MathDB

2023 Czech-Polish-Slovak Junior Match

Part of Czech-Polish-Slovak Junior Match

Subcontests

(6)
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S<- S+1/S

Bartek patiently performs operations on fractions. In each move, he adds its inverse to the current result, obtaining a new result. Bartek starts with the number 11: after the first move, he receives the result 2, after the second move, the result is 52\frac{5}{2}, after the third move 2910\frac{29}{10}, etc. After 300300 moves, Bartek receives the result xx. Determine the largest integer not greater than xx.

x,y,z -> ..., xy + yz + zx = 1000, digits related

Mazo performs the following operation on triplets of non-negative integers: If at least one of them is positive, it chooses one positive number, decreases it by one, and replaces the digits in the units place with the other two numbers. It starts with the triple xx, yy, zz. Find a triple of positive integers xx, yy, zz such that xy+yz+zx=1000xy + yz + zx = 1000 (*) and the number of operations that Mazo can subsequently perform with the triple x,y,zx, y, z is
(a) maximal (i.e. there is no triple of positive integers satisfying (*) that would allow him to do more operations);
(b) minimal (i.e. every triple of positive integers satisfying (*) allows him to perform at least so many operations).
4
2

coloring 1x1 in a nxn array

Each field of the n×nn \times n array has been colored either red or blue, with the following conditions met: ∙\bullet if a row and a column contain the same number of red fields, the field at their intersection is red; ∙\bullet if a row and a column contain different numbers of red cells, the field at their intersection is blue. Prove that the total number of blue cells is even.

2 tangents of different circles intersect on a line

In triangle ABCABC, the points MM and NN are the midpoints of the sides ABAB and ACAC, respectively. The bisectors of interior angles ∠ABC\angle ABC and ∠BCA\angle BCA intersect the line MNMN at points PP and QQ, respectively. Let pp be the tangent to the circumscribed circle of the triangle AMPAMP passing through point PP, and qq be the tangent to the circumscribed circle of the triangle ANQANQ passing through point QQ. Prove that the lines pp and qq intersect on line BCBC.
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Q symmetric of P wrt BH wanted, orthocenter related

Given is an acute triangle ABCABC. Point PP lies inside this triangle and lies on the bisector of angle ∠BAC\angle BAC. Suppose that the point of intersection of the altitudes HH of triangle ABPABP lies inside triangle ABCABC. Let QQ be the intersection of the line APAP and the line perpendicular to ACAC passing through HH. Prove that QQ is the point symmetrical to PP wrt the line BHBH.

n persons a a party, everyone dislikes only 1

nn people met at the party, with n≥2n \ge 2. Each person dislikes exactly one other person present at the party (but not necessarily reciprocal, i.e. it may happen that AA dislikes BB even though BB does not dislike AA) and likes all others. Prove that guests can be seated at three tables in such a way that each guest likes all the people at his table.
2
2

number of positive divisors of n is second largest divisor of n.

For a positive integer nn, let d(n)d(n) denote the number of positive divisors of nn. Determine all positive integers nn for which d(n)d(n) is the second largest divisor of nn.

permutations of numbers 1-2023

The numbers 1,2,...,20231, 2,..., 2023 are written on the board in this order. We can repeatedly perform the following operation with them: We select any odd number of consecutively written numbers and write these numbers in reverse order. How many different orders of these 20232023 numbers can we get?
Example: If we start with only the numbers 1,2,3,4,5,61, 2, 3, 4, 5, 6, we can perform the following steps 1,2,3,4,5,6→3,2,1,4,5,6→3,6,5,4,1,2→3,6,1,4,5,2→...1, 2, 3, 4, 5, 6 \to 3, 2, 1,4, 5, 6 \to 3, 6, 5, 4, 1, 2 \to 3, 6, 1, 4, 5, 2 \to ...
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