2018 Tajikistan Team Selection Test

Part of Tajikistan Team Selection Test

Subcontests

(6)

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TST TAJIKISTAN Day 3 Problem 3

Problem 9. The numbers 1,2,…,〖97〗^2 are written in the cells of a 97×97 board. In the center of each cell, there is a tower with the height equal to the number of that cell. Is it possible to see the top of any tower from the top of any other tower? (one point A can see the other point B, iff there is no other point on the segment AB).

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TST TAJIKISTAN 2018 Day 3

Problem 8. For every non-negative integer n, define an n-variable function K_n (x_1,x_2,…,x_n ) as follows: K_0=1 K_1 (x_1 )=〖x_1〗^2 K_(n+2) (x_1,x_2,…,x_(n+2) )=〖x_(n+2)〗^2.K_(n+1) (x_1,x_2,…,x_(n+1) )+(x_(n+2)+x_(n+1))K_n (x_1,x_2,…,x_n ) Prove that: K_n (x_1,x_2,…,x_n )=K_n (x_n,…〖,x〗_2,x_1 )

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TST TAJIKISTAN 2018 Day 3 Problem 1

Problem 7. On the board, Sabir writes 10 consecutive numbers. For each number, Salim writes the sum of its digits on his paper, and Sabrina writes the number of its divisors on her paper. Is it possible for Sabrina’s 10 numbers to be exactly the same as Salim’s 10 numbers in some order? (the repetitions of the numbers should also be the same)

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TST TAJIKISTAN Day 2 Problem 3

Problem 6. Let H be orthocenter of an acute-angled triangle ABC. Points E,F are on the segments AB,AC respectively, such that BE=BH,CF=CH. The lines EH,FH meet BC in X,Y respectively. Draw the perpendicular HZ from H to EF. Prove that the circumcircle of triangle XYZ is tangent to the circle with diameter BC.

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TST Tajikistan 2018 Day 1 Problem 2

Problem 2. Prove that for every n≥3, there exists a convex polygon with n sides, such that one can divide it into n-2 triangles that are all similar, but pairwise non-congruent.
[color=#00f]Moved to HSO. ~ oVlad

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TST 2018 TAJIKISTAN Day 1 Problem 1

Problem 1. Let ω be the incircle of triangle ABC which is tangent to BC,CA,AB at points D,E,F, respectively. The altitudes of triangle DEF with respect to E,F meet AB,AC at points X,Y, respectively. Prove that the second intersection of the circumcircles of triangles AEX,AFY lies on the circle ω.