MMO Preparation Worksheet-1

1. Find all primes p such that 7p + 1 is a perfect square.

2. Prove that if n is an odd integer then 24|nn − n.
3. Solve x2 − 4x − 1 = 2y in Z.
4. Find all n ∈ N such that 7|2n + n2 .
5. Find all b ∈ N, for which there exists a ∈ N satisfying

b|a2 + 1 and b|a3 − 1.

6. Solve x2 + y 2 + z 2 = 2010xyz in Z.
7. Find all n ∈ N for which n2012 + n + 1 is a prime number.
8. Solve x2 + 8xy + 25y 2 = 225 in Z.
9. Prove that 90|x2 + xy + y 2 =⇒ 900|xy.
10. Find all primes p, q, r and n ∈ N such that
1 1 1 1
+ + = .
p q r n

11. Let a be a given natural number. Prove that there exist infinitely many
integers (b, c) such that ab + 1, bc + 1, ac + 1 are perfect squares.
1 1 1
12. Let a, b, c ∈ N such that a + b = c and gcd(a, b, c) = d. Prove that abcd
is a perfect square.
13. Solve x2 + y 2 + z 2 = xy + yz + zx + 3 in Z.
14. Solve x3 + y 3 + z 3 = 3xyz + 5 in Z.
15. Solve x3 + y 3 = (x + y)2 in Z.
p p
16. Find all primes p, q such that p2 + 14pq + q 2 + p2 + 7pq + q 2 ∈ N.
17. Find all functions f : R → R such that for all x, y ∈ R:

f (x2 − y 2 ) = (x − y)(f (x) + f (y)).

18. Find all functions f : R → R such that for all x, y ∈ R:

f (x3 + y) = x2 f (x) + f (y).

19. Find all functions f : R → R such that for all x, y ∈ R:

f (x2 − y 2 ) = xf (x) − yf (y).

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20. Find all functions f : R → R such that for all x, y ∈ R:

f (x + y)(f (x) − y) = xf (x) − yf (y).

21. Find all functions f : R → R such that for all x, y ∈ R:

f (f (x) + y) = 2x + f (f (y) − x).

22. Find all functions f : Z → Z such that for all x, y ∈ Z:

f (x + f (y)) − f (x) = y.

23. Find all functions f : N → N such that for all m, n ∈ N:

f (f (m) + f (n)) = m + n + 3.

24. Find all functions f : N → N such that for all m, n ∈ N:

f (f (m) + f (n)) = m + n.

25. Find all functions f : R → R such that for all x, y ∈ R:

f (x − f (y)) = 2f (x) + x + f (y).

26. Find all functions f : Q → Q such that for all x, y ∈ Q:

f (2f (x) + f (y)) = 2x + y.

27. Let a, b, c be positive real numbers such that abc = 1. Prove that
b+c c+a a+b √ √ √
√ + √ + √ ≥ a + b + c + 3.
a b c

28. If the equation x4 + ax3 + 2x2 + bx + 1 = 0 has at least one real root, then
a2 + b2 ≥ 8.
29. Let a, b, c be positive real numbers. Prove that
a b c 9
+ + ≥ .
(b + c)2 (c + a)2 (a + b)2 4(a + b + c)

30. Let a, b, c be positive real numbers such that abc = 2. Prove that
√ √ √
a3 + b3 + c3 ≥ a b + c + b c + a + c a + b.

31. Prove that for any a, b, c ∈ (1, 2) the inequality

√ √ √
b a c b a c
√ √ + √ √ + √ √ ≥ 1.
4b c − c a 4c a − a b 4a b − b c

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32. Let a, b, c be positive reals such that abc ≤ 1. Prove that
a b c
+ + ≥ a + b + c.
b c a

33. Let a, b, c be positive real numbers. Prove that

a3 b3 c3 a2 b2 c2
2
+ 2+ 2 ≥ + + .
b c a b c a

34. Prove that if n > 3 and x1 , x2 , · · · , xn > 0 such that x1 x2 · · · xn = 1, then

1 1 1
+ + ··· + > 1.
1 + x1 + x1 x2 1 + x2 + x2 x3 1 + xn + xn x1

35. Let x, y, z > −1. Prove that

1 + x2 1 + y2 1 + z2
2
+ 2
+ ≥ 2.
1+y+z 1+z+x 1 + x + y2

36. Let a, b, c > 0 with a + b + c = 1. Prove that

a2 + b a2 + b a2 + b
+ + ≥ 2.
b+c b+c b+c

37. Let a, b, c be positive real numbers. Prove that

ab bc ca a+b+c
+ + ≤ .
a + b + 2c b + c + 2a c + a + 2b 4

38. Let a, b, c be positive real numbers such that ab + bc + ca = 1. Prove that

1 − a2 1 − b2 1 − c2 3
+ + ≤
2 + b2 + c 2 2 + c2 + a2 2 + a2 + b2 4

39. Let a, b, c be positive real numbers. Prove that

a2 − bc b2 − ca c2 − ab
+ 2 + 2 ≤ 0.
2a2 + ab + ac 2b + bc + ba 2c + ca + cb

40. Let a, b, c be positive real numbers such that a2 + b2 + c2 = 3. Prove that

1 1 1 15 14 14 14
+ + + ≥ 2 + 2 + 2 .
b+c b+c b+c 4 a +7 b +7 c +7

41. Let a, b, c be positive real numbers. Prove that

ab bc ca a+b+c
+ + ≤ .
a + 3b + 2c b + 3c + 2a c + 3a + 2b 6