Leonhard Euler
Problems of the Semester
2023
Ali Onat
1 Introductory
Problem 1.1. Calculate
Problem 1.2. 100 students participated in an exam with 5 questions. If every question was solved by exactly 50 students, what is the minimum number of students who solved less than 3 questions?
Problem 1.3. For the sum S = 5 + 55 + 555 + 5555 + ... + 5555...555 where the last term has 50 digits, what is the remainder when S is divided by 9?
2 Intermediate
Problem 2.1. In the American Mathematics Competition (AMC), a students receives 6 points for a correct answer, 1.5 points for a blank and 0 points for an incorrect answer out of a total of 25 questions. If every student at AAU got different scores, what is the maximum possible number of students?
Problem 2.2. How many solutions does the following equation have?
|||||x^2 + 3x − 5| − 4| − 3| − 2| − 1| = 3x − 15
Problem 2.3. How many ordered positive integer tuples satisfy the inequality
Problem 2.4. 1234 people live in Cartcurtistan. If there is exactly one pair of people who have the same number of friends, how many different values can this number take?
3 Olympiad
Problem 3.1. a, b, c, d, e, f are real numbers such that
a + b + c + d + e + f = 20
and
(a + 1)^2 + (b + 1)^2 + (c + 1)^2 + (d + 1)^2 + (e + 1)^2 + (f + 1)^2 = 12
Find the maximum possible value of f.
Problem 3.2. Find all solutions to the equation
p^2 − p + 1 = a^3
where p is a prime number and a is a positive integer.
Problem 3.3. Find, with proof, all positive integer solutions to the equation
3^x = z^2 + 5^y
*You may directly send your solutions to aonat26@my.uaa.k12.tr
Past Problems
Problems of May
2023
1) Introduction
Welcome Traveler!
2) Problems
2.1) Problem 1:
Ali has two rectangular prisms both with side lengths a, a, b. Ali stacks these rectangular prisms on top of each other such that the first rectangular prism has side a as its height, and the rectangular prism on top has side b as its height. Ali wants to fit these in a room that has 20m between the roof and floor. Find the largest possible value for the sum of their volumes.
2.2) Problem 2:
There is a sequence that has every 6 digit number including the digits 1, 2, 3, 4, 5, 6 as its elements
arranged in decreasing order. Let S be the sum obtained by adding the terms of this sequence with
alternating signs. Find S.
S = 654321 − 654312 + 654231 − 654213 + ... + 123465 − 123456
Ali Onat
Problems of February
2023
1) Ali gets lost while looking for his ten horses. He embarks on his journey from node 1 and travels through the edges of length 1. If Ali travels exactly 9 units and ends up at node 1, how many paths could he have taken?
2) Dr. Eggman wants to eat eggs. He can only eat eggs with all of five distinct types of spices. He has 10 rooms in his house, and every morning he can randomly check 6 rooms for spice jars before he collapses out of hunger. How many spice jars does Dr. Eggman need to buy to guarantee to find all 5 types of spices in these 6 rooms?
3) Sanem is controlling a visual piece of modern art made up of 14 consecutive lights that are either on or off. Every day, he can change the states of the lights and create a pattern that has never been seen before. However, three consecutive there cannot be patterns that include three consecutive turned-on lights because it disturbs the eyes of the visitors. For how many days can Sanem continue finding new patterns, before running out of patterns?
4) There is a group of 10 people going to Sanji’s sea restaurant. Each member of the group is either male and female with equal probability. A female eats 1 duck and a male eats 2 ducks. What is the expected value for the amount of ducks eaten?
5) 6 people including Rügzö and Fıstık are standing on a circle. There are 2 people between Rügzö and Fıstık. Rügzö has an alarm set for 7 seconds later, and no one in the line wants to hear his awful ringtone. Every person passes the clock to either the person on their right or left with equal probability in 1 second. When the alarm is set off, the people holding the clock and the people next to him hear the alarm. What is the probability that Fıstık hears Rügzö’s insufferable ringtone?
Ali Onat
6) Jamal the cotton picker has a 10mx9m cotton farm. He divides his farm into 90 1mx1m squares and plants cotton on each of these squares. However, some rats dig out holes on some of the squares. Jamal wants to run a water pipe through his farm without running into the holes starting from the bottom left square ending on the top right square, and for the sake of efficiency, the pipe will only go through 18 squares. In how many ways can Jamal construct the pipeline?
Sanem Naz Kafalı & Ali Onat
7) If there are n people waiting to get their nails done in a nail salon, and each person takes an average of x minutes to be served, but every y minutes such that y>x, a new person enters the line. How many minutes will it take for the salon to be empty for the first time, assuming that the line does not move any faster or slower?
8) When a city’s population is a six digit number abc def and the sum of a, b and c is equal to the sum of d, e and f, it is called a perfect city. We are given the information that UAALAND is a perfect city. Sanem finds all the possible values for the population of the city UAALAND and adds them up. What is the remainder if Sanem divides this sum by 13?
Sanem Naz Kafalı
Problems of March
In the census, the following conversation occurs between a census official and a mathematician:
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"How many daughters do you have?"
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"3."
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"What are their ages?"
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"The product of their age is 72, and the sum is the house number of my house, which is 14."
The census official looks at the door and after a little thought,
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"This information is not enough."
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"My oldest daughter is a chess player."
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"Now it's done. Thanks"
What is the age of the girls?
Selin Deniz Akdoğan
Problems of October
1) Eight fairies fly in the shape of a circle. Each fairy will cast a spell on another. Find the probability that two fairies never cast a spell on each other. In other words, if fairy A casts a spell on fairy B, fairy B should not cast a spell on fairy A yet choose a different fairy.
2) In 13 integers, show that there exist 7 integers of which combination is divisible by 7. Also, show that 13 is the minimum integer for this condition to be satisfied.
Eda Toprak
Problems of July
1) Once upon a time, in a distant galaxy, Emperor Mehmet Ekin was bored and wanted to have some fun with his peasants. Emperor Mehmet Ekin designed a problem, and if the peasants give a wrong answer to the problem, they all are going to suffer under Mehmet Ekin’s egoist narcissism. However, if they give a correct answer, Emperor Mehmet Ekin will free them, and they will live happily ever after. The problem is considerably simple:
A 8 by 8 chess board is given. The peasants will start from A1(lower left corner) square of the chessboard, and the goal is to reach to the H8 (upper right corner) square of the chessboard. The goal is to determine how many possible ways there are to complete this process. The peasants are only allowed to move up and right on this chessboard.
However, in the middle of the problem, because Emperor Mehmet Ekin is bored, he adds another rule to the game. He decides that the problem was easy, and therefore, he adds the following addition:
The players are not allowed to pass the A1-H8 diagonal passing through the chess board.
Can you help the peasants and save them from the ruthless Emperor Mehmet Ekin?
Hüseyin Yağız Devre
Problems of June
1) What is the maximum number of regions created within a circle when n points on the circle's circumference are chosen to be connected to form chords within the circle? (The maximum number of regions means that at one place, a maximum of two chords could pass through a point of intersection.)
Eda Toprak
Problems of May
1) For positive integers n define , where m is the greatest integer with . Given a positive integer , define a sequence by taking
For what do we have constant for sufficiently large ? (B-1, Putnam 1991)
2) A pair of adult rabbits produces a pair of baby rabbits once each month. Each pair of baby rabbits requires one month to grow to be adults and subsequently produces one pair of baby rabbits each month thereafter. Determine the number of pairs of adult and baby rabbits after some number of months. The rabbits are immortal.
Selin Deniz Akdoğan
Problems of February
1) 97 unit squares of a 14 x14 chessboard will be marked. In how many different ways can it be made so that there are no two unit squares marked that share a common edge?
Güney Baver Gürbüz
2) The rate of population growth of Hoatzins is 5.4% per year. The carrying capacity for the population on an island is 15,547. The initial population is 3000. Calculate the population in 5 years.
Model how the population would change if it somehow managed to get to 17,000.
Eda Toprak
Taylor series, possible passwords and the probability, number of ways to arrange card...
You may directly send your solutions to our email mathchronicles21@gmail.com.