BD-BDMO 2022

Find all solutions for real $x$ , $$\lfloor x\rfloor^3 -7 \lfloor x+\frac{1}{3} \rfloor=-13.$$

In $\triangle ABC, \angle BAC$ is a right angle. $BP$ and $CQ$ are bisectors of $\angle B$ and $\angle C$ respectively, which intersect $AC$ and $AB$ at $P$ and $Q$ respectively. Two perpendicular segments $PM$ and $QN$ are drawn on $BC$ from $P$ and $Q$ respectively. Find the value of $\angle MAN$ with proof.

Prove that if the numbers $3,4,5, \dots ,3^5$ are partitioned into two disjoint sets, then in one of the sets the number $a,b,c$ can be found such that $ab=c.$ ( $a,b,c$ may not be pairwise distinct)

Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minimum integer $x$ with proof such that if $n-m\geq x,$ then Pratyya's number will be larger than Payel's number everyday.

In an acute triangle $\triangle ABC$ , the midpoint of $BC$ is $M$ . Perpendicular lines $BE$ and $CF$ are drawn respectively on $AC$ from $B$ and on $AB$ from $C$ such that $E$ and $F$ lie on $AC$ and $AB$ respectively. The midpoint of $EF$ is $N.$ $MN$ intersects $AB$ at $K.$ Prove that, the four points $B,K,E,M$ lie on the same circle.

About $5$ years ago, Joydip was researching on the number $2017$ . He understood that $2017$ is a prime number. Then he took two integers $a,b$ such that $0<a,b <2017$ and $a+b\neq 2017.$ He created two sequences $A_1,A_2,\dots ,A_{2016}$ and $B_1,B_2,\dots, B_{2016}$ where $A_k$ is the remainder upon dividing $ak$ by $2017$ , and $B_k$ is the remainder upon dividing $bk$ by $2017.$ Among the numbers $A_1+B_1,A_2+B_2,\dots A_{2016}+B_{2016}$ count of those that are greater than $2017$ is $N$ . Prove that $N=1008.$

Sabbir noticed one day that everyone in the city of BdMO has a distinct word of length $10$ , where each letter is either $A$ or $B$ . Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word $ABB$ is located consecutively, then these letters can be changed to $BBA$ or if $BBA$ is located somewhere in the word consecutively, then these letters can be changed to $ABB$ (if wanted, the word can be kept as it is, without making this change.) For example $AABBA$ can be transformed into $AAABB$ (the opposite is also possible.) Now Sabbir made a team of $N$ citizens where no one is friends with anyone. What is the highest value of $N.$

Solve the following problems -

A) Find any $158$ consecutive integers such that the sum of digits for any of the numbers is not divisible by $17.$ B) Prove that, among any $159$ consecutive integers there will always be at least one integer whose sum of digits is divisible by $17.$