MathDB

2002 Vietnam Team Selection Test

Part of Vietnam Team Selection Test

Subcontests

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Play the Vietnam TST 2002 Game!

On a blackboard a positive integer n0n_0 is written. Two players, AA and BB are playing a game, which respects the following rules: −- acting alternatively per turn, each player deletes the number written on the blackboard nkn_k and writes instead one number denoted with nk+1n_{k+1} from the set \left\{n_k-1, \dsp \left\lfloor\frac {n_k}3\right\rfloor\right\}; −- player AA starts first deleting n0n_0 and replacing it with n_1\in\left\{n_0-1, \dsp \left\lfloor\frac {n_0}3\right\rfloor\right\}; −- the game ends when the number on the table is 0 - and the player who wrote it is the winner. Find which player has a winning strategy in each of the following cases: a) n0=120n_0=120; b) n_0=\dsp \frac {3^{2002}-1}2; c) n_0=\dsp \frac{3^{2002}+1}2.

Q(x)=(x^2+6x+10) * P^2(x) - 1

Find all polynomials P(x)P(x) with integer coefficients such that the polynomial Q(x)=(x2+6x+10)⋅P2(x)−1 Q(x)=(x^2+6x+10) \cdot P^2(x)-1 is the square of a polynomial with integer coefficients.
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interior angle bisector of BC intersects the trisectors

Find all triangles ABCABC for which ∠ACB\angle ACB is acute and the interior angle bisector of BCBC intersects the trisectors (AX,(AY(AX, (AY of the angle ∠BAC\angle BAC in the points N,PN,P respectively, such that AB=NP=2DMAB=NP=2DM, where DD is the foot of the altitude from AA on BCBC and MM is the midpoint of the side BCBC.

Half of the elements in the array are colored in red

Let n≥2n\geq 2 be an integer and consider an array composed of nn rows and 2n2n columns. Half of the elements in the array are colored in red. Prove that for each integer kk, 1kk rows such that the array of size k×2nk\times 2n formed with these kk rows has at least k!(n−2k+2)(n−k+1)(n−k+2)⋯(n−1) \frac { k! (n-2k+2) } {(n-k+1)(n-k+2)\cdots (n-1)} columns which contain only red cells.