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Math Olympiad
Math Olympiad Problem Solving
Stanford University EPGY Summer Institutes 2008
Problem Set: Proof by Contradiction
3 is irrational.
(b) Prove that 6 is irrational.
(c) If you attempt to prove that 49 is irrational by using the same argument
as in (a) and (b), where does the argument break down?
1. (a) Prove that
2. The product of 34 integers is equal to 1. Show that their sum cannot be 0.
3. Prove that the sum of two odd squares cannot be a square.
4. Let a1 , a2 , . . . , a2000 be natural numbers such that
+ ··· +
= 1.
a1 a2
Prove that at least one of the ak ’s is even. Hint: clear the denominators.
5. Prove that log2 3 is irrational.
6. A palindrome is an integer whose decimal expansion is symmetric, e.g. 1, 2, 11, 121,
15677651 (but not 010, 0110) are palindromes. Prove that there is no positive
palindrome which is divisible by 10.
7. Let 0 < α < 1. Prove that α > α.
8. In 4ABC, ∠A > ∠B. Prove that BC > AC.
9. Show that if a is rational and b is irrational, then a + b is irrational.
10. Prove that the cube root of 2 is irrational.
11. Prove that there is no smallest positive real number.
12. Prove that there are no positive integer solutions to the equation
x2 − y 2 = 10.
13. Given that a, b, c are odd integers, prove that the equation ax2 + bx + c = 0
cannot have a rational root.
14. Prove that there do not exist positive integers a, b, c and n such that
a2 + b2 + c2 = 2n abc.
Summer 2008
Problem Set: Proof by Contradiction
Math Olympiad
15. Show that the equation
b2 + b + 1 = a2
has no positive integer solutions a, b.
16. Let a, b, c be integers satisfying a2 + b2 = c2 . Show that abc must be even.
17. Let P (x) = xn + an−1 xn−1 + · · · + a1 x + a0 be a polynomial with integral
coefficients. Suppose that there exist four distinct integers a, b, c, d with P (a) =
P (b) = P (c) = P (d) = 5. Prove that there is no integer k with P (k) = 8.
18. Let p(x) be a polynomial with integer coefficients satisfying p(0) = p(1) = 1999.
Show that p has no integer zeros.
Summer 2008
Problem Set: Proof by Contradiction