* Merkel acknowledges rich-poor and urban-rural divides
* Says there are concerns about crime and immigration
* Promises more spending on health and education
* Schaeuble says a minority government is possible (Adds comment from Siemens CEO, political expert)
By Andrea Shalal
BERLIN, Dec 31 (Reuters) - German Chancellor Angela Merkel, under mounting pressure after three months of failing to form a new conservative-led coalition, vowed on Sunday to address growing social divisions.
In her annual New Year's address, Merkel promised increased investment in security and defence, improved health care and education, and efforts to overcome growing urban-rural disparities.
"The world won't wait for us. We must create the conditions now that ensure that Germany continues to prosper in 10, 15 years," she said. "And Germany will only prosper if its success serves all people and improves and enriches our lives."
Merkel, seeking a fourth term in office, is trying to persuade the centre-left Social Democrats (SPD) to extend the "grand coalition" that has ruled for the past four years, even though both blocs suffered big losses in September's election.
Party leaders will meet on Jan. 3, with exploratory talks scheduled from Jan. 7 to 12.
Former finance minister Wolfgang Schaeuble, now president of the lower house of parliament, on Saturday said he could not rule out a minority government if no deal emerged.
The mainstream parties fear that a new election could hand further gains to the far-right Alternative for Germany (AfD), which has entered parliament for the first time amid concerns about Merkel's decision in 2015 to welcome over a million mainly Muslim migrants to Germany.
Merkel's popularity has waned since her efforts to form a coalition with two smaller parties collapsed in November, and a poll released on Saturday suggested that nearly two-thirds of voters want her to resign if talks with the SPD also fail.
Herfried Muenkler, professor at Berlin's Humboldt University, said even if Merkel assembled a new coalition, she was unlikely to see out her term, because there would either be a new elections or she would make way for a successor. "We are in a transition phase," he told the Welt am Sonntag newspaper.
The chancellor said that, while many took pride in Germany's prosperity and diverse and open society, others feared crime and violence, complained about a lack of doctors in rural areas, and worried about how to manage the influx of migrants.
"Both are realities in our country: the success and the optimism, but also the fears and the doubts. For me, each is a spur to act," she said.
Joe Kaeser, chief executive of the industrial group Siemens , said the lack of a German government was being noticed around the world, but backed Merkel.
"Stability, predictability and reliability are extremely important components of 'Made in Germany'," he told the Frankfurter Allgemeine Sonntagszeitung in an interview published on Sunday. "The chancellor stands for Germany as an anchor of stability in Europe and in the world."
But he said he was missing a clear vision from policy-makers on how to tackle challenges such as digitalisation, structural change, education, and equal opportunity.
Merkel said the decisive issue in coming years would be to strengthen the European Union and its economic success while also bolstering its external boundaries and security.
She said she hoped the new year strengthened the ties that united Germans, including mutual respect and the inviolable dignity of all people - a core tenet of the German constitution framed after the end of World War Two. (Reporting by Andrea Shalal and Christoph Steitz; Editing by Kevin Liffey)