You are given a high-school competition-level math problem that requires answer-construction. Your task is to formalize the problem in Lean 4 (v4.23.0).

Your output should follow this template:

1. Start with imports and namespaces such as import Mathlib and open Real.

2. Define the ground-truth answer as an abbrev: 
   `abbrev <problem_name>_answer : β := <expression>`, where β is the type for answer variable. 
   In case the answer depends on context variables, use a lambda function over the context type:
   `abbrev <problem_name>_answer : α → β := fun x => <expression>` Here, α is the type for context variable.
   Do not include input variable bindings (like `(n : ℕ)`) before `: α → β` — instead encode those inside the function body `fun x => <expression>`

3. Write the formal theorem statement using this template or similar way, and leave the proof of the theorem statement by sorry placeholder:
   theorem <problem_name> (x : α) (hypotheses about x) (y : β): P(x, y) ↔ y = <problem_name>_answer x := by sorry

Other Instructions:
- In case of `optimal answers` (e.g. minimum, maximum), you need to encode them via `IsLeast`, `IsGreatest`, etc.
- In case of `multiple answers` (i.e. answer set), you can replace β with Set β, and replace P(x, y) ↔ y = <problem_name>_answer x with P(x, y) ↔ y ∈ <problem_name>_answer x.
- You should NEVER expose the actual ground-truth answer in the theorem statement. Only define it inside abbrev `<problem_name>_answer`.
- Output your lean code by using delimiter <<<your lean code>>>. 

Example 1:

Problem:
Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the line $y=x$.  You may assume that a triangle of minimum perimeter exists.
Answer:
M(a,b) = $\sqrt{2a² + 2b²}$

Formalization:
<<<
import Mathlib

open Set Function Metric

abbrev putnam_1998_b2_answer : ℝ → ℝ → ℝ := 
   fun a b => Real.sqrt (2*a^2 + 2*b^2)

theorem putnam_1998_b2
(a b : ℝ)
(hab : 0 < b ∧ b < a)
: sInf {d : ℝ | ∃ (c : ℝ) (x : ℝ), d = Real.sqrt ((a - c)^2 + (b - 0)^2) +  Real.sqrt ((c - x)^2 + (0 - x)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) ∧
  Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((c - x)^2 + (0 - x)^2) > Real.sqrt ((a - x)^2 + (b - x)^2) ∧
  Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) > Real.sqrt ((c - x)^2 + (0 - x)^2) ∧
  Real.sqrt ((c - x)^2 + (0 - x)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) > Real.sqrt ((a - c)^2 + (b - 0)^2)}
 = putnam_1998_b2_answer a b := by sorry
>>>


