The 27th Day

In the middle of the Cold War, aliens show up and give weapons capable of destroying all life on Earth to five random individuals: an American writer, an English actress, a German scientist, a Russian soldier, and a Chinese peasant. The aliens tell them that if the human race destroys itself with the weapons, they'll take over Earth, however if humanity manages to survive 27 days without unleashing Armageddon upon themselves, the aliens will leave in peace.

The five humans decide to go with the simple solution: don't tell anyone about the boxes. Which would have worked out just fine except that the aliens then make their presence know to the world and tell everyone about the five.

Tropes used in The 27th Day include:

Alien Abduction: The five are taken non-violently in order to give them the weapons.
Ambition Is Evil: Averted. Ambition is good, it's aggression that causes evil.
America Saves the Day: Averted. The scientist who figures everything out is German, and the American forces actually give into the Soviet demand to pull out of Europe (granted, it's just a stalling tactic).
Apocalypse How: The aliens are expecting a class 3a, The Leader is aiming for a Class 0 in North America.
Can't Argue with Elves: Nobody seems to think that attempting to engineer xenocide by giving humans the means to destroy themselves makes the aliens a little less noble and peaceful than they say they are. In fact one clue as to the true purpose of the boxes is that Klaus realizes that no species as wonderful and peaceful could have build a weapon designed to wipe out a planet.
Defrosting Ice Queen: Eve is quite hostile towards Jonathan in the beginning.
Dirty Commies: The real threat comes from the USSR getting their hands on one of the boxes.
Distant Finale: The epilogue takes place ten years into the future.
Eagle Land: Flavor 1, America is the good guy in this situation.
Evil Overlord: The Leader, who runs the USSR.
Fair for Its Day: While the USSR is a hellhole and run by evil commies trying to destroy capitalism and freedom, Godofsky is portrayed as honest and noble. He only hands over the weapons after being brainwashed, and even tries to stop the Leader in the end.
Faster-Than-Light Travel: Use by the aliens for travel and to create Time Dilation.
The Federation: The organization that runs the galaxy and consists of 30,000 members.
Acceptable Feminine Goals: In the epilogue, Jonathan, Klaus, and Ivan have important jobs with celebrity status. Eve has a bunch of kids.
The Film of the Book: A very faithful adaptation, written by the author of the book.
First Contact: Which leads to mass-panic, not surprisingly.
Heroic Sacrifice: Ivan nearly dies trying to stop the Leader, and one of the scientists volunteers to be the test subject on the weapon.
Humans Are the Real Monsters: What the aliens are banking on.
Kill'Em All: The Leaders plan.
A Million Is a Statistic: Discussed - since suffering is personal, it can only be understood on a personal level. That's why thousands starving on the other side of the world doesn't bother people as much as a one person in danger.
The Nameless: The Leader is never referred to as anything else, even in other countries.
No Celebrities Were Harmed: Nikita Khrushchev is not the one in charge.
Sci-Fi Writers Have No Sense of Scale: Averted - Due to the sheer size and number of habitable planets, it's impossible for any one species to explore the entire galaxy.
Science Marches On & Artistic License: Astronomy has advanced quite a bit since the book was written in the 1950's.
Special Effects Failure: During the cut to make the one capsule vanish from the professor's hand doesn't line up perfectly.
Stock Footage: The alien ship is lifted from Earth vs. the Flying Saucers.
Sugar Bowl: What the world ends up becoming.
Year Inside, Hour Outside: The alien has an extended conversation with the five chosen, but no time has passed when they are returned.