MathDB
Concatenating powers

Source: Kyiv City MO 2024 Round 2, Problem 10.1

2/4/2024
For some positive integer nn, Katya wrote on the board next to each other numbers 2n2^n and 14n14^n (in this order), thus forming a new number AA. Can the number A−1A - 1 be prime?
Proposed by Oleksii Masalitin
number theoryDigits
Combinatorics from IMO 3rd absolute place

Source: Kyiv City MO 2024 Round 2, Problem 10.4

2/4/2024
There are n≥1n \geq 1 notebooks, numbered from 11 to nn, stacked in a pile. Zahar repeats the following operation: he randomly chooses a notebook whose number kk does not correspond to its location in this stack, counting from top to bottom, and returns it to the kkth position, counting from the top, without changing the location of the other notebooks. If there is no such notebook, he stops.
Is it guaranteed that Zahar will arrange all the notebooks in ascending order of numbers in a finite number of operations?
Proposed by Zahar Naumets
combinatoricspermutations
Comeback of inequalities (I'm sorry)

Source: Kyiv City MO 2024 Round 2, Problem 10.2

2/4/2024
For any positive real numbers a,b,c,da, b, c, d, prove the following inequality: (a2+b2)(b2+c2)(c2+d2)(d2+a2)≥64abcd∣(a−b)(b−c)(c−d)(d−a)∣(a^2+b^2)(b^2+c^2)(c^2+d^2)(d^2+a^2) \geq 64abcd|(a-b)(b-c)(c-d)(d-a)| Proposed by Anton Trygub
inequalitiesalgebra
Diameter configuration

Source: Kyiv City MO 2024 Round 2, Problem 10.3

2/4/2024
Let AHA,BHB,CHCAH_A, BH_B, CH_C be the altitudes of the triangle ABCABC. Points A1A_1 and C1C_1 are the projections of the point HBH_B onto the sides ABAB and BCBC, respectively. B1B_1 is the projection of BB onto HAHCH_AH_C. Prove that the diameter of the circumscribed circle of △A1B1C1\triangle A_1B_1C_1 is equal to BHBBH_B.
Proposed by Anton Trygub
circumcirclegeometry
System of equations: hard

Source: Kyiv City MO 2024 Round 2, Problem 11.1

2/4/2024
Solve the following system of equations in real numbers: {x2=y2+z2,x2024=y2024+z2024,x2025=y2025+z2025.\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2024}=y^{2024}+z^{2024},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right. Proposed by Mykhailo Shtandenko, Anton Trygub, Bogdan Rublov
algebrasystem of equations
Interval fasting

Source: Kyiv City MO 2024 Round 2, Problem 11.3

2/4/2024
For a given positive integer nn, we consider the set MM of all intervals of the form [l,r][l, r], where the integers ll and rr satisfy the condition 0≤l<r≤n0 \leq l < r \leq n. What largest number of elements of MM can be chosen so that each chosen interval completely contains at most one other selected interval?
Proposed by Anton Trygub
combinatorics
Fantastic number theory

Source: Kyiv City MO 2024 Round 2, Problem 11.2

2/4/2024
Mykhailo wants to arrange all positive integers from 11 to 20242024 in a circle so that each number is used exactly once and for any three consecutive numbers a,b,ca, b, c the number a+ca + c is divisible by b+1b + 1. Can he do it?
Proposed by Fedir Yudin
number theoryDivisibility
Game of stones

Source: Kyiv City MO 2022 Round 2, Problem 7.4

1/30/2022
Fedir and Mykhailo have three piles of stones: the first contains 100100 stones, the second 101101, the third 102102. They are playing a game, going in turns, Fedir makes the first move. In one move player can select any two piles of stones, let's say they have aa and bb stones left correspondently, and remove gcd(a,b)gcd(a, b) stones from each of them. The player after whose move some pile becomes empty for the first time wins. Who has a winning strategy?
As a reminder, gcd(a,b)gcd(a, b) denotes the greatest common divisor of a,ba, b.
(Proposed by Oleksii Masalitin)
combinatoricsgameGCD
Problem rejected from IMO, EGMO, USAMO. But I still like it!

Source: Kyiv City MO 2024 Round 2, Problem 11.4

2/4/2024
Let ABCABC be an acute triangle with circumcenter OO and orthocenter HH. Rays AOAO, COCO intersect sides BC,BABC, BA in points A1,C1A_1, C_1 respectively, KK is the projection of OO onto the segment A1C1A_1C_1, MM is the midpoint of ACAC. Prove that ∠HMA=∠BKC1\angle HMA = \angle BKC_1.
Proposed by Anton Trygub
geometry
Easy NT on LCMs

Source: Kyiv City MO 2022 Round 2, Problem 7.1

1/30/2022
a) Do there exist positive integers aa and dd such that [a,a+d]=[a,a+2d][a, a+d] = [a, a+2d]?
b) Do there exist positive integers aa and dd such that [a,a+d]=[a,a+4d][a, a+d] = [a, a+4d]?
Here [a,b][a, b] denotes the least common multiple of integers a,ba, b.
number theoryLCM
Hedgehog recoloring points on circle

Source: Kyiv City MO 2022 Round 2, Problem 9.2, 10.2

1/30/2022
20222022 points are arranged in a circle, one of which is colored in black, and others in white. In one operation, The Hedgehog can do one of the following actions:
1) Choose two adjacent points of the same color and flip the color of both of them (white becomes black, black becomes white)
2) Choose two points of opposite colors with exactly one point in between them, and flip the color of both of them
Is it possible to achieve a configuration where each point has a color opposite to its initial color with these operations?
(Proposed by Oleksii Masalitin)
combinatoricsColoring
LCM genius problem from our favorite author

Source: Kyiv City MO 2022 Round 2, Problem 8.1

1/30/2022
Find all triples (a,b,c)(a, b, c) of positive integers for which a+[a,b]=b+[b,c]=c+[c,a]a + [a, b] = b + [b, c] = c + [c, a].
Here [a,b][a, b] denotes the least common multiple of integers a,ba, b.
(Proposed by Mykhailo Shtandenko)
number theoryleast common multiple
Trains go wild

Source: Kyiv City MO 2022 Round 2, Problem 7.2

1/30/2022
There is a central train station in point OO, which is connected to other train stations A1,A2,…,A8A_1, A_2, \ldots, A_8 with tracks. There is also a track between stations AiA_i and Ai+1A_{i+1} for each ii from 11 to 88 (here A9=A1A_9 = A_1). The length of each track AiAi+1A_iA_{i+1} is equal to 11, and the length of each track OAiOA_i is equal to 22, for each ii from 11 to 88.
There are also 88 trains B1,B2,…,B8B_1, B_2, \ldots, B_8, with speeds 1,2,…,81, 2, \ldots, 8 correspondently. Trains can move only by the tracks above, in both directions. No time is wasted on changing directions. If two or more trains meet at some point, they will move together from now on, with the speed equal to that of the fastest of them.
Is it possible to arrange trains into stations A1,A2,…,A8A_1, A_2, \ldots, A_8 (each station has to contain one train initially), and to organize their movement in such a way, that all trains arrive at OO in time t<12t < \frac{1}{2}?
(Proposed by Bogdan Rublov)
combinatorics
Bogdan and Monica playing weird games...

Source: Kyiv City MO 2022 Round 2, Problem 8.2, 11.1

1/30/2022
Monica and Bogdan are playing a game, depending on given integers n,kn, k. First, Monica writes some kk positive numbers. Bogdan wins, if he is able to find nn points on the plane with the following property: for any number mm written by Monica, there are some two points chosen by Bogdan with distance exactly mm between them. Otherwise, Monica wins.
Determine who has a winning strategy depending on n,kn, k.
(Proposed by Fedir Yudin)
combinatoricsgamegeometry
Very cute algebra

Source: Kyiv City MO 2022 Round 2, Problem 8.3, 9.3

1/30/2022
Nonzero real numbers x1,x2,…,xnx_1, x_2, \ldots, x_n satisfy the following condition:
x1−1x2=x2−1x3=…=xn−1−1xn=xn−1x1x_1 - \frac{1}{x_2} = x_2 - \frac{1}{x_3} = \ldots = x_{n-1} - \frac{1}{x_n} = x_n - \frac{1}{x_1}
Determine all nn for which x1,x2,…,xnx_1, x_2, \ldots, x_n have to be equal.
(Proposed by Oleksii Masalitin, Anton Trygub)
algebra
GCD genius problem from our favorite author

Source: Kyiv City MO 2022 Round 2, Problem 9.1

1/30/2022
Find all triples (a,b,c)(a, b, c) of positive integers for which a+(a,b)=b+(b,c)=c+(c,a)a + (a, b) = b + (b, c) = c + (c, a).
Here (a,b)(a, b) denotes the greatest common divisor of integers a,ba, b.
(Proposed by Mykhailo Shtandenko)
number theorygreatest common divisor
Easy algebra

Source: Kyiv City MO 2022 Round 2, Problem 10.1

1/30/2022
Positive reals x,y,zx, y, z satisfy xy+1x+1=yz+1y+1=zx+1z+1\frac{xy+1}{x+1} = \frac{yz+1}{y+1} = \frac{zx+1}{z+1}
Do they all have to be equal?
(Proposed by Oleksii Masalitin)
algebrainequalitiesUkraine
Equal ratio implies tangency

Source: Kyiv City MO 2022 Round 2, Problem 10.3

1/30/2022
Let AHA,BHB,CHCAH_A, BH_B, CH_C be the altitudes of triangle ABCABC. Prove that if HBCAC=HCAAB\frac{H_BC}{AC} = \frac{H_CA}{AB}, then the line symmetric to BCBC with respect to line HBHCH_BH_C is tangent to the circumscribed circle of triangle HBHCAH_BH_CA.
(Proposed by Mykhailo Bondarenko)
geometryratio
NT: Smallest degree of polynomial

Source: Kyiv City MO 2022 Round 2, Problem 10.4

1/30/2022
Prime p>2p>2 and a polynomial QQ with integer coefficients are such that there are no integers 1≤i<j≤p−11 \le i < j \le p-1 for which (Q(j)−Q(i))(jQ(j)−iQ(i))(Q(j)-Q(i))(jQ(j)-iQ(i)) is divisible by pp. What is the smallest possible degree of QQ?
(Proposed by Anton Trygub)
number theorypolynomialalgebra
Prefix sums of permutation

Source: Kyiv City MO 2022 Round 2, Problem 11.3

1/30/2022
Find the largest kk for which there exists a permutation (a1,a2,…,a2022)(a_1, a_2, \ldots, a_{2022}) of integers from 11 to 20222022 such that for at least kk distinct ii with 1≤i≤20221 \le i \le 2022 the number a1+a2+…+ai1+2+…+i\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i} is an integer larger than 11.
(Proposed by Oleksii Masalitin)
number theorypermutationsalgebra
Crazy geometry on incircles

Source: Kyiv City MO 2022 Round 2, Problem 11.4

1/30/2022
Let ABCDABCD be the cyclic quadrilateral. Suppose that there exists some line ll parallel to BDBD which is tangent to the inscribed circles of triangles ABC,CDAABC, CDA. Show that ll passes through the incenter of BCDBCD or through the incenter of DABDAB.
(Proposed by Fedir Yudin)
geometry
Computer making crazy operations with polynomials

Source: Kyiv City MO 2022 Round 2, Problem 11.2

1/30/2022
Initially memory of computer contained a single polynomial x2−1x^2-1. Every minute computer chooses any polynomial f(x)f(x) from its memory and writes f(x2−1)f(x^2-1) and f(x)2−1f(x)^2-1 to it, or chooses any two distinct polynomials g(x),h(x)g(x), h(x) from its memory and writes polynomial g(x)+h(x)2\frac{g(x) + h(x)}{2} to it (no polynomial is ever erased from its memory). Can it happen that after some time, memory of computer contains P(x)=11024(x2−1)2048−1P(x) = \frac{1}{1024}(x^2-1)^{2048} - 1?
(Proposed by Arsenii Nikolaiev)
algebrapolynomial
No right triangle on a grid

Source: Kyiv City MO 2023 7.4

5/14/2023
For n≥2n \ge 2 consider n×nn \times n board and mark all n2n^2 centres of all unit squares. What is the maximal possible number of marked points that we can take such that there don't exist three taken points which form right triangle? Proposed by Mykhailo Shtandenko
combinatorics
Rectangle cut into 6 squares

Source: Kyiv City MO 2023 Round 1, Problem 7.1

12/16/2023
The rectangle is cut into 6 squares, as shown on the figure below. The gray square in the middle has a side equal to 1. What is the area of the rectangle?
https://i.ibb.co/gg1tBTN/Kyiv-MO-2023-7-1.png
geometryrectangle
Best algebra in the history of MO

Source: Kyiv City MO 2023 Round 1, Problem 9.1

12/16/2023
Find the integer which is closest to the value of the following expression:
((3+1)2023−(13−1)2023)⋅((3+2)2023−(13−2)2023)⋅…⋅((3+8)2023−(13−8)2023)\left((3 + \sqrt{1})^{2023} - \left(\frac{1}{3 - \sqrt{1}}\right)^{2023} \right) \cdot \left((3 + \sqrt{2})^{2023} - \left(\frac{1}{3 - \sqrt{2}}\right)^{2023} \right) \cdot \ldots \cdot \left((3 + \sqrt{8})^{2023} - \left(\frac{1}{3 - \sqrt{8}}\right)^{2023} \right)
algebra