MathDB

2005 Italy TST

Part of Italy TST

Subcontests

(3)
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2

Multiplicative function for coprime m,n

The function ψ:N→N\psi : \mathbb{N}\rightarrow\mathbb{N} is defined by ψ(n)=∑k=1ngcd⁡(k,n)\psi (n)=\sum_{k=1}^n\gcd (k,n).
(a)(a) Prove that ψ(mn)=ψ(m)ψ(n)\psi (mn)=\psi (m)\psi (n) for every two coprime m,n∈Nm,n \in \mathbb{N}. (b)(b) Prove that for each a∈Na\in\mathbb{N} the equation ψ(x)=ax\psi (x)=ax has a solution.

Alberto and Barbara play a game

Let NN be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing 11, and thereafter if a player has written nn on a certain move, his adversary is allowed to write n+1n+1 or 2n2n as long as he/she does not obtain a number greater than NN. The player who writes NN wins. (a)(a) Determine which player has a winning strategy for N=2005N=2005. (b)(b) Determine which player has a winning strategy for N=2004N=2004. (c)(c) Find for how many integers N≤2005N\le 2005 Barbara has a winning strategy.
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2

Distances between centroid and the sides of a triangle

(a)(a) Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality. (b)(b) Determine the points in a triangle that minimize the sum of the distances to the sides.

F,C,H are collinear

The circle Γ\Gamma and the line ℓ\ell have no common points. Let ABAB be the diameter of Γ\Gamma perpendicular to ℓ\ell, with BB closer to ℓ\ell than AA. An arbitrary point C≠AC\not= A, BB is chosen on Γ\Gamma. The line ACAC intersects ℓ\ell at DD. The line DEDE is tangent to Γ\Gamma at EE, with BB and EE on the same side of ACAC. Let BEBE intersect ℓ\ell at FF, and let AFAF intersect Γ\Gamma at G≠AG\not= A. Let HH be the reflection of GG in ABAB. Show that F,CF,C, and HH are collinear.
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Two fixed points of a function

Suppose that f:{1,2,…,1600}→{1,2,…,1600}f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\} satisfies f(1)=1f(1)=1 and f^{2005}(x)=x \text{for}\ x=1,2,\ldots ,1600. (a)(a) Prove that ff has a fixed point different from 11. (b)(b) Find all n>1600n>1600 such that any f:{1,…,n}→{1,…,n}f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\} satisfying the above condition has at least two fixed points.

Least number of conspiring students

A stage course is attended by n≥4n \ge 4 students. The day before the final exam, each group of three students conspire against another student to throw him/her out of the exam. Prove that there is a student against whom there are at least (n−1)(n−2)3\sqrt[3]{(n-1)(n- 2)} conspirators.