To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints.

Let's denote:
- $x_1$ as the number of consoles sold,
- $x_2$ as the number of discs sold.

The objective function aims to maximize profit. Given that each console is sold for a profit of $200 and each disc for a profit of $30, the objective function can be written as:
\[ \text{Maximize:} \quad 200x_1 + 30x_2 \]

The constraints are as follows:
1. The store owner can spend at most $30,000 on consoles and discs. Given that a console costs $300 and a disc costs $30:
\[ 300x_1 + 30x_2 \leq 30000 \]
2. A minimum of 20 but at most 50 consoles are sold each month:
\[ 20 \leq x_1 \leq 50 \]
3. The number of discs sold is at most five times the number of consoles sold:
\[ x_2 \leq 5x_1 \]

Thus, the symbolic representation of the problem can be summarized as:
```json
{
  'sym_variables': [('x1', 'number of consoles'), ('x2', 'number of discs')],
  'objective_function': '200*x1 + 30*x2',
  'constraints': [
    '300*x1 + 30*x2 <= 30000',
    '20 <= x1 <= 50',
    'x2 <= 5*x1'
  ]
}
```

To solve this problem using Gurobi in Python, we can write the following code:
```python
from gurobipy import *

# Create a new model
m = Model("Video_Game_Store")

# Define variables
x1 = m.addVar(name='consoles', vtype=GRB.INTEGER)
x2 = m.addVar(name='discs', vtype=GRB.INTEGER)

# Set objective function
m.setObjective(200*x1 + 30*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(300*x1 + 30*x2 <= 30000, name='budget')
m.addConstr(x1 >= 20, name='min_consoles')
m.addConstr(x1 <= 50, name='max_consoles')
m.addConstr(x2 <= 5*x1, name='discs_vs_consoles')

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Consoles: {x1.x}")
    print(f"Discs: {x2.x}")
    print(f"Max Profit: {m.objVal}")
else:
    print("No optimal solution found")
```