Problem solving involves more than intelligence there

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Problem solving involves more than intelligence. There is always some luck involved. There must be a genuine, deep-rooted interest. Great thinkers must have mental toughness. Positive thinking is necessary for clear thinking. Fostering a constructive atmosphere is critical. Education is good iff it promotes exploration. Problem solving is fun.
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The Paper Research How to Solve It  (George P ó lya) The Art and Craft of Problem Solving  (Paul Zeitz) 5 Solved Problems Contest and Journal Problems Domestic and International Contests Investigation and Reflections
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Problems 1. Let  k    1 be an integer. Show there are  exactly 3 k -1  integers  n  such that: n  has  k  digits, all of the digits are odd, n  is divisible by 5, and m  =  n / 5  has  k  odd digits. Austrian-Polish Mathematics Competition 1996
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Problems 2.  We  call  an  integer  m  “retrievable”  if  for  some integers  x  and  y m  = 3 x 2  + 4 y 2 . Show  that  if  m  is  retrievable,  then  13 m  is  retrievable.  AMTNYS , Jan. 2007
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Problems 3. At ABC University, the mascot does as many pushups  after  each  ABCU  score  as  the  team  has  accumulated.  The team always makes extra points after touchdowns,  so  it  scores  only  in  increments  of  3  and  7.  For  each  sequence  a 1 a 2 , …,  a n  where each  a k  = 3 or 7, let  P ( a 1 a 2 ,  …,  a n )  denote  the  total  number  of  pushups  the  mascot does for the scoring sequence  a 1 a 2 , …,  a n . For  example,  P (3,7,3) = 3 + (3 + 7) + (3 + 7 + 3) = 26. Call  a  positive  integer  k  accessible  if  there  is  a  scoring  sequence   a 1 a 2 , …,   a n  such that  P ( a 1 a 2 , …,   a n ) =  k . Is  there a number  K  such that for all  t     K t  is accessible?  If not, prove it, and if so, find  K.   Pi Mu Epsilon , Spring 2007
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Problems 4. Players 1, 2, 3, …,  n  are seated around a table and  each has a single penny.  Player 1 passes a penny to 
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