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Answer
Cities 1: Capital of the Tatarstan region.
Kazan
Cities 2: Coldest large city in the world.
Yakutsk
Cities 3: City situated on the Golden Horn Bay that hosts the Russian Pacific Fleet.
Vladivostok
Cities 4: "Third capital" of Russia
Yekaterinburg
Artificial Landmarks 1: Iconic cathedral situated in the Red Square.
Saint Basil's Cathedral
Artificial Landmarks 2: Many concerts, such as the one for "Swan Lake," were performed at this Moscow venue.
Bolshoi Theater
Artificial Landmarks 3: One of the largest art museums in the world, begun as Catherine the Great's private collection
Hermitage
Artificial Landmarks 4: Island known for its wooden structures
Kizhi Island
Natural Landmarks 1: The oldest and deepest lake in the world, and home to the only freshwater seal species.
Lake Baikal
Natural Landmarks 2: Rock formation located on an eponymous river.
Lena Pillars
Natural Landmarks 3: These seven rock pillars, ranging from 30-42 meters tall, are frequently said to be the work of magic.
Manpupuner
Natural Landmarks 4: Standing at the division between Europe and Asia is this mountain in the Caucasus, the highest in Europe.
Elbrus
People 1: "Terrible" tsar who rose Russia to prominence, but who also committed deeds such as murdering his own son. (In this context, "terrible" does not necessarily refer to his terribleness, but his ability to instill terror and awe)
Ivan the Terrible
People 2: Inventor of the modern periodic table.
Dmitri Mendeleev
People 3: Last leader of the Soviet Union
Mikhail Gorbachev
People 4: Behavioral psychologist known for his dogs.
Ivan Pavlov
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Answer
History 1: Defeat of Germanic Crusaders by Novgorodians, led by Alexander Nevsky, in 1242
Battle on the Ice
History 2: Liberal coup d'etat that led to many prominent noblemen and writers being exiled to Irkutsk.
Decembrist Revolt
History 3: First satellite to be launched into space
Sputnik-1
History 4: One of the most important battles of WWII, in which Soviet forces stopped Nazi advance into Russia
Battle of Stalingrad
Culture 1: Famously long Tolstoy novel centering on Napoleon's invasion of Moscow
War and Peace
Culture 2: In this Dostoevsky novel, a bitter man tells of his encounter with the prostitute Liza while attacking deterministic thought.
Notes from the Underground
Culture 3: Movie directed by Sergei Einsenstein about a major mutiny in the Revolution of 1905
Battleship Potemkin
Culture 4: In this movie directed by Andrei Tarkovsky, one can find a place known as "The Zone" that will grant their innermost desires...
Stalker
Composers 1: "The Nutcracker," "Romeo and Juliet"
Pyotr Ilyich Tchaikovsky
Composers 2: "Rhapsody on a Theme of Paganini," "The Miserly Knight" (< creator of the musical compositions in the opera, NOT the book itself)
Sergei Rachmaninoff
Composers 3: "The Firebird," "The Rite of Spring"
Igor Stravinsky
Composers 4: "Lady MacBeth of Mtsensk," "Waltz No.2"
Only 18/32; I expect that I shall be sent to the Gulag. Mind you, so should you if you continue to demand the word "cathedral" in the answer to no 5 even though it is in the question.
Vladivostok is not on an island, though. St. Basil's still doesn't work either; maybe you haven't pressed Submit.
Both of those should be fixed by now.