MathDB

1985 IMO Longlists

Part of IMO Longlists

Subcontests

(74)
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Constant sequence (Bothered me a lot to type it)

Let f1=(a1,a2,…,an),n>2f_1 = (a_1, a_2, \dots , a_n) , n > 2, be a sequence of integers. From f1f_1 one constructs a sequence fkf_k of sequences as follows: if fk=(c1,c2,…,cn)f_k = (c_1, c_2, \dots, cn), then fk+1=(ci1,ci2,ci3+1,ci4+1,...,cin+1)f_{k+1} = (c_{i_{1}}, c_{i_{2}}, c_{i_{3}} + 1, c_{i_{4}} + 1, . . . , c_{i_{n}} + 1), where (ci1,ci2,…,cin)(c_{i_{1}}, c_{i_{2}},\dots , c_{i_{n}}) is a permutation of (c1,c2,…,cn)(c_1, c_2, \dots, c_n). Give a necessary and sufficient condition for f1f_1 under which it is possible for fkf_k to be a constant sequence (b1,b2,…,bn),b1=b2=⋯=bn(b_1, b_2,\dots , b_n), b_1 = b_2 =\cdots = b_n, for some k.k.
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Partitioning a deck with 2 cards in n types into pairs

A collection of 2n2n letters contains 22 each of nn different letters. The collection is partitioned into nn pairs, each pair containing 22 letters, which may be the same or different. Denote the number of distinct partitions by unu_n. (Partitions differing in the order of the pairs in the partition or in the order of the two letters in the pairs are not considered distinct.) Prove that un+1=(n+1)un−n(n−1)2un−2.u_{n+1}=(n+1)u_n - \frac{n(n-1)}{2} u_{n-2}.
Similar Problem :
A pack of 2n2n cards contains nn pairs of 22 identical cards. It is shuffled and 22 cards are dealt to each of nn different players. Let pnp_n be the probability that every one of the nn players is dealt two identical cards. Prove that 1pn+1=n+1pn+n(n−1)2pn−2.\frac{1}{p_{n+1}}=\frac{n+1}{p_n} + \frac{n(n-1)}{2p_{n-2}}.
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Euclidean plane

Let OO be a point on the oriented Euclidean plane and (i,j)(\mathbf i, \mathbf j) a directly oriented orthonormal basis. Let CC be the circle of radius 11, centered at OO. For every real number tt and non-negative integern n let MnM_n be the point on CC for which ⟨i,OMn→⟩=cos⁡2nt.\langle \mathbf i , \overrightarrow{OM_n} \rangle = \cos 2^n t. (or OMn→=cos⁡2nti+sin⁡2ntj\overrightarrow{OM_n} =\cos 2^n t \mathbf i +\sin 2^n t \mathbf j). Let k≥2k \geq 2 be an integer. Find all real numbers t∈[0,2π)t \in [0, 2\pi) that satisfy
(i) M0=MkM_0 = M_k, and
(ii) if one starts from M0M0 and goes once around CC in the positive direction, one meets successively the points M0,M1,…,Mk−2,Mk−1M_0,M_1, \dots,M_{k-2},M_{k-1}, in this order.
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