In a convex quadrilateral $A B C D, \angle A B D=30^{\circ}, \angle B C A=75^{\circ}, \angle A C D=25^{\circ}$ and
$C D=C B$. Extend $C B$ to meet the circumcircle of triangle $D A C$ at $E$. Prove that $C E=B D .$
Suppose $r\geq 2$ is an integer, and let $m_{1},n_{1},m_{2},n_{2} \cdots ,m_{r},n_{r}$ be $2r$ integers such that
$$|m_{i}n_{j}−m_{j}n_{i}|=1$$
for any two integers $i$ and $j$ satisfying $1\leq i <j <r$. Determine the maximum possible value of $r$.
Solution:
Let us consider the case for $r =2$.
Then $|m_{1}n_{2} - m_{2}n_{1}| =1$.......(1)
Let us take $m_{1} =1, n_{2} =1, m_{2} =0, n_{1} =0$. Then, clearly the condition holds for $r =2$.
Now, let us check the case for $r =3$. Then if the conditions were to hold then, we should have, $$i =1, j =2,3$$
$$i =2, j =3$$
Then,
$$| m_{1} n_{2}-m_{2} n_{1}| =1\cdots(2) \\|m_{1} n_{3}-m_{3} n_{1}| =1 \cdots(3) \\ |m_{2} n_{3} - m_{3} n_{2}| =1 \cdots(4)$$ respectively.
Then, let us try to choose the value of $m_{1},n_{1}; m_{2},n_{2}; m_{3},n_{3}$ such that the conditions (2) ,(3) and (4) holds.
Then, let us take $m_{1} =1,n_{2}=1,m_{2}=0,n_{1}=0,m_{3}=1,n_{3}=1,m_{3} =1,n_{3} =0$ respectively.
Therefore the results holds for $r =3$.
Now, let us consider the case where $r\geq 4$
Let $c_{i}$ and $d_{i}$ be the remainders when $m_{i}$ and $n_{i}$ are divided by $2$ respectively, that is,
$c_{i} \equiv m_{i}$ (mod 2), and $d_{i} \equiv n_{i}$ (mod 2)
Therefore $c_{i}, d_{i} \in \{0,1\}$, since these are the only possible remainders when something is divided by $2$.
Now, the parities of both $m_{i}$ and $n_{i}$ cannot be the even , as in that case for any $j$, we have, $$|m_{i} n_{j} - m_{j} n_{i}| \neq 1$$, as it would be clearly even.
therefore, the possible values of the orderes pair $(c_{i},d_{i})$ would be $(0,1), (1,0)$ or $(1,1)$ respectively.
now, we see that if the parity patterns of $(m_{i},n_{i})$ and $(m_{j},n_{j})$ be the same, that is the parities of $m_{i}$ and $m_{j}$ ; $n_{i}$ and $n_{j}$ are the same, then,
$$ |m_{i} n_{j} - m_{j}n_{i}| = \text{ even } \neq 1$$
Therefore , the parity patterns of two pairs cannot be same.
Now, there are $4$ pairs $(m_{1}, n_{1}); (m_{2}, n_{2}) ; (m_{3}, n_{3})$ and $(m_{4}, n_{4})$ respectively.
therefore by pigeon hole principle at last two of these four pairs should have the same parity pattern, leading to a contradiction that we just discussed.
Therefore the conditions are not satisfied for $r =4$.
Therefore the maximum value of $r$ would be $3$.
This is a work in progress. Please come back soon for more updates. We are adding problems, solutions and discussions on INMO (Indian National Math Olympiad 2021)
INMO 2021, Problem 1
Suppose $r \geq 2$ is an integer, and let $m_{1}, n_{1}, m_{2}, n_{2}, \cdots, m_{r}, n_{r}$ be $2 r$ integers such that
$$
|m_{i} n_{j}-m_{j} n_{i}|=1
$$
for any two integers $i$ and $j$ satisfying $1 \leq i<j<r$. Determine the maximum possible value of $r$.
INMO 2021, Problem 2
Find all pairs of integers $(a, b)$ so that each of the two cubic polynomials
$$
x^{3}+a x+b \text { and } x^{3}+b x+a
$$
has all the roots to be integers.
INMO 2021, Problem 3
Betal marks 2021 points on the plane such that no three are collinear, and draws all possible line segments joining these. He then chooses any 1011 of these line segments, and marks their midpoints. Finally, he chooses a line segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using only a straightedge. Can Vikram always complete this challenge?
Note: A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points.
INMO 2021, Problem 4
A Magician and a Detective play a game. The Magician lays down cards numbered from 1 to 52 face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.
Prove that the Detective can guarantee a win if and only if she is allowed to ask at least 50 questions.
INMO 2021, Problem 5
In a convex quadrilateral $A B C D, \angle A B D=30^{\circ}, \angle B C A=75^{\circ}, \angle A C D=25^{\circ}$ and
$C D=C B$. Extend $C B$ to meet the circumcircle of triangle $D A C$ at $E$. Prove that $C E=B D .$
INMO 2021, Problem 6
Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients, and let deg $P$ denote the degree of a nonzero polynomial $P .$ Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions: