Running Race
Gold medal winner: Cheetah

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Record: 101 km/h
Cheetah is the fastest running animal in the world that can run at a speed of 101 km/h, which is 2½ times as fast as humans. A human can run at the speed of 40 km/h. An average human athlete can complete a 100 meters running race in 9 to 10 seconds, whereas cheetah can reach the same distance within 3.5 seconds. Cheetah can finish the 400 meters running race within 16 seconds whereas human takes ~45 seconds to finish.
Silver medal winner: Pronghorns antelope Deer

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Record: 98 km/h
Bronze medal winner: Kangaroo and Ostrich

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Record: 72 km/h
Weightlifting
Gold medal winner: Rhinoceros beetle

Image credit: Extreme science
Record: The rhinoceros beetle can carry 850 times its own weight, even an African elephant can carry up to 25% of its own weight. That would be like a person of 70 kilograms carrying 59,000 kilograms weight.
Silver medal winner: Leaf-cutter ant

Image credit: Bristol Zoo garden
Record: Leaf-cutter ant can carry pieces of leaf that weigh 50 times its own weight. When compared to human, this is equivalent to pulling of a car for 8 kilometers with his teeth.
Bronze medal winner: Trapdoor spider

Image credit: Wikimedia commons
Record: Trapdoor spider can carry 40 times its own weight.
High Jump
Gold medal winner: Froghopper

Image credit: Wikimedia commons
Record: This small insect is a world champion in high jump. This can hop 100 times more than its size (hurl upwards 70 centimeters into the air). To compete with insect a human would have to jump 210 meters. This insect accelerates from the ground with a force that is 400 times greater than gravity whereas humans can only jump with a force that is three times that of gravity.
Silver medal winner: Locust

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Record: Locusts can jump 70 cm that is equivalent to a human jumping 18 meters.
Bronze medal winner: Fleas

Image credit: How stuff works
Record: Though it cannot fly it can jump 7 to 10 inches high that is many times to their height.
100 meters free style swimming
Gold medal winner: Sailfish

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Record: The speed of sailfish is 110km/h which is the highest speed reported in a fish. Sailfish can swim 100 meters within 3.6 seconds. The previous Olympic record of human for 100 meters free style swimming is 47.24 seconds.
Silver medal winner: Shark

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Record: Shark swims 100 meters in 7.2 seconds while hunting at the speed of 50km/h.
Bronze medal Winner: Blue whales

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Record: Blue whales can swim 100 meters within 7.5 seconds at the speed of 48km/h.
Long Jump
Gold medal winner: Flea

Image credit: How stuff works
Record: Flea can jump over thirteen inches long, that is about 150 times its body length that is equivalent to a person jumping nearly a 1000 feet. The acceleration of a flea when jumping is 50 times faster than a space shuttle. If fleas were the size of humans, they would be able to jump over the St. Paul's Cathedral in London 600 times in 3 days.
Silver medal winner: Grasshopper

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Record: Grasshopper can jump 30 inches that is up to 20 times the length of its body. This is equivalent to a person jumping an entire football field in a single bound.
Bronze medal winner: Cricket
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Record: Crickets are able to jump about 3 feet in to the air that is up to 20 times their body length.
Marathon
Winner: Arctic Tern bird

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Record: Arctic Tern bird migrates from Arctic region to Antarctic region every year. That means it travels 43,200 km every year. This is the longest regular migration by any known animal. It can travel 200 km continuously at a time.
After looking at all these I wouldn't be very surprised if people start conducting Olympics for animals too.