Vilmundur studied at Menntaskólinn í Reykjavík, an Icelandic Junior College, from 1964 to 1968.
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The party won four seats in the elections that year.
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When he returned to Iceland in 1973, he received a job as a teacher of history at Menntaskólinn í Reykjavík and wrote articles in the Icelandic newspapers.
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In the 1978 elections Vilmundur was a candidate in Reykjavík and was voted into office.