MathDB
NT for beginners

Source: Kyiv City MO 2023 7.3

5/14/2023
Prove that there don't exist positive integer numbers kk and nn which satisfy equation nn+(n+1)n+1+(n+2)n+2=2023kn^n+(n+1)^{n+1}+(n+2)^{n+2} = 2023^k. Proposed by Mykhailo Shtandenko
number theory
Equal signs needed easy

Source: Kyiv City MO 2023 Round 1, Problem 7.2

12/16/2023
You are given n3n \geq 3 distinct real numbers. Prove that one can choose either 33 numbers with positive sum, or 22 numbers with negative sum.
Proposed by Mykhailo Shtandenko
algebra
Close integer

Source: Kyiv City MO 2023 Round 1, Problem 8.1

12/16/2023
Find the integer which is closest to the value of the following expression:
((7+48)2023+(748)2023)2((7+48)2023(748)2023)2((7 + \sqrt{48})^{2023} + (7 - \sqrt{48})^{2023})^2 - ((7 + \sqrt{48})^{2023} - (7 - \sqrt{48})^{2023})^2
algebraexpression
Hedgehogs going WILD

Source: Kyiv City MO 2023 Round 1, Problem 8.3

12/16/2023
A hedgehog is a circle without its boundaries. The diameter of the hedgehog is the diameter of the corresponding circle. We say that the hedgehog sits at the at the point where the center of the circle is located.
We are given a triangle with sides a,b,ca, b, c, with hedgehogs sitting at its vertices. It is known that inside the triangle there is a point from which you can reach any side of the triangle by walking along a straight line without hitting any hedgehog. What is the largest possible sum of the diameters of these hedgehogs?
Proposed by Oleksiy Masalitin
geometry
Easy algebra for 8-graders

Source: Kyiv City MO 2023 8.2

5/14/2023
Positive integers kk and nn are given such that 3kn3 \le k \le n.Prove that among any nn pairwise distinct real numbers one can choose either kk numbers with positive sum, or k1k-1 numbers with negative sum. Proposed by Mykhailo Shtandenko
algebra
Similar numbers

Source: Kyiv City MO 2023 Round 1, Problem 8.4

12/16/2023
Let's call a pair of positive integers a1a2ak\overline{a_1a_2\ldots a_k} and b1b2bk\overline{b_1b_2\ldots b_k} kk-similar if all digits a1,a2,,ak,b1,b2,,bka_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k are distinct, and there exist distinct positive integers m,nm, n, for which the following equality holds:
a1m+a2m++akm=b1n+b2n++bkna_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n
For which largest kk do there exist kk-similar numbers?
Proposed by Oleksiy Masalitin
number theoryalgebraDigits
Convex quadrilaterals don't exist??

Source: Kyiv City MO 2023 Round 1, Problem 8.5

12/16/2023
You are given a square n×nn \times n. The centers of some of some mm of its 1×11\times 1 cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer n3n \geq 3, find the largest value of mm for which it is possible.
Proposed by Oleksiy Masalitin, Fedir Yudin
combinatoricscombinatorial geometrysquare grid
Inequality with number $2023$

Source: Kyiv City MO 2023 Round 1, Problem 10.1

12/16/2023
Find all positive integers nn that satisfy the following inequalities:
46202346n46n-46 \leq \frac{2023}{46-n} \leq 46-n
inequalities
Equal signs needed

Source: Kyiv City MO 2023 9.2

5/14/2023
Non-zero real numbers a,ba, b and cc are given such that ab+bc+ac=0ab+bc+ac=0. Prove that numbers a+b+ca+b+c and 1a+b+1b+c+1a+c\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{a+c} are either both positive or both negative. Proposed by Mykhailo Shtandenko
algebra
Integer convex equilateral 2023-gon

Source: Kyiv City MO 2023 Round 1, Problem 9.5

12/16/2023
Does there exist on the Cartesian plane a convex 20232023-gon with vertices at integer points, such that the lengths of all its sides are equal?
Proposed by Anton Trygub
geometrynumber theory
Arc midpoints in the right triangle

Source: Kyiv City MO 2023 Round 1, Problem 9.3

12/16/2023
You are given a right triangle ABCABC with ACB=90\angle ACB = 90^\circ. Let WA,WBW_A , W_B respectively be the midpoints of the smaller arcs BCBC and ACAC of the circumcircle of ABC\triangle ABC, and NA,NBN_A , N_B respectively be the midpoints of the larger arcs BCBC and ACAC of this circle. Denote by PP and QQ the points of intersection of segment ABAB with the lines NAWBN_AW_B and NBWAN_BW_A, respectively. Prove that AP=BQAP = BQ.
Proposed by Oleksiy Masalitin
geometrycircumcircle
System error

Source: Kyiv City MO 2023 Round 1, Problem 10.2

12/16/2023
For any given real a,b,ca, b, c solve the following system of equations: {ax3+by=cz5,az3+bx=cy5,ay3+bz=cx5.\left\{\begin{array}{l}ax^3+by=cz^5,\\az^3+bx=cy^5,\\ay^3+bz=cx^5.\end{array}\right. Proposed by Oleksiy Masalitin, Bogdan Rublov
algebrasystem of equations
Cutting grid into corners squared

Source: Kyiv City MO 2023 Round 1, Problem 10.4

12/16/2023
Positive integers m,nm, n are such that mnmn is divisible by 99 but not divisible by 2727. Rectangle m×nm \times n is cut into corners, each consisting of three cells. There are four types of such corners, depending on their orientation; you can see them on the figure below. Could it happen that the number of corners of each type was the exact square of some positive integer?
Proposed by Oleksiy Masalitin
https://i.ibb.co/Y8QSHyf/Kyiv-MO-2023-10-4.png
gridcombinatorics
This one was misplaced

Source: Kyiv City MO 2023 Round 1, Problem 11.3

12/16/2023
Let II be the incenter of triangle ABCABC with AB<ACAB < AC. Point XX is chosen on the external bisector of ABC\angle ABC such that IC=IXIC = IX. Let the tangent to the circumscribed circle of BXC\triangle BXC at point XX intersect the line ABAB at point YY. Prove that AC=AYAC = AY.
Proposed by Oleksiy Masalitin
circumcirclegeometry
Pairs with close sums

Source: Kyiv City MO 2023 Round 1, Problem 11.2

12/16/2023
You are given n4n\geq 4 positive real numbers. Consider all n(n1)2\frac{n(n-1)}{2} pairwise sums of these numbers. Show that some two of these sums differ in at most 2n2\sqrt[n-2]{2} times.
Proposed by Anton Trygub
algebra
Grid geometry

Source: Kyiv City MO 2023 Round 1, Problem 10.3

12/16/2023
Consider all pairs of distinct points on the Cartesian plane (A,B)(A, B) with integer coordinates. Among these pairs of points, find all for which there exist two distinct points (X,Y)(X, Y) with integer coordinates, such that the quadrilateral AXBYAXBY is convex and inscribed.
Proposed by Anton Trygub
analytic geometrygridgeometrycyclic quadrilateral
Compare A and B

Source: Kyiv City MO 2023 Round 1, Problem 11.1

12/16/2023
Which number is larger: A=19:120233A = \frac{1}{9} : \sqrt[3]{\frac{1}{2023}}, or B=log202391125B = \log_{2023} 91125?
algebraCompare
AoPS suggested tag 3D geometry for this number theory

Source: Kyiv City MO 2023 Round 1, Problem 11.4

12/16/2023
Find all pairs (m,n)(m, n) of positive integers, for which number 2n13m2^n - 13^m is a cube of a positive integer.
Proposed by Oleksiy Masalitin
number theory
Graph wars

Source: Kyiv City MO 2023 Round 1, Problem 11.5

12/16/2023
In a galaxy far, far away there are 225225 inhabited planets. Between some pairs of inhabited planets there is a bidirectional space connection, and it is possible to reach any planet from any other (possibly with several transfers). The influence of a planet is the number of other planets with which this planet has a direct connection. It is known that if two planets are not connected by a direct space flight, they have different influences. What is the smallest number of connections possible under these conditions?
Proposed by Arsenii Nikolaev, Bogdan Rublov
combinatoricsgraph theory
equal circumradii, diagonals of quadr. (Kyiv City Olympiad 2003 9.4)

Source:

6/27/2021
The diagonals of a convex quadrilateral divide it into four triangles. The radii of the circles circumscribed around these triangles are equal. Can such a property have a quadrilateral other than: a) parallelogram, b) rhombus?
(Sharygin Igor)
geometryparallelogramrhombusinradius
angle chasing, isosceles and angle bisectors (2015 Kyiv City MO 8.3)

Source:

9/2/2020
In the isosceles triangle ABCABC, (AB=BC) (AB = BC) the bisector ADAD was drawn, and in the triangle ABDABD the bisector DEDE was drawn. Find the values of the angles of the triangle ABCABC, if it is known that the bisectors of the angles ABDABD and AEDAED intersect on the line ADAD.
(Fedak Ivan)
geometryangle bisectoranglesisoscelesAngle Chasing
trapezoid construction (Kyiv City Olympiad 2003 8.5)

Source:

6/27/2021
Three segments 22 cm, 55 cm and 1212 cm long are constructed on the plane. Construct a trapezoid with bases of 22 cm and 55 cm, the sum of the sides of which is 1212 cm, and one of the angles is 60o60^o.
(Bogdan Rublev)
geometrytrapezoidconstruction
angles in isosceles wanted, AM=2ВК, 24^o (Kyiv City Olympiad 2004 8.7 )

Source:

6/27/2021
In an isosceles triangle ABCABC with base ACAC, on side BCBC is selected point KK so that BAK=24o\angle BAK = 24^o. On the segment AKAK the point MM is chosen so that ABM=90o\angle ABM = 90^o, AM=2BKAM=2BK. Find the values ​​of all angles of triangle ABCABC.
anglesgeometryisosceles
&lt;DOE=90^o if BD=AD, CB=CE, right triangle (Kyiv City Olympiad 2004 7.3)

Source:

6/27/2021
Given a right triangle ABCABC (A<45o\angle A <45^o,C=90o \angle C = 90^o), on the sides ACAC and ABAB which are selected points D,ED,E respectively, such that BD=ADBD = AD and CB=CECB = CE. Let the segments BDBD and CECE intersect at the point OO. Prove that DOE=90o\angle DOE = 90^o.
geometryright triangleright angle
triangle construction given 3 points (Kyiv City Olympiad 2004 9.7)

Source:

6/27/2021
The board depicts the triangle ABCABC, the altitude AHAH and the angle bisector ALAL which intersectthe inscribed circle in the triangle at the points MM and N,PN, P and QQ, respectively. After that, the figure was erased, leaving only the points H,MH, M and QQ. Restore the triangle ABCABC.
(Bogdan Rublev)
geometryconstruction