MathDB

1996 Polish MO Finals

Part of Polish MO Finals

Subcontests

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two n-tuples and inequality

ai,xia_i, x_i are positive reals such that a1+a2+...+an=x1+x2+...+xn=1a_1 + a_2 + ... + a_n = x_1 + x_2 + ... + x_n = 1. Show that 2∑i<jxixj≤n−2n−1+∑aixi21−ai 2 \sum_{i<j} x_ix_j \leq \frac{n-2}{n-1} + \sum \frac{a_ix_i ^2}{1-a_i} When do we have equality?

Find all natural numbers n such that p_n &gt; \frac{1}{3}

From the set of all permutations ff of {1,2,...,n}\{1, 2, ... , n\} that satisfy the condition: f(i)≥i−1f(i) \geq i-1 i=1,...,ni=1,...,n one is chosen uniformly at random. Let pnp_n be the probability that the chosen permutation ff satisfies f(i)≤i+1f(i) \leq i+1 i=1,...,ni=1,...,n Find all natural numbers nn such that pn>13p_n > \frac{1}{3}.
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Area apef/area abp does not depend on p

Let PP be a point inside a triangle ABCABC such that ∠PBC=∠PCA<∠PAB\angle PBC = \angle PCA < \angle PAB. The line PBPB meets the circumcircle of triangle ABCABC at a point EE (apart from BB). The line CECE meets the circumcircle of triangle APEAPE at a point FF (apart from EE). Show that the ratio ∣APEF∣∣ABP∣\frac{\left|APEF\right|}{\left|ABP\right|} does not depend on the point PP, where the notation ∣P1P2...Pn∣\left|P_1P_2...P_n\right| stands for the area of an arbitrary polygon P1P2...PnP_1P_2...P_n.

Find all n such that x_n = 111111

Let p(k)p(k) be the smallest prime not dividing kk. Put q(k)=1q(k) = 1 if p(k)=2p(k) = 2, or the product of all primes <p(k)< p(k) if p(k)>2p(k) > 2. Define the sequence x0,x1,x2,...x_0, x_1, x_2, ... by x0=1x_0 = 1, xn+1=xnp(xn)q(xn)x_{n+1} = \frac{x_np(x_n)}{q(x_n)}. Find all nn such that xn=111111x_n = 111111
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