the numbers 
are integers, then 
is divisible by an odd prime.
equal sectors. On each circle 
sectors are painted white and the other are painted black. The smaller circle is then placed on top of the larger circle, so that their centers coincide. Show that one can rotate the small circle so that the sectors on the two circles line up and at least sectors on the small circle lie over sectors of the same color on the big circle.
of positive integers for which![\[\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}.\]](http://latex.artofproblemsolving.com/2/3/2/232bffeaf3bc1927e2cde4db377f8936d50e1671.png)
is a complex matrix with 
prove that is self-adjoint, i.e., that 
and a positive real number 
, prove that there exist infinitely many positive integers 
for which we can find pairwise coprime integers 
less than satisfying![\[\text{gcd}(\varphi(n_1), \varphi(n_2), \cdots, \varphi(n_k)) \geq n^{1-\varepsilon}.\]](http://latex.artofproblemsolving.com/a/c/9/ac9ae436ed238197dc6b8a2d6ac4357827703547.png)
Proposed by Cheng Jiang from Tsinghua University
Let ![$\ x,y\in U[0,1]$](http://latex.artofproblemsolving.com/7/1/7/71796e02c7717bf7e1e3837ff68bf97d66deb90d.png)
. Find the probability that the nearest integer to the ratio 
is odd. Let 
. Find the probability that the nearest integer to the ratio is odd.
be a Chebyshev polynomial of the second kind. If n>2 and x > 2 is a integer, Could 
be a prime number?
be a Chebyshev polynomial of the second kind. If n>2 and x > 2 is a integer, Could be a prime number?
is even .
odd we can check that 
for 
and 
is divisible with the minimal polynomial of 
and 
.