o(Qi1 + Qi2 + ... + Qin) ≥ o(Qi1).
xn+1 = xn(xn + 1/n).
Prove that there exists exactly one value of x1 which gives 0 < xn < xn+1 < 1 for all n.
| A1. A circle has center on the side AB of the cyclic quadrilateral ABCD. The other three sides are tangent to the circle. Prove that AD + BC = AB. |
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| A2. Let n and k be relatively prime positive integers with k < n. Each number in the set M = {1, 2, 3, ... , n-1} is colored either blue or white. For each i in M, both i and n-i have the same color. For each i in M not equal to k, both i and |i-k| have the same color. Prove that all numbers in M must have the same color. |
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A3. For any polynomial P(x) = a0 + a1x + ... + akxk with integer coefficients, the number of odd coefficients is denoted by o(P). For i = 0, 1, 2, ... let Qi(x) = (1 + x)i. Prove that if i1, i2, ... , in are integers satisfying 0 ≤ i1 < i2 < ... < in, then:
o(Qi1 + Qi2 + ... + Qin) ≥ o(Qi1). |
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| B1. Given a set M of 1985 distinct positive integers, none of which has a prime divisor greater than 23, prove that M contains a subset of 4 elements whose product is the 4th power of an integer. |
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| B2. A circle center O passes through the vertices A and C of the triangle ABC and intersects the segments AB and BC again at distinct points K and N respectively. The circumcircles of ABC and KBN intersect at exactly two distinct points B and M. Prove that ∠OMB is a right angle. |
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B3. For every real number x1, construct the sequence x1, x2, ... by setting:
xn+1 = xn(xn + 1/n). Prove that there exists exactly one value of x1 which gives 0 < xn < xn+1 < 1 for all n. |
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