AoPS Academy
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.
with full rank. Suppose that has at least two positive eigenvalues and at least one negative eigenvalue. Show that for all 
such that 
, there exists 
such that 
.
![$f : [0, 1] \to \mathbb{R}$](http://latex.artofproblemsolving.com/c/b/e/cbe179b391a30215ad8b35bc89c1e533c478a544.png)
be a function with continuous derivative such that 
and 
. Show that there exists a real number 
such that 
and 
for all 
such that 
.
be the sequence such that 
and for 
![\[x_{n+1}=\ln(e^{x_n}-x_n)\]](http://latex.artofproblemsolving.com/8/3/a/83a48a694713230a7dbb59952fed3915fe9824b4.png)
(as usual, the function 
is the natural logarithm). Show that the infinite series![\[x_0+x_1+x_2+\cdots\]](http://latex.artofproblemsolving.com/4/c/e/4cefbe5d286f9702fa86ab1651c30059022909c7.png)
converges and find its sum.
be an integer, and let 
be non-constant integer-coefficient polynomials with positive leading coefficients. A positive integer 
is called good if 
such that for any integer 
, the number of good integers among 
does not exceed 
be positive integers, and 
be integers greater than 1 satisfying![\[x_1! \cdot x_2! \cdots x_n! = m!.\]](http://latex.artofproblemsolving.com/2/2/4/22406c872a4589595ad726954f64d023dd2f634c.png)
Prove that: (1) 
; (2) There exists a positive real constant such that for sufficiently large 
,![\[\max\{x_1, x_2, \cdots, x_n\} > m - c \cdot \frac{\ln m}{\ln \ln m}.\]](http://latex.artofproblemsolving.com/7/c/8/7c86de44a841cf32e9d6142754a28b6ec523ff23.png)
Created by Mucong Sun, Tianjin Experimental Binhai School
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
and 
.
over all triples 
of distinct real numbers such that
is called perfect if the sum of its positive divisors 
is twice , that is, 
. For example, 
is a perfect number since the sum of its positive divisors is 
, which is twice . Prove that if is a positive perfect integer, then:![\[
\sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1}
\]](http://latex.artofproblemsolving.com/0/c/0/0c08f527e87f7aae4ca5226f19bd084249551fe3.png)
where the sums are taken over all prime divisors 
of .
![\[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\]](http://latex.artofproblemsolving.com/3/f/0/3f0b8424645a6111de6a657a2bedf02ceced6942.png)
(Here lcm denotes the least common multiple, and 
denotes the greatest integer 
.)
be a convex, continuously differentiable function from 
to 
such that 
for all 
. Prove that the improper integral 
is convergent if and only if the series 
is convergent for all 
.
,![\[
-\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{\sqrt{6}}{9}
\quad \text{for } \frac{\pi}{n} \leq \theta \leq \pi - \frac{\pi}{n},
\]](http://latex.artofproblemsolving.com/d/e/3/de361adc16c54cc154fda34c66436fd15978e55b.png)
![\[
-\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{1}{5}
\quad \text{for } \frac{\pi}{n} \leq \theta \leq \frac{\pi}{2}.
\]](http://latex.artofproblemsolving.com/9/f/d/9fd0aeb1cf4958d14a076656fd034c33cdc16ead.png)
