AoPS Academy
holds.![\[
\left\{\, (\pm1,\,\pm1),\; (\pm3,\,\pm5),\; (\pm7,\,\pm23) \,\right\}
\]](http://latex.artofproblemsolving.com/0/0/f/00f82923feb669d822498ab16e96c3911e29d69a.png)
Note: I have faced similar type of equation before 
which leads to 
and got a solution from pythagorian parametrization but I dont have a clue about this one.
amount of balloons, where 
. Then, on his first turn, Bob takes 
amount of ballons where 
. After first turn, Alice and Bob alternately takes as many balloons as his/her partner has. Is it possible for Bob to take amount of balloons at first, such that after a finite amount of turns, one of them have a number of balloons that is a multiple of 
?
and have answers
and 
be circles with centres 
and 
, respectively, such that the radius of is less than the radius of . Suppose and intersect at two distinct points 
and 
. Line 
intersects at 
and at 
, so that 
lie on in that order. Let 
be the circumcentre of triangle 
. Line 
meets again at 
and meets again at 
. Let 
be the orthocentre of triangle 
. parallel to is tangent to the circumcircle of triangle 
.
+4 w 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
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 
.)
is large enough, in every 
square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by 
.
is large enough, in every square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by .
, be the matrix corresponding to entries in the square.
, where 
. over a sub-square, 
, equals the sum 
.
be a colouring of the lattice. If is sufficiently large, then by Gallai's theorem there exists 
monochromatic lattice points that are vertices of a square. These correspond to four entries of such that 
. This is exactly what we wanted to prove.
,![\[
-\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{\sqrt{6}}{9}
\quad \text{for } \frac{\pi}{n} \leq \theta \leq \pi - \frac{\pi}{n},
\]](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)
