A pair of positive integers $(m,n)$ is called 'steakmaker' if they maintain the equation 1 + 2$^m$ = n$^2$. For which values of m and n, the pair $(m,n)$ are steakmaker, find the sum of $mn$

Consider rectangle $ABCD$.$ E$ is the mid-point of $AD$ and $F$ is the mid-point of $ED$. $CE$ cuts $AB$ in $G$ and $BF$ cuts $CD$ in $H$ point. We can write ratio of areas of $BCG$ and $BCH$ triangles as $\frac{m}{n}$. Find the value of $10m + 10n + mn$.

Prottasha has a 10 sided dice. She throws the dice two times and sum the numbers she gets. Which number has the most probability to come out?

Once in a restaurant Dr. Strange found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or the item he didn’t tried on the 1st day. Similarly, on the 3rd day, he eats either item 4 or the item he didn’t tried on the 2nd day. If someday he's not able to choose items that way, he eats the item that remained uneaten from the menu. In how many ways can he eat the items for 12 days?

For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$. $z$ is the maximum real number such that [$\frac{5}{z}$] + [$\frac{6}{z}$] = 7. Find the value of$ 20z$.

Point $P$ is taken inside the square $ABCD$ such that $BP + DP=25$, $CP - AP = 15$ and $\angle$ABP = $\angle$ADP. What is the radius of the circumcircle of $ABCD$?

Tiham is trying to find 6 digit positive integers$ PQRSTU$ (where $PQRSTU $are not necessarily distinct). But he only wants the numbers where the sum of the 3 digit number$ PQR$, and the 3 digit number $STU$ is divisible by 37. How many such numbers Tiham can find?

Let $ABC$ be a triangle where$\angle$B=55 and $\angle$ C = 65. D is the mid-point of BC. Circumcircle of ACD and ABD cuts AB and AC at point F and E respectively. Center of circumcircle of AEF is O. $\angle$FDO = ?

You have 2020 piles of coins in front Of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th pile contains 2020 coins. Guess a positive integer k, in which piles contain at least k coins, take away exact k coins from these piles. Find the minimum number of turns you need to take way all of these coins?

Sokal da tries to find out the largest positive integer n such that if n transforms to base-7, then it looks like twice of base-10. $156$ is such a number because $(156)_{10}$ = $(312)_7$ and 312 = 2$\times$156. Find out Sokal da's number.

A prime number$ q $is called 'Kowai' number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one 'Kowai' number can be found. Find the summation of all 'Kowai' numbers.

$2^{2921}$ has $581$ digits and starts with a $4$. How many $2^n$'s starts with a $4$, where $0$ is the last digit?