2006
A6: 9/32 * sqrt(2)
C2: 1003
C5: {(n,k) in Z+^2 | exists t: 2^t | n and k < 2^t}
C7: A(v,e,f) = v-1
G5: (90, 90)
N1: {(0,2), (0,-2), (4,23), (4,-23)}
N5: emptyset
N6: the set is {+-(ax - by) | 0 < x < b, x + y = a+b // 2}, its size is b-1 if both a,b are odd, otherwise 2(b-1)
2007
A2: 1,2,...,2008
A4: { (f(x) = 2x) }
A5: 3n
C1: only solution: n*[0] + (n-1)*[0]+[1] + (n-2)*[0]+2*[1] + ... + [0]+(n-1)*[1] + n*[1] (python notation for list arithmetics)
C3: { 69,84 }
G6: 1
N1: {(2,4)}
N5: { (f(n) = n) }
2008:
A1: { (f(x) = x), (f(x) = 1/x) }
A3: ( NO, YES )
A7: { (a,b,c,d) in R^4 | a=b and c=d }
C1: 6
C2: if n=1 then 1, if n=2 then 2, otherwise 3*2^(n-2)
C3: 180180
C4: 2^(k-n)
C5: unique solution: f(n) = p1^(p1^e1 - 1) * p2^(p2^e2 - 1) * ... * pn^(pn^en - 1), where p1^e1 * p2^e2 * ... * pn^en is the unique factorization of n
2009:
A1: 1
A3: { (f(x) = x) }
A7: { (f(x) = x, f(x) = -x) }
C1: ( YES, NO )
C2: 2n // 3 + 1
C4: (m+1) 2^(m+2)
C5: NO
C6: 998^2-4 = 996000
G1: { 60, 90 }
N4: { 1,2,3,4 }
2010:
A1: { (f(x) = C) | C = 0 or 1 <= C < 2 }
A3: 25/2
A5: { (f(x) = 1/x) }
C1: YES
C2: 2^(N-2) + 1
C3: 0
C4: YES
N1: 39
N2: { (6,3). (9,3), (9,5), (54,5) }
N3: 5
N4: (1, -51^2)
N5: { f(n) = n+C | C in N_0 }
2011:
A1: { { d, 5d, 7d, 11d }, { d, 11d, 19d, 29d } }
A2: x1=1, x2=x3=...=x2011 = 2023065
A3: { (f(x) = x, g(x) = x) } + { (f(x) = x^2+C, g(x) = x) | C in R }
A4: { (f(n) = n, g(n) = 1) }
C1: 1*3*5*...*(2n-1)
C4: 3
C5: 3m/2 - 1
C7: 3986729
N3: { (f(x) = e*x^d + c) | e in {-1,1}, c in Z, d is a positive divisor of n }
N4: { 1,3,5 }
2012:
A1: { (f(x) = 0) } + { (f(x) = k) | k in Z-{0} } + { (f(x) = 0 if x is even, else k) | k in Z-{0} } +
    { (f(x) = 0 if x mod 4 = 0, or 4k if x mod 4 = 2, else k) | k in Z-{0} }
A2: ( YES, NO )
A5: { (f(x) =  x-1) }
C2: (2n-1) // 5
C3: 4*999^4 / 27 = 147556443852
C4: 4022
N1: { (m,n) | gcd(m,n) = 1 }
N2: { (2,251,252) }
N3: the set of all prime numbers
N4: NO
N5: { (f(x) = a*x^m) | a,m in N_0 }
N7: { n in N_0 | n mod 4 = 1 or n mod 4 = 2 }
2013
A5: { (f(n) = n+1),
      (f(n) = n+5 if n mod 4 = 4, or n-3 if n mod 4 = 3, else n+1) }
A6: { (P(x) = t*x) | t in R }
C1: 2n-1
C2: 2013
C8: NO
N1: { (f(n) = n) }
N4: NO
N6: { (f(x) = floor(x)), (f(x) = ceil(x)) } + { f(x) = C | C in Z }
2014:
A3: 2
A4: { (f(n) = 2n+1007) }
A5: { x in R | x < 0 or x = 1 }
A6: { (f(n) = n+1), (f(n) = n+1 if n > 0, or 0 if n = 0, else -n+1) } +
    { (f(n) = n+1 if n > -1, else -n+1) | a in Z+ }
C3: floor(sqrt(n-1))
C6: 100
C8: all moves except taking the empty cards
N1: (n-2)*2^n + 1
N5: {(3,2,5),(3,5,2)} + {(2,n,2^k-n) | 0 < n < 2^k}
2015:
A2: { (f(x) = -1), (f(x) = x+1) }
A3: n(n-1)
A4: { (f(x) = x), (f(x) = 2-x) }
A5: for d in Z+, for odd integers l_0, ..., l_(d-1), all functions given by
    f(m*d + i) = 2k*m*d + l_i * d
C2: { n in Z | n >= 3 and n is odd }
C3: 3024
C4: draw game if n in {1,2,4,6}, else B wins
G4: { sqrt(2) }
N1: { M in Z | M >= 2 }
N5: { (2,2,2), (2,2,3), (2,3,2), (2,6,11), (2,11,6), (3,2,2), (3,5,7), (3,7,5),
      (5,3,7), (5,7,3), (6,2,11), (6,11,2), (7,3,5), (7,5,3), (11,2,6), (11,6,2) }
N7: { k in Z | k >= 2 }
N8: { (f(x) = ax+b) | b in Z, a in Z, mho(a) = 0 } (mho is a function from the problem statement)
2016:
A2: 1/2
A3: { n in Z | n is odd and n >= 3 }
A4: { (f(x) = 1/x) }
A6: 2016
A7: { (f(x) = -1), (f(x) = x-1) }
A8: 4/9
C1: 2 if n = 2k, else 1
C2: { 1 }
C4: { n in Z+ | n mod 9 = 0 }
C5: n-2 if n is even, else n-3
C8: 2n
N1: { (P(x) = x), (P(x) = 1),  (P(x) = 2),  (P(x) = 3),  (P(x) = 4),  (P(x) = 5),  (P(x) = 6),  (P(x) = 7),  (P(x) = 8), (P(x) = 9) }
N2: { n in Z | n >= 2 and n is not an odd prime number }
N3: 6
N6: { (f(n) = n^2) }
N8: { (P(x) = a*(rx+s)^d) | a,r,s in Z, a != 0, r >= 1, gcd(r,s) = 1 }
2017:
A2: { -2,0,2 }
A5: -(n-1)/2
A6: { (f(x) = 0), (f(x) = x-1), (f(x) = 1-x) }
C3: 2*sum_(j=0)^8 (n choose j) - 1
C5: NO
C6: n(n+1)(2n+1)/6
C8: n^2+1
G8: 6048
N1: { 3n | n in Z+ }
N3: the set of prime numbers
N4: 807
N5: { (3,2) }
N6: 3
2018:
A1: { (f(x) = 1) }
A2: { 3n | n in Z+ }
A4: 2016 / (2017^2)
A5: { (f(x) = C1*x + C2/x) | C1,C2 in R+ }
A7: 8 / 7^(1/3)
C2: 100
C4: NO
C5: k(4k^2 + k - 1)/2
N1: { (n,k) | n does not divide k and k does not divide n }
N5: NO