MathDB
Hardest algebra ever

Source: Kyiv City MO 2022 Round 1, Problem 9.2

1/23/2022
For any reals x,yx, y, show the following inequality:
(x+4)2+(y+2)2+(x−5)2+(y+4)2≤(x−2)2+(y−6)2+(x−5)2+(y−6)2+20\sqrt{(x+4)^2 + (y+2)^2} + \sqrt{(x-5)^2 + (y+4)^2} \le \sqrt{(x-2)^2 + (y-6)^2} + \sqrt{(x-5)^2 + (y-6)^2} + 20
(Proposed by Bogdan Rublov)
algebrainequalities
Present fraction as a difference of smaller fractions

Source: Kyiv City MO 2022 Round 1, Problem 7.1

1/23/2022
Represent 12021\frac{1}{2021} as a difference of two irreducible fractions with smaller denominators.
(Proposed by Bogdan Rublov)
constructionFractionalgebra
Subsets with sum equal to one

Source: Kyiv City MO 2022 Round 1, Problem 7.3, 8.2

1/23/2022
You are given nn not necessarily distinct real numbers a1,a2,…,ana_1, a_2, \ldots, a_n. Let's consider all 2n−12^n-1 ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to 11?
For example, if a=[−1,2,2]a = [-1, 2, 2], then we got 33 once, 44 once, 22 twice, −1-1 once, 11 twice, so the total number of ones here is 22.
(Proposed by Anton Trygub)
algebracombinatorics
Magic country and businessman Victor

Source: Kyiv City MO 2022 Round 1, Problem 7.4

1/23/2022
In some magic country, there are banknotes only of values 33, 2525, 8080 hryvnyas. Businessman Victor ate in one restaurant of this country for 20242024 days in a row, and each day (except the first) he spent exactly 11 hryvnya more than the day before (without any change). Could he have spent exactly 10000001000000 banknotes?
(Proposed by Oleksii Masalitin)
number theory
Sums not divisible by differences

Source: Kyiv City MO 2022 Round 1, Problem 8.4

1/23/2022
What's the largest number of integers from 11 to 20222022 that you can choose so that no sum of any two different chosen integers is divisible by any difference of two different chosen integers?
(Proposed by Oleksii Masalitin)
number theory
Geometry from legendary author

Source: Kyiv City MO 2022 Round 1, Problem 8.3

1/23/2022
In triangle ABCABC ∠B>90∘\angle B > 90^\circ. Tangents to this circle in points AA and BB meet at point PP, and the line passing through BB perpendicular to BCBC meets the line ACAC at point KK. Prove that PA=PKPA = PK.
(Proposed by Danylo Khilko)
geometry
Relating mean and gcd

Source: Kyiv City MO 2022 Round 1, Problem 8.1

1/23/2022
Consider 55 distinct positive integers. Can their mean be
a)Exactly 33 times larger than their largest common divisor?
b)Exactly 22 times larger than their largest common divisor?
number theoryalgebragreatest common divisor
Tournament went wrong

Source: Kyiv City MO 2022 Round 1, Problem 8.5

1/23/2022
20222022 teams participated in an underwater polo tournament, each two teams played exactly once against each other. Team receives 2,1,02, 1, 0 points for win, draw, and loss correspondingly. It turned out that all teams got distinct numbers of points. In the final standings the teams were ordered by the total number of points.
A few days later, organizers realized that the results in the final standings were wrong due to technical issues: in fact, each match that ended with a draw according to them in fact had a winner, and each match with a winner in fact ended with a draw. It turned out that all teams still had distinct number of points! They corrected the standings, and ordered them by the total number of points.
Could the correct order turn out to be the reversed initial order?
(Proposed by Fedir Yudin)
combinatoricsTournament
Easiest algebra ever

Source: Kyiv City MO 2022 Round 1, Problem 9.1

1/23/2022
What's the smallest possible value of (x+y+∣x−y∣)2xy\frac{(x+y+|x-y|)^2}{xy} over positive real numbers x,yx, y?
algebrainequalities
Tournament still broken

Source: Kyiv City MO 2022 Round 1, Problem 9.5

1/23/2022
n≥2n\ge 2 teams participated in an underwater polo tournament, each two teams played exactly once against each other. A team receives 2,1,02, 1, 0 points for a win, draw, and loss correspondingly. It turned out that all teams got distinct numbers of points. In the final standings, the teams were ordered by the total number of points.
A few days later, organizers realized that the results in the final standings were wrong due to technical issues: in fact, each match that ended with a draw according to them in fact had a winner, and each match with a winner in fact ended with a draw. It turned out that all teams still had distinct number of points! They corrected the standings and ordered them by the total number of points.
For which nn could the correct order turn out to be the reversed initial order?
(Proposed by Fedir Yudin)
Tournamentcombinatorics
Selecting divisors of square-free integers

Source: Kyiv City MO 2022 Round 1, Problem 9.4

1/23/2022
Let's call integer square-free if it's not divisible by p2p^2 for any prime pp. You are given a square-free integer n>1n>1, which has exactly dd positive divisors. Find the largest number of its divisors that you can choose, such that a2+ab−na^2 + ab - n isn't a square of an integer for any a,ba, b among chosen divisors.
(Proposed by Oleksii Masalitin)
number theory
Bisector and circles

Source: Kyiv City MO 2022 Round 1, Problem 9.3

1/23/2022
Let ALAL be the inner bisector of triangle ABCABC. The circle centered at BB with radius BLBL meets the ray ALAL at points LL and EE, and the circle centered at CC with radius CLCL meets the ray ALAL at points LL and DD. Show that AL2=AE×ADAL^2 = AE\times AD.
(Proposed by Mykola Moroz)
geometry
Fractions everywhere

Source: Kyiv City MO 2022 Round 1, Problem 10.1

1/23/2022
Does there exist a quadratic trinomial ax2+bx+cax^2 + bx + c such that a,b,ca, b, c are odd integers, and 12022\frac{1}{2022} is one of its roots?
algebraquadratic trinomial
Selecting pairs with composite sums

Source: Kyiv City MO 2022 Round 1, Problem 10.2, 11.2

1/23/2022
You are given 2n2n distinct integers. What's the largest integer CC such that you can always form at least CC pairs from them, so that no integer is in more than one pair, and the sum of integers in each pair is a composite number?
(Proposed by Anton Trygub)
number theory
Same multisets of sums

Source: Kyiv City MO 2022 Round 1, Problem 11.1

1/23/2022
The teacher wrote 55 distinct real numbers on the board. After this, Petryk calculated the sums of each pair of these numbers and wrote them on the left part of the board, and Vasyl calculated the sums of each triple of these numbers and wrote them on the left part of the board (each of them wrote 1010 numbers). Could the multisets of numbers written by Petryk and Vasyl be identical?
algebra
Cute board game with tokens

Source: Kyiv City MO 2022 Round 1, Problem 10.5

1/23/2022
There is a black token in the lower-left corner of a board m×nm \times n (m,n≥3m, n \ge 3), and there are white tokens in the lower-right and upper-left corners of this board. Petryk and Vasyl are playing a game, with Petryk playing with a black token and Vasyl with white tokens. Petryk moves first.
In his move, a player can perform the following operation at most two times: choose any his token and move it to any adjacent by side cell, with one restriction: you can't move a token to a cell where at some point was one of the opponents' tokens.
Vasyl wins if at some point of the game white tokens are in the same cell. For which values of m,nm, n can Petryk prevent him from winning?
(Proposed by Arsenii Nikolaiev)
combinatoricsgame
Nobody solved this inequality!

Source: Kyiv City MO 2022 Round 1, Problem 10.4

1/23/2022
For any nonnegative reals x,yx, y show the inequality x2y2+x2y+xy2≤x4y+x+y4x^2y^2 + x^2y + xy^2 \le x^4y + x + y^4.
inequalities
Cutting square into squares of two sizes

Source: Kyiv City MO 2022 Round 1, Problem 11.5

1/23/2022
Find the smallest integer nn for which it's possible to cut a square into 2n2n squares of two sizes: nn squares of one size, and nn squares of another size.
(Proposed by Bogdan Rublov)
combinatoricscutting the paper
Pairwise products form arithmetic progression

Source: Kyiv City MO 2022 Round 1, Problem 11.4

1/23/2022
You are given n≥4n\ge 4 positive real numbers. It turned out that all n(n−1)2\frac{n(n-1)}{2} of their pairwise products form an arithmetic progression in some order. Show that all given numbers are equal.
(Proposed by Anton Trygub)
algebraArithmetic Progressionarithmetic sequence
Square cutting

Source: Kyiv City MO 2024 Round 1, Problem 7.1

1/28/2024
Square ABCDABCD is cut by a line segment EFEF into two rectangles AEFDAEFD and BCFEBCFE. The lengths of the sides of each of these rectangles are positive integers. It is known that the area of the rectangle AEFDAEFD is 3030 and it is larger than the area of the rectangle BCFEBCFE. Find the area of square ABCDABCD.
Proposed by Bogdan Rublov
geometryrectangle
Parity disparity

Source: Kyiv City MO 2024 Round 1, Problem 7.2

1/28/2024
Is it possible to write the numbers from 11 to 100100 in the cells of a of a 10×1010 \times 10 square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original 10×1010 \times 10 square, the numbers in them must have the same parity.
The figure below shows examples of such pairs of cells, in which the numbers must have the same parity.
https://i.ibb.co/b3P8t36/Kyiv-MO-2024-7-2.png
Proposed by Mykhailo Shtandenko
arrangingsquareParitycombinatorics
Intersection of bisector and altitude

Source: Kyiv City MO 2024 Round 1, Problem 9.2/10.2

1/28/2024
Let BL,ADBL, AD be the bisector and the altitude correspondingly of an acute triangle ABC. They intersect at point TT. It turned out that the altitude LKLK of △ALB\triangle ALB is divided in half by the line ADAD. Prove that KT⊥BLKT \perp BL.
Proposed by Mariia Rozhkova
geometryangle bisector
Easy numbergame theory

Source: Kyiv City MO 2024 Round 1, Problem 7.3

1/28/2024
Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from 11 to 20242024 that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by 20242024 loses. Who wins if every player wants to win?
Proposed by Mykhailo Shtandenko
gamenumber theoryDivisibility
Make S nonnegative

Source: Kyiv City MO 2024 Round 1, Problem 7.4

1/28/2024
For real numbers a1,a2,…,a200a_1, a_2, \ldots, a_{200}, we consider the value S=a1a2+a2a3+…+a199a200+a200a1S = a_1a_2 + a_2a_3 + \ldots + a_{199}a_{200} + a_{200}a_1. In one operation, you can change the sign of any number (that is, change aia_i to −ai-a_i), and then calculate the value of SS for the new numbers again. What is the smallest number of operations needed to always be able to make SS nonnegative?
Proposed by Oleksii Masalitin
algebracombinatoricsOperation
Sum and product equal to 8

Source: Kyiv City MO 2024 Round 1, Problem 8.1

1/28/2024
Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to 88.
number theoryDigits