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Find the sum of all even numbers from 5 to 15.

Let $x$, $y$, $z$ be three odd numbers such that $x! \times y! \times z! = 10!$. Find the value of $x+y+z$.

Find the value of the 100th term of the sequence 2, 5, 8, 11, \dots.

A digital clock in 24-hour format (HH:MM:SS) provides 6 independent outputs in 6 cells, where the outputs in cells 1, 2, 3, 4, 5, and 6 correspond to the functions $a^2 - 2a + 3$, $2b^2 - 3b + 4$, $2c^2 - c + 4$, $2d - d^2 + 1$, $1 + 4e - 2e^2$, and $2f^2 - 4f + 5$, respectively. If $1 \leq a,b,c,d,e,f \leq 12$ and each of $a,b,c,d,e,f$ is an integer, how many distinct valid times can be displayed on this clock?

A digit '5' is written before a 3-digit number. The new 4-digit number is double the original 3-digit number multiplied by 82. Find the 3-digit number.

Imon has a large solid cylinder with a radius of 512 units. He wants to cut the cylinder from the outside in such a way that only one solid cylinder remains. The outer diameters of the resulting pipes are halved sequentially. If the radius of the pipes and the cylinder are integers, what is the maximum number of pipes that can be obtained?

Sadia's keyboard has buttons for numbers 0 to 9, but the '3' button is broken. Her friend's WiFi password contains no characters other than digits. The password is four digits long. How many possible passwords cannot be typed on her keyboard?

Using only the digit 16, create the number 61. You can use addition, subtraction, multiplication, division, and square roots. What is the minimum number of 16's needed to achieve this? (Ignore negative numbers)

Payel and Protyay are playing a number game. In each turn, Payel chooses a positive integer and writes down the last digit of its square in a secret notebook. What is the minimum number of turns after which Protyay can be sure that at least one digit has been written multiple times in Payel's secret notebook?

Faizal has some notepads. He can evenly distribute his notepads among 4 friends, 12 friends, 15 friends, and 24 friends. What is the minimum number of notepads that Faizal has?

Tahmid eats an apple at noon. After a 10-minute break, he eats the next apple. Then before each apple, he takes double the previous break. How many apples can he eat before midnight?

Moma has some chocolates. She can evenly distribute her chocolates among 6 friends, 8 friends, 12 friends, and 18 friends. What is the minimum number of chocolates Moma has?

Let $n$ be a positive integer. The LCM of $n$ and 80 is 1200 and the GCD of $n$ and 80 is 20. Find the value of $n$.

List all subsets of the set $S = \{2, 6, 12, 24, 25, 30\}$ that contain 3 elements each. What is the sum of the maximum numbers of each subset?

In a regular hexagon ABCDEF, points P, Q, R are the midpoints of AB, AF, and BC, respectively. If the area of the hexagon is 2024 square units, what is the area of the pentagon EQPRD?

Payel makes mistakes 3 times out of 6 attempts, Shakur makes mistakes 3 times out of 7 attempts, and Tiham makes mistakes once out of 8 attempts. If they work together on one task, what is the percentage chance that the task will be completed correctly?

In the Prothom Alo office, there are 1000 employees. Their ID numbers range from 1 to 1000. Those with exactly 3 odd digits work in Dhaka, those with exactly 2 odd digits work in Rajshahi, and everyone else works elsewhere. How many employees work elsewhere?

If $x > 2$, what is the minimum integer value of $x$ for which the expression $x^2 - 5x + 3$ leaves a remainder of 8 when divided by 11?

The area of triangle $\Delta ABC$ is 6 square units. The base of triangle $\Delta DEF$ is 6 times that of triangle $\Delta ABC$, and the height of triangle $\Delta DEF$ is equal to that of triangle $\Delta ABC$. What is the area of triangle $\Delta DEF$?

What will fit in the blank in the sequence below? B, 3, E, 2, G, 9, P, __, V, 4, Z

Let $X$ and $Y$ be two prime numbers. If the difference of their squares is 40, what is the sum of these two numbers?

What is the sum of all integers $k$ between 1 and 200 for which the product of all divisors can be expressed as "$k^{\frac{9}{2}}$"?

Majedur wanted to do something fun using letters and numbers. He assigned numbers from 1 to 26 to all English letters, such that A = 1, B = 2, ..., Z = 26. Then he converted his name (MAJEDUR) into numbers. Using the digits of the converted number, he formed a 5-digit palindrome number. What is the minimum value of the palindrome number? (Note: Each digit can be used as many times as it appears. For example, if digit 6 appears 3 times, it can be used up to 3 times. Not all digits need to be used.)

Calculate the value of: \[ \frac{(4 \times 7 + 2)(6 \times 9 + 2)(8 \times 11 + 2) \dots (100 \times 103 + 2)} {(5 \times 8 + 2)(7 \times 10 + 2)(9 \times 12 + 2) \dots (99 \times 102 + 2)} \]

A building has 7 floors, numbered 1st, 2nd, 3rd, \dots, 7th. Each floor has $2^n$ rooms, where $n$ is the floor number. It takes $(2^n - 1)$ flowers to decorate each room. How many flowers are needed to decorate all the rooms?

Tahmid, Niloy, and Majed love to fish in the river. However, since the river is far away, only one of them goes fishing on Tuesdays. They fish in the order Tahmid, Majed, Niloy, and then again Tahmid. Given that January 2, 2024, is a Tuesday and Niloy goes fishing that day, how many times will Tahmid go fishing that year?

Four cardamoms are used to cook one dish of kacchi biriyani, which is shared equally among Imon, Moma, and Sadia. If the probability that Moma has three cardamoms on her plate is $\frac{p}{q}$, find the minimum value of $p+q$.

Let $f(x + 1, y) - f(x, y) = x$, $f(x, y) - f(x, y + 1) = y$, and $f(0, 0) = 0$. If $f(a, b) = 101$, find the minimum value of $a + b$.

Shithil loves juice. He has mango juice, orange juice, and lychee juice in three separate glasses. He has 600 ml of mango juice, 360 ml of orange juice, and 200 ml of lychee juice, and he wants to create a mixture in the ratio of 15:9:6. How much additional lychee juice should he add?

On a circular path, Majedur jogs at a speed of 8 m/s and runs 200 m on a straight path. At the same time, his friend Tahmid jogs at a speed of 6 m/s and walks 150 m in the same time frame. If they start running at the same time, how much faster will Majedur cover the distance compared to Tahmid?

Two numbers are added, producing a result. If the result is doubled, it exceeds the smaller number by 10. Find the smaller number.

A tree produces 2 seeds every year. It takes 2 years for each new seed to become a tree that can produce seeds for the first time. After 10 years, how many seed-producing trees will there be?

In a book exchange program at Majed's school, each of the 8 students brings one book. Each student will give their book to another student, and no student will receive more than one book. How many different ways can this book exchange be completed?

Determine the value of $2 + 4 + 6 + 8 + \dots + 80$.

In this year's Math Olympiad, each participant received an equal number of chocolates. Just before the exam started, 20 more participants joined. However, the Math Olympiad committee had only 100 extra chocolates. Even though the number of chocolates per person decreased, it was possible to distribute the chocolates equally among everyone. What was the maximum number of participants this year?

The coordinates of two endpoints of the diagonal of a square are $(11,11)$ and $(2024,2024)$. How many points with integer coordinates lie on the line $x = y + 11$ that are located inside or above the square?

How many numbers can be formed in the form of $rac{P}{Q}$ using any two prime numbers $P$ and $Q$ from 1 to 200, such that the value is less than 1?

$F(x + 1) = \frac{1}{F(x) - 1}$; $F(1) = 0$; Determine the value of \[ \prod_{i = 2}^{19} F(i) = \frac{a}{b} \] if $a$ and $b$ are coprime.

The product of two positive integers is 216, and their GCD is not equal to either of the numbers. What is the maximum sum of the two numbers?

Determine the largest two-digit prime factor of $inom{100}{50}$.

Fuad has learned multiplication and division in his class. He is thinking of expressing 2016 as a product of three integers. He found some numbers. At one point, he noticed that his second number is one more than 5 times the first number, and the third number is 10 more than double the second number. If all three guessed numbers are positive integers, what could be the sum of those three numbers?

Tahmid went to a strange country where everyone uses a 3-digit password. If the password user is female, the first digit will be 2, and if male, it will be 1. The remaining digits cannot be 0, 1, or 2. How many different passwords can be used in that country?

Two cells in a 6 × 6 grid are randomly colored black. The two adjacent cells are at a distance of 1 unit. The distance between the two black cells refers to the distance between their centers. What is the probability that the two black cells are at least 3 units apart, represented as $rac{p}{q}$, where $p$ and $q$ are coprime? Find the value of $p + q$.

A pentagon is a geometric shape bounded by five sides and five angles. If the angles of a pentagon are consecutive integers, what is the maximum angle?

The lengths of the sides of a triangle are 7, 10, and 11 units, respectively. The length of the unequal side of another isosceles triangle is twice that of the equal sides. If the perimeters of both triangles are the same, what is the length of the equal sides of the isosceles triangle?

If $f(x+y)=f(x)+f(y)$ and $f(1) = 3$, determine the value of $f(30)$.

Niloy is reading a magazine that used a total of 2046 digits for its page numbers. Determine the number of pages in the magazine (starting from page number 1).

What is the largest two-digit positive integer whose digits multiply to 24?

Imagine a digital clock displaying hours, minutes, and seconds. It is set for a 24-hour format, meaning at 2:30 PM, the display shows 14:30:00. How many different times can be displayed on that clock in a day?

The number $20 imes 14$ has some even factors and some odd factors. How many more even factors does it have compared to odd factors?

A clock behaves strangely. After 9 PM, every time the hour hand makes a complete rotation, its speed decreases by half. At what time will it take more than 2024 minutes to register one minute passing on the clock?

Niloy's birthday was on Tuesday, July 22, 2008. In which of the following years will his birthday next fall on a Saturday?

Antpur is a vast area where water is supplied through tanks. There is a large tank in the middle of the area and several smaller tanks. Water is supplied in such a way that each tank (excluding the large one) receives water from just one other tank, and each tank either supplies water to seven other tanks or none at all. If there are a total of 2024 tanks in the area, how many tanks do not supply water?

Tahmid has 295 Taka. He spends 30 Taka on a juice and 10 Taka on a packet of biscuits. If a packet of chips costs 15 Taka, how many packets of chips can he buy with the remaining money?

Payel wants to draw three equilateral polygons on the three sides of an equilateral triangle. If each polygon shares one side with another, how many sides does each polygon have?

Two positive integers have a GCD of 10. Find the minimum sum of those two numbers.

What is the maximum number of distinct integers that can have a GCD of 2024?

Majed went to a shop to buy chocolates using only 7 and 8 Taka coins. He likes a type of chocolate that costs 9 Taka each. What is the minimum number of chocolates he must buy so that he doesn't get any change back from the shopkeeper?

How many positive integers are there that are factors of at least one of $12^{10}, 14^{12},$ and $18^9$?

For any integer n, the expression $n^4 - kn^3 - n^2 + 2n$ is divisible by 4, where k is a positive integer. What is the minimum value of k?

Determine the value of $4 + 6 + 8 + \ldots + 30$.

A new infection of a microorganism has been detected in Emoland. The number of microorganisms doubles every day if their number is 100 or less. If their number is between 101 and 1000, it increases by 170 every day, and if it exceeds 1000, it increases by 130 every day. Scientists have discovered a biocide that can destroy 210 microorganisms at the end of each day, after the population growth. Today, the number of microorganisms is 4048. How many days will it take to make Emoland microorganism-free?

Momar has 32 apples. How many ways can he distribute the apples among Sadia and Mithila so that each of the three receives at least 5 apples?

In a 20-over cricket match, Bangladesh scored a total of 160 runs. How many runs did they score on average per over?

If we write the digits of a positive integer x in reverse, we denote it as rev(x). For example, rev(123) = 321. If for two different positive integers x and y, x + rev(y) = rev(x) + y, we say they have a friendship. What is the maximum number of friendships possible among two-digit numbers? (Here, consider (a,b) and (b,a) as the same pair.)

The lengths of the sides of triangle ABC are given as: AB = 10, BC = 20, CA = 15. The incenter of the triangle is K. Points D and E are taken on AB and BC respectively such that AC || DE and DE passes through point K. What is the perimeter of triangle BDE?

The current time on a 12-hour format digital clock is 11:57 AM. The product of the sum of the hour and minute digits is 24; ((1+1) × (5+7)). What is the probability that this product is an even number at any time of the day, represented as a/b, where a and b are coprime? Determine the value of a + b.

The numbers from 1 to 64 are placed randomly on an 8 × 8 chessboard. Then, Ishraq replaces each number with the remainder when divided by 32. He then cuts a square without looking, ensuring at least one number appears twice among the numbers in the square. What is the minimum area of the square, expressed as a/b of the entire board, where a and 8 are coprime? Determine the value of a + b.

In the 2023 World Cup, Bangladesh cricketer Najmul Hossain Shanto scored a total of 213 runs. Meanwhile, Tawhid Hridoy scored a total of 115 runs. Tanzid Hasan Tamim scored a total of 97 runs. Shanto played 9 matches, Hridoy 5 matches, and Tanzid 9 matches. The Bangladesh team scored a total of 2125 runs in the World Cup. If the combined runs of Shanto, Hridoy, and Tanzid is a/b of the team's total runs, what is the value of a + 7b - 17?

Imon has some tiles with a length-to-width ratio of 3:2. He wants to create a square using only these tiles. What is the minimum number of tiles he needs?

a, b, and c are three distinct integers between 1 and 2024 such that a^b + b^c + c^a is an even number. What is the maximum possible value of a + b + c?

Samira wanted to do something fun with letters and numbers. So, she assigned numbers from 1 to 26 for all English letters, such that A = 1, B = 2, ..., Z = 26. Then she converted her name (SAMIRA) into numbers. Using the digits of the converted number, she creates a 7-digit palindrome. What is the minimum value of the palindrome? [Note: A digit can be used as many times as it appears at most. For example, if the digit 6 appears 3 times, it can be used at most 3 times. Not every digit needs to be used.]

Determine the value of (2^{2024} - 2^{2020}) / (2^{2020} - 2^{2016}).

If the product of three prime numbers is a three-digit number, find the difference between the maximum and minimum values of these primes.

Sadia cut a square birthday cake in such a way that she gets the maximum number of pieces. (The pieces of the cake do not have to be equal, and there are no restrictions on cutting the cake in the middle.) She saw that after several cuts, there is one piece remaining after giving each guest one piece. After cutting it 3 more times, she found that everyone received 2 pieces. What is the minimum number of guests at the birthday party?

The number \overline{a55bc} is divisible by 8, 9, and 11, where a, b, and c are three different digits. Determine the value of 2a + 3b + 4c.

Rimi asked her father how many chocolates he brought from abroad. Rimi's father then wrote the letter equation below, where each letter represents a single digit, and said that R + I + M + I is the total number of chocolates, \overline{RI} × \overline{MI} = \overline{AAA}. What is the total number of chocolates?

How many maximum points can 12 straight lines intersect at if 4 lines are parallel?

If n = 3! + 5! + 7! + ... + 2023!, what is the remainder when n is divided by 11? Here, k! represents the factorial of k; that is, k! = 1 × 2 × 3 × ... × (k - 1) × k.

What is the sum of all divisors of 13?

How many integers of the form n^5 divide the integer 9! × 10! × 11!, where n is a positive integer?

What is the next term in the sequence 11, 11, 12, 14, 17...?

If a × b × c = 30 and (a, b, c) and (b, c, a) are distinct, how many solutions are possible for the given equation with positive integers?

Majed has the number 117!. How many times can he divide this number by 2 so that it has the maximum number of trailing zeros?

The number \overline{3bcd} is a four-digit odd number whose all divisors are odd. The number is divisible by 3 and 7. If c + d = b, determine the number.

Jyoti gave a book to Sadia. Sadia can read a maximum of 4 pages in a day and 6 pages in total over two consecutive days. If she reads 1 page on the first day, how many pages can she read at most after 7 days?

The sum of the page numbers of a chapter in a book is 75. If there are 5 pages in the chapter, on which page does the chapter end?

In a quiz competition, each small question has 6 points and each big question has 10 points. How many different numbers of questions can be answered to score 2024 points?

$f(x + 1) = f(x) + x^2 + x + 1$; $f(1) = 1$; What is $f(55)$?

Payel went on vacation to Dinajpur. She stayed there for 2 weeks and 3 days. How many days did she stay in Dinajpur in total?

abc is a three-digit number where $a+b=c$. How many such numbers can be found?

In a village, there are 6 houses labeled 1, 2, 3, 4, 5, and 6. Each house is connected by a road, and any two houses are connected by only one road. The construction cost of each road is equal to the larger label value of the two houses it connects. Calculate the total cost to build all the roads in the village.

If the largest angle of a triangle is 70°, determine the minimum possible value of the smallest angle.

Four times a fraction is \(\frac{8}{3}\). What will be three times that fraction?

How many positive integers smaller than 2024 can be expressed in the form of $2^x - 2^y$, where $x$ and $y$ are non-negative integers? (For example, $2^4 - 2^1 = 14$, so 14 is one such number)

A math club is organizing a math festival. They invite 16 students from each of the 15 schools to the festival. The math club wants to order enough pizza so that each student gets 2 slices. If one pizza has 12 slices, how many pizzas should the math club order?

In a room, there are 200 boxes and 30 people. The first person puts one ball in each box, the second person puts one ball in every 2nd, 4th, 6th... box, the third person puts one ball in every 3rd, 9th, 12th... box. In this way, the nth person will put one ball in every multiple of n. How many balls will be in the 64th box?

Tuna has $8n-1$ apples and $5n+1$ oranges. He wants to distribute the apples and oranges equally among some friends, so that each gets $p$ apples and $q$ oranges. Find the minimum value of $n$ for which the value of $p + q$ is determined.

You bought some chocolates and ice creams, where each chocolate costs 270 Taka and each ice cream costs 180 Taka. Your expenses on chocolates and ice creams are equal. On average, how much did you spend on each item?

A two-digit number is 24. It is divisible by both of its digits 2 and 4. Find the sum of all even two-digit numbers that are divisible by their two digits.