Revenge on the 2018 TSTST Part 2
by yayups, Nov 29, 2018, 5:26 AM
Day 2 as promised: I got 771 in test. I actually have really bad memory from real tests - I can't reconstruct any of my JMO solutions and these TSTST problems took me a while even though I solved two of them before.
Problem 4 wrote:
For an integer 
, denote by 
the set of integers 
for which the polynomial 
has an integer root.
Ivan Borsenco
, denote by 
the set of integers 
for which the polynomial 
has an integer root.
- Let
denote the set of integers for which contains two consecutive integers. Show that is infinite but ![\[ \sum_{n \in S} \frac 1n \le 1. \]](//latex.artofproblemsolving.com/9/c/d/9cd13bd7a3f104733545be791b61283368a19792.png)
- Prove that there are infinitely many positive integers
such that contains three consecutive integers.
![\[ \sum_{n \in S} \frac 1n \le 1. \]](http://latex.artofproblemsolving.com/9/c/d/9cd13bd7a3f104733545be791b61283368a19792.png)
Ivan Borsenco
and 
,
.
,
.
and 
,![\[b+\frac{n}{b}=a+\frac{n}{a}+1\implies n(1/a-1/b)=b-a+1\implies n=ab\left(1+\frac{1}{b-a}\right).\]](http://latex.artofproblemsolving.com/6/e/6/6e62c3135360463d055e1ba06fdeb74217167b18.png)
We need 
,
,
.
,
and 
for some 
.![\[n=\ell^2k(k-1)(1+1/\ell)=\ell(\ell+1)k(k-1).\]](http://latex.artofproblemsolving.com/4/c/7/4c7ebdf614379a9eae54f802fd5f996e2e436d48.png)
Thus, 
for positive integers 
.
,![\[\sum_{n\in S}\frac{1}{n}\le\sum_{x\ge 1}\sum_{y\ge 1}\frac{1}{x(x+1)}=\left(\sum_{x\ge 1}\frac{1}{x(x+1)}\right)^2=1,\]](http://latex.artofproblemsolving.com/e/c/0/ec0d39556f0a4326c198e539d926d0652541f7b8.png)
as desired.
to work as well, then bashing out 
is a square, difference of squares, and then factor.
if and only if 
.
.
as well. It is a straightforward computation to verify that![\[(2xy+x+y-1)^2 - 4x(x+1)y(y+1) = \left(\frac{x(x+1)}{2}-4\right)^2\]](http://latex.artofproblemsolving.com/2/8/0/2805b89d06abdea96d5830ddf7e97f7b105b25a7.png)
where 
,Problem 5 wrote:
Let 
be an acute triangle with circumcircle 
, and let 
be the foot of the altitude from 
to 
. Let 
and 
be the points on with 
and 
. The tangent to at intersects lines 
and 
at 
and 
respectively; the tangent to at intersects lines and at 
and 
respectively. Show that the circumcircles of 
and 
are congruent, and the line through their centers is parallel to the tangent to at .
Ankan Bhattacharya and Evan Chen
be an acute triangle with circumcircle 
, and let 
be the foot of the altitude from 
to 
. Let 
and 
be the points on with 
and 
. The tangent to at intersects lines 
and 
at 
and 
respectively; the tangent to at intersects lines and at 
and 
respectively. Show that the circumcircles of 
and 
are congruent, and the line through their centers is parallel to the tangent to at .Ankan Bhattacharya and Evan Chen
![[asy]
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[/asy]](http://latex.artofproblemsolving.com/0/a/0/0a0c7b444f168cb165c10c3e18d37db95465af5c.png)
is the perpendicular bisector of 
,
shows that it is the same line as 
.
is cyclic because 
,
is cyclic with diameter 
.
is 
.
,
.![\[\angle E_1OF_1=\pi-\angle OE_2P-\angle OF_1P=\pi-\angle A=\angle E_1AF_1,\]](http://latex.artofproblemsolving.com/b/5/b/b5bc02b2d7ba8b5456c8bbe68bbd16ae2745d054.png)
so 
.
and 
,![\[\angle OE_1E_2=\angle OPQ=\angle OQP=\angle OE_2E_1,\]](http://latex.artofproblemsolving.com/9/9/f/99f797a404b00cfda8a79341214d3875b90c193c.png)
so 
.
,
,
.
Problem 6 wrote:
Let 
, and for every positive integer define ![\[ T_n = \left\{ (a_1, \dots, a_n) \in S^n \mid a_1 + \dots + a_n \equiv 0 \pmod{100} \right\}. \]](//latex.artofproblemsolving.com/d/b/9/db9c2e99b54b653a4d3c8dc7fad0bef698208a5e.png)
Determine which have the following property: if we color any 
elements of red, then at least half of the -tuples in 
have an even number of coordinates with red elements.
Ray Li
, and for every positive integer 
define ![\[ T_n = \left\{ (a_1, \dots, a_n) \in S^n \mid a_1 + \dots + a_n \equiv 0 \pmod{100} \right\}. \]](http://latex.artofproblemsolving.com/d/b/9/db9c2e99b54b653a4d3c8dc7fad0bef698208a5e.png)
Determine which have the following property: if we color any 
elements of 
red, then at least half of the -tuples in 
have an even number of coordinates with red elements.Ray Li
be 
if the randomly chosen 
is blue, and 
if it is red. Note that if ![$I\subsetneq[n]=\{1,\ldots,n\}$](http://latex.artofproblemsolving.com/2/a/7/2a77bd3a3462a6b5c619abcbca2eb44b79bf99b3.png)
,
has a uniform random distribution (note that ![$I\not=[n]$](http://latex.artofproblemsolving.com/a/d/e/adea2c9d0e9b323bf0384c95cb93654df4556ab0.png)
)
.![\begin{align*}
p_e &= \sum_{\substack{I\subset[n]\\|I|\text{ even}}}\mathrm{Pr}\left(b_i=0\iff i\in I\right)\\
&= \sum_{\substack{I\subset[n]\\|I|\text{ even}}}\mathrm{Pr}\left(\prod_{i\in I}(1-b_i)\prod_{i\not\in I}b_i=1\right) \\
&= \sum_{\substack{I\subset[n]\\|I|\text{ even}}}\mathbb{E}\left(\prod_{i\in I}(1-b_i)\prod_{i\not\in I}b_i\right),
\end{align*}](http://latex.artofproblemsolving.com/7/f/4/7f475ad525573fe0a548b59b45ca17400c37b6f1.png)
where the last equality follows since 
is a random variable that takes values in 
.![\[p_o = \sum_{\substack{I\subset[n]\\|I|\text{ odd}}}\mathbb{E}\left(\prod_{i\in I}(1-b_i)\prod_{i\not\in I}b_i\right).\]](http://latex.artofproblemsolving.com/3/d/5/3d5ae39dda93b8a2398912539d8134a8e1e182eb.png)
One can see then that![\[p_e-p_o = \mathbb{E}\sum_{I\subset[n]}\left(\prod_{i\in I}(b_i-1)\prod_{i\not\in I}b_i\right)=\mathbb{E}f(b_1,\ldots,b_n),\]](http://latex.artofproblemsolving.com/6/c/b/6cbaf87c823c8c82ede3f59c02d96a977afb4425.png)
where ![$f(b_1,\ldots,b_n)=\sum_{I\subset[n]}\left(\prod_{i\in I}(b_i-1)\prod_{i\not\in I}b_i\right)$](http://latex.artofproblemsolving.com/0/e/7/0e7afef1c209ca3c97857d6271f7e60eb36034bd.png)
.
by factoring. We noted that all squarefree monmials in the 
have expectation 
,![\[p_e-p_o=\mathbb{E}f(b_1,\ldots,b_n) = 2^n\mathbb{E}(b_1\cdots b_n)+(2\cdot(1/4)-1)^n-(2\cdot(1/4))^n,\]](http://latex.artofproblemsolving.com/6/5/4/65434adddc3fe6edc3a09ae6344334631db2e32c.png)
or that![\[p_e-p_o=\frac{1}{2^n}((-1)^n-1)+2^n\mathbb{E}(b_1\cdots b_n).\]](http://latex.artofproblemsolving.com/d/1/6/d16d923cbc371fbf648c5a969cf0c29b8d5592a2.png)
Since the condition of the problem is equivalent to 
,![\[\frac{\text{\# of elements of }T_n\text{that are all blue}}{|T_n|}>\frac{1}{4^n}(1-(-1)^n).\]](http://latex.artofproblemsolving.com/a/c/8/ac854eae3ce7d84a5091aef56c5f168bd1de1c2c.png)
If 
blue if and only if 
,
.
April 2021
September 2020