Difference between revisions of "2010 IMO Problems/Problem 6"

(Solution)
(Solution)
Line 11: Line 11:
 
:Let  
 
:Let  
 
:<math>w1 = \frac{a_{1}}{1}, w2 = \frac{a_{2}}{2}, ......, ws = \frac{a_{s}}{s}</math>
 
:<math>w1 = \frac{a_{1}}{1}, w2 = \frac{a_{2}}{2}, ......, ws = \frac{a_{s}}{s}</math>
:<math>\text{We claim the right choice of l is the one maximizing wl.}</math>
+
:<math>\text{We claim the right choice of l is the one maximizing} w_{l}.</math>
 
:<math>\text{Our plan is to view each } a_{n} \text{ as a linear combination of the weights } w_{1}\text{, ..., } w_{s} \text{ and track their coefficients}.</math>
 
:<math>\text{Our plan is to view each } a_{n} \text{ as a linear combination of the weights } w_{1}\text{, ..., } w_{s} \text{ and track their coefficients}.</math>
 
:<math>\text{To this end, let’s define an n-type to be a vector T = ⟨t1,...,ts⟩ of nonnegative integers such that}</math>
 
:<math>\text{To this end, let’s define an n-type to be a vector T = ⟨t1,...,ts⟩ of nonnegative integers such that}</math>

Revision as of 11:29, 6 October 2022

Problem

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\]

Author: Morteza Saghafiyan, Iran

Solution

So for solving This Problem, we need to take a assumption that,

Let
$w1 = \frac{a_{1}}{1}, w2 = \frac{a_{2}}{2}, ......, ws = \frac{a_{s}}{s}$
$\text{We claim the right choice of l is the one maximizing} w_{l}.$
$\text{Our plan is to view each } a_{n} \text{ as a linear combination of the weights } w_{1}\text{, ..., } w_{s} \text{ and track their coefficients}.$
$\text{To this end, let’s define an n-type to be a vector T = ⟨t1,...,ts⟩ of nonnegative integers such that}$
  • $n = t_{1}+...+t_{s}$
  • $t_{i} \text{is divisibe by i for every i}$

See Also

2010 IMO (Problems) • Resources
Preceded by
Problem 5
1 2 3 4 5 6 Followed by
Last Question
All IMO Problems and Solutions

Let w1 = a1 1 , w2 = a2 2 , . . . , ws = as s . (The choice of the letter w is for “weight”.) We claim the right choice of ℓ is the one maximizing wℓ . Our plan is to view each an as a linear combination of the weights w1, . . . , ws and track their coefficients. To this end, let’s define an n-type to be a vector T = ⟨t1, . . . , ts⟩ of nonnegative integers such that • n = t1 + · · · + ts; and • ti is divisible by i for every i. We then define its valuation as v(T) = Ps i=1 witi . Now we define a n-type to be valid according to the following recursive rule. For 1 ≤ n ≤ s the only valid n-types are T1 = ⟨1, 0, 0, . . . , 0⟩ T2 = ⟨0, 2, 0, . . . , 0⟩ T3 = ⟨0, 0, 3, . . . , 0⟩ . . . Ts = ⟨0, 0, 0, . . . , s⟩ for n = 1, . . . , s, respectively. Then for any n > s, an n-type is valid if it can be written as the sum of a valid k-type and a valid (n − k)-type, componentwise. These represent the linear combinations possible in the recursion; in other words the recursion in the problem is phrased as an = max T is a valid n-type v(T). In fact, we have the following description of valid n-types: Claim — Assume n > s. Then an n-type ⟨t1, . . . , ts⟩ is valid if and only if either • there exist indices i < j with i + j > s, ti ≥ i and tj ≥ j; or • there exists an index i > s/2 with ti ≥ 2i. Proof. Immediate by forwards induction on n > s that all n-types have this property. The reverse direction is by downwards induction on n. Indeed if P i ti i > 2, then we may subtract off on of {T1, . . . , Ts} while preserving the condition; and the case P i ti i = 2