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1989 APMO Problems

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Problem 1

Let x_1, x_2, x_3, \dots , x_n be positive real numbers, and let

S=x_1+x_2+x_3+\cdots +x_n.

Prove that

(1+x_1)(1+x_2)(1+x_3)\cdots (1+x_n)\leq 1+S+\dfrac{S^2}{2!}+\dfrac{S^3}{3!}+\cdots +\dfrac{S^n}{n!}.

Solution

Problem 2

Prove that the equation

6(6a^2+3b^2+c^2)=5n^2

has no solutions in integers except a=b=c=n=0.

Solution

Problem 3

Let A_1,A_2,A_3 be three points in the plane, and for convenience, let A_4=A_1, A_5=A_2. For n=1, 2, and 3, suppose that B_n is the midpoint of A_nA_{n+1}, and suppose that C_n is the midpoint of A_nB_n. Suppose that A_nC_{n+1} and B_nA_{n+2} meet at D_n, and that A_nB_{n+1} and C_nA_{n+2} meet at E_n. Calculate the ratio of the area of triangle D_1D_2D_3 to the area of triangle E_1E_2E_3.

Solution

Problem 4

Let S be a set consisting of m pairs (a,b) of positive integers with the property that 1\leq a<b\leq n. Show that there are at least

4m\cdot \dfrac{\left(m-\dfrac{n^2}{4}\right)}{3n}

triples (a,b,c) such that (a,b), (a,c), and (b,c) belong to S.

Solution

Problem 5

Determine all functions f from the reals to the reals for which

(1) f(x) is strictly increasing,

(2) f(x)+g(x)=2x for all real x,

where g(x) is the composition inverse function to f(x). (Note: f and g are said to be composition inverses if f(g(x))=x and g(f(x))=x for all real x.)

Solution

See also

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