Two separate movie-related contests are currently underway at the official movie website:

The DVD Present Campaign

The "Mewtwo Special Pack", which contains all Mewtwo-related parts of the anime (the new movie "Extremespeed Genesect: The Awakening of Mewtwo", its related TV special "Mewtwo: The Prologue to Its Awakening", the very first Pocket Monsters movie, "Mewtwo Strikes Back", its TV special sequel "Mewtwo! I Am Here" and the only episode of the TV anime to prominently feature Mewtwo, "Tokiwa Gym! The Final Badge!"), as well as the regular DVD release Pocket Monsters the Movie - Extremespeed Genesect: The Awakening of Mewtwo / Pikachu and its Eievui Friends will be given away to 5 winners each.

An application form has been posted on the official site. Entries must be submitted before February 28th, 2014 12am Japan time, and require a Japanese shipping address. A list of winners will be posted in mid-March.

The Triple Illustration Contest

Like previous years, a drawing contest is held through various Japanese children's magazines. Three separate basic illustrations, with submission forms attached, will be provided through a total of 10 different magazine issues. One illustration features Yveltal, the other features Pikachu and Cleffi, and the third features Diancie and Serena.

Participants will have to cut out this illustration, color it and add details around the basic illustration, and send it in. These illustrations with their submission forms can also be downloaded from the official movie site, so no purchase of the physical magazines is required.

The participating magazines, the issue where the illustration and submission form can be found, and their release dates, are as follows.
Pokémon Fan issue 34, on sale February 20th
Corocoro Ichiban, April issue, on sale February 21st
TV-kun, April issue, on sale March 1st
Kindergartner Magazine, April issue, on sale March 1st
First Grader Magazine, April issue, on sale March 1st
Second Grader Magazine, April issue, on sale March 1st
Ciao, April issue, on sale March 3rd
Weekly Shonen Sunday issue 14/2014, on sale March 5th
Corocoro Comic, April issue, on sale March 5th
Pucchigumi, April issue, on sale March 5th

All entries must be in by April 18th, 2014.

The prizes are as following:

Grand Prize
*The winner from each magazine will receive either a Nintendo 3DS LL Pocket Monsters X Pack Premium Gold or a Nintendo 3DS LL Pocket Monsters Y Pack Premium Gold.
*Their winning picture will also be displayed alongside their name in the movie's ending credits.
*The winners from Ciao and Pucchigumi will also receive a copy of their submitted picture redrawn in the anime style.

Runner-Up Prizes
*Five winners from both TV-kun, Second Grader Magazine, Pokémon Fan, Corocoro Comic, Corocoro Ichiban and Shonen Sunday will receive a Pokémon Tretta gift set consisting of a Xerneas/Yveltal Design Tretta Report, a Tretta File 3 and a Super Ball Colored Collection Suitcase.
*Five winners from both Kindergartner Magazine and First Grader Magazine will receive a Monster Collection Pokémon Center XY playset.
*Ten winners from both Pucchigumi and Ciao will receive an Energetically Responding Talking Fokko plushie from the Talking Plushies toyline.

Lucky Prizes
*100 random entrant from each magazine will receive 4 packs of cards from the "Wild Blaze" Pocket Monsters Card Game series.

Pokémon with You Wagon Prize
*An undetermined number of submissions from the different magazines will be used for the next design of the Pokémon with You Wagon. This design will also be shown during the movie's end credits.



The downloadable submission form for Pokémon Fan, featuring the Yveltal illustration, can be seen here

UPDATE: Today's episode of Pokémon Get TV, The Rocket Gang's Romantic and Youthful Origin / In This 6th Installment of 'The Pokémon Zukan Completion Journey', Abareru-kun Aims to Achieve a Strict Quota in His Home Prefecture Fukushima presented this contest in their "Pokémon Morning News" segment. Anchor Shoko-tan presented the three different base drawing, and explained the basic rules to the contest.



Shoko-tan then presented her own example drawing, made from the Yveltal base.



Sources: http://www.pokemon-movie.jp/presents_dvd/ and http://www.pokemon-movie.jp/illustration_contest/