MathDB

P3

Part of 2022/2023 Tournament of Towns

Problems(7)

Two circles and a chord

Source: 44th International Tournament of Towns, Senior A-Level P3, Fall 2022

2/16/2023
Consider two concentric circles Ω\Omega and ω\omega. Chord ADAD of the circle Ω\Omega is tangent to ω\omega. Inside the minor disk segment ADAD of Ω\Omega, an arbitrary point PP{} is selected. The tangent lines drawn from the point PP{} to the circle ω\omega intersect the major arc ADAD of the circle Ω\Omega at points BB{} and CC{}. The line segments BDBD and ACAC intersect at the point QQ{}. Prove that the line segment PQPQ passes through the midpoint of line segment ADAD.
Note. A circle together with its interior is called a disk, and a chord XYXY of the circle divides the disk into disk segments, a minor disk segment XYXY (the one of smaller area) and a major disk segment XYXY.
geometryTournament of Towns
Would the Baron ever lie?

Source: 44th International Tournament of Towns, Junior A-Level P3, Fall 2022

2/16/2023
Baron Munchausen claims that he has drawn a polygon and chosen a point inside the polygon in such a way that any line passing through the chosen point divides the polygon into three polygons. Could the Baron’s claim be correct?
combinatoricsgeometryTournament of TownsBaron Munchausen
Colorful segments

Source: 44th International Tournament of Towns, Senior O-Level P3, Fall 2022

2/16/2023
There are 2022 marked points on a straight line so that every two adjacent points are the same distance apart. Half of all the points are coloured red and the other half are coloured blue. Can the sum of the lengths of all the segments with a red left endpoint and a blue right endpoint be equal to the sum of the lengths of all the segments with a blue left endpoint and a red right endpoint?
combinatoricsequal segmentsTournament of Towns
Easy pentagon geo

Source: 44th International Tournament of Towns, Junior O-Level P3, Fall 2022

2/16/2023
A pentagon ABCDEABCDE is circumscribed about a circle. The angles at the vertices AA{}, CC{} and EE{} of the pentagon are equal to 100∘100^\circ. Find the measure of the angle ∠ACE\angle ACE.
Tournament of TownsgeometrypentagonAngle Chasing
Roots of polynomial

Source: 44th International Tournament of Towns, Senior A-Level P3, Spring 2023

4/4/2023
P(x)P(x) is polynomial with degree n>5n>5 and integer coefficients have nn different integer roots. Prove that P(x)+3P(x)+3 have nn different real roots.
algebrapolynomialnumber theory
Pedestrian integers

Source: 44th International Tournament of Towns, Junior A-Level P3, Spring 2023

1/9/2024
Let us call a positive integer pedestrian if all its decimal digits are equal to 0 or 1. Suppose that the product of some two pedestrian integers also is pedestrian. Is it necessary in this case that the sum of digits of the product equals the product of the sums of digits of the factors?
Viktor Kleptsyn, Konstantin Knop
number theorydecimal representation
Two lines are perpendicular

Source: 44th International Tournament of Towns, Senior O-Level P3, Spring 2023

1/9/2024
Let II{} be the incenter of triangle ABC.ABC{}. Let NN{} be the foot of the bisector of angle B.B{}. The tangent line to the circumcircle of triangle AINAIN at AA{} and the tangent line to the circumcircle of triangle CINCIN{} at CC{} intersect at D.D{}. Prove that lines ACAC{} and DIDI are perpendicular.
Mikhail Evdokimov
geometry