Converting smart pointers

Here's a problem:

1. You have a class named some_class:

struct base {
    virtual void some_methods() = 0;
    virtual ~base();
};

struct derived: public base {
    void some_methods() /*override*/;
    virtual void derived_method() const;

    ~derived() /*override*/;
};

2. You have a third-party API that returns constructed derived by shared pointer to base and accepts shared pointer to const derived in other functions:

#include <boost/shared_ptr.hpp>
boost::shared_ptr<const base> construct_derived();
void im_accepting_derived(boost::shared_ptr<const derived> p);

3. You have to make the following code work:

void trying_hard_to_pass_derived() {
    boost::shared_ptr<const base> d = construct_derived();

    // Oops! Compile time error:
    // ‘const struct base; has no member named ‘derived_method;.
    d->derived_method();

    // Oops! Compile time error:
    // could not convert ‘d; to ‘boost::shared_ptr<const derived>;.
    im_accepting_derived...