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$2^{20} \times 3^{10} \times 5^{8}$ How many perfect cube divisors does this number have?

In cosmic quidditch, 26 players play, 13 per team. Half are red, and the other half are black. Each uses a spaceship numbered from 1 to 26. In a match, 3 players get injured, and the sum of their spaceship numbers is 39. If the probability that all of them are from the red team is $\frac{a}{b}$, where $a$ and $b$ are coprime, find $a + b$.

There is a large area called Antarapur, supplied with water through tanks. The central tank is large, and the others are small. The water flows sequentially from the large tank to smaller tanks. Each small tank either supplies water to 7 others or none at all. If there are 2024 tanks in total, how many tanks do not supply water?

The product of two positive integers is 15, and their GCD is 1. What is the LCM of these two numbers?

Let $p$ and $q$ be two prime numbers such that $p^3 + 1 = q^2$. Find the value of $p + q$.

Niloy can fit either 100 pencils or 120 pens in his bag. After placing 60 pens in the bag, what is the maximum number of pencils he can still fit?

You have an infinite number of 1, 2, and 3 unit currency notes. How many ways can you use these notes to pay exactly 12 units for a pen?

Pratyay stood two sticks of lengths 40 and 60 parallel to each other at a fixed distance apart. He tied a string from the top of each stick to the midpoint of the other. If he wants to place a third stick such that the intersection of the strings touches the top of the third stick, what must the height of the third stick be?

Payel and Pratyay are playing with a sequence of numbers. They start with 3, and to get the next number, they multiply the previous number by 2 and then subtract 1. For example, starting with 3, the next number is $(3 \times 2) - 1 = 5$. What is the 10th number in the sequence?

The sum of two three-digit positive integers is divisible by 3. What is the maximum possible difference between these two numbers?

Majed has two magic stones. Every time he rubs the stones, each one produces another stone. How many times must he rub the stones to have 100 stones in total?

$1^3 + 2^3 + 3^3 + \dots + n^3$ is divisible by $n + 3$. What is the largest possible value of $n$?

If November 21, 2023 is a Tuesday, what is the date of the first Tuesday in February 2024?

Given $f(x,y) = y - x^2$ and $|x| + |y| \leq 21$, find the difference between the maximum and minimum values of $f(x, y)$ where both $x$ and $y$ are integers.

Niloy, Tahmid, and Jyoti each have a six-sided die, colored red, blue, and yellow, respectively. In how many ways can the sum of the numbers rolled on the three dice be 11?

$y = 10^{2024} - x$, where $x$ is a prime number. For what is the smallest value of $x$ such that $y$ is divisible by 9?

A triangular number is a number that can form an equilateral triangle with that many dots. For example, 1, 3, and 6 are triangular numbers. What is the sum of the first 50 triangular numbers?

A cubic box can just fit a football with a radius of 20 meters. The volume of the empty space in the box can be expressed as $a^3 (1 - \frac{\pi}{b})$, where $a$ and $b$ are positive integers. Find the value of $a + b$.

What is the smallest positive integer that, when added to $1^{14} + 2^{14} + 3^{14} + \dots + 2024^{14}$, makes the result divisible by 7?

An ant is standing at point (1,1) on a grid. It needs to touch the line $y = 0$ once, the line $y = 8$ once, and finally reach the point (6,5). What is the minimum distance the ant can travel to reach its destination?

${2024}P{1430}$ is expressed as $k \cdot 11^x$, where $x$ and $k$ are non-negative integers. What is the maximum possible value of $x$? (${n}P{r} = \frac{n!}{(n - r)!}$)

The distance between Payel's and Pratyay's houses is 21 km. Payel started walking, and some time later Pratyay started walking as well. Payel covered $\frac{5}{7}$ of the total distance when they met. How far did Pratyay walk?

$ABCD$ is a cyclic trapezium where $AB \parallel CD$, $AB = 4$, $CD = 6$, and the height between $AB$ and $CD$ is 5. If the radius of the circumcircle of $ABCD$ is $\sqrt{x}$, what is the value of $x$?

The sum of three consecutive integers is 216. What is the largest of these three integers?

Jyoti has a polygon with 2024 sides. She wants to draw straight lines from each vertex to every other vertex. How many straight lines will she draw?

How many integer solutions $(p,q)$ exist such that $pq + q^2 + 2024 = (p - q)^2$?

In a regular hexagon ABCDEF, points P, Q, R are the midpoints of AB, AF, BC respectively. If the area of the hexagon is 4860 square units, what is the area of the pentagon EQPRD?

Let $f$ and $g$ be two functions where $f(n) = \sqrt[7]{n^3}$, $g(n) = \sqrt[5]{n^2}$ and $f_1 = f(n)$, $f_2 = f(f(n))$, $f_3 = f(f(f(n)))$. Similarly, $g_1 = g(n)$, $g_2 = g(g(n))$, $g_3 = g(g(g(n)))$. Find the value of $g_1(f_1.f_2 \dots f_\infty(25))g_2(f_1.f_2 \dots f_\infty(25)) \dots g_\infty(f_1.f_2 \dots \dots f_\infty(25))$.

There is a magic box near the station where if a certain amount of money is kept, at the end of the day, 2 Taka more is returned. One day, he kept 2 Taka in the box and received 4 Taka at the end of the first day, 6 Taka at the end of the second day, and so on. After 35 days, he has a total amount of $x$. He deposits all the money back in the box. The next day, he receives an amount equal to the LCM of $x$ and 600. How much money does he have now?

At Payel\'s wedding, the lanterns are arranged such that there are exactly 3 red lanterns between every two green lanterns. If there are a total of 2026 lanterns (red and green combined) and the 74th lantern is green, what is the maximum number of green lanterns there?

Using the digits 0, 1, 2, 7, 8, 9 only once, how many three-digit numbers can be formed?

Four friends, Protyay, Pial, Bindu, and Fuad, went camping in the forest. They have a tent that can only accommodate one person at a time. After dark, they decided that every hour, any one of them would sleep while the other three would keep watch until it gets light. It was found that Protyay kept watch for 7 hours, which is more than each of the others. Fuad kept watch for 4 hours, which is less than each of the others. If each person\'s watch time is an integer, how long was it dark?

Consider a positive integer \( n \) such that \( n! \) is the smallest number divisible by \( 2^{2024} \), \( 3^{2024} \), and \( 5^{2024} \). Determine the value of \( n \).

$2024 = a^3 \times b \times c$ where $b$ and $c$ are two different two-digit prime numbers. How many new numbers can be formed by changing $b$ and $c$ like $2024$ that can be expressed in the form $a^3 \times b \times c$? (Both changed $b$ and $c$ must be different two-digit prime numbers.)

What is the smallest positive integer that must be added to 2024 for the sum to be a perfect square?

Using the digits $0, 1, 3, 5, 7, 9$ only once, how many four-digit numbers can be formed?

Determine the maximum value of \( x + y \) such that $x$ and $y$ are positive integers and \( 11x + 13y = 738 \).

The number $\overline{3a5b7c8d}$ is divisible by 3, where a, b, c, d are four different digits. What is the minimum sum of a, b, c, d?

Tahmid has three jars of digestive candies. The first jar has 12 candies, the second jar has 18 candies, and the third jar has 24 candies. He wants to distribute all the candies equally among his friends. What is the maximum number of friends Tahmid can have?

A stick of length $y$ meters is standing upright $x$ meters in front of a wall. When a light is lit 5 meters above the ground, the shadow of the stick on the ground is 3 meters long. If the light is raised by 1 meter, the shadow of the stick becomes 1 meter shorter. What is the value of $x + y$?

What is the remainder when $20232024 + 20242025$ is divided by 3?

A wooden cube has a side length of $n$ units. All its faces are painted red, and it is cut into $n^3$ unit cubes. The total surface area of the unit cubes that are painted red is exactly one-eighth of the total surface area. If the length of the diagonal of the larger cube is $a$, what is the value of $\sqrt{3} a$?

In a certain country, 10 runs are scored for a six at a distance of 80 meters and 12 runs for a six at a distance of 100 meters. If every ball in a 48-ball over is valid, how many runs can be scored in an over if three sixes are hit, assuming all runs are scored only from fours and sixes?

If all six angles of a hexagon are odd integers in a sequence, what is the maximum angle?

In an exam, the average score of 12 students is 57. Due to a correction in the score of one student, the average score of the students decreased by 1. How much was that student\'s score reduced?

Jeba has a mystery number. When she multiplies it by 5 and then adds 3, she gets 18. What is Jeba\'s mystery number?

What is the sum of the first three digits from the left of the sum of the series $11 + 181 + 1881 + 18881 + \dots$ for the first 20 terms?

What is the largest six-digit palindromic number that is divisible by 8?

If $x$ and $y$ are two prime numbers and $x^2 - y^2 = 72$, find the minimum value of $x + y$.

Two candidates are running for the class captain election. There are 60 students in the classroom. What is the minimum number of votes a candidate must receive to be certain of winning the election? (Assume that the candidates cannot vote for themselves.)

There are 7 students in a classroom. A committee of 5 will be formed from them. However, there are some issues while forming the committee. Imon and Mazed do not want to be in the committee together. Tiham said that if Shakur is not included in the committee, he will not be in it either. Jyoti said that she will be in the committee only if Tahmid or Niloy is included in the committee. However, if both Tahmid and Niloy are included, Jyoti will not be in the committee. How many ways can the committee be formed respecting the above conditions?

A palindrome number is one that remains the same when written backwards, such as 2112 or 66. How many numbers from 1 to 2024 can be made into a palindrome number by placing the first digit of the number at the end? (For example, placing the first digit 1 of the number 133 at the end makes it 1331, which is a palindrome number.)

Tahmid, Jyoti, and Niloy are playing a fun card game where names from A to J are written. Each name card has 4 different colors, making a total of 40 cards. According to the rules of the game, everyone will take 2 cards, and points will be scored if the 2 cards are of the same name. Initially, Tahmid and Niloy took 2 cards each and found that all 4 cards were different. Now, the probability of Jyoti scoring points can be expressed in the form of m/n, where m and n are co-prime numbers. What is the value of m + n?

Shakur created a sequence where the nth term is the sum of the first n natural numbers. What is the sum of the first 99 terms of Shakur's sequence?

n! represents the product of all integers from 1 to n. For example: 5! = 1 × 2 × 3 × 4 × 5. Niloy has the number 117!. His friend Tahmid has an infinite number of 5s. How many minimum 5s should Niloy take from Tahmid so that when multiplied with 117!, the product has the maximum number of trailing zeros?

If \overline{abba} is divided by \overline{cc}, the quotient is \overline{aaa}. What is the maximum possible value of a + b + c for all numbers that satisfy this condition?

Imon has some tiles with lengths and widths of 3 and 2 units respectively. He wants to make a square using only those tiles. What is the area of the smallest square he can make?

The sum of two prime numbers is 25. What is the value of the larger of the two numbers?

Find the next number in the pattern: 50, 49, 46, 41, ......

A wooden cube with one side of n units is painted red on all its faces and cut into n^3 unit cubes. One-eighth of the total surface area of the unit cubes is red. How many additional faces of the unit cubes need to be painted red so that one-fourth of the total faces are red?

One day you felt like you wanted to drink date juice, but you had to go to the village to get it. The only way to travel from your house to the village is by bus. One day you left and after drinking 4 glasses of juice, you came back home to find that you spent 132 taka. Another day you went with a friend, drank a total of 10 glasses of juice, and came back to find that you spent 294 taka. How much is the bus fare from your house to the village?

A fun nonlinear web series named 'Dream' has 16 episodes, and the rules for watching it are to start from the second episode and watch the remaining episodes in any order but finish with the last episode. The condition is that all prime-numbered episodes must be watched together and all even-numbered episodes must be watched together (after 2, any episode can be watched). What is the minimum possible value of a + b if the number of ways to watch the series can be expressed as (a!)^3 × b, where a and b are integers?

What is the largest six-digit palindrome number that is divisible by 6?

Shithil loves juice very much. He has three glasses with mango juice, orange juice, and lychee juice separately. He prepared a mixture with 600 ml of mango juice, 320 ml of orange juice, and 300 ml of lychee juice. If he wants to make the ratio of mango juice, orange juice, and lychee juice in the mixture 12:7:6, how much more orange juice in milliliters needs to be added?

Tunnah has 8n - 1 apples and 5n + 1 oranges. He wants to distribute apples and oranges equally among his friends, ensuring that everyone gets a whole number of fruits. Find the sum of all possible integer values of n less than 100.

Jihan went to the shop to buy paper and pens. The price of paper is 9 taka and the price of pens is 4 taka. If Jihan has 97 taka, what is the maximum number of papers he can buy so that he has 12 taka left?

Niloy can carry 100 pencils or 120 pens or 150 erasers in his bag. After keeping 25 pencils and 30 pens in his bag, how many maximum erasers can he carry?

Niloy's birthday was on Tuesday, July 21, 2020. When will his birthday next fall on a Saturday?

If (2 + 1)(2^2 + 1)(2^{2^2} + 1) ... (2^{2^{10}} + 1) + 1 = 2^n, what is the value of n?

In a four-digit number, the first and last digits are 1 and 4 respectively. The product of the two middle digits is an even two-digit number, and their sum is an odd single-digit number. Also, the product of the two digits is double their sum. None of the digits in this four-digit number can be repeated. Find the largest number.

Each cell of an 8 × 8 chessboard has an arrow indicating any direction. An arrow indicates a direction that moves 1 meter in the opposite direction. Determine the minimum value of a + b when the board exceeds its original position by a^2√b distance.

The average of three different positive integers is 5. What is the maximum possible value among the three numbers?

Imon has 5 storybooks with an average page number of 485. He bought two more books with 380 and 665 pages respectively. If he can read a maximum of 25 pages in a day, what is the minimum number of days required to finish all the books?

How many three-digit numbers exist where the sum of the digits equals 5?

Niloy is reading a magazine that uses a total of 2049 digits for the page numbers. Determine the total number of pages in the magazine. (The first page number starts from 1)

A paper has some straight lines drawn on it, none of which are parallel to each other, and there are no intersection points where more than two lines cross. What is the minimum number of lines needed to ensure that the paper is divided into at least 2024 regions?

The sum of two prime numbers is 55. Find the value of the larger of the two numbers.

In an $n \times n$ grid, the total number of unit squares is divisible by both 11 and 25. Determine the minimum value of $n$.

Chinku has created a machine that outputs a sequence of numbers based on the input of an English word (meaningful or not). For example, if 'abc' is input, the output is 123, and for 'dydx', the output is 425424. How many distinct words can Chinku input such that the output is 21121221?

If $\sqrt[16]{100!} = a \sqrt{ b \sqrt{c \sqrt{d \sqrt{e}}}}$ and $b$, $c$, $d$, $e$ are positive integers that do not produce a perfect square, what is the number of factors of $abc$?

The length of a side of a rhombus is 5 meters. If the length of the side is doubled, what will be the new perimeter of the rhombus?

There are 100 cards in a box, each with a different integer written on it from 1 to 100. If any two cards are drawn and their numbers are multiplied to get a perfect cube, how many different products are possible? (The two cards are returned to the box after being drawn)

A 6-digit number of the form \( \overline{ABCABC} \) is called a Tahmid magic number, where \( A \), \( B \), and \( C \) are digits. What is the highest common factor among all the magic numbers?

You have some chocolates that you want to distribute evenly among your friends. You notice that if distributed between two people, 1 chocolate remains, if distributed among four people, 3 chocolates remain, if among six people, 5 remain, and if among eight, 7 remain. What is the minimum number of chocolates you could possibly have?

Find the sum of all possible positive integers $x$ such that $x^2 - 23x + 127$ is a perfect square.

A rhombus has a side length of 13 meters. If the length of the side is doubled, determine the new perimeter of the rhombus.

If $p$, $q$, $r$ are any three different prime numbers greater than 3, what is the largest positive integer that will always divide $(p - q)(q - r)(r - p)$?

If the largest angle of a triangle is 65 degrees, what is the minimum possible value of the smallest angle of the triangle?

In a dark room, there are 90, 80, 70, 60, and 50 red, pink, green, black, and blue socks respectively. If a pair is defined as two socks of the same color, how many socks must be drawn to ensure that at least 20 pairs of socks have been drawn?

How many integers can be found between 1 and 2024 that can be expressed simultaneously in the forms $4a + 19$ and $6b + 21$, where $a$, $b$ are integers?

In triangle ABC, points P and Q are on sides AB and AC respectively such that AP = $\frac{3}{7}$ * AB and AQ = $\frac{4}{9}$ * AC. Lines BQ and CP intersect at point O. The extended line AO intersects BC at point R. Express $\frac{BR}{BC}$ in the form $\frac{x}{y}$ where $x$, $y$ are coprime integers, and find the value of $x + y$.

$$ S_n = \sum_{k=1}^{50} \frac{(-1)^k k}{4k^2 - 1} $$ Express $S_n$ in the form $ -\frac{a}{b} $, where $a$ and $b$ are coprime. Find the value of $a + b$.

Given \(f(n) = 3f(n + 1) - 2f(n - 1)\), \(f(0)=0\) and \(f(1)=1\), find the value of $2f(2023) + 3f(2024)$.

There are some students in a classroom, each having the same number of multiple chocolates. The total number of chocolates in the classroom is between 300 and 400. If the total number is given, how many students could there be and how many chocolates would each have?

A bullet strikes a concrete wall creating an elliptical hole. The distance between the two farthest points of the ellipse is 26 mm. If the diameter of the bullet is 10 mm and it strikes the wall at an angle of $\tan^{-1}\frac{a}{b}$, what is the value of $a + b$?

Any number with four or fewer digits can be expressed in the form $1000a + 100b + 10c + d$, where $a$, $b$, $c$, $d$ are single-digit non-negative integers. If you transform such a four-digit or fewer number $x$ using the rule $f(x) = a^4 + b^3 + c^2 + d$, find the difference between the largest and smallest values of $x$ such that $f(2024) = f(x)$.

How many pairs of distinct prime numbers less than 100 exist such that the difference of their squares is a perfect square? (Here, $(a,b)$ and $(b,a)$ are considered the same pair.)

Anupam has a magic box that transforms any number placed inside into its square the next day. He loves to play with this box. On the first day, he puts 23 in the box, and the number he gets the next day discards all but the last two digits and puts the remaining number back in the box. If this process continues daily, what number will he place in the box on the 2026th day?

When Shan asked Swarga to write the integers from 1 to 2024, Swarga mistakenly wrote \( 123456789101112131415 \dots \dots \dots \dots \dots \dots 2024 \) in sequence. Shan got angry and told Swarga, 'Swarga, now add all the digits of the large number you obtained. After that, if the sum is not a single digit, add those digits again until a single-digit sum is obtained.' What is the final sum that Swarga arrived at?

Among any four prime numbers, if subtracting 2 from the product of three of the primes yields the fourth prime, which is greater than 100. What is the minimum possible sum of the first three primes?