MathDB

2002 Moldova Team Selection Test

Part of Moldova Team Selection Test

Subcontests

(4)
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(Im)possible algebra

The sequence Pn (x), n ∈ N of polynomials is defined as follows: P0 (x) = x, P1 (x) = 4x³ + 3x Pn+1 (x) = (4x² + 2)Pn (x) − Pn−1 (x), for all n ≥ 1 For every positive integer m, we consider the set A(m) = { Pn (m) | n ∈ N }. Show that the sets A(m) and A(m+4) have no common elements.

ΣP_iA^{2p}_i is an integer, where P_i projections of points A_i of an arc on OA

Let CC be the circle with center O(0,0)O(0,0) and radius 11, and A(1,0),B(0,1)A(1,0), B(0,1) be points on the circle. Distinct points A1,A2,....,An−1A_1,A_2, ....,A_{n-1} on CC divide the smaller arc ABAB into nn equal parts (n≥2n \ge 2). If PiP_i is the orthogonal projection of AiA_i on OAOA (i=1,...,n−1i =1, ... ,n-1), find all values of nn such that P1A12p+P2A22p+...+Pn−1An−12pP_1A^{2p}_1 +P_2A^{2p}_2 +...+P_{n-1}A^{2p}_{n-1} is an integer for every positive integer pp.

real parts of roots of P(x) < n-1/2, (x-n+1) does not divide P(x), P(n) is prime

Let P(x)P(x) be a polynomial with integer coefficients for which there exists a positive integer n such that the real parts of all roots of P(x)P(x) are less than n−12n- \frac{1}{2} , polynomial x−n+1x-n+1 does not divide P(x)P(x), and P(n)P(n) is a prime number. Prove that the polynomial P(x)P(x) is irreducible (over Z[x]Z[x]).
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There exists a partition of the set

Prove that there exists a partition of the set A={13,23,…,20003}A = \{1^3, 2^3, \ldots , 2000^3\} into 1919 nonempty subsets such that the sum of elements of each subset is divisible by 200122001^2.

Moldova TST

Let AA be a set containing 4k4k consecutive positive integers, where k≥1k \geq 1 is an integer. Find the smallest kk for which the set A can be partitioned into two subsets having the same number of elements, the same sum of elements, the same sum of the squares of elements, and the same sum of the cubes of elements.

Moldova Team Selection Test

Let S={a1,…,an}S= \{ a_1, \ldots, a_n\} be a set of n≥1n\geq 1 positive real numbers. For each nonempty subset of SS the sum of its elements is written down. Show that all written numbers can be divided into nn classes such that in each class the ratio of the greatest number to the smallest number is not greater than 22.
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Three circles externally tangent - M, N, and L are collinear

The circles W1,W2,W3W_1, W_2, W_3 in the plane are pairwise externally tangent to each other. Let P1P_1 be the point of tangency between circles W1W_1 and W3W_3, and let P2P_2 be the point of tangency between circles W2W_2 and W3W_3. AA and BB, both different from P1P_1 and P2P_2, are points on W3W_3 such that ABAB is a diameter of W3W_3. Line AP1AP_1 intersects W1W_1 again at XX, line BP2BP_2 intersects W2W_2 again at YY, and lines AP2AP_2 and BP1BP_1 intersect at ZZ. Prove that X,YX, Y, and ZZ are collinear.

midpoints of 2 arcs, point on arc,2 incenters, similar triangles wanted

A triangle ABCABC is inscribed in a circle GG. Points MM and NN are the midpoints of the arcs BCBC and ACAC respectively, and DD is an arbitrary point on the arc ABAB (not containing CC). Points I1I_1 and I2I_2 are the incenters of the triangles ADCADC and BDCBDC, respectively. If the circumcircle of triangle DI1I2DI_1I_2 meets GG again at PP, prove that triangles PNI1PNI_1 and PMI2PMI_2 are similar.

locus of points M so BM x CM / MA_1 is minimal, A_1 = AM \cap (ABC)

A triangle ABCABC is inscribed in a circle GG. For any point MM inside the triangle, A1A_1 denotes the intersection of the ray AMAM with GG. Find the locus of point MM for which BM⋅CMMA1\frac{BM\cdot CM}{MA_1} is minimal, and find this minimum value.
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Inequality on n numbers - sum is geq 1/n(alpha+beta)

Positive numbers \alpha ,\beta , x_1, x_2,\ldots, x_n (n≥1n \geq 1) satisfy x1+x2+⋯+xn=1x_1+x_2+\cdots+x_n = 1. Prove that ∑i=1nxi3αxi+βxi+1≥1n(α+β).\sum_{i=1}^{n} \frac{x_i^3}{\alpha x_i+\beta x_{i+1}} \geq \frac{1}{n(\alpha+\beta)}. Note. xn+1=x1x_{n+1}=x_1.

The triangular numbers ans the sequences

Consider the triangular numbers Tn=n(n+1)2,n∈NT_n = \frac{n(n+1)}{2} , n \in \mathbb N. (a) If ana_n is the last digit of TnT_n, show that the sequence (an)(a_n) is periodic and find its basic period.
(b) If sns_n is the sum of the first nn terms of the sequence (Tn)(T_n), prove that for every n≥3n \geq 3 there is at least one perfect square between sn−1ands_{n-1} and s_n$.

Polynomial with lots of powers of 2

Prove that for every positive integer n, there exists a polynomial p(x) with integer coefficients such that p(1), p(2),..., p(n-1), p(n) are distinct powers of 2.