BD-BDMO 2021

For a positive integer $n$ , let $A(n)$ be the equal to the remainder when $n$ is divided by $11$ and let $T(n)=A(1)+A(2)+A(3)+ \dots + A(n)$ . Find the value of $$A(T(2021))$$

Let $u, v$ be real numbers. The minimum value of $\sqrt{u^2+v^2} +\sqrt{(u-1)^2+v^2}+\sqrt {u^2+ (v-1)^2}+ \sqrt{(u-1)^2+(v-1)^2}$ can be written as $\sqrt{n}$ . Find the value of $10n$ .

Let $ABC$ be a triangle with incenter $I$ . Points $E$ and $F$ are on segments $AC$ and $BC$ respectively such that, $AE=AI$ and $BF=BI$ . If $EF$ is the perpendicular bisector of $CI$ , then $\angle{ACB}$ in degrees can be written as $\frac{m}{n}$ where $m$ and $n$ are co-prime positive integers. Find the value of $m+n$ .

$P(x)$ is a polynomial in $x$ with non-negative integer coefficients. If $P(1)=5$ and $P(P(1))=177$ , what is the sum of all possible values of $P(10)$ ?

How many ways can you roll three 20-sided dice such that the sum of the three rolls is exactly $42$ ? Here the order of the rolls matter. (Note that a 20-sided die is is very much like a regular 6-sided die other than the fact that it has $20$ faces instead of $6$ )

Let $ABC$ be an acute-angled triangle. The external bisector of $\angle BAC$ meets the line $BC$ at point $N$ . Let $M$ be the midpoint of $BC$ . $P$ and $Q$ are two points on line $AN$ such that, $\angle PMN=\angle MQN=90^{\circ}$ . If $PN=5$ and $BC=3$ , then the length $QA$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are co-prime positive integers. What is the value of $(a+b)$ ?

A binary string is a word containing only $0$ s and $1$ s. In a binary string, a $1-$ run is a non extendable substring containing only $1$ s. Given a positive integer $n$ , let $B(n)$ be the number of $1-$ runs in the binary representation of $n$ . For example, $B(107)=3$ since $107$ in binary is $1101011$ which has exactly three $1-$ runs. What is the following expression equal to? $$B(1)+B(2)+B(3)+ \dots + B(255)$$

Shakur and Tiham are playing a game. Initially, Shakur picks a positive integer not greater than $1000$ . Then Tiham picks a positive integer strictly smaller than that.Then they keep on doing this taking turns to pick progressively smaller and smaller positive integers until some one picks $1$ . After that, all the numbers that have been picked so far are added up. The person picking the number $1$ wins if and only if this sum is a perfect square. Otherwise, the other player wins. What is the sum of all possible values of $n$ such that if Shakur starts with the number $n$ , he has a winning strategy?

A positive integer $n$ is called nice if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is nice because its largest proper divisors are $3,2,1$ and $6=3+2+1$ . Find the number of nice integers not greater than $3000$ .

$A_1A_2A_3A_4A_5A_6A_7A_8$ is a regular octagon. Let $P$ be a point inside the octagon such that the distances from $P$ to $A_1A_2, A_2A_3$ and $A_3A_4$ are $24, 26$ and $27$ respectively. The length of $A_1A_2$ can be written as $a \sqrt{b} -c$ , where $a,b$ and $c$ are positive integers and $b$ is not divisible by any square number other than $1$ . What is the value of $(a+b+c)$ ?

How many quadruples of positive integers $(a,b,m,n)$ are there such that all of the following statements hold?

1. $a,b<5000$ 2. $m,n<22$ 3. $gcd(m,n)=1$ 4. $(a^2+b^2)^m=(ab)^n$

A function $g: \mathbb{Z} \to \mathbb{Z}$ is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers $m$ and $n$ . Let $f$ be an adjective function such that the value of $f(1)+f(2)+\dots+f(30)$ is minimized. Find the smallest possible value of $f(25)$ .